Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples...
Transcript of Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples...
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ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Families of varieties of generaltype: the Shafarevich conjecture
and related problems
Sándor Kovács
University of Washington
![Page 2: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/2.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Motto
"To me,algebraic geometry
is algebra with a kick"–Solomon Lefschetz
![Page 3: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/3.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 Curves
2 Families
3 Shafarevich’s Conjecture
4 The Proof of Shafarevich’s Conjecture
5 Generalizations
![Page 4: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/4.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry
![Page 5: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/5.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Curves
Smooth ComplexProjective Curve
=Compact Riemann
Surface
![Page 6: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/6.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Curves
Smooth ComplexProjective Curve
=Compact Riemann
Surface
![Page 7: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/7.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry
![Page 8: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/8.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
![Page 9: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/9.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
![Page 10: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/10.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
![Page 11: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/11.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
![Page 12: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/12.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Genus
genus 0
genus 1
genus 2, ...
![Page 13: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/13.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Outline
1 CurvesRiemann SurfacesGenusTopology, Arithmetic and Differential Geometry
![Page 14: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/14.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 15: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/15.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 16: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/16.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
![Page 17: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/17.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
![Page 18: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/18.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Smooth Projective Curves
Smooth curves C come in three types:
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
![Page 19: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/19.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
Smooth curves C come in three types, and...
• C ' P1 (rational)
• C is a plane cubic (elliptic)
• C has genus ≥ 2 (general type)
![Page 20: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/20.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 21: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/21.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1 π1 is trivial
• C is a plane cubic
• C has genus ≥ 2
![Page 22: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/22.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1 π1 is trivial
• C is a plane cubic π1 is abelian
• C has genus ≥ 2
![Page 23: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/23.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Topology of Curves
...their fundamental groups come in three types:
• C ' P1 π1 is trivial
• C is a plane cubic π1 is abelian
• C has genus ≥ 2 π1 is non-commutative
![Page 24: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/24.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
Smooth curves C come in three types, and...
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 25: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/25.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 26: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/26.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 lots
• C is a plane cubic
• C has genus ≥ 2
![Page 27: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/27.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 non-finitely generated
• C is a plane cubic
• C has genus ≥ 2
![Page 28: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/28.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 non-finitely generated
• C is a plane cubic finitely generated
• C has genus ≥ 2
![Page 29: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/29.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Arithmetic of Curves
...rational points on them come in three types:
• C ' P1 non-finitely generated
• C is a plane cubic finitely generated
• C has genus ≥ 2 finite
![Page 30: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/30.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
Smooth curves C come in three types, and...
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 31: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/31.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1
• C is a plane cubic
• C has genus ≥ 2
![Page 32: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/32.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 positively curved
• C is a plane cubic
• C has genus ≥ 2
![Page 33: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/33.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 positively curved
• C is a plane cubic flat
• C has genus ≥ 2
![Page 34: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/34.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 positively curved
• C is a plane cubic flat
• C has genus ≥ 2 negatively curved
![Page 35: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/35.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 parabolic
• C is a plane cubic
• C has genus ≥ 2
![Page 36: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/36.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 parabolic
• C is a plane cubic elliptic
• C has genus ≥ 2
![Page 37: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/37.jpg)
ShafarevichConjecture
Sándor Kovács
CurvesRiemann Surfaces
Genus
Topology, Arithmetic andDifferential Geometry
Families
Shafarevich’sConjecture
The Proof
Generalizations
Differential Geometry of Curves
...their curvatures come in three types:
• C ' P1 parabolic
• C is a plane cubic elliptic
• C has genus ≥ 2 hyperbolic
![Page 38: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/38.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Outline
2 FamiliesExamples and DefinitionsIsotrivial and Non-isotrivial Families
![Page 39: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/39.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
![Page 40: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/40.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
![Page 41: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/41.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
![Page 42: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/42.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
![Page 43: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/43.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
![Page 44: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/44.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
X
y
π
A1λ
![Page 45: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/45.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
A family of curves of general type
y2 = x5 − 5λx + 4λ defines a smooth family ofcurves parametrized by λ 6= 0, 1.
X = (y2 = x5 − 5λx + 4λ) ⊆A3x ,y ,λ
y
π
A1λ
A2x ,y ⊇ Xλ = π−1(λ) ⊆ X
y
π
λ ∈ A1λ
![Page 46: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/46.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
![Page 47: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/47.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
![Page 48: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/48.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
![Page 49: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/49.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Notation
B – smooth projective curve of genus g,∆ ⊆ B – finite set of points.
A family over B \ ∆, f : X → B \ ∆, is asurjective (flat) map with equidimensional,connected fibers.
A family is smooth if Xb = f−1(b) is smooth for∀b ∈ B.
![Page 50: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/50.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Outline
2 FamiliesExamples and DefinitionsIsotrivial and Non-isotrivial Families
![Page 51: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/51.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Isotrivial Family
![Page 52: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/52.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Isotrivial Family
All smooth fibers are isomorphic to P1.
![Page 53: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/53.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Isotrivial Family
All smooth fibers are isomorphic to P1.
![Page 54: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/54.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
![Page 55: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/55.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
![Page 56: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/56.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
Warm-Up Question: Let q ∈ Z. How manyisotrivial families of curves of genus q over Bare there?
![Page 57: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/57.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
FamiliesExamples and Definitions
Isotrivial and Non-isotrivial
Shafarevich’sConjecture
The Proof
Generalizations
Non-isotrivial families
The family defined by y2 = x5 − 5λx + 4λ is notisotrivial, because
Xλ1 6' Xλ2
for λ1 6= λ2.
A family f : X → B is isotrivial if there exists afinite set ∆ ⊂ B such that Xb ' Xb′ for allb, b′ ∈ B \ ∆.
Warm-Up Question: Let q ∈ Z. How manyisotrivial families of curves of genus q over Bare there? ∞
![Page 58: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/58.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Outline
3 Shafarevich’s ConjectureStatement and DefinitionsResults and Connections
![Page 59: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/59.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
![Page 60: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/60.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
![Page 61: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/61.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
![Page 62: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/62.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
![Page 63: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/63.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture (1962)
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q over B \ ∆.These will be called “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
Note:
2g(B) − 2 + #∆ ≤ 0 ⇔
{
g(B) = 0 & #∆ ≤ 2g(B) = 1 & ∆ = ∅
![Page 64: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/64.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
![Page 65: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/65.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
![Page 66: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/66.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
![Page 67: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/67.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic implies that
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
![Page 68: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/68.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
![Page 69: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/69.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
An algebraic variety X is algebraicallyhyperbolic if
![Page 70: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/70.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
An algebraic variety X is algebraicallyhyperbolic if
• ∀ A1 \ {0} → X regular map is constant
![Page 71: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/71.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity
A complex analytic space X is Brody hyperbolicif every C → X holomorphic map is constant.X Brody hyperbolic is equivalent to
• ∀ C∗ → X holomorphic map is constant,• ∀ T → X holomorphic map is constant, for any
complex torus T = Cn/Z2n.
An algebraic variety X is algebraicallyhyperbolic if
• ∀ A1 \ {0} → X regular map is constant, and• ∀ A → X regular map is constant, for any
abelian variety (projective algebraic group) A.
![Page 72: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/72.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
![Page 73: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/73.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
![Page 74: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/74.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
![Page 75: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/75.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
![Page 76: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/76.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Hyperbolicity: Examples
B – smooth projective curve,∆ ⊆ B – finite subset.
• If g(B) ≥ 2, thenB \ ∆ is hyperbolic for arbitrary ∆.
• If g(B) = 1, thenB \ ∆ is hyperbolic ⇔ ∆ 6= ∅.
• If g(B) = 0, thenB \ ∆ is hyperbolic ⇔ #∆ ≥ 3.
B \ ∆ is hyperbolic ⇔ 2g(B) − 2 + #∆ > 0.
![Page 77: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/77.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆. “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
![Page 78: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/78.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆. “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
![Page 79: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/79.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
![Page 80: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/80.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
![Page 81: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/81.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Shafarevich’s Conjecture: Take Two
Fix B, ∆ as before, q ∈ Z, q ≥ 2. Then
(I) there exists only finitely many non-isotrivialsmooth families of curves of genus q overB \ ∆, “admissible families”.
(II) If2g(B) − 2 + #∆ ≤ 0,
then there aren’t any admissible families.
(II∗) If there exists any admissible families, thenB \ ∆ is hyperbolic.
![Page 82: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/82.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Outline
3 Shafarevich’s ConjectureStatement and DefinitionsResults and Connections
![Page 83: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/83.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 84: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/84.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 85: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/85.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 86: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/86.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 87: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/87.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 88: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/88.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 89: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/89.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
History
The Shafarevich Conjecture was confirmed byPARSHIN (1968) for ∆ = ∅, and byARAKELOV (1971) in general.INTERMEZZO: The number field case.
• Mordell Conjecture (1922):F – a number field (a finite extension field of Q),C – a smooth projective curve of genus ≥ 2
defined over F .Then C has only finitely many F -rational points.This was proved by FALTINGS (1983).
• The Shafarevich Conjecture has a number fieldversion as well.
![Page 90: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/90.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
![Page 91: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/91.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
![Page 92: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/92.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN (1968). using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
![Page 93: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/93.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjectureStatements
Results and Connections
The Proof
Generalizations
Geometric Mordell Conjecture
Let f : X → B be a non-isotrivial family ofprojective curves of genus ≥ 2.Then there are only finitely many sections of f ,i.e., σ : B → X , such that f ◦ σ = idB.
This was proved by MANIN (1963).
And again by PARSHIN (1968) using
“Parshin’s Covering Trick” to prove that
The Shafarevich Conjectureimplies
The Mordell Conjecture.
![Page 94: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/94.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Outline
4 The Proof of Shafarevich’s ConjectureDeformation TheoryThe Arakelov-Parshin method
![Page 95: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/95.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations
A deformation of an algebraic variety X is afamily F : X → T such that there exists a t ∈ Tthat X ' Xt .
![Page 96: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/96.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations
A deformation of an algebraic variety X is afamily F : X → T such that there exists a t ∈ Tthat X ' Xt .
X
T
f
Xt
t
![Page 97: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/97.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type
Two algebraic varieties X1 and X2 have thesame deformation type if there exists a familyF : X → T , t1, t2 ∈ T such that X1 ' Xt1 andX2 ' Xt2 .
![Page 98: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/98.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type
Two algebraic varieties X1 and X2 have thesame deformation type if there exists a familyF : X → T , t1, t2 ∈ T such that X1 ' Xt1 andX2 ' Xt2 .
X
T
ft1
t2
Xt1
Xt2
![Page 99: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/99.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations of Families
A deformation of a family X → B is a familyF : X → B × T such that there exists a t ∈ Tthat (X → B) ' (Xt → B × {t}), whereXt = F−1(B × {t}).
![Page 100: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/100.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformations of Families
A deformation of a family X → B is a familyF : X → B × T such that there exists a t ∈ Tthat (X → B) ' (Xt → B × {t}), whereXt = F−1(B × {t}).
F
X
T×B
Xt
{ }B t×
![Page 101: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/101.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type of Families
Two families X1 → B and X2 → B have thesame deformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1 → B) ' (Xt1 → B × {t}) and(X2 → B) ' (Xt2 → B × {t}).
![Page 102: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/102.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Deformation Type of Families
Two families X1 → B and X2 → B have thesame deformation type if there exists a familyF : X → B × T , t1, t2 ∈ T such that(X1 → B) ' (Xt1 → B × {t}) and(X2 → B) ' (Xt2 → B × {t}).
F
XXt
1
2{ }B t×
1{ }B t×
Xt2
T×B
![Page 103: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/103.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
Outline
4 The Proof of Shafarevich’s ConjectureDeformation TheoryThe Arakelov-Parshin method
![Page 104: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/104.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:
![Page 105: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/105.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.
![Page 106: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/106.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.
![Page 107: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/107.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.
Note:• (B) and (R) together imply (I).
![Page 108: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/108.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.(H) Hyperbolicity: If there exist admissible families,
then B \ ∆ is hyperbolic.
Note:• (B) and (R) together imply (I).
![Page 109: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/109.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The ProofDeformation Theory
Arakelov-Parshin method
Generalizations
The Arakelov-Parshin method
ARAKELOV and PARSHIN reformulated theShafarevich conjecture the following way:(B) Boundedness: There are only finitely many
deformation types of admissible families.(R) Rigidity: Admissible families do not admit
non-trivial deformations.(H) Hyperbolicity: If there exist admissible families,
then B \ ∆ is hyperbolic.
Note:• (B) and (R) together imply (I).• (H) = (II∗) and hence is equivalent to (II).
![Page 110: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/110.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 111: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/111.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
![Page 112: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/112.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
![Page 113: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/113.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
![Page 114: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/114.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
![Page 115: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/115.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional fibers
First order of business:
Generalize the setup.
Find a replacement for “genus” and “q ≥ 2”.
• Hilbert polynomial, h.• Kodaira dimension, κ = −∞, 0, 1, . . . , dim.
![Page 116: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/116.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 117: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/117.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
![Page 118: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/118.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
P1 deg = 1, 2 g = 0 κ = −∞
![Page 119: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/119.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
P1 deg = 1, 2 g = 0 κ = −∞
elliptic deg = 3 g = 1 κ = 0
![Page 120: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/120.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Plane curves:Kodaira
type degree genus dimension
P1 deg = 1, 2 g = 0 κ = −∞
elliptic deg = 3 g = 1 κ = 0
general deg ≥ 4 g ≥ 2 κ = 1type
![Page 121: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/121.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
![Page 122: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/122.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
deg < n + 1 κ = −∞
![Page 123: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/123.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
deg < n + 1 κ = −∞
deg = n + 1 κ = 0
![Page 124: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/124.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
deg < n + 1 κ = −∞
deg = n + 1 κ = 0
deg > n + 1 κ = dim
![Page 125: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/125.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
deg = n + 1 κ = 0
deg > n + 1 κ = dim
![Page 126: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/126.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
Calabi-Yau deg = n + 1 κ = 0
deg > n + 1 κ = dim
![Page 127: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/127.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
Calabi-Yau deg = n + 1 κ = 0
general type deg > n + 1 κ = dim
![Page 128: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/128.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Kodaira dimension
Hypersurfaces in Pn:Kodaira
type degree dimension
Fano deg < n + 1 κ = −∞
Calabi-Yau deg = n + 1 κ = 0
general type deg > n + 1 κ = dim
Products:
dim(X × Y ) = dim X + dim Y
κ(X × Y ) = κ(X ) + κ(Y )
![Page 129: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/129.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 130: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/130.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
![Page 131: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/131.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
![Page 132: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/132.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
For curves, the Hilbert polynomial is determinedby the genus, so this is a natural generalization.
![Page 133: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/133.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Varieties of General Type
X is of general type if κ(X ) = dim X .
Recall: A curve C is of general type iff g(C) ≥ 2.
Genus is replaced with the Hilbert polynomial.
For curves, the Hilbert polynomial is determinedby the genus, so this is a natural generalization.
![Page 134: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/134.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
![Page 135: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/135.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
![Page 136: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/136.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
![Page 137: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/137.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
![Page 138: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/138.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
![Page 139: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/139.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
The Shafarevich Conjecture
Setup: Fix B a smooth projective curve, ∆ ⊆ Ba finite subset, and h a polynomial.f : X → B is an admissible family if
• f is non-isotrivial• for all b ∈ B \ ∆, Xb is a smooth projective
variety, canonically embedded with Hilbertpolynomial h.
“Canonically embedded” means that it isembedded by the global sections of ωm
Xb.
In particular, Xb is of general type.
![Page 140: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/140.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 141: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/141.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
![Page 142: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/142.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
![Page 143: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/143.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
(R) Rigidity: Admissible families do not admitnon-trivial deformations.
![Page 144: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/144.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
(R) Rigidity: Admissible families do not admitnon-trivial deformations.
(H) Hyperbolicity: If there exist admissible families,then B \ ∆ is hyperbolic.
![Page 145: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/145.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Shafarevich Conjecture
(B) Boundedness: There are only finitely manydeformation types of admissible families.
(R) Rigidity: Admissible families do not admitnon-trivial deformations.
(H) Hyperbolicity: If there exist admissible families,then B \ ∆ is hyperbolic.
(WB) Weak Boundedness:If f : X → B is an admissible family, thendeg f∗ωm
X/B is bounded in terms of B, ∆, h, m.
![Page 146: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/146.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
![Page 147: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/147.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(WB) ⇒ (H)
![Page 148: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/148.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(K , 2002) (WB) ⇒ (H)
![Page 149: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/149.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(K , 2002) (WB) ⇒ (H)
Hence Weak Boundedness is the key statementtowards proving (B) and (H).
![Page 150: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/150.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Weak Boundedness
Moduli Theory + (WB) ⇒ (B)
(K , 2002) (WB) ⇒ (H)
Hence Weak Boundedness is the key statementtowards proving (B) and (H).
(WB) Weak Boundedness:If f : X → B is an admissible family, thendeg f∗ωm
X/B is bounded in terms of B, ∆, h, m.
![Page 151: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/151.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 152: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/152.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History
![Page 153: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/153.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History
“modern” = last 25 years
![Page 154: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/154.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History
“modern” ≈ last 10 years
![Page 155: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/155.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
![Page 156: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/156.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
BEAUVILLE (1981):(H) holds for dim(X/B) = 1.
![Page 157: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/157.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
BEAUVILLE (1981):(H) holds for dim(X/B) = 1.
CATANESE-SCHNEIDER (1994):(H) can be used for giving upper bounds on thesize of automorphisms groups of varieties ofgeneral type.
![Page 158: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/158.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
BEAUVILLE (1981):(H) holds for dim(X/B) = 1.
CATANESE-SCHNEIDER (1994):(H) can be used for giving upper bounds on thesize of automorphisms groups of varieties ofgeneral type.
SHOKUROV (1995):(H) can be used to prove quasi-projectivity ofmoduli spaces of varieties of general type.
![Page 159: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/159.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
![Page 160: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/160.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
![Page 161: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/161.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
K (1997):(H) holds for dim(X/B) = 2.
![Page 162: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/162.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
K (1997):(H) holds for dim(X/B) = 2.
K (2000):(H) holds for dim(X/B) arbitrary.
![Page 163: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/163.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History: Hyperbolicity
MIGLIORINI (1995):(H) holds for dim(X/B) = 2 and g(B) = 1.
K (1996):(H) holds for dim(X/B) arbitrary and g(B) = 1.
K (1997):(H) holds for dim(X/B) = 2.
K (2000):(H) holds for dim(X/B) arbitrary.
VIEHWEG AND ZUO (2002):Brody hyperbolicity holds as well.
![Page 164: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/164.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):
![Page 165: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/165.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and
![Page 166: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/166.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
![Page 167: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/167.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
As a byproduct of their proof they also obtainedthat (H) holds in these cases.
![Page 168: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/168.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
As a byproduct of their proof they also obtainedthat (H) holds in these cases.
K (2002), VIEHWEG AND ZUO (2002):(WB) holds under more general assumptions.
![Page 169: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/169.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Modern History:(Weak) Boundedness
BEDULEV AND VIEHWEG (2000):• (B) holds for dim(X/B) = 2, and• (WB) holds for dim(X/B) arbitrary.
As a byproduct of their proof they also obtainedthat (H) holds in these cases.
K (2002), VIEHWEG AND ZUO (2002):(WB) holds under more general assumptions.
K (2002):(WB) implies (H).
![Page 170: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/170.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
![Page 171: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/171.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
![Page 172: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/172.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
![Page 173: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/173.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
![Page 174: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/174.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
More History
...more related results by
FALTINGS (1983)
ZHANG (1997)
OGUISO AND VIEHWEG (2001)
..and more.
![Page 175: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/175.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 176: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/176.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
![Page 177: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/177.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,
![Page 178: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/178.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
![Page 179: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/179.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption.
![Page 180: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/180.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.
![Page 181: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/181.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
![Page 182: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/182.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Many details change:
Instead of ∆ being a finite subset,it is a 1-codimensional subvariety of B.
A family being non-isotrivial is no longer a goodassumption. It has to be replaced by a familyhaving maximal variation in moduli.
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.That is, the log-Kodaira dimension of B ismaximal: κ(B, ∆) = dim B.
![Page 183: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/183.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
![Page 184: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/184.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
K (2003), VIEHWEG-ZUO, (2003):Viehweg’s conjecture holds for B = Pn.
![Page 185: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/185.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Higher dimensional bases
Viehweg’s Conjecture (2001): If there exists anadmissible family with maximal variation inmoduli, then (B, ∆) is of log general type.
K (2003), VIEHWEG-ZUO, (2003):Viehweg’s conjecture holds for B = Pn.
K (1997):If f : X → B is admissible and B is an abelianvariety, then ∆ 6= ∅.
![Page 186: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/186.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
But, what about Rigidity?
![Page 187: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/187.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Outline
5 GeneralizationsHigher Dimensional FibersKodaira DimensionAdmissible FamiliesWeak BoundednessRecent ResultsHigher Dimensional BasesRigidity
![Page 188: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/188.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
![Page 189: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/189.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and
![Page 190: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/190.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
![Page 191: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/191.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
Let Y → B be an arbitrary non-isotrivial familyof curves of genus ≥ 2, and
C a smooth projective curve of genus ≥ 2.
C
f
B
X Y C= ×
CC
b bX Y C= ×
YbY C
b
C
![Page 192: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/192.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
f : X = Y × C → B is an admissible family,and
C
f
B
X Y C= ×
CC
b bX Y C= ×
YbY C
b
C
![Page 193: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/193.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity – An Example
f : X = Y × C → B is an admissible family,and
any deformation of C gives a deformation of f .
C
f
B
X Y C= ×
CC
b bX Y C= ×
YbY C
b
C
![Page 194: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/194.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
![Page 195: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/195.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
Question: Under what additional conditionsdoes (R) hold?
![Page 196: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/196.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
Question: Under what additional conditionsdoes (R) hold?
K (2004), VIEHWEG-ZUO, (2004):Rigidity holds for strongly non-isotrivial families.
![Page 197: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/197.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Rigidity
Therefore, (R) fails.
Question: Under what additional conditionsdoes (R) hold?
K (2004), VIEHWEG-ZUO, (2004):Rigidity holds for strongly1non-isotrivial families.
1unfortunately the margin is not wide enough to define
this term.
![Page 198: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/198.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Current Work
Work in progress:
![Page 199: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/199.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Current Work
Work in progress:
• Geometric description of strongly non-isotrivialfamilies.
![Page 200: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/200.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
GeneralizationsHigher Dimensional Fibers
Kodaira Dimension
Admissible Families
Weak Boundedness
Recent Results
Higher Dimensional Bases
Rigidity
Where Next?
Current Work
Work in progress:
• Geometric description of strongly non-isotrivialfamilies.
• Joint work with Stefan Kebekus (Köln):Families over two-dimensional bases.
![Page 201: Families of varieties of general type: the Shafarevich ...kovacs/pdf/colloquium...Families Examples and Definitions Isotrivial and Non-isotrivial Shafarevich’s Conjecture The Proof](https://reader035.fdocuments.net/reader035/viewer/2022071408/6100c889a155cd08e7156f19/html5/thumbnails/201.jpg)
ShafarevichConjecture
Sándor Kovács
Curves
Families
Shafarevich’sConjecture
The Proof
Generalizations
Acknowledgement
This presentation was made using thebeamertex LATEX macropackage of Till Tantau.http://latex-beamer.sourceforge.net