Post on 20-Nov-2021
Batyrev Mirror Symmetry
Wanlong Zheng
Sept 9th, 2021
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 1 / 28
Table of Contents
1 Recap on polytopes and toric geometry
2 Calabi-Yau toric hypersurfaces
3 Quintic threefold example
4 What’s next from here
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 2 / 28
Polytopes
Definition
A polytope ∆ ⊂ MR is the convex hull of a finite set of points. It isintegral (or a lattice polytope) if the vertices lie inside M.
Definition
The polar polytope of ∆ is a polytope in the dual lattice
∆ := v ∈ NR | 〈m, v〉 ≤ 1 for all vertices m of ∆.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 3 / 28
Polytopes
Definition
A polytope ∆ ⊂ MR is the convex hull of a finite set of points. It isintegral (or a lattice polytope) if the vertices lie inside M.
Definition
The polar polytope of ∆ is a polytope in the dual lattice
∆ := v ∈ NR | 〈m, v〉 ≤ 1 for all vertices m of ∆.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 3 / 28
Polar polytope examples
∆
∆
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 4 / 28
Polar polytope examples
∆ ∆
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 4 / 28
Polytopes
Definition
An integral polytope ∆ is reflexive if origin is in the interior of ∆, and ∆ isalso integral.
Lemma
If ∆ is reflexive, then the only integral point in the interior is origin. Also ∆
is reflexive.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 5 / 28
Polytopes
Definition
An integral polytope ∆ is reflexive if origin is in the interior of ∆, and ∆ isalso integral.
Lemma
If ∆ is reflexive, then the only integral point in the interior is origin. Also ∆
is reflexive.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 5 / 28
Toric varieties from polytopes
1 Each lattice point m = (a1, . . . , ad) in M = Zd represents a monomial
ta11 . . . tadd .
2 Let ∆∩Zd = m0, . . . ,ms . There is a map
(C∗)d → (C∗)s+1 → Ps
t = (t1, . . . , td) 7→ (tm0 , . . . , tms ) 7→ [tm0 : · · · : tms ]
3 X∆ is defined to be the closure of the image.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 6 / 28
Toric varieties from polytopes
1 Each lattice point m = (a1, . . . , ad) in M = Zd represents a monomial
ta11 . . . tadd .
2 Let ∆∩Zd = m0, . . . ,ms . There is a map
(C∗)d → (C∗)s+1 → Ps
t = (t1, . . . , td) 7→ (tm0 , . . . , tms ) 7→ [tm0 : · · · : tms ]
3 X∆ is defined to be the closure of the image.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 6 / 28
Toric varieties from polytopes
1 Each lattice point m = (a1, . . . , ad) in M = Zd represents a monomial
ta11 . . . tadd .
2 Let ∆∩Zd = m0, . . . ,ms . There is a map
(C∗)d → (C∗)s+1 → Ps
t = (t1, . . . , td) 7→ (tm0 , . . . , tms ) 7→ [tm0 : · · · : tms ]
3 X∆ is defined to be the closure of the image.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 6 / 28
Toric varieties from polytopes
Proposition
The fan of the toric variety X∆ is the normal fan of ∆.
Previous example:
give rises to P1 ×P1 → P8, while gives XP∼= P1 ×P1 → P3.
(Line bundle L⊗2 vs L here)
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 7 / 28
Toric varieties from polytopes
Fact: the dualizing sheaf of a toric variety X is
OX (−∑ρ
Dρ).
Then X is Gorenstein iff∑ρDρ is Cartier, and is Fano iff
∑ρDρ is Cartier
and ample.
(Recall X is Gorenstein iff the dualizing sheaf is a line bundle; X is furtherFano if the dual of the dualizing sheaf is ample)
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 8 / 28
Toric varieties from polytopes
Proposition
∆ is reflexive iff X∆ is Fano.
Example: Pn, P1 ×P1.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 9 / 28
Toric varieties from polytopes
Proposition
∆ is reflexive iff X∆ is Fano.
Example: Pn, P1 ×P1.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 9 / 28
Table of Contents
1 Recap on polytopes and toric geometry
2 Calabi-Yau toric hypersurfaces
3 Quintic threefold example
4 What’s next from here
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 10 / 28
Intuitions of what’s going on
(Assume everything is smooth)
Reflexive polytope ∆→ Fano toric variety X∆ and a general divisor V ∈ |−KX |.
A nice anticanonical hypersurface in X usually are CY (by adjunctionKV = 0).
Idea: taking dual of everything. Consider dual polytope ∆, a generaldivisor V ∈ |−KX∆ |. Then V and V should be mirrors of each other.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 11 / 28
Intuitions of what’s going on
(Assume everything is smooth)
Reflexive polytope ∆→ Fano toric variety X∆ and a general divisor V ∈ |−KX |.
A nice anticanonical hypersurface in X usually are CY (by adjunctionKV = 0).
Idea: taking dual of everything. Consider dual polytope ∆, a generaldivisor V ∈ |−KX∆ |. Then V and V should be mirrors of each other.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 11 / 28
Intuitions of what’s going on
(Assume everything is smooth)
Reflexive polytope ∆→ Fano toric variety X∆ and a general divisor V ∈ |−KX |.
A nice anticanonical hypersurface in X usually are CY (by adjunctionKV = 0).
Idea: taking dual of everything. Consider dual polytope ∆, a generaldivisor V ∈ |−KX∆ |. Then V and V should be mirrors of each other.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 11 / 28
Controlling singularities
Subdivisions (blowups), with some imposed conditions.
Definition
Let ∆ be a reflexive polytope. A projective subdivision of its normal fan isanother fan Σ such that:
1 Generators of the rays of Σ are in ∆ ∩N \ 0, and
2 XΣ is projective and simplicial (i.e. Σ is polytopal and simplicial).
There exists maximal projective subdivisions.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 12 / 28
Controlling singularities
Subdivisions (blowups), with some imposed conditions.
Definition
Let ∆ be a reflexive polytope. A projective subdivision of its normal fan isanother fan Σ such that:
1 Generators of the rays of Σ are in ∆ ∩N \ 0, and
2 XΣ is projective and simplicial (i.e. Σ is polytopal and simplicial).
There exists maximal projective subdivisions.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 12 / 28
Controlling singularities
Subdivisions (blowups), with some imposed conditions.
Definition
Let ∆ be a reflexive polytope. A projective subdivision of its normal fan isanother fan Σ such that:
1 Generators of the rays of Σ are in ∆ ∩N \ 0, and
2 XΣ is projective and simplicial (i.e. Σ is polytopal and simplicial).
There exists maximal projective subdivisions.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 12 / 28
Controlling singularities
Subdivisions correspond to birational morphisms f : XΣ → X∆.
Lemma
If Σ is a projective subdivision, then
XΣ is a Gorenstein orbifold.
∆ is the polytope associated to −KXΣ.
−KXΣis semi-ample.
f ∗(KX∆) = KXΣ
.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 13 / 28
Controlling singularities
Subdivisions correspond to birational morphisms f : XΣ → X∆.
Lemma
If Σ is a projective subdivision, then
XΣ is a Gorenstein orbifold.
∆ is the polytope associated to −KXΣ.
−KXΣis semi-ample.
f ∗(KX∆) = KXΣ
.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 13 / 28
Controlling singularities
Proposition
If ∆ is a reflexive polytope of dimension n, then a general memberV ∈ |−KX∆
| is CY of dimension n− 1.
If Σ is a projective subdivision, then a general member V ∈ |−KXΣ| is
a CY orbifold.
Special case when V is a 3-fold. If Σ is a maximal projective subdivision,then V ⊂ XΣ and V turn out to be smooth.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 14 / 28
Controlling singularities
Proposition
If ∆ is a reflexive polytope of dimension n, then a general memberV ∈ |−KX∆
| is CY of dimension n− 1.
If Σ is a projective subdivision, then a general member V ∈ |−KXΣ| is
a CY orbifold.
Special case when V is a 3-fold. If Σ is a maximal projective subdivision,then V ⊂ XΣ and V turn out to be smooth.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 14 / 28
Batyrev mirror
Let ∆ be a n-dimensional reflexive polytope, Σ a maximal projectivesubdivision of the normal fan of ∆, and V a general anticanonicalhypersurface of XΣ. Let ∆ and V be the Batyrev mirror.
Theorem
The hodge numbers are related by
h1,1(V ) = hn−2,1(V ) and hn−2,1(V ) = h1,1(V ).
i.e. for 3-folds, this suffices; but ≥ 4-folds, it’s not enough.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 15 / 28
Batyrev mirror
Let ∆ be a n-dimensional reflexive polytope, Σ a maximal projectivesubdivision of the normal fan of ∆, and V a general anticanonicalhypersurface of XΣ. Let ∆ and V be the Batyrev mirror.
Theorem
The hodge numbers are related by
h1,1(V ) = hn−2,1(V ) and hn−2,1(V ) = h1,1(V ).
i.e. for 3-folds, this suffices; but ≥ 4-folds, it’s not enough.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 15 / 28
Table of Contents
1 Recap on polytopes and toric geometry
2 Calabi-Yau toric hypersurfaces
3 Quintic threefold example
4 What’s next from here
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 16 / 28
Quintic threefold example
Let ∆ be the convex hull of−1−1−1−1
,
4−1−1−1
,
−14−1−1
,
−1−14−1
,
−1−1−14
.
This is the standard simplex Conv(0, e1, e2, e3, e4) dilated by 5 times, thensubtracted by (−1,−1,−1,−1). It has 125 lattice points.
This is reflexive, and X∆ = P4. So V is an arbitrary quintic threefold in P4.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 17 / 28
Quintic threefold example
Let ∆ be the convex hull of−1−1−1−1
,
4−1−1−1
,
−14−1−1
,
−1−14−1
,
−1−1−14
.
This is the standard simplex Conv(0, e1, e2, e3, e4) dilated by 5 times, thensubtracted by (−1,−1,−1,−1). It has 125 lattice points.
This is reflexive, and X∆ = P4.
So V is an arbitrary quintic threefold in P4.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 17 / 28
Quintic threefold example
Let ∆ be the convex hull of−1−1−1−1
,
4−1−1−1
,
−14−1−1
,
−1−14−1
,
−1−1−14
.
This is the standard simplex Conv(0, e1, e2, e3, e4) dilated by 5 times, thensubtracted by (−1,−1,−1,−1). It has 125 lattice points.
This is reflexive, and X∆ = P4. So V is an arbitrary quintic threefold in P4.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 17 / 28
Quintic threefold example
Hodge diamond of a quintic threefold:
1
0 0
0 1 0
1 101 101 1
0 1 0
0 0
1
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 18 / 28
Quintic threefold example
If mirror pair, then hp,q(V ) = h3−p,q(V ).
Since h1,1(V ) = 1, we want
dimH1(V ,TV ) = dimH1(V ,ΩV ) = 1.
So V should be a one-parameter family.
On the other hand, dim of moduli for quintic threefolds is
h0(OP4(5)) − 1 − dimPGL(5, C) = 126 − 1 − 24 = 101.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 19 / 28
Quintic threefold example
If mirror pair, then hp,q(V ) = h3−p,q(V ).
Since h1,1(V ) = 1, we want
dimH1(V ,TV ) = dimH1(V ,ΩV ) = 1.
So V should be a one-parameter family.
On the other hand, dim of moduli for quintic threefolds is
h0(OP4(5)) − 1 − dimPGL(5, C) = 126 − 1 − 24 = 101.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 19 / 28
Quintic threefold example
If mirror pair, then hp,q(V ) = h3−p,q(V ).
Since h1,1(V ) = 1, we want
dimH1(V ,TV ) = dimH1(V ,ΩV ) = 1.
So V should be a one-parameter family.
On the other hand, dim of moduli for quintic threefolds is
h0(OP4(5)) − 1 − dimPGL(5, C) = 126 − 1 − 24 = 101.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 19 / 28
Quintic threefold example
If mirror pair, then hp,q(V ) = h3−p,q(V ).
Since h1,1(V ) = 1, we want
dimH1(V ,TV ) = dimH1(V ,ΩV ) = 1.
So V should be a one-parameter family.
On the other hand, dim of moduli for quintic threefolds is
h0(OP4(5)) − 1 − dimPGL(5, C) = 126 − 1 − 24 = 101.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 19 / 28
Quintic threefold example
The polar polytope of ∆ is
∆ = Conv(e1, e2, e3, e4, (−1,−1,−1,−1)).
It has 6 lattice points.
X∆ is identified with the quotient P4/G where
G = (a1, . . . , a5) ∈ (Z/5)5 |∑
ai = 0/(Z/5),
where quotient is by the diagonal subgroup, and G acts by multiplication ofroots of unity.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 20 / 28
Quintic threefold example
The polar polytope of ∆ is
∆ = Conv(e1, e2, e3, e4, (−1,−1,−1,−1)).
It has 6 lattice points.
X∆ is identified with the quotient P4/G where
G = (a1, . . . , a5) ∈ (Z/5)5 |∑
ai = 0/(Z/5),
where quotient is by the diagonal subgroup, and G acts by multiplication ofroots of unity.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 20 / 28
Quintic threefold example
Anticanonicals in X∆ are defined by
x50 + x51 + x52 + x53 + x54 +ψx0x1x2x3x4 = 0
for coordinates (xi ) of P4.
i.e. pre-controlling singularity, V
is the quotient of the above hypersurfacein P4 by action of G (generically hypersurface is smooth, but quotient isnot).
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 21 / 28
Quintic threefold example
Anticanonicals in X∆ are defined by
x50 + x51 + x52 + x53 + x54 +ψx0x1x2x3x4 = 0
for coordinates (xi ) of P4.
i.e. pre-controlling singularity, V
is the quotient of the above hypersurfacein P4 by action of G (generically hypersurface is smooth, but quotient isnot).
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 21 / 28
Quintic threefold example
Anticanonicals in X∆ are defined by
x50 + x51 + x52 + x53 + x54 +ψx0x1x2x3x4 = 0
for coordinates (xi ) of P4.
i.e. pre-controlling singularity, V
is the quotient of the above hypersurfacein P4 by action of G (generically hypersurface is smooth, but quotient isnot).
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 22 / 28
Quintic threefold example
Anticanonicals in X∆ are defined by
x50 + x51 + x52 + x53 + x54 +ψx0x1x2x3x4 = 0
for coordinates (xi ) of P4.
i.e. pre-controlling singularity, V
is the quotient of the above hypersurfacein P4 by action of G (generically hypersurface is smooth, but quotient isnot).
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 22 / 28
Quintic threefold example
Finally, pick a maximal projective subdivision and consider the propertransform of V
.
Concretely, the singular locus of the quotient consists of 10 curves
e.g. C01 = x0 = x1 = 0, x52 + x53 + x54 = 0/G
and 10 points
e.g. p012 = x0 = x1 = x2 = 0, x53 + x54 = 0/G .
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 23 / 28
Quintic threefold example
Finally, pick a maximal projective subdivision and consider the propertransform of V
.
Concretely, the singular locus of the quotient consists of 10 curves
e.g. C01 = x0 = x1 = 0, x52 + x53 + x54 = 0/G
and 10 points
e.g. p012 = x0 = x1 = x2 = 0, x53 + x54 = 0/G .
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 23 / 28
Quintic threefold example
Fact: resolving Cij gives 4× 10 new exceptionals, and pijk 6× 10, giving atotal of 100 new divisors. Thus h1,1(V ) = 101.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 24 / 28
Table of Contents
1 Recap on polytopes and toric geometry
2 Calabi-Yau toric hypersurfaces
3 Quintic threefold example
4 What’s next from here
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 25 / 28
Toric complete intersections
Start with an (r + d)-dimensional reflexive polytope ∆, together with adecomposition as a Minkowski sum
∆ = ∆1 + · · ·+∆r
where each ∆i is a lattice polytope containing origin. This is known asa nef-partition.
Each ∆i has a generic hypersurface Vi , and⋂Vi is a d-dimensional
complete intersection CY variety, that needs to be resolved.
Instead of taking ∆, one defines using information from all ∆i :
∇i = v ∈ NR |〈v , a〉 ≤ 1 for all a ∈ ∆i
and 〈v , b〉 ≤ 0 for all b ∈ ∆j 6=i .
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 26 / 28
Toric complete intersections
Start with an (r + d)-dimensional reflexive polytope ∆, together with adecomposition as a Minkowski sum
∆ = ∆1 + · · ·+∆r
where each ∆i is a lattice polytope containing origin. This is known asa nef-partition.
Each ∆i has a generic hypersurface Vi , and⋂Vi is a d-dimensional
complete intersection CY variety, that needs to be resolved.
Instead of taking ∆, one defines using information from all ∆i :
∇i = v ∈ NR |〈v , a〉 ≤ 1 for all a ∈ ∆i
and 〈v , b〉 ≤ 0 for all b ∈ ∆j 6=i .
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 26 / 28
Toric complete intersections
Start with an (r + d)-dimensional reflexive polytope ∆, together with adecomposition as a Minkowski sum
∆ = ∆1 + · · ·+∆r
where each ∆i is a lattice polytope containing origin. This is known asa nef-partition.
Each ∆i has a generic hypersurface Vi , and⋂Vi is a d-dimensional
complete intersection CY variety, that needs to be resolved.
Instead of taking ∆, one defines using information from all ∆i :
∇i = v ∈ NR |〈v , a〉 ≤ 1 for all a ∈ ∆i
and 〈v , b〉 ≤ 0 for all b ∈ ∆j 6=i .
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 26 / 28
Gross-Siebert
Given a CY manifold X , degenerate it into a union of toric varietiesglued along toric strata.
Take the dual intersection complex, and dualize similar to taking polarpolytopes.
Glue back stuff.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 27 / 28
Gross-Siebert
Given a CY manifold X , degenerate it into a union of toric varietiesglued along toric strata.
Take the dual intersection complex, and dualize similar to taking polarpolytopes.
Glue back stuff.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 27 / 28
Gross-Siebert
Given a CY manifold X , degenerate it into a union of toric varietiesglued along toric strata.
Take the dual intersection complex, and dualize similar to taking polarpolytopes.
Glue back stuff.
Wanlong Zheng Batyrev Mirror Symmetry Sept 9th, 2021 27 / 28
Reference
Victor Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yauhypersurfaces in toric varieties.
Cox, Katz Chapter 4.
Mattia Talpo. Batyrev mirror symmetry.
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