Geometric Lang-Vojta's conjecture in P2aturchet/Index Folder... · Geometric Lang-Vojta’s...

Geometric Lang-Vojta’s conjecture in P2

Amos Turchet

Universita degli studi di Udine

On Lang and Vojta’s conjecturesCentre International de Rencontres Mathematiques

March 6th 2014

Amos Turchet (University of Udine) Geometric Lang-Vojta’s conjecture in P2 March 6th 2014 1 / 36

Table of contents

1 Hyperbolicity

2 Geometric Lang-Vojta’s conjecture

3 Logarithmic Geometry

Hyperbolicity Brody Hyperbolicity

Basic Definition

DefinitionA complex manifold M is said to be (Brody)-hyperbolic if every holomorphicmap f : C→ M is constant.

Brody Theorem implies that for compact manifolds Brody-hyperbolicity isequivalent to being hyperbolic in the sense of Kobayashi.

Fundamental problemDescribe geometric attributes of X that implies hyperbolicity.

Basic Definition

Hyperbolicity Riemann Surfaces

Hyperbolicity for RS

In dimension 1, i.e. for Riemann surfaces, the following easy fact holds:

TheoremIf X is a Riemann surface of genus at least 2, every holomorphic map C→ Xis constant. Therefore X is hyperbolic.

The theorem follows essentially from Liouville’s Theorem. In particular thegeometric condition genus ≥ 2 implies hyperbolicity.

The theorem follows essentially from Liouville’s Theorem.

In particular thegeometric condition genus ≥ 2 implies hyperbolicity.

The theorem follows essentially from Liouville’s Theorem. In particular thegeometric condition genus ≥ 2 implies hyperbolicity.

Hyperbolicity for affine curves

There is an analogous results for affine curves, i.e. complements of a finiteset of points in a RS.

TheoremIf X is a Riemann surface of genus g(X ), and S a finite set of points of X .Then

X \ S is hyperbolic ⇐⇒ 2g(X )− 2 +#S > 0

Follows essentially from Picard’s Little Theorem. Notice how the geometriccondition 2g(X )− 2 +#S > 0 now encodes information about the pointsat infinity of X .

Q: What happens in higher dimensions?

Follows essentially from Picard’s Little Theorem.

Notice how the geometriccondition 2g(X )− 2 +#S > 0 now encodes information about the pointsat infinity of X .

Hyperbolicity Hyperbolicity in Higher Dimensions

Hyperbolicity in higher dimensions

Complete geometric description of hyperbolic manifolds in dimension greaterthan one is still out of reach.

Kobayashi proposed a famous conjecture for manifold embedded in projec-tive space

Conjecture (Kobayashi)(a) a very general hypersurface D ⊂ Pn+1 with n ≥ 2 of degree deg D ≥

2n + 1 is hyperbolic;(b) Pn \D with n ≥ 2 is hyperbolic for a very general hypersurface D ⊂ Pn

of degree deg D ≥ 2n + 1.

Results in many important cases: works by Y.T. Siu, Demailly, El Goul,McQuillan, Diverio, Merker, Rousseau, Pacienza...

Problems:

Achieve the conjecture’s bound is really hard;

Full description of hyperbolic manifold not embedded in Pn seems verydifficult to discuss;

Strategy: Lower the expectation.

Problems:

Achieve the conjecture’s bound is really hard;

Full description of hyperbolic manifold not embedded in Pn seems verydifficult to discuss;

Strategy: Lower the expectation.

Hyperbolicity Algebraic Hyperbolicity

Hyperbolicity for Algebraic Varieties

Instead of complex manifolds we focus on algebraic projective varieties.

Theorem (Demailly)Let X be a compact complex hyperbolic algebraic variety. Then there existsan ε > 0 such that every compact irreducible curve C ⊂ X satisfies:

(?) − χ(C) = 2g(C)− 2 ≥ ε deg C

where C is the normalization of the curve C, and the degree is calculatedrespect to an ample divisor in X.

Property (?) is defined in a pure algebraic way.

(?) − χ(C) = 2g(C)− 2 ≥ ε deg C

Algebraic Hyperbolicity

Definition (Demailly)A projective algebraic variety X is said to be algebraically hyperbolic if (?)holds.

Hyperbolicity implies algebraic hyperbolicity; hence alg. hyperbolicity canbe used to test whether a projective variety can be hyperbolic.

Algebraic Hyperbolicity is directly related to Vojta conjectures.

Algebraic Hyperbolicity for affine varieties

DefinitionGiven a projective algebraic variety X and a normal crossing divisor D, X \Dis said to be algebraically hyperbolic if there exists an ε > 0 such that everycompact irreducible curve C ⊂ X satisfies:

−χ(C) = 2g(C)− 2 +#S ≥ ε deg C

where S is the set ν−1(D) and ν : C → C is the normalization.

We call X \D weakly algebraically hyperbolic if the previous bound reads

max1, 2g(C)− 2 +#S ≥ ε deg C

Algebraic Hyperbolicity for affine varieties

DefinitionGiven a projective algebraic variety X and a normal crossing divisor D, X \Dis said to be algebraically hyperbolic if there exists an ε > 0 such that everycompact irreducible curve C ⊂ X satisfies:

−χ(C) = 2g(C)− 2 +#S ≥ ε deg C

where S is the set ν−1(D) and ν : C → C is the normalization.

We call X \D weakly algebraically hyperbolic if the previous bound reads

max1, 2g(C)− 2 +#S ≥ ε deg C

Geometric Lang-Vojta’s conjecture Log general type surfaces

The conjecture

Conjecture (Geometric Lang-Vojta)

Let X be a smooth projective algebraic surface with canonical divisor KXand let D be a reduced normal crossing divisor on X . Let X = X \D be thecomplement of the support of D. If X is of log-general type, i.e. if D + KXis big, then X is weakly algebraically hyperbolic.

Previous conjecture can be seen as a geometric version of the famous Lang-Vojta’s conjecture predicting degeneracy of integral points on log-generaltype surfaces.

The “weakly” cannot be suppressed: there are examples of log general typesurface which contains Gm

The conjecture

Log general type surface, not AH

Geometric Lang-Vojta’s conjecture The P2 case

The case of P2

When X = P2 previous conjecture takes the form:

ConjectureLet D be a reduced plane curve with normal crossing and let X = P2 \D.If deg D ≥ 4, then X is weakly algebraically hyperbolic.

As in example before: in the case deg D = 4 the complement X is not(algebraic) hyperbolic!

The case of P2

When X = P2 previous conjecture takes the form:

ConjectureLet D be a reduced plane curve with normal crossing and let X = P2 \D.If deg D ≥ 4, then X is weakly algebraically hyperbolic.

As in example before: in the case deg D = 4 the complement X is not(algebraic) hyperbolic!

Known results

Lang-Vojta Conjecture for P2 is known for complements of very generalcurves of degree at least 5 (Xi Chen, Pacienza and Rousseau).

When the degree of D is 4 the conjecture is known for generic D having atleast three components:

The four line case follows from an extension of Mason’s ABC theorem(Brownawell and Masser);

The three component case can be reduced to a S-unit gcd problem(Corvaja and Zannier).

All these results have structural obstruction for extensions to the remainingcases, namely D with less than three irreducible components and deg D = 4.

Known results

Geometric Lang-Vojta’s conjecture Ideas for extensions

Goal and ideas

Goal: prove Geometric LV for the complement in P2 of a (very) genericquartic (even irreducible!).

Idea: consider the three component case as a deformation of the irreduciblecase.

A similar idea is used in Xi Chen’s proof of LV Conjecture: he degeneratethe boundary D to a union of hyperplanes and then applying the knownresults for P2 \ ∪i=1,5Hi .

His argument is involved and requires the degree of D to be at least 5 towork.

In our case there are problems that need to be addressed.

Goal and ideas

An example

Dt1Dt2

DtsCt0Ct2

Problems

No control on the variations of the number of points of intersection.

Curves can become reducible.

A component of the curve could coincide with an irreducible componentof the divisor.

Solution: use logarithmic Geometry (take care of multiplicities of in-tersection).

Problems

Main result

Theorem (T. - WAH for complements of quartics)P2 \D is weakly algebraic hyperbolic for every simple normal crossing divisorD of degree 4 which flattly and log smoothly deforms to a conic and twolines.

Restrictions:

Need to consider only simple normal crossing divisors (possible exten-sions are announced).

Deformations need to be logarithmically smooth which is stronger thanflat.

Argument requires to work in logarithmic category.

Main result

Restrictions:

Main result

Restrictions:

Main result

Restrictions:

Sketch of the argument

We start from Corvaja and Zannier’s result (CZ in the sequel) for the com-plement of a conic and two lines that read as follows

Theorem (Corvaja and Zannier, 2008)Let X = P2 \D, where D is a quartic consisting of the union of a smoothconic and two lines in general position. Let C be a smooth complete al-gebraic curve and S ⊂ C a finite set of points. Then for every morphismf : C → P2 such that f −1(D) ⊂ S the following holds:

deg f (C) ≤ 215 · 35 ·max1,χS(C)

The main idea is to reformulate this results in terms of vanishing of certainmoduli spaces of log stable maps KΓ(P

Then use the properness of the stack KΓ(P2,D) to extend the

result for very general D deforming flattly to a conic and two lines.

deg f (C) ≤ 215 · 35 ·max1,χS(C)

result for very general D deforming flattly to a conic and two lines.

deg f (C) ≤ 215 · 35 ·max1,χS(C)

result for very general D deforming flattly to a conic and two lines.Amos Turchet (University of Udine) Geometric Lang-Vojta’s conjecture in P2 March 6th 2014 19 / 36

Logarithmic Geometry

Why Logarithmic Geometry

If in the deformation argument multiplicities of intersection are fixed,S can be controlled.

Log Geometry gives a way to define point of intersection even for irre-ducible components coinciding with a component of the divisor.

Each curve C has a “natural” log structure coming from the set S.

Logarithmic Geometry Basic definitions

Fundamental example

X a smooth variety, D a normal crossing divisor on X . Consider the followingsheaf:

M(V ) = f ∈ OX (V ) : f |V \D ∈ OX (V \D)∗

M has the structure of sheaf of monoids on X (sum of invertiblefunctions needs not to be invertible);

There exists a well-defined map α : M → OX which is the identitywhen restricted to O∗X ;

Derivatives of sections of M generates H0(X ,Ω1X (D)) where Ω1

X (D)is the sheaf of differential forms with logarithmic poles along D (namelogarithmic geometry).

The normal crossing condition can be rephrased in local terms ifone considers the etale topology instead of the Zariski topology.

Fundamental example

M(V ) = f ∈ OX (V ) : f |V \D ∈ OX (V \D)∗

Fundamental example

M(V ) = f ∈ OX (V ) : f |V \D ∈ OX (V \D)∗

Fundamental example

M(V ) = f ∈ OX (V ) : f |V \D ∈ OX (V \D)∗

Fundamental example

M(V ) = f ∈ OX (V ) : f |V \D ∈ OX (V \D)∗

Fundamental example

M(V ) = f ∈ OX (V ) : f |V \D ∈ OX (V \D)∗

Logarithmic Scheme

DefinitionA logarithmic scheme is a couple (X ,M) where X is a scheme andM is asheaf of monoid on the etale site of X , called a log structure together witha morphisms of sheaves of monoids α :M→ OX such that α−1O∗X → O∗Xis an isomorphism.

A morphism of log schemes (X ,MX ) → (Y ,MY ) is a couple (f , f [)where f : X → Y is a morphisms of schemes and f [ : f ∗MY →MX is amorphism of log structures on X .

Here f ∗MX is the logarithmic structure associated to the map f −1(MY )→f −1OY → OX , called the inverse image.

Logarithmic Scheme

DefinitionA logarithmic scheme is a couple (X ,M) where X is a scheme andM is asheaf of monoid on the etale site of X , called a log structure together witha morphisms of sheaves of monoids α :M→ OX such that α−1O∗X → O∗Xis an isomorphism.

A morphism of log schemes (X ,MX ) → (Y ,MY ) is a couple (f , f [)where f : X → Y is a morphisms of schemes and f [ : f ∗MY →MX is amorphism of log structures on X .

Here f ∗MX is the logarithmic structure associated to the map f −1(MY )→f −1OY → OX , called the inverse image.

Logarithmic Geometry Log Stable maps

Log curves are prestable

Logarithmic geometry appears naturally when considering stable curves.

Every log curve is naturally a pointed nodal curve.

Every pointed and at most nodal curve carries a “canonical” log struc-ture

Logarithmic Geometry Log stable maps and WAH

Back to P2,D

Goal: extend CZ result for D a conic and two lines to log-stable curves.

First we need to address the problem of curves C with more than one irre-ducible components. How to properly extend S = C ∩D?

We need to take into account the nodes, i.e. S will be the set of distinguishedpoints (marked points and nodes).

Moreover we would like to have a well defined notion of S in each irreduciblecomponent. Therefore nodes should count as two points (this can be madeprecise using log charts).

Back to P2,D

Logarithmic Geometry An example

Marked points

log stability

Consider ϕ : C → P2: log-structures are given respectively by S and D.Log stability in this case is equivalent to usual stability plus the followingconditions:

1 For every irreducible component Ci of C such that Ci maps to a degreeone irreducible component of D, SCi contains at least three points.

2 For every irreducible component Cj of C such that Cj maps to the degreetwo irreducible component of D, SCj contains at least four points.

log stability

Extension to log stable maps

With previous definition the following extension of CZ holds

PropositionGiven C,S,D as above, let ϕ : C → P2 be a non-constant stable log-morphism such that ϕ−1(D) ⊂ S. Then the degree of the image ϕ(C)verifies:

deg(ϕ(C)) ≤ A ·max1,χ(C \ S)

Here A is the same constant appearing in CZ Theorem.

Extension to log stable maps

With previous definition the following extension of CZ holds

PropositionGiven C,S,D as above, let ϕ : C → P2 be a non-constant stable log-morphism such that ϕ−1(D) ⊂ S. Then the degree of the image ϕ(C)verifies:

deg(ϕ(C)) ≤ A ·max1,χ(C \ S)

Here A is the same constant appearing in CZ Theorem.

Moduli interpretation

Pivotal observation: previous Proposition can be rephrased in the followingway:

Given a log-stable curve C as before, the degree of the image of every log-stable map to (P2,MD) corresponds to the degree of the Chow class ofthe image curve.

In particular for a map f : (C,MC) → (P2,MD) if f∗([C]) = β forβ ∈ A1(P2) then deg f (C) = deg β.

Hence once β ∈ A1(P2) g = g(β) and n= #S are fixed a log-stable mapsf : (C,MC) → (P2,MD) from a genus g , n-marked curve exists only ifdeg β verifies the inequality of Proposition.

The moduli space of log stable maps

Previous remark can be made precise using the moduli space KΓ(P2,MD)

for a discrete data Γ.

Definition (Discrete data)Let Γ = (β, g , n,~c) be a fourple consisting of the following data:

β ∈ H2(X ,Z) is a curve class;n, g are two non-negative integers (marked points and genus);~c is a n-vector of non-negative integers (multiplicities) that verify:

∑i=1

ci = c1(D) ∩ β

We denote by KΓ(P2,MD) the moduli space of log-stable map to

(P2,MD) of type Γ.

∑i=1

ci = c1(D) ∩ β

(P2,MD) of type Γ.

∑i=1

ci = c1(D) ∩ β

(P2,MD) of type Γ.

CZ equivalent to KΓ((P2,MD), β) = ∅

PropositionGiven a curve B ⊂ X such that S = B ∩ D and deg(B) > Aχs(B), letβ denote the corresponding element of A1(P2). Then the moduli spaceKΓ((P

2,MD), β) is empty for g = g(B), n = #S and every vector ofmultiplicities.

PropositionSuppose that for every plane curve B with degB > AχSBB, where SB =B ∩ D, KΓ((P

2,MD), β) are empty if g = g(B) and n = #SB. Thenevery log-stable map f from a genus g curve C to X with log-structuref ∗MD (and S = SC = f −1(D)) verifies

deg f (C) ≤ AχS(C)

Using properness of KΓ((P2,MD), β)

We have reduced the weak algebraic hyperbolicity problem to the emptinessof a stack. Now we use the following Theorem:

Theorem (Chen, Abramovich-Chen, Gross-Siebert)KΓ((P

2,MD), β) is a proper DM Stack.

The Theorem implies that the condition KΓ((P2,MD), β) = ∅ extends

to a very generic D of degree 4 over a flat family which is logarithmicallysmooth.

With previous propositions this gives the conclusion.

Final comments

Theorem uses in an essential way that MD comes for a simple normalcrossing divisor on D (i.e. P2,MD is a Deligne-Faltings pair).

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