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

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 ∆.

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 ∆.

Polar polytope examples

∆ ∆

Polytopes

Definition

An integral polytope ∆ is reflexive if origin is in the interior of ∆, and ∆ isalso integral.

If ∆ is reflexive, then the only integral point in the interior is origin. Also ∆

is reflexive.

Polytopes

Definition

An integral polytope ∆ is reflexive if origin is in the interior of ∆, and ∆ isalso integral.

If ∆ is reflexive, then the only integral point in the interior is origin. Also ∆

is reflexive.

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.

ta11 . . . tadd .

(C∗)d → (C∗)s+1 → Ps

t = (t1, . . . , td) 7→ (tm0 , . . . , tms ) 7→ [tm0 : · · · : tms ]

ta11 . . . tadd .

(C∗)d → (C∗)s+1 → Ps

t = (t1, . . . , td) 7→ (tm0 , . . . , tms ) 7→ [tm0 : · · · : tms ]

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)

Fact: the dualizing sheaf of a toric variety X is

OX (−∑ρ

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)

Proposition

∆ is reflexive iff X∆ is Fano.

Example: Pn, P1 ×P1.

Proposition

∆ is reflexive iff X∆ is Fano.

Example: Pn, P1 ×P1.

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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.

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.

Definition

Subdivisions correspond to birational morphisms f : XΣ → X∆.

If Σ is a projective subdivision, then

XΣ is a Gorenstein orbifold.

∆ is the polytope associated to −KXΣ.

−KXΣis semi-ample.

f ∗(KX∆) = KXΣ

Subdivisions correspond to birational morphisms f : XΣ → X∆.

If Σ is a projective subdivision, then

XΣ is a Gorenstein orbifold.

∆ is the polytope associated to −KXΣ.

−KXΣis semi-ample.

f ∗(KX∆) = KXΣ

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.

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.

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.

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.

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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.

4−1−1−1

−14−1−1

−1−14−1

−1−1−14

This is reflexive, and X∆ = P4.

So V is an arbitrary quintic threefold in P4.

4−1−1−1

−14−1−1

−1−14−1

−1−1−14

This is reflexive, and X∆ = P4. So V is an arbitrary quintic threefold in P4.

Hodge diamond of a quintic threefold:

1 101 101 1

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.

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.

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.

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).

x50 + x51 + x52 + x53 + x54 +ψx0x1x2x3x4 = 0

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 .

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 .

Fact: resolving Cij gives 4× 10 new exceptionals, and pijk 6× 10, giving atotal of 100 new divisors. Thus h1,1(V ) = 101.

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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 .

∆ = ∆1 + · · ·+∆r

∇i = v ∈ NR |〈v , a〉 ≤ 1 for all a ∈ ∆i

and 〈v , b〉 ≤ 0 for all b ∈ ∆j 6=i .

∆ = ∆1 + · · ·+∆r

∇i = v ∈ NR |〈v , a〉 ≤ 1 for all a ∈ ∆i

and 〈v , b〉 ≤ 0 for all b ∈ ∆j 6=i .

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.

Gross-Siebert

Glue back stuff.

Gross-Siebert

Glue back stuff.

Reference

Victor Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yauhypersurfaces in toric varieties.

Cox, Katz Chapter 4.

Mattia Talpo. Batyrev mirror symmetry.

Batyrev Mirror Symmetry

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