laurent polynomials in mirror symmetryvassilev/goldensands/przyjalkowski.pdf · in homological...

Laurent polynomials in Mirror Symmetry

Victor Przyjalkowski

November 21, 2011

Table of contents

1 Mirror Symmetry of variations of Hodge structuresA sideB side

2 Optimistic picture

3 Rank 1 Fano threefolds

Usual picture of Mirror Symmetry for Fano varieties

Fano variety XLandau–Ginzburg model

Y → C

Symplectic properties(A side)


Algebraic properties(B side)


Homological Mirror Symmetry: derived categories


Y → C

Fukaya category Fuk(X )Fukaya–Seidel category

FS(Y )

Db(Coh − X )Orlov’s category of

singularities Dbsing (Y )

Usual way to study


Y → C





We study, on a quantitative level


Y → C





Idea of Mirror Symmetry of variations of Hodge structures

General idea

QH∗(X ) periods of Y → C

Scheme of construction

Fano variety X with Pic (X ) = Z



Gromov–Witten invariants (of genus 0), numbers that countrational curves lying on X




Quantum cohomology QH∗(X ), deformation of cohomologyring H∗(X )





1st Dubrovin’s connection over C[t, t−1], or quantumD-module LX






Definition of differentiation in a trivial H∗(X )-bundle

t∂

∂t(γ) = KX ⋆ γ






2nd Dubrovin’s connection, or regularized quantum D-moduleLX






2nd Dubrovin’s connection, or regularized quantum D-moduleLX

Example of regularization: “trick with factorials”

If IX = 1 +∑

ai ti is a solution of LX I = 0, then

IX = 1 +∑

i !aiti is a solution of LX I = 0.

Example

Solutions of quantum D-modules

IP3=

∑t4k

(k!)4, I 3−quart =

∑ (4k)!(k!)5

tk

Solutions of regularized quantum D-modules

IP3=

∑ (4k)!(k!)4

t4k , I 3−quart =∑ k!(4k)!

(k!)5tk =

∑ (4k)!(k!)4

tk

Remark

IP3(t) = I 3−quart(t4)

Some of known cases

Givental

Complete intersections.

Hori–Vafa, Bertram–Ciocan-Fontanine–Kim

Grassmannians.

Kim, . . .: Quantum Lefschetz theorem

Complete intersections in studied varieties.

P.

Rank 1 Fano threefolds.

Picard–Fuchs differential equation

Definition

Consider a family Y → C with fibers Yt . Let {ωt} be a smoothfamily of dimC Yt-forms on Yt and let {∆t} be a smooth family ofdimC Yt-cycles on Yt . Then a Picard–Fuchs differential equation isan equation with solutions of type

∫∆t

ωt .

Mirror Symmetry conjecture of variations of Hodge

structures

Conjecture

For any Fano variety X there exists a family Y → C such thatLX = PFY .

How to find periods?

Let Y = (C∗)n. Then f : Y → C can be represented by a Laurentpolynomial f . Let φi be a constant term of f i . Let

Φf = 1 + φ1t + φ2t2 + φ3t

3 + . . .

Theorem (see, for instance, P., CNTP’08, arXiv:0707.3758)

PFY (Φf ) = 0

Very weak Landau–Ginzburg models

Definition

A very weak Landau–Ginzburg model for Fano variety X is aLaurent polynomial f such that LX Φf = 0.


Definition


Conjecture (Mirror Symmetry of variations of Hodge structures)

It always exists.


Definition


Conjecture (Mirror Symmetry of variations of Hodge structures)

It always exists.

Example

A Laurent polynomial fP3 = x + y + z + 1xyz

is a very weak

Landau–Ginzburg model for P3.

If {f = t} is a very weak Landau–Ginzburg model for P3, then

{f = t4} (if it can be represented by Laurent polynomial) is a veryweak Landau–Ginzburg model for quartic threefold.

Some other known cases

Going back to Hori–Vafa

Complete intersections.

Eguchi–Hori–Xiong

Grassmannians.

P.

Rank 1 Fano threefolds.

London group (Coates, Corti, Galkin, Golyshev, Kasprzyk, . . .)

Fano threefolds of any rank.

Idea–hope

Compactification principle

Fiberwise compactification of appropriate very weakLandau–Ginzburg for Fano variety is a (B side of)Landau–Ginzburg model from Homological Mirror Symmetryviewpoint.

Idea–hope



Questions

1 How to choose “correct” very weak Landau–Ginzburg models?

Idea–hope



Questions

1 How to choose “correct” very weak Landau–Ginzburg models?

2 What do they know about Fano varieties?

Weak Landau–Ginzburg models

In Homological Mirror Symmetry, Landau–Ginzburg model is aC-parameterized family of Calabi–Yau varieties. So according tocompactification principle, we put an additional condition onLaurent polynomials.

Definition

A very weak Landau–Ginzburg model f is called a weak one if foralmost all λ ∈ C the fiber {f = λ} is birationally Calabi–Yau.

Toric weak Landau–Ginzburg models

Let us have a Laurent polynomial f and a toric variety T

A Newton polytope ∆f of f is a convex hull of its non-zeroterms.

A fan polytope ∆T is a convex hull of vectors that generate afan of T . This is not a polytope defining T , this is a dual one!


Batyrev: let X degenerate to terminal Gorenstein toric variety T .Let vi ’s generate rays of T ’s fan. Then Batyrev’s suggestion forLandau–Ginzburg model for X is

∑

i

xvi .


Batyrev: let X degenerate to terminal Gorenstein toric variety T .Let vi ’s generate rays of T ’s fan. Then Batyrev’s suggestion forLandau–Ginzburg model for X is

∑

i

xvi .

Generalization

Let Fano variety degenerate to toric variety T . Then there exists aweak Landau–Ginzburg model f such that ∆T = ∆f . Suchpolynomial is called toric weak Landau–Ginzburg model for X .

Example

P3 and quartic threefold

A Laurent polynomial fP3 = x + y + z + 1xyz

is a very weak

Landau–Ginzburg model for P3. A Laurent polynomial

f3−quart = (x+y+z)4

xyzis a very weak Landau–Ginzburg model for

quartic threefold. Compactifications of their generic fibers arequartics in P

3 with Du Val singularities, so they are birational toK3 surfaces. P

3 is toric itself, quartic threefold degenerates toquartic T = {x1x2x3x4 = x4

0} with ∆T = ∆f3−quart. Thus both

these Laurent polynomials are toric weak Landau–Ginzburg models.

Optimistic picture

The following data is in 1-in-1 correspondence for given Fanovariety X .

Toric degenerations of X .

Toric weak Landau–Ginzburg models for X .

Compactifications of these toric Landau–Ginzburg models are (Bside of) Landau–Ginzburg models for HMS for X .

Question

Is it true that all Landau–Ginzburg models (of the samedimension) for HMS go back to toric weak ones?

Examples

Fano threefolds of Rank 1.

Smooth complete intersections in (weighted) projectivespaces.

The first example satisfy compactification principle.

What is known?

Iskovskikh Classification. There are 17 families of them.

P. Examples of their weak Landau–Ginzburg models.

Ilten–Lewis–P. These weak Landau–Ginzburg models are toric. Inparticular, there exist toric degenerations of rank 1Fano threefolds.

Dolgachev–Nikulin duality for K3 surfaces

Consider a Fano threefold X of degree deg X = (−KX )3 and indexiX . Put n = deg X/2i3X . The anticanonical section of X is a〈2n〉-polarized K3 surface S .Its Dolgachev–Nikulin dual K3 surface (mirror dual) is a K3 surfaceS ′ with

Pic (S ′) = H ⊕ E8(−1) ⊕ E8(−1) ⊕ 〈−2n〉,

where H is a rank 2 lattice(

0 11 0

).

Good weak Landau–Ginzburg models

Fibers of Landau–Ginzburg model are expected to be mirror dualto anticanonical sections of the initial Fano variety.

Definition

A weak Landau–Ginzburg model for Fano variety X is called good

if its general fiber is a K3 surface with Picard lattice

H ⊕ E8(−1) ⊕ E8(−1) ⊕ 〈−2n〉,

where 2n = deg X/i3X .

Remark

In particular, fibers of good weak Landau–Ginzburg models for P3

and quartic threefold have the same lattices.

Shioda–Inose surfaces

Theorem (Katzarkov–Lewis–P.)

Weak Landau–Ginzburg models from the list are good.

Remark

Fibers of good weak Landau–Ginzburg models are Shioda–InoseK3 surfaces. Families of these surfaces for given Picard lattice areparameterized by P

1.

Uniqueness

Corollary

Good weak Landau–Ginzburg models are unique, up to birationaltransforms and coverings.

Remark

Periods control coverings.

Corollary

LG (P3)4:1

LG (3 − quart).

Uniqueness

Theorem (follows from Hanamura, Batyrev, Kontsevich, . . .)

Compactifications of good weak Landau–Ginzburg models for givenFano variety differ by flops. Motives of their fibers coincide(noticed by Galkin).

Corollary

Up to flops we get correct Landau–Ginzburg models from HMSpoint of view.

Comments

Remark (in progress)

Flops control deformations of the initial Fano varieties andstructures related by them (stability conditions, etc.).

Question

Is this uniqueness property hold for higher dimension/rank of Fanovarieties?

Prediction of Betti numbers

From uniqueness property and flops property it follows that allcompactified good weak Landau–Ginzburg models are birational incodimension 1. In particular, numbers of components of theirfibers are fixed.

Theorem (P.)

Let kX be a number of (a unique) fiber that is (possible) reducible.Then kX = h12(X ) + 1.

Possible answers on two questions

Question 1

How to choose “correct” very weak Landau–Ginzburg models?

Conjectural answer

We need to choose toric weak ones.

Question 2

What do they know about Fano varieties?

Conjectural answer

Everything what Landau–Ginzburg models for B side of HMS knowplus toric degenerations.

laurent polynomials in mirror symmetryvassilev/goldensands/przyjalkowski.pdf · in homological...

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