Toric Landau–Ginzburg models

  title={Toric Landau–Ginzburg models},
  author={Victor Przyjalkowski},
  journal={Russian Mathematical Surveys},
  pages={1033 - 1118}
  • V. Przyjalkowski
  • Published 24 September 2018
  • Mathematics
  • Russian Mathematical Surveys
This review of the theory of toric Landau–Ginzburg models describes an effective approach to mirror symmetry for Fano varieties. It focuses mainly on the cases of dimensions and , as well as on the case of complete intersections in weighted projective spaces and Grassmannians. Conjectures that relate invariants of Fano varieties and their Landau– Ginzburg models, such as the Katzarkov–Kontsevich–Pantev conjectures, are also studied. Bibliography: 89 titles. 

Fibers over infinity of Landau-Ginzburg models

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this

On Calabi-Yau compactifications of Landau-Ginzburg models for coverings of projective spaces

We suggest the procedure that constructs a log Calabi–Yau compactification of weak Landau–Ginzburg model of a Fano variety. We apply the suggestion for del Pezzo surfaces and coverings of projective

Projecting Fanos in the mirror

In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was

Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds.

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.

Threefolds fibred by mirror sextic double planes

Abstract We present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces.

P=W Phenomena

In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a cohomology of a log Calabi--Yau manifold to the perverse Leray filtration on the

Graph potentials and moduli spaces of rank two bundles on a curve

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode

Graph potentials and symplectic geometry of moduli spaces of vector bundles

. We give the rst examples of Fano manifolds with multiple optimal tori, i.e. we construct monotone Lagrangian tori 𝐿 , such that the weighted number of holomorphic Maslovindextwodiscswithboundaryon

Laurent polynomials in Mirror Symmetry: why and how?

We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau–Ginzburg

On singular log Calabi-Yau compactifications of Landau-Ginzburg models

We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of



On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

It is well known that Givental’s toric Landau–Ginzburg models for Fano complete intersections admit Calabi–Yau compactifications. We give an alternative proof of this fact. As a consequence of this

Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

We prove that smooth Fano threefolds have toric Landau- Ginzburg models. More precisely, we prove that their Landau-Ginzburg models, represented as Laurent polynomials, admit compactifications to

Landau-Ginzburg Hodge numbers for mirrors of del Pezzo surfaces

Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau–Ginzburg models for Fano complete intersections in Grassmannians similar to Givental’s construction for

Weak Landau–Ginzburg models for smooth Fano threefolds

We consider Landau–Ginzburg models for smooth Fano threefolds of the principal series and prove that they can be represented by Laurent polynomials. We check that these models can be compactified to

Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models

In this paper we prove the smoothness of the moduli space of Landau-Ginzburg models. We formulate and prove a Tian-Todorov theorem for the deformations of Landau-Ginzburg models, develop the

Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models

Abstract For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and

On Hodge numbers of complete intersections and Landau--Ginzburg models

We prove that the Hodge number $h^{1,N-1}(X)$ of an $N$-dimensional ($N\geqslant 3$) Fano complete intersection $X$ is less by one then the number of irreducible components of the central fiber of

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Abstract.Given a smooth projective toric variety X, we construct an A∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is