# Toric Landau–Ginzburg models

@article{Przyjalkowski2018ToricLM, title={Toric Landau–Ginzburg models}, author={Victor Przyjalkowski}, journal={Russian Mathematical Surveys}, year={2018}, volume={73}, pages={1033 - 1118} }

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.

## 11 Citations

### Fibers over infinity of Landau-Ginzburg models

- Mathematics
- 2020

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

- Mathematics
- 2021

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

- Mathematics
- 2019

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.

- Mathematics
- 2018

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

### Threefolds fibred by mirror sextic double planes

- MathematicsCanadian Journal of Mathematics
- 2020

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

- Mathematics
- 2019

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

- Mathematics
- 2020

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

- Mathematics
- 2022

. 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?

- MathematicsProyecciones (Antofagasta)
- 2022

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

- MathematicsSbornik: Mathematics
- 2022

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…

## References

SHOWING 1-10 OF 96 REFERENCES

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

- Mathematics
- 2017

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

- Mathematics
- 2016

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…

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

- Mathematics
- 2014

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

- Mathematics
- 2009

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

- Mathematics
- 2014

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

- MathematicsCanadian Journal of Mathematics
- 2016

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

- Mathematics
- 2013

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

- Mathematics
- 2006

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…