# Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models

@article{Katzarkov2014BogomolovTianTodorovTF, title={Bogomolov-Tian-Todorov theorems for Landau-Ginzburg models}, author={Ludmil Katzarkov and Maxim Kontsevich and Tony Pantev}, journal={arXiv: Algebraic Geometry}, year={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 necessary Hodge theory for varieties with potentials, and prove a double degeneration statement needed for the unobstructedness result. We discuss the various definitions of Hodge numbers for non-commutative Hodge structures of Landau-Ginzburg type and the role they play in mirror symmetry. We also…

## 81 Citations

### Degenerations, fibrations and higher rank Landau-Ginzburg models

- Mathematics
- 2021

We study semi-stable degenerations of quasi-Fano varieties to unions of two pieces. We conjecture that the higher rank Landau-Ginzburg models mirror to these two pieces can be glued together to lower…

### Hodge–Tate Conditions for Landau–Ginzburg Models

- MathematicsPublications of the Research Institute for Mathematical Sciences
- 2018

We give a sufficient condition for a class of tame compactified Landau-Ginzburg models in the sense of Katzarkov-Kontsevich-Pantev to satisfy some versions of their conjectures. We also give examples…

### Toric Landau–Ginzburg models

- MathematicsRussian Mathematical Surveys
- 2018

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…

### 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…

### 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…

### 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…

### Fibers over infinity of Landau–Ginzburg models

- MathematicsCommunications in Number Theory and Physics
- 2022

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 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…

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