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

@article{Przyjalkowski2016CalabiYauCO, title={Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds}, author={Victor Przyjalkowski}, journal={Sbornik: Mathematics}, year={2016}, volume={208}, pages={992 - 1013} }

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 families of K3 surfaces, and we describe their fibres over infinity. We also give an explicit construction of Landau-Ginzburg models for del Pezzo surfaces and any divisors on them. Bibliography: 40 titles.

## 20 Citations

### 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### 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…

### Calabi–Yau threefolds fibred by high rank lattice polarized K3 surfaces

- MathematicsMathematische Zeitschrift
- 2019

We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by…

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

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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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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…

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

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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…

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

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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…

### Birational geometry via moduli spaces.

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In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau-Ginzburg models. Based on that we suggest a…

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Abstract We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent…

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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…

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We classify Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system has no base points but does not give an embedding, and we classify anticanonically embedded Fano…