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

@article{Przyjalkowski2017OnTC, title={On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections}, author={Victor Przyjalkowski}, journal={Mathematical Notes}, year={2017}, volume={103}, pages={104-110} }

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 proof, we obtain a description of the fibers over infinity of the compactified toric Landau–Ginzburg models.

## 11 Citations

### 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…

### 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…

### 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…

### 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 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…

### String topology with gravitational descendants, and periods of Landau-Ginzburg potentials

- Mathematics
- 2018

This paper introduces new operations on the string topology of a smooth manifold: gravitational descendants of its cotangent bundle, which are augmentations of the Chas-Sullivan $L_\infty$ algebra…

### Nef partitions for codimension 2 weighted complete intersections

- MathematicsANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2019

We prove that a smooth well-formed Fano weighted complete intersection of codimension 2 has a nef partition. We discuss applications of this fact to
Mirror Symmetry. In particular we list all nef…

### Hodge level for weighted complete intersections

- Mathematics
- 2018

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge level, that is, the maximal distance between non-trivial Hodge…

## References

SHOWING 1-10 OF 28 REFERENCES

### 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…

### Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

- Mathematics, Physics
- 2013

### 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…

### 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…

### 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…

### 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…

### On Landau--Ginzburg models for Fano varieties

- Mathematics
- 2007

We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12.

### Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models

- Mathematics
- 2010

We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.