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

@article{Doran2015ToricDA, title={Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models}, author={Charles F. Doran and Andrew Harder}, journal={Canadian Journal of Mathematics}, year={2015}, volume={68}, pages={784 - 815} }

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 expressions for Givental's Landau–Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau–Ginzburg models can be expressed as…

## 24 Citations

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

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- 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…

### Mutation equivalence of toric Landau-Ginzburg models

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- 2020

Given a Fano complete intersection defined by sections of a collection nef line bundles $L_1,\ldots, L_c$ on a Fano toric manifold $Y$, a construction of Givental/Hori-Vafa provides a mirror-dual…

### Toric Landau–Ginzburg models

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- 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…

### Projecting Fanos in the mirror

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- 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…

### Birationality of mirror models for Fano toric complete intersections

- MathematicsBulletin of the London Mathematical Society
- 2021

Given a Fano complete intersection defined by sections of a collection nef line bundles L1,…,Lc on a Fano toric manifold Y , a construction of Givental/Hori–Vafa provides a mirror‐dual…

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

- MathematicsMathematical Notes
- 2018

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…

### 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…

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

- MathematicsProceedings of the Steklov Institute of Mathematics
- 2015

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…

### 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…

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