On Hodge numbers of complete intersections and Landau--Ginzburg models

@article{Przyjalkowski2013OnHN,
  title={On Hodge numbers of complete intersections and Landau--Ginzburg models},
  author={Victor Przyjalkowski and Constantin Shramov},
  journal={arXiv: Algebraic Geometry},
  year={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 (any) Calabi--Yau compactification of Givental's Landau--Ginzburg model for $X$. 

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References

SHOWING 1-10 OF 44 REFERENCES

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

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

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 of Fano Varieties and Constructing Mirror Manifolds

For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

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

Quantum cohomology of smooth complete intersections in weighted projective spaces and in singular toric varieties

Givental's theorem for complete intersections in smooth toric varieties is generalized to Fano varieties. The Gromov-Witten invariants are found for Fano varieties of dimension ≥3 that are complete

On weak Landau–Ginzburg models for complete intersections in Grassmannians

A common way to construct Fano varieties in higher dimensions is to represent them as complete intersections in familiar varieties such as toric varieties or Grassmannians. The well-known Givental

Gromov-Witten classes, quantum cohomology, and enumerative geometry

The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic

An introduction to motivic integration

By associating a `motivic integral' to every complex projective variety X with at worst canonical, Gorenstein singularities, Kontsevich proved that, when there exists a crepant resolution of