Hodge numbers of Landau–Ginzburg models

Andrew Harder

doi:10.1016/j.aim.2020.107436

Hodge numbers of Landau–Ginzburg models

Andrew Harder

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the Hodge numbers f^p,q of Landau–Ginzburg models as defined by Katzarkov, Kontsevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y,w) so that h^p,q(X)=f^3−q,p(Y,w).

Original language	English (US)
Article number	107436
Journal	Advances in Mathematics
Volume	378
DOIs	https://doi.org/10.1016/j.aim.2020.107436
State	Published - Feb 12 2021

Keywords

Algebraic geometry
Hodge theory
Mirror symmetry
Toric varieties

ASJC Scopus subject areas

Mathematics(all)

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10.1016/j.aim.2020.107436

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title = "Hodge numbers of Landau–Ginzburg models",

abstract = "We study the Hodge numbers fp,q of Landau–Ginzburg models as defined by Katzarkov, Kontsevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y,w) so that hp,q(X)=f3−q,p(Y,w).",

keywords = "Algebraic geometry, Hodge theory, Mirror symmetry, Toric varieties",

author = "Andrew Harder",

note = "Funding Information: I would like to thank Valery Lunts for many valuable suggestions and comments. I would also like to thank Charles Doran, Ludmil Katzarkov, and Victor Przyjalkowski for useful conversations. I was partially supported by an NSERC postgraduate scholarship and the Simons Collaboration in Homological Mirror Symmetry during the preparation of this paper. ",

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N2 - We study the Hodge numbers fp,q of Landau–Ginzburg models as defined by Katzarkov, Kontsevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y,w) so that hp,q(X)=f3−q,p(Y,w).

AB - We study the Hodge numbers fp,q of Landau–Ginzburg models as defined by Katzarkov, Kontsevich, and Pantev. First we show that these numbers can be computed using ordinary mixed Hodge theory, then we give a concrete recipe for computing these numbers for the Landau–Ginzburg mirrors of Fano threefolds. We finish by proving that for a crepant resolution of a Gorenstein toric Fano threefold X there is a natural LG mirror (Y,w) so that hp,q(X)=f3−q,p(Y,w).

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KW - Hodge theory

KW - Mirror symmetry

KW - Toric varieties

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