Toric Landau–Ginzburg models

@article{Przyjalkowski2018ToricLM,
  title={Toric Landau–Ginzburg models},
  author={Victor Przyjalkowski},
  journal={Russian Mathematical Surveys},
  year={2018},
  volume={73},
  pages={1033 - 1118}
}
  • V. Przyjalkowski
  • Published 24 September 2018
  • Mathematics
  • Russian Mathematical Surveys
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 case of complete intersections in weighted projective spaces and Grassmannians. Conjectures that relate invariants of Fano varieties and their Landau– Ginzburg models, such as the Katzarkov–Kontsevich–Pantev conjectures, are also studied. Bibliography: 89 titles. 

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Projecting Fanos in the mirror

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Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds.

We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.

Threefolds fibred by mirror sextic double planes

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P=W Phenomena

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Graph potentials and moduli spaces of rank two bundles on a curve

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. These graphs encode degenerations of curves to rational curves, and graph potentials encode

Graph potentials and symplectic geometry of moduli spaces of vector bundles

. We give the rst examples of Fano manifolds with multiple optimal tori, i.e. we construct monotone Lagrangian tori 𝐿 , such that the weighted number of holomorphic Maslovindextwodiscswithboundaryon

Laurent polynomials in Mirror Symmetry: why and how?

We survey the approach to mirror symmetry via Laurent polynomials, outlining some of the main conjectures, problems, and questions related to the subject. We discuss: how to construct Landau–Ginzburg

On singular log Calabi-Yau compactifications of Landau-Ginzburg models

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

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