Generalized Homological Mirror Symmetry and cubics

Katzarkov, Ludmil; Przyjalkowski, Victor

doi:10.1134/s0081543809010118

Cited by 12 publications

(14 citation statements)

References 9 publications

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“…Monodromy of Landau-Ginzburg models, gap in spectra is at most 2. This conjecture was partially verified in [KP09], [IKP11], [IKS]. We record our findings in Table 6.…”

Section: Examplessupporting

confidence: 63%

“…This was first observed in [KP09], [IKP11], and [IKS]. Applying the theory of Orlov's spectra to the case of four-dimensional cubic, four-dimensional X 10 , and Kuechle manifolds we arrive at the following conjecture suggested by Hasset-Kuznetsov-Tschinkel program.…”

Section: Examplesmentioning

confidence: 61%

“…The main idea in this paper is that proving HMS is like studying birational transformations (noncommutative included) on A and B side of HMS. Later new ideas were introduced in [HKK], [KKP], [KP09], [DKK12a], [DKK12b], [BFK10]. These ideas can be summarized as follows:…”

Section: Moduli Approach To Birational Geometrymentioning

confidence: 99%

“…On derived category level a lot of fundamental work was done by Kuznetsov and then Addington and Thomas. On the Landau-Ginzburg side calculations were done by [KP09] and [IKS]. We extend the approach we have undertaken in the case of three-dimensional Fano threefolds to the case of some fourfolds.…”

Section: High Dimensional Fano'smentioning

confidence: 99%

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Birational Geometry via Moduli Spaces

Cheltsov

Katzarkov

Przyjalkowski

2013

Birational Geometry, Rational Curves, and Arithmetic

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Summary. In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau-Ginzburg models. Based on that we suggest a generalization of Kawamata's categorical approach to birational geometry enhancing it via geometry of moduli spaces of Landau-Ginzburg models. We suggest a conjectural application to Hasset-Kuznetsov-Tschinkel program based on new nonrationality "invariants" we consider -gaps and phantom categories. We make several conjectures for these invariants in the case of surfaces of general type and quadric bundles.

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“…Monodromy of Landau-Ginzburg models, gap in spectra is at most 2. This conjecture was partially verified in [KP09], [IKP11], [IKS]. We record our findings in Table 6.…”

Section: Examplessupporting

confidence: 63%

Section: Examplesmentioning

confidence: 61%

Section: Moduli Approach To Birational Geometrymentioning

confidence: 99%

Section: High Dimensional Fano'smentioning

confidence: 99%

See 2 more Smart Citations

Birational Geometry via Moduli Spaces

Cheltsov

Katzarkov

Przyjalkowski

2013

Birational Geometry, Rational Curves, and Arithmetic

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“…In particular h 13 = 1 and h 22 = 21. The central fiber of Calabi-Yau compactification of Hori-Vafa type toric Landau-Ginzburg model has 2 components intersecting by particular Picard rank 20 K3 surface (see [KP09]).…”

Section: Reconstructing Hodge Numbersmentioning

confidence: 99%

Weak Landau-Ginzburg models for smooth Fano threefolds

Przyjalkowski¹

2013

Izv. Math.

108

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Abstract. We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactifications. We state most of known methods of finding Landau-Ginzburg models in terms of Laurent polynomials. We discuss Laurent polynomial representation of Landau-Ginzburg models of Fano varieties and state some problems related to it.

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Beyond large complex structure: quantized periods and boundary data for one-modulus singularities

Bastian,

van de Heisteeg,

Schlechter

2024

J. High Energ. Phys.

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We study periods in an integral basis near all possible singularities in one-dimensional complex structure moduli spaces of Calabi-Yau threefolds. Near large complex structure points these asymptotic periods are well understood in terms of the topological data of the mirror Calabi-Yau manifold. The aim of this work is to characterize the period data near other boundaries in moduli space such as conifold and K-points. Using results from Hodge theory, we provide the general form of these periods in a quantized three-cycle basis. Based on these periods we compute the prepotential and related physical couplings of the underlying supergravity theory. Moreover, we elucidate the meaning of the model-dependent coefficients that appear in these expressions: these can be identified with certain topological and arithmetic numbers associated to the singular geometry at the moduli space boundary. We illustrate our findings by studying a wide set of examples.

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Generalized Homological Mirror Symmetry and cubics

Abstract: Abstract-We discuss an approach to studying Fano manifolds based on Homological Mirror Symmetry. We consider some classical examples from a new point of view.

Cited by 12 publications

References 9 publications

Birational Geometry via Moduli Spaces

Birational Geometry via Moduli Spaces

Weak Landau-Ginzburg models for smooth Fano threefolds

Beyond large complex structure: quantized periods and boundary data for one-modulus singularities

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