Toric Landau–Ginzburg models

Przyjalkowski, Victor

doi:10.1070/rm9852

Cited by 12 publications

(10 citation statements)

References 40 publications

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“…Rank of Picard group. If X is a smooth Fano threefold such that rk Pic(X) = 1, then (♥) in Main Theorem is already established in [Prz13,Prz18], and (♦) in Main Theorem follows from the proof of [ILP13, Theorem 4.1]. Thus, we will always assume that rk Pic(X) 2.…”

Section: The Proofmentioning

confidence: 99%

Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds

Cheltsov¹,

Przyjalkowski²

2018

Preprint

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We verify Katzarkov-Kontsevich-Pantev conjecture for Landau-Ginzburg models of smooth Fano threefolds.P {x},{y},{t} : quadratic term (x + t)(y + t); P {x},{t},{y,z} : quadratic term z(x + t); P {y},{z},{x,t} : quadratic term z(x + t).By Corollary 1.12.2, we have D −2 P = 0 for every P ∈ Σ, so that [f −1 (−2)] = 2. Thus, we see that (♥) in Main Theorem holds in this case, since h 1,2 (X) = 1.1.13. Curves on singular quartic surfaces. We will prove (♦) in Main Theorem by computing the intersections form of the curves C 1 , . . . , C r on a general surface in the pencil S. To do this, let k = C(λ), let S k be the quartic surface in P 3 k that is given by (1.4.1), and let ν : S k → S k be the minimal resolution of singularities of the surface S k .Lemma 1.13.1. Suppose that λ is a general element of C. Then the surface S λ is singular, and it has du Val singularities. Let M be the r × r matrix with entries M ij ∈ Q that are given by M ij = C i • C j , where C i • C j is the intersection of the curves C i and C j on the surface S λ . Then the right hand side of (♦) is equal toProof. Let F be a general fiber of the morphism f. Then H 2 (F, R) ∼ = Z 22 , since F is a smooth K3 surface. This easily implies the required assertion.3 x2 3 ž3 −3 ť5 3 x3 ž2 3 +6 ť6 3 x2 3 ž3 −3 ť6 3 x3 3 −3 ť6 3 x3 ž2 3 +3 ť6 3 x3 3 ž3 − ť6 3 x4 3 −3 ť6 3 x2 3

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Section: The Proofmentioning

confidence: 99%

Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds

Cheltsov¹,

Przyjalkowski²

2018

Preprint

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“…Any smooth Fano threefold has toric Landau-Ginzburg model (see [Prz18] for details), that is a Landau-Ginzburg model whose total space is an algebraic torus (C * ) 3 and which satisfies certain conditions. In [Prz17] their log Calabi-Yau compactifications are constructed.…”

Section: Connections To the Conjectures Of [Kkp17]mentioning

confidence: 99%

P=W Phenomena

Harder¹,

Katzarkov²,

Przyjalkowski³

2019

Preprint

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In this paper, we describe recent work towards the mirror P=W conjecture, which relates the weight filtration on a cohomology of a log Calabi-Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi-Yau manifold, taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi-Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [CHM12], and de Cataldo and Migliorini [CM10]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror P=W conjecture provides a possible bridge between the mirror P=W conjecture and the well-known P=W conjecture in nonabelian Hodge theory.

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“…In this section we define a toric Landau-Ginzburg model, the main object of our study. For details and examples see [5,8,19,20,40,64,65,67] and references therein.…”

Section: Toric Landau-ginzburg Modelsmentioning

confidence: 99%

“…In the threefold case, this duality is called the Dolgachev-Nikulin-Pinkham duality [27,60], and can be formulated in terms of orthogonal Picard lattices. In [29,67] the uniqueness of compactified toric Landau-Ginzburg models satisfying these conditions is proved for rank one Fano threefolds, and so the theorem holds for all Dolgachev-Nikulin toric Landau-Ginzburg models.…”

Section: Given a Laurent Polynomialmentioning

confidence: 99%

Projecting Fanos in the mirror

Kasprzyk,

Katzarkov,

Przyjalkowski

et al. 2019

Preprint

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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 introduced; using Mirror Symmetry these connections were transferred to the side of Landau-Ginzburg models. In the paper mentioned above a nice way to connect of Picard rank one Fano threefolds was found. We apply this approach to all smooth Fano threefolds, connecting their degenerations by toric basic links. In particular, we find a lot of Gorenstein toric degenerations of smooth Fano threefolds we need. We implement mutations in the picture as well. It turns out that appropriate chosen toric degenerations of the Fanos are given by toric basic links from a few roots. We interpret the relations we found in terms of Mirror Symmetry.

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Toric Landau–Ginzburg models

Cited by 12 publications

References 40 publications

Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds

Katzarkov-Kontsevich-Pantev Conjecture for Fano threefolds

P=W Phenomena

Projecting Fanos in the mirror

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