A category of kernels for equivariant factorizations and its implications for Hodge theory

Ballard, Matthew; Favero, David; Katzarkov, Ludmil

doi:10.1007/s10240-013-0059-9

Cited by 80 publications

(140 citation statements)

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“…The paper is based on examples we have analyzed in [KP11], [KNS], [FIK12], [BFK11], [KK]. All these suggest a direct connection between monodromy of Landau-Ginzburg models, spectra and wall crossings in the moduli space of stability conditions, which was partially explored in [IKS].…”

Section: Introductionmentioning

confidence: 99%

“…This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following [BFK10] and [BFK11] we enhance constructions from [Kuz09] by introducing NoetherLefschetz spectra -an interplay between Orlov spectra [Ol94] and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where above techniques might apply.…”

mentioning

confidence: 99%

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Double Solids, Categories and Non-Rationality

Iliev

Katzarkov²,

Przyjalkowski

2013

Proceedings of the Edinburgh Mathematical Society

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Abstract. This paper suggests a new approach to questions of rationality of threefolds based on category theory. Following [BFK10] and [BFK11] we enhance constructions from [Kuz09] by introducing NoetherLefschetz spectra -an interplay between Orlov spectra [Ol94] and Hochschild homology. The main goal of this paper is to suggest a series of interesting examples where above techniques might apply. We start by constructing a sextic double solid X with 35 nodes and torsion in H 3 (X, Z). This is a novelty -after the classical example of Artin and Mumford (1972), this is the second example of a Fano threefold with a torsion in the 3-rd integer homology group. In particular X is non-rational. We consider other examples as well -V 10 with 10 singular points and double covering of quadric ramified in octic with 20 nodal singular points.After analyzing the geometry of their Landau Ginzburg models we suggest a general non-rationality picture based on Homological Mirror Symmetry and category theory. IntroductionThis paper suggests a new approach to questions of rationality of threefolds based on category theory. It was inspired by recent work of V. Shokurov and by A. Kuznetsov's idea about the Griffiths component (see [Kuz08]). This work is a natural continuation of ideas developed in [Ka09], [GKKN11] and of ideas of Kawamata and his school.We first extend classical example of Artin and Mumford to construct a sextic double solid X with 35 nodes and torsion in H 3 (X, Z). The construction is based on an approach by M. Gross and suggests close relation between Artin and Mumford example and the sextic double solid X with 35 nodes. This example, a novelty on its own, opens a possibility of series of interesting examples -V 10 with 10 singular points and double covering of quadric ramified in octic with 20 nodal singular points.In this paper we start investigating these examples from the point of view of Homological Mirror Symmetry (HMS). We consider the mirrors of the sextic double solid X with 35 nodes, of the Fano variety V 10 with 10 singular points in general position and of the double covering of quadric ramified in octic with 20 nodal singular points. We note that the monodromy around the singular fiber over zero of the Landau-Ginzburg models is strictly unipotent in all these examples, which suggests that the categorical behavior should be very similar to the one of the Artin-Mumford example. We conjecture that the reason for categorical similarity in all these examples is that they contained the category of an Enriques surface as a semiorthogonal summand in their derived categories. This is done in Section 5, where we introduce Landau-Ginzburg models and compare their singularities.In Section 6 we introduce several new rationality invariants coming out of the notions of spectra and enhanced Noether-Lefschetz spectra of categories. We give a conjectural categorical explanation of the examples from Sections 2, 3, 4, 5. The novelty (conjecturally) is that non-rationality of these examples cannot be picked by Orlov spectr...

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Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

Double Solids, Categories and Non-Rationality

Iliev

Katzarkov²,

Przyjalkowski

2013

Proceedings of the Edinburgh Mathematical Society

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“…The kernel of φ is Z 3 ⊕ Z 3 , a finite group of order 9 generated by (1, −1, 0) and (1, 0, −1). By Theorem 7.5 DGrB(w, N ) is an admissible subcategory of D b (coh X 3 ) where X 3 is the cubic sevenfold defined by w. Hence, by Corollary 7.6 we obtain: 6 By [BFK1] there is an equivalence of categories,…”

Section: Categorical Covers and Griffiths Groupsmentioning

confidence: 90%

“…When considering the categories, DGrB(w, M ) and DGrB(w, N ), the following abstract situation occurs (see [BFK1] for a more complete discussion).…”

Section: Categorical Covers and Griffiths Groupsmentioning

confidence: 99%

“…For certain families, the noncommutative CY 3-folds appearing above can be related back to algebraic varieties with trivial canonical class by utilizing the categorical covering picture from [BFK1]. For example, if the functions f (x 0 , x 1 , x 2 ), g(x 3 , x 4 , x 5 ), h(x 6 , x 7 , x 8 ) all define smooth elliptic curves, E 1 , E 2 , E 3 respectively, then the product E 1 × E 2 × E 3 is a Z 3 × Z 3 -cover of the noncommutative CY 3-fold corresponding to the cubic sevenfold, X 3 , defined by f + g + h.…”

Section: Theorem 11 Suppose X Is a Smooth Fano-calabi-yau Complete mentioning

confidence: 99%

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On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

Favero

Iliev

Katzarkov

2014

Pure and Applied Mathematics Quarterly

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A deep result of Voisin asserts that the Griffiths group of a general non-rigid Calabi-Yau (CY) 3-fold is infinitely generated. This theorem builds on an earlier method of hers which was implemented by Albano and Collino to prove the same result for a general cubic sevenfold. In fact, Voisin's method can be utilized precisely because the variation of Hodge structure on a cubic 7-fold behaves just like the variation of Hodge structure of a Calabi-Yau 3-fold. We explain this relationship concretely using Kontsevitch's noncommutative geometry. Namely, we show that for a cubic 7-fold, there is a noncommutative CY 3-fold which has an isomorphic Griffiths group. This serves as partial confirmation of seminal work of Candelas, Derrick, and Parkes describing a cubic 7-fold as a mirror to a rigid CY 3-fold.Similarly, one can consider other examples of Fano manifolds with with the same type of variation of Hodge structure as a Calabi-Yau threefold (FCYs). Among the complete intersections in weighted projective spaces, there are only three classes of smooth FCY manifolds; the cubic 7-fold X 3 , the fivefold quartic double solid X 4 , and the fivefold intersection of a quadric and a cubic X 2.3 . We settle the two remaining cases, following Voisin's method to demonstrate that the Griffiths group for a general complete intersection FCY manifolds, X 4 and X 2.3 , is also infinitely generated.Received December 11, 2012. David Favero, Atanas Iliev and Ludmil KatzarkovIn the case of X 4 , we also show that there is a noncommutative CY 3-fold with an isomorphic Griffiths group. Finally, for X 2.3 there is a noncommutative CY 3-fold, B, such that the Griffiths group of X 2.3 surjects on to the Griffiths group of B. We finish by discussing some examples of noncommutative covers which relate our noncommutative CYs back toq honest algebraic varieties such as products of elliptic curves and K3-surfaces.

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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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A category of kernels for equivariant factorizations and its implications for Hodge theory

Cited by 80 publications

References 69 publications

Double Solids, Categories and Non-Rationality

Double Solids, Categories and Non-Rationality

On the Griffiths groups of Fano manifolds of Calabi-Yau Hodge type

Birational Geometry via Moduli Spaces

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