Orlov spectra: bounds and gaps

Ballard, Matthew; Favero, David; Katzarkov, Ludmil

doi:10.1007/s00222-011-0367-y

Cited by 56 publications

(86 citation statements)

References 44 publications

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“…1.1]) could then be restated as follows: a cubic fourfold is rational if and only if it is categorically representable in codimension 2. Finally, we can argue some conjectural relation between categorical representability and the existence of gaps in the Orlov spectrum defined in [BFK10].…”

Section: Conjecture 11 ([Orl05]) Let X and Y Be Smooth Projective Vmentioning

confidence: 92%

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Categorical representability and intermediate Jacobians of Fano threefolds

Bernardara

Bolognesi

2013

Derived Categories in Algebraic Geometry

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Abstract. We define, basing upon semiorthogonal decompositions of D b (X), categorical representability of a projective variety X and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of X by providing examples and stating open questions.

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Section: Conjecture 11 ([Orl05]) Let X and Y Be Smooth Projective Vmentioning

confidence: 92%

“…Another recent theory is based on Orlov spectra and their gaps [BFK10]. Let us refrain even to sketch a definition of it, but just notice that [BFK10, Conj.…”

Section: Categorical Representability and Rationality: Further Develomentioning

confidence: 99%

Categorical representability and intermediate Jacobians of Fano threefolds

Bernardara

Bolognesi

2013

Derived Categories in Algebraic Geometry

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“…Let us recall the definitions of the Orlov spectrum and discuss some of the main results in [BFK10]. Let T be a triangulated category.…”

Section: Theorem 54 Let X Be a Smooth Fano Variety Let Lg(x) Be Itmentioning

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%

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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“…Example 6.3. The "shift" version of Proposition 4.6 mentioned in Section 5.6 implies [13] that if a collection of spherical objects S 1 , . .…”

Section: Orlov Spectrummentioning

confidence: 99%