Exceptional collections on some fake quadrics

Lee, Kyoung-Seog; Shabalin, Timofey

doi:10.48550/arxiv.1410.3098

Cited by 2 publications

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“…Recently, interesting new categories in the derived categories of surfaces of general type with p g = q = 0 were discovered (cf. [9,8,21,49,50,51]). Their Grothendieck groups are finite torsion and their Hochschild homology groups vanish.…”

Section: K3 Surfacesmentioning

confidence: 99%

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Fano visitors, Fano dimension and orbifold Fano hosts

Kiem,

Lee

2015

Preprint

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In [35], the authors proved that every complete intersection smooth projective variety Y is a Fano visitor, i.e. its derived category D b (Y ) is equivalent to a full triangulated subcategory of the derived category D b (X) of a smooth Fano variety X, called a Fano host of Y . They also introduced the notion of Fano dimension of Y as the smallest dimension of a Fano host X and obtained an upper bound for the Fano dimension of each complete intersection variety.In this paper, we provide a Hodge-theoretic criterion for the existence of a Fano host which enables us to determine the Fano dimensions precisely for many interesting examples, such as low genus curves, quintic Calabi-Yau 3folds and general complete intersection Calabi-Yau varieties.Next we initiate a systematic search for more Fano visitors. We generalize the methods of [35] to prove that smooth curves of genus at most 4 are all Fano visitors and general curves of genus at most 9 are Fano visitors. For surfaces and higher dimensional varieties, we find more examples of Fano visitors and raise natural questions.We also generalize Bondal's question and study triangulated subcategories of derived categories of Fano orbifolds. We proved that there are Fano orbifolds whose derived categories contain derived categories of orbifolds associated to quasi-smooth complete intersections in weighted projective spaces, Jacobians of curves, generic Enriques surfaces, some families of Kummer surfaces, bielliptic surfaces, surfaces with κ = 1, classical Godeaux surfaces, product-quotient surfaces, holomorphic symplectic varieties, etc.An interesting recent discovery is the existence of quasi-phantom subcategories in derived categories of some surfaces of general type with pg = q = 0 ([8, 9, 21, 36, 49, 50, 51]). But no examples of Fano with quasi-phantom have been found. From the above constructions, we found Fano orbifolds whose derived categories contain quasi-phantom categories or phantom categories.Proposition 1.3. (Proposition 4.7) Let Y be a Fano visitor and X be a Fano host of Y . Then we have the inequality of Hodge numbers p−q=i h p,q (Y ) ≤ p−q=i h p,q (X) for all i.As a direct consequence, we obtain the following.Corollary 1.4. (Corollary 4.9) If h n,0 (Y ) = 0 for n = dim Y > 0, then the Fano dimension of Y is at least n + 2.Combining this corollary with the Fano host construction in [35], we obtain the following.

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Section: K3 Surfacesmentioning

confidence: 99%

“…An interesting recent discovery is the existence of quasi-phantom subcategories in derived categories of some surfaces of general type with p g = q = 0 ( [8,9,21,49,50,51]). But no examples of Fano with quasi-phantom have been found.…”

Section: Introductionmentioning

confidence: 99%

Fano visitors, Fano dimension and orbifold Fano hosts

Kiem,

Lee

2015

Preprint

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“…Let S = (C ×D)/G be a surface isogenous to a higher product with p g = q = 0. It is easy to see that the maximal possible length of an exceptional collection is less than or equal to 4(see [11,19,22,23,24] for more details). In this section we construct exceptional collections of line bundles of maximal length 4 on S = (C × D)/G where G = G(32, 27).…”

Section: Derived Categories Of Surfaces Isogenous To a Higher Product...mentioning

confidence: 99%

“…Quasiphantom categories are surprising new subcategories in the derived categories of algebraic varieties first discovered by Böhning, Bothmer and Sosna in [7]. Their discovery provides new perspectives on the study of derived categories of algebraic varieties and recently many examples of quasiphantom categories were constructed by many authors(see [1,6,7,10,11,15,16,18,19,20,22,23,24] for more details). However their structures are quite mysterious and we do not know whether every surface of general type with p g = q = 0 has a quasiphantom category in its derived category.…”

Section: Introductionmentioning

confidence: 98%

“…In particular, they proved that the possible list of groups (16,3), G(32, 27), A 5 . In [11,19,22,23,24], the authors constructed quasiphantom categories in derived categories of surfaces isogenous to a higher product except G = G(32, 27), A 5 cases. It is very natural to expect that derived categories of all surfaces isogenous to a higher product with p g = q = 0 have quasiphantom categories in their derived categories.…”

Section: Introductionmentioning

confidence: 99%

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Quasiphantom categories on a family of surfaces isogenous to a higher product

Kim

Lee

2017

Journal of Algebra

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We construct exceptional collections of line bundles of maximal length 4 on S = (C ×D)/G which is a surface isogenous to a higher product with p g = q = 0 where G = G(32, 27) is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.A surface S which is isomorphic to (C ×D)/G where C, D are curves of genus ≥ 2 and G is a finite group acting on C × D freely is called a surface isogenous to a higher product. Surfaces isogenous to a higher product are interesting and important classes of surfaces of general type. They play an important role in the study of moduli spaces of general type surfaces. Bauer, Catanese and Grunewald classified surfaces isogenous to a higher product of unmixed type with p g = q = 0 in [4]. In particular, they proved that the possible list ofA 5 cases. It is very natural to expect that derived categories of all surfaces isogenous to a higher product with p g = q = 0 have quasiphantom categories in their derived categories. However quasiphantom categories in derived categories of surfaces had not been constructed for G = G(32, 27), A 5 cases.In this paper we construct exceptional collections of line bundles of maximal length 4 on S = (C × D)/G which is a surface isogenous to a higher product with p g = q = 0 and G is G(32, 27). From these exceptional collections we can obtain new examples of quasiphantom categories. They are obtained as the orthogonal complements of the exceptional collections of maximal length 4 on S.Acknowledgement. We are grateful to Changho Keem, Young-Hoon Kiem for their words of encouragement. We thank Fabrizio Catanese for answering many questions and helpful discussions. We also thank Alexander Kuznetsov for his suggestion to use a computer program to compute the derived categories of the family of surfaces considered in this paper. Part of this work was done when the third named author was a research fellow of Korea Institute for Advanced Study. He thanks KIAS for wonderful working condition and kind hospitality.This work was also supported by IBS-R003-Y1. Last but not least, we thank the referees for their valuable comments which helped us to improve the manuscript.Notations. We will work over C. Derived category of a variety will mean the

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Exceptional collections on some fake quadrics

Cited by 2 publications

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Fano visitors, Fano dimension and orbifold Fano hosts

Fano visitors, Fano dimension and orbifold Fano hosts

Quasiphantom categories on a family of surfaces isogenous to a higher product

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