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The K3 category of a cubic fourfold

Part of: Surfaces and higher-dimensional varieties (Co)homology theory

Published online by Cambridge University Press:  02 March 2017

Daniel Huybrechts
Daniel Huybrechts*
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany email huybrech@math.uni-bonn.de
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Abstract

Smooth cubic hypersurfaces $X\subset \mathbb{P}^{5}$ (over $\mathbb{C}$) are linked to K3 surfaces via their Hodge structures, due to the work of Hassett, and via a subcategory ${\mathcal{A}}_{X}\subset \text{D}^{\text{b}}(X)$, due to the work of Kuznetsov. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this paper, both aspects are studied further and extended to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of ${\mathcal{A}}_{X}$ for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.

Keywords

cubic fourfoldK3 surfacesHodge theoryderived categories

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces 14F05: Sheaves, derived categories of sheaves and related constructions
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 3 , March 2017 , pp. 586 - 620
Copyright
© The Author 2017 

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