The \(K3\) category of a cubic fourfold. (English) Zbl 1440.14180

Summary: 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 B. Hassett [Compos. Math. 120, No. 1, 1–23 (2000; Zbl 0956.14031)], and via a subcategory \({\mathcal{A}}_{X}\subset \text{D}^{\text{b}}(X)\), due to the work of A. Kuznetsov [Prog. Math. 282, 219–243 (2010; Zbl 1202.14012)]. The relation between these two viewpoints has recently been elucidated by N. Addington and R. Thomas [Duke Math. J. 163, No. 10, 1885–1927 (2014; Zbl 1309.14014)]. 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.


14J28 \(K3\) surfaces and Enriques surfaces
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
Full Text: DOI arXiv


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