The K3 category of a cubic fourfold

By D. Huybrechts. Compositio Mathematica, 2015

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a5551528849cbc5382d84f24084cef03ec7fcc52

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In Geometry and Topology

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Citations in the corpus

Stability conditions on Kuznetsov components (2017, background): “Kuznetsov components of cubic fourfolds, as well as the basic existence result for moduli spaces of stable complexes in [Tod08]); to the extension of results by Addington-Thomas [AT14] and Huybrechts [Huy17], thereby giving a Hodge-theoretic characterisation of the locus where Ku(X) is derived equiv”
Lectures on K3 Surfaces (2016, background): “Lectures on K3 Surfaces, first published by Cambridge University Press, (c) Daniel Huybrechts, 2015 Cambridge University Press’s catalogue entry can be found at: www.cambridge.org/9781107153042 Contents Chapter 1.”
The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, background): “Recently, CY2 categories have attracted a great deal of attention due to their connections to birational geometry, Hodge theory, and the construction of hyperkähler varieties [1, 39, 40, 7, 61, 6, 57, 76].”