Chow groups and derived categories of \(K3\) surfaces (Q2918460)

Let \(X\) be a complex \(K3\) surface. Three algebraic objects can be associated to \(X\) to study its geometrical properties: the singular cohomology groups \(H^*(X,\mathbb{Z})\), endowed with the intersection pairing and the Hodge structure, the rational Chow ring \(\mathrm{CH}^*(X)\) of algebraic cycles, and the bounded derived category \(D(X)\), which is a linear triangulated category. The latter two objects can be defined over arbitrary fields, and contain a big amount of geometrical information. For example, in the complex case a Theorem of Mumford proves that \(\mathrm{CH}^2(X)\) is infinite dimensional, while Bloch conjecture would imply that, if one works over a number field or over \(\bar{\mathbb{Q}}\), then \(\mathrm{CH}^2(X) = \mathbb{Q}\).NEWLINENEWLINEThe category \(D(X)\) has been recently considered by many authors to study the geometrical properties of \(X\). This paper is a nice introduction to the questions that relate the structure of the derived category \(D(X)\) and the the cohomology and the Chow ring of a a \(K3\) surface. Bloch-Beilinson filtration and the Bloch conjecture, Bridgeland conjecture on the structure of \(\mathrm{Aut}(D(X))\), spherical objects and the recent author's result on their interplay with relevant subgroups of \(\mathrm{CH}^*(X)\) are the main topics.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].