Isak Sundelius: Derived Autoequivalences of the Product of a K3 Surface and an Elliptic Curve

Abstract.

In this project I aim to present a survey of the tools, as well as some of the surrounding theory, necessary for proving the main result of my thesis:

If \(S\) is a K3 surface and \(E\) an elliptic curve, then the group \(\operatorname{Aut} D^b(S\times E)\) of autoequivalences of the bounded derived category of \(\operatorname{Coh}(S\times E)\), the category of coherent sheaves over the product \(S\times E\), is isomorphic to \(\operatorname{Aut} D^b(S) \times \operatorname{Aut}D^b(E)/\mathbb{Z}\).

Due to a result by Orlov, an autoequivalence of \(D^b(X)\), with \(X\) a smooth projective variety, is isomorphic to a so-called Fourier-Mukai transform (FMT). Due to this result, Fourier-Mukai transforms, and most importantly their kernels, are the main focus of this survey. Orlov's result easily yields that any element of \(\operatorname{Aut} D^b(S) \times \operatorname{Aut}D^b(E)\) produces an element of \(\operatorname{Aut} D^b(S\times E)\).

Semi-homogeneous sheaves over elliptic curves are presented and exemplified. They are defined in terms of the dimension of the stabilisers resulting from an action \(E\times\hat{E}\curvearrowright \operatorname{Coh}(E)\). An FMT over an elliptic curve yields an induced action on the product \(E\times\hat{E}\) and this action is compared to the induced action on cohomology.

An analogous situation is carried over to the product \(S\times E\) in order to simplify the study of the FMTs over \(S\times E\) and, in particular, whether these FMTs are elements in \(\operatorname{Aut} D^b(S) \times \operatorname{Aut}D^b(E)\). In particular, it is presented how the main result holds in full generality if FMTs over \(S\times E\) with trivial induced action on \(E\times\hat{E}\) are elements of the product \(\operatorname{Aut} D^b(S) \times \operatorname{Aut}D^b(E)\). A technical assumption is made in order to further utilise the properties of the spaces involved and to get closer to showing the main result.