Linearisations of triangulated categories with respect to finite group actions

Given an action of a finite group on a triangulated category, we investigate under which conditions one can construct a linearised triangulated category using DG-enhancements. In particular, if the group is a finite group of automorphisms of a smooth projective variety and the category is the bounded derived category of coherent sheaves, then our construction produces the bounded derived category of coherent sheaves on the smooth quotient variety resp. stack. We also consider the action given by the tensor product with a torsion canonical bundle and the action of a finite group on the category generated by a spherical object.Comment: 14 pages; comments welcom

Derived equivalent conjugate K3 surfaces

We show that there exist a complex projective K3 surface $X$ and an automorphism of the complex numbers $\sigma$ such that the conjugate K3 surface $X^\sigma$ is a non-isomorphic Fourier-Mukai partner of $X$.Comment: 12 pages; v2: minor changes, mostly correction of typos; v3: added application to Hilbert scheme

Fourier-Mukai partners of canonical covers of bielliptic and Enriques surfaces

We prove that the canonical cover of an Enriques surface does not admit non-trivial Fourier-Mukai partners. We also show that the canonical cover of a bielliptic surface has at most one non-isomorphic Fourier-Mukai partner. The first result is then applied to birational Hilbert schemes of points and the second to birational generalised Kummer varieties. An appendix establishes that there are no exceptional or spherical objects in the derived category of a bielliptic surface.Comment: 10 pages; v2: corrected a reference and some typo

Derived categories and scalar extensions

This thesis consists of three parts all of which deal with questions related to scalar extensions and derived categories. In the first part we consider the question whether the conjugation of a complex projective K3 surface X by an automorphism of the complex numbers can produce a non-isomorphic Fourier-Mukai partner of X. The answer is affirmative. The conjugate surface is thus in particular a moduli space of locally free sheaves on X. We use our result to give higher-dimensional examples of derived equivalent conjugate varieties. We furthermore prove that a similar result holds for abelian surfaces. The topic of the second part is the behaviour of stability conditions under scalar extensions. Namely, given a smooth projective variety X over some field K and its bounded derived category, one can associate to it a complex manifold of stability conditions. Given a finite Galois extension we compare the stability manifolds of X and of the base change scheme in general and under the additional assumption that the numerical Grothendieck group does not change under the scalar extension. In the third and last part we consider the following question: Can one naturally define an L-linear triangulated category if a K-linear triangulated category and a field extension are given? We propose a construction and prove that our definition gives the expected result in the geometric case. It also gives the anticipated result when applied to the derived category of an abelian category with enough injectives and with generators. We furthermore prove that in the just mentioned cases the dimension of the triangulated category in question does not change for finite Galois extensions

On the dynamical degrees of reflections on cubic fourfolds

We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will provide numerical evidence for potential restrictions on tuples of dynamical degrees realisable on general cubic fourfolds which can be violated on the projective four-space.Comment: 28 pages, 3 figure