Every other Hodge isometry of the cohomology of a K3 surface is
induced by an autoequivalence of the derived category
pdf-file
Unpublished preprint FU Berlin (2002). This material is included in my thesis.
The same argument appeared in the
paper
"Autoequivalences of Derived Category of A K3 Surface and
Monodromy Transformations" by Hosono, Lian, Oguiso and Yau.
Fourier-Mukai transforms and stable bundles on elliptic curves
pdf-file
Contributions to Algebra and Geometry (Beiträge zur Algebra und Geometrie)
Vol. 46, No. 2 (2005), 423-434.
Joint work with Georg Hein (2003).
See also Chapter 14 in the book "Abelian Varieties, Theta Functions and the
Fourier Transform" by Polishchuk;
and see
this
and
this
paper by Burban and Kreussler for the case of singular curves.
Groups of autoequivalences of derived categories of smooth projective varieties
pdf-file
This is my PhD thesis, handed in January 31, 2005. My supervisor was
Daniel Huybrechts.
Equivariant autoequivalences for finite group actions
pdf-file
Advances in Mathematics 216 (2007), 62-74.
This stems from chapter 3 of my thesis but contains major improvements. The arXiv
version can be found
here.
If reading the article, it is valuable to also look at section 4 of this
paper by Dolgachev.
Postnikov-Stability for Complexes on Curves and Surfaces
pdf-file
Accepted by International Journal of Mathematics.
Joint work with Georg Hein.
Here we provide a new notion of (semi)stability for complexes (or objects in a
triangulated category). The starting point is that semistable bundles on a
curve always have a non-trivial orthogonal complement in the derived sense. In
particular, it is a different type of generalisation that Bridgeland's notion
of a stability condition which start from the Harder-Narashiman filtrations.
The arXiv version can be found
here.
McKay correspondence for the Poincare series of Kleinian and Fuchsian singularities
pdf-file
Mathematischen Annalen 347 (2010), 689-702.
Joint work with Wolfgang Ebeling.
The arXiv version is
here.
Also see
this
article by Kajiura, Saito, Takahashi about a different lift of the Milnor lattice of a
Kleinian singularity, to a hereditary category (be sure to also look at the appendix
by Ueda). And see
this
paper by the same authors about a lift to a hereditary category of some Fuchsian
hypersurface singularities coming from weight systems. Note that the categorical approach
of the above article works for a much larger class of Fuchsian singularities.
Postnikov-stability versus Semistability of Sheaves
pdf-file
Joint with Georg Hein. A sequel to the first article on Postnikov stability.
We show that Postnikov-stability encompasses both Giesker and slope semistability,
in arbitrary dimension. Available on the arXiv
here.
Poincare series and Coxeter functors for Fuchsian singularities
pdf-file
Advances in Mathematics 225 (2010), 1387-1398.
Joint work with Wolfgang Ebeling.
The arXiv version is
here.
This article introduces a notion of Coxeter element for Fuchsian singularities of higher
genus, where the Milnor lattice does not have a root basis anymore. We justify this
definition by showing that it generalises earlier results about Poincare series and
proceed to lift the Coxeter element to a functor.
Autoequivalencs of toric surfaces
pdf-file
Accepted by Proceedings of the AMS.
Joint work with Nathan Broomhead.
The arXiv version is
here.
A geometric construction of Coxeter-Dynkin diagrams of bimodal singularities
pdf-file
Joint work with Wolfgang Ebeling.
The arXiv version is
here.
Comparing Coxeter functors made from exceptional and spherical objects
pdf-file
Joint work with Chris Brav. This is a preprint, feedback is welcome. The first two sections
(including the statement and proof of the main theorem) are finished.
How not to hang a picture on a wall — Topology in school
pdf-file
This is not mathematical research. I wrote this article because I got positive feedback about my short
notes for some Kangaroo sessions .