Computing the derived category I

We begin to fill in the details on the B-side of the mirror by ‘computing’ the derived category. We first recall some algebra (garnered from Huybrechts’s books ‘Complex Geometry’ and ‘Fourier-Mukai Transforms in Algebraic Geometry’ (go and buy them! Or at least borrow them from your local library: they’re both great if you’re into this kind of thing)). Then we use that algebra to present an argument of Beilinson giving a finite generating set for the derived category of coherent sheaves on projective space. Finally we show that the derived category of a smooth projective variety is split-generated by the restrictions of these sheaves from the ambient space. In the next post on computing the derived category we will see the relevance of these facts for the computation on the B-side.

HMS for the quartic surface, I

This is the first in a sequence of blog posts about Seidel’s paper (http://arxiv.org/abs/math/0310414). The paper proves the homological mirror symmetry conjecture for the quartic K3 surface. The purpose of this post is two-fold: on the one hand to give a rough idea of what mirror symmetry is and why we are reading the paper; on the other to give an outline of the argument used in the paper in which, at a hefty 122 pages, it is easy to lose the thread.

Homological Mirror Symmetry for the Quartic Surface

This blog will serve as an online outlet for our 3-person reading group on Seidel’s paper “Homological mirror symmetry for the quartic surface” (http://arxiv.org/abs/math/0310414) and whatever else we decide to post. Preliminary reading for those interested in following the posts on mirror symmetry include:

1. Richard Thomas’s notes on derived categories: http://arxiv.org/abs/math/0001045
2. Andrei Caldararu’s notes on derived categories: http://arxiv.org/abs/math/0501094

Enjoy!