derived categories of twisted sheaves on calabi-yau manifolds

DERIVED CATEGORIES OF TWISTED SHEAVES ON CALABI-YAU
MANIFOLDS
Andrei Horia Căldăraru, Ph.D.
Cornell University 2000
This dissertation is primarily concerned with the study <strong>of</strong> <strong>derived</strong> <strong>categories</strong> <strong>of</strong>
<strong>twisted</strong> <strong>sheaves</strong> on Calabi-Yau manifolds. Twisted <strong>sheaves</strong> occur naturally in a
variety <strong>of</strong> problems, but the most important situation where they are relevant
is in the study <strong>of</strong> moduli problems <strong>of</strong> semistable <strong>sheaves</strong> on varieties. Although
universal <strong>sheaves</strong> may not exist as such, in many cases one can construct them
as <strong>twisted</strong> universal <strong>sheaves</strong>. In fact, the twisting is an intrinsic property <strong>of</strong> the
moduli problem under consideration.
A fundamental construction due to Mukai associates to a universal sheaf a
transform between the <strong>derived</strong> category <strong>of</strong> the original space and the <strong>derived</strong> category
<strong>of</strong> the moduli space, which <strong>of</strong>ten turns out to be an equivalence. In the
present work we study what happens when the universal sheaf is replaced by a
<strong>twisted</strong> one. Under these circumstances we obtain a transform between the <strong>derived</strong>
category <strong>of</strong> <strong>sheaves</strong> on the original space and the <strong>derived</strong> category <strong>of</strong> <strong>twisted</strong>
<strong>sheaves</strong> on the moduli space.
The dissertation is divided into two parts. The first part presents the main
technical tools: the Brauer group, <strong>twisted</strong> <strong>sheaves</strong> and their <strong>derived</strong> category, as
well as a criterion for checking whether an integral transform is an equivalence
(a so-called Fourier-Mukai transform). When this is the case we also obtain results
regarding the cohomological transforms associated to the ones on the level <strong>of</strong>
<strong>derived</strong> <strong>categories</strong>.
In the second part we apply the theoretical results <strong>of</strong> the first part to a large
set <strong>of</strong> relevant examples. We study smooth elliptic fibrations and the relationship
between the theory <strong>of</strong> <strong>twisted</strong> <strong>sheaves</strong> and Ogg-Shafarevich theory, K3 surfaces,
and elliptic Calabi-Yau threefolds. In particular, the study <strong>of</strong> elliptic Calabi-Yau
threefolds leads us to an example which is likely to provide a counterexample to the
generalization <strong>of</strong> the Torelli theorem from K3 surfaces to threefolds. A similarity
between the examples we study and certain examples considered by Vafa-Witten
and Aspinwall-Morrison shows up, although we can only guess the relationship
between these two situations at the moment.