Description. The Fourier-Mukai transform is an equivalence between the derived categories
of coherent sheaves on an abelian variety, and on its dual variety. Introduced
by Mukai, it has found many recent applications in the work of Chen-Hacon
and Pareschi-Popa, and has become a fundamental tool in studying the geometry of abelian
varieties, and more generally, of irregular varieties. Furthermore, similar equivalences
play a major role in studying the birational geometry of algebraic varieties through
derived categories (see work of Bondal-Orlov, Bridgeland and Kawamata). In this workshop
we will focus on applications of the Fourier-Mukai transform to regularity of sheaves on
abelian varieties and more generally of objects in derived categories, and on connections
with the Schottky problem, with syzygies of abelian varieties, and with the Generic
Vanishing theorems orginating in work of Green-Lazarsfeld. Plenty of concrete
applications will be provided.
Relevant papers.
1. G. Pareschi and M. Popa, M-regularity and the Fourier-Mukai transform, Pure and Applied Mathematics
Quarterly 4 (2008), no. 3, 587-611. F. Bogomolov Special Issue, math.AG/0512645 2. G. Pareschi and M. Popa, Generic vanishing and minimal cohomology classes
on abelian varieties, Math. Ann. 340 (2008), no. 1, 209-222, math/0610166 3. G. Pareschi and M. Popa,
GV-sheaves, Fourier-Mukai transform, and generic vanishing, math/0608127 4. G. Pareschi and M. Popa, Regularity on abelian varieties, III: relationship with Generic Vanishing
and applications, to appear in the Proceedings of the Clay Institute Workshop ``Aspects of vector bundles
5. G. Pareschi and M. Popa, Strong generic vanishing and a higher dimensional Castelnuovo-de Franchis
References. Some useful reference texts:
[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry.
[Ha] R. Hartshorne, Algebraic geometry.
[Hu] D. Huybrechts, Fourier-Mukai transforms in Algebraic Geometry.
[La] R. Lazarsfeld, Positivity in Algebraic Geometry.
[Mu] D. Mumford, Abelian varieties.
[Th] R. Thomas, Derived categories for the working mathematician.
Background. The lectures will assume familiarity with basic Algebraic Geometry
as in [Ha], as well as some knowledge of abelian varieties, homological techniques,
and vanishing theorems. While there will be two introductory lectures on abelian varieties
and derived categories, we hope that people have seen at least part of this stuff before.
Here is some suggested reading:
1. For abelian varieties:
- Chapter II in [Mu], as well as at least the statements in Sections 16 and 17 in Chapter III.
- As a review (mostly in the complex case) one could use Section 9.1 in [Hu].
- At least the definitions of Albanese and Picard varieties (see, for example, p.331 in [GH]).
2. For derived categories:
- The basic facts about derived categories, as covered in Chapters I-III of [Hu]. An alternative first reading
would be the paper [Th].
- Some more advanced topics that will come up in the lectures are covered in Chapters 5 and 9 of [Hu].
It would be good to have a look at these topics before the workshop.
3. General facts about:
- (Statements of the) basic vanishing theorems (see Chapter 4 in [La].
- Castelnuovo-Mumford regularity (see Section 1.8 A in [La])
- Syzygies (see Section 1.8 D in [La]).
Attending the workshop. If you would like to attend the workshop, please contact Mircea Mustaţă
at
mmustata@umich.edu. Some limited funding is available for graduate students and postdocs.
Deadline to register: March 18.
Financial support. Support for the workshop is provided by the NSF (through an RTG grant) and by
the David and Lucile Packard Foundation (through a Packard Fellowship).
Related links.
Here is a map of the central campus. The Mathematics Department is in the East Hall,
at the intersection of South University and Church Street.