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Abstract:
After the overview lecture of December, we start going through the material in more detail:
 
We introduce the Hilbert scheme of points on a surface, define moduli spaces of stable sheaves, and state the Vafa-Witten formula for the Euler numberts of moduli spaces of rank 2 sheaves.   The moduli spaces of sheaves are in general singular, of a dimension different from the expected dimension. We introduce perfect obstruction theories and the virtual fundamental class. Integrating cohomology classes against this virtual fundamental class allows to define virtual versions of enumerative invariants, which behave similar to the invariants of smooth varieties, among them a virtual version of the Euler number.
We conjecture that the Vafa-Witten formula computes this virtual Euler number.

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The seminar will consist of lecture series by Alina Marian and by Lothar Goettsche (of which the abstracts are below), and talks by Postdocs and Faculty on their research.
 
Toward the cohomology and Chow rings of moduli spaces of sheaves (Alina Marian)
 
A seminar direction will examine the problem of understanding Lie algebra actions on the cohomology and Chow rings of moduli spaces of sheaves interms of the Chern classes of the universal sheaf. One classical action on cohomology is the Lefschetz sl(2) associated with an ample divisor class on a projective variety. For moduli spaces of sheaves over curves and surfaces, Grothendieck's standard conjectures are often known to hold, in particular Lefschetz sl(2) actions should be expressible via algebraic correspondences. Nevertheless there are very few explicit algebraic constructions of the Lefschetz operators. Progress in this direction would have important applications, not least to holomorphic symplectic geometry; the seminar will explain this circle of ideas.
 
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Virtual invariants of moduli spaces of sheaves on surfaces and Vafa-Witten invariants (Lothar Goettsche)
 
Another direction of the seminar will be virtual invariants of moduli spaces of sheaves on surfaces and Vafa-Witten invariants. Moduli spaces of sheaves on surfaces with $p_g>0$ tend to be singular, but they carry a so-called perfect obstruction theory, which allows to define virtual versions of the standard topological invariants of smooth varieties, e.g. the virtual Euler number. In 1994 Vafa and Witten gave a formula for "Euler numbers" of moduli spaces of rank 2 sheaves on surfaces. Recently a mathematical definition of these Vafa-Witten invariants was given in arbitrary rank by Tanaka and Thomas in terms of moduli spaces of Higgs pairs. We will review the definition of these invariants and their relation to virtual Euler numbers of moduli spaces of sheaves on surfaces, and show computations of these invariants leading to conjectural generating functions in terms of modular functions.