ICTP/IGAP seminar on Algebraic Geometry
 
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.
 
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The abstract for the first talk on Thursday 9 December, at 16:00:
 
Lothar Goettsche (ICTP)
 
Virtual invariants of moduli spaces of sheaves on surfaces and Vafa-Witten invariants I, Introduction and overview

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Abstract: This will be the first talk in the series of Goettsche's lectures. It is an introduction and overview of the material that will be covered in more detail in the lectures. We introduce Hilbert schemes of points, moduli spaces of sheaves on surfaces, and introduce Vafa-Witten moduli spaces, and give an overview of results on their (virtual) Euler numbers, related to S-duality conjectures from Physics.