Using arguments from theoretical physics, Vafa and Witten gave a generating function for the Euler numbers of moduli spaces of rank 2 coherent sheaves on algebraic surfaces. Sheaves can be viewed as generalizations of vector bundles. Moduli spaces are algebraic varieties that parametrize the objects in which we are interested. While the VafaWitten formula cannot be literally true as a formula for the Euler numbers in general, we give an interpretation of it in terms of virtual topological invariants, and confirm it in many cases. We extend the results to finer topological invariants and also to rank 3. Such virtual invariants occur everywhere in modern enumerative geometry, inspired by physics, like GromovWitten invariants and Donaldson Thomas invariants, when attempting to make sense of the predictions from physics. Basically the idea is that the moduli spaces have very bad properties, but they "want to be" smooth, and the virtual invariants are those that they would have if they were smooth. We will try to give elementary explanations of all these concepts and their background, as much as is possible. Most of the time will be spent to just trying to explain the problem and the result. If there is time, some hints will be given about the methods.
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Virtual topological invariants of moduli spaces
Lothar Göttsche
Institution:
ICTP
Location:
Luigi Stasi Seminar Room, ICTP
Schedule:
Thursday, March 22, 2018  14:00
Abstract:
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Upcoming events

Alberto Maspero
Birkhoff coordinates for the Toda lattice in the limit of infinitely many particles with an application to FPU
Monday, March 19, 2018  10:00

Koen van den Dungen
The Kasparov product on open manifolds
Monday, March 19, 2018  14:00 to 15:30

Simone Di Marino
Sobolev spaces and nonlocal functionals in metric measure spaces
Tuesday, March 20, 2018  16:00

Lothar Göttsche
Virtual topological invariants of moduli spaces
Thursday, March 22, 2018  14:00
Today's Lectures

Stefano Bianchini
09:00 to 11:00

Marcello Porta (University of Tubingen)
11:00 to 13:00

Nicola Gigli
14:00 to 16:00
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