ICTP/IGAP Seminar on Algebraic Geometry - Lecture Series - Thursday, 9 December, at 16:00 (Lothar GOETTSCHE) - Online and Presence

[ICTP Math Section]

*_ICTP/IGAP seminar on Algebraic Geometr_*y

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*

Register in advance for this meeting:
https://zoom.us/meeting/register/tJ0kdumhqT8tG9B4jZokHzs0M6Qjbd-zfBa-
After registering, you will receive a confirmation email containing 
information about joining the meeting.

*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.

/This will be a _*hybrid seminar*_. All are very welcome to join either 
online or in person (if provided with a green pass). *Venue:* Luigi 
Stasi seminar room, for those wishing to attend in person./

http://indico.ictp.it/event/9758/


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