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Matei Toma: Lectures

January 10, 2018
TITLE: Moduli spaces of semistable sheaves with respect to Kähler polarizations

ABSTRACT: For a compact Kähler manifold (X,\omega) the Kobayashi-Hitchin correspondence gives homeomorphisms between moduli spaces of irreducible Hermite-Einstein connections and moduli spaces of stable vector bundles on X. Whereas gauge theoretical compactifications for these spaces are known to exist by work of Donaldson, Uhlenbeck and Tian, the question of constructing modular compactifications in complex geometry is still open in the above setting.

In this talk we report on some recent progress in this direction obtained by two different methods jointly with Daniel Greb and Julius Ross and with Daniel Greb and Peter Heinzner respectively. We deal with the case when X is projective and \omega is an arbitrary Kähler class, which arises in wall crossing phenomena in algebraic geometry. Unlike the first one, the second method is GIT-free and it is likely to extend to the general situation.