# Moduli spaces of semistable sheaves; an alternative construction method

##
Matei Toma

created by risa on 23 Nov 2018

29 nov 2018
-- 14:30

Aula 211, Dip. Matematica, Università "Roma Tre", Roma

**Abstract.**

When $X$ is a projective manifold and $\omega$ is a rational ample class,
modular compactifications of the moduli space of stable vector bundles on $X$
have been constructed in Algebraic Geometry by putting appropriate classes
of semistable sheaves at the boundary. These compactifications appear as
global quotients. No similar constructions are known over a general compact
Kaehler manifold $(X,\omega)$. In this talk we present an alternative
construction method using "local quotients" which covers the case when
$\omega$ is an arbitrary Kaehler class on a projective manifold $X$. This is
the subject of joint recent work with Daniel Greb. Essential use is made of
the notion introduced by Jarod Alper of a good moduli space of an algebraic
stack. Besides solving a wall-crossing issue appearing in the context of
projective manifolds, this alternative construction method is likely to
extend to the general case of Kaehler manifolds.