| 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. |