Abstract

There are several approaches for extending (semi)stability from sheaves to complexes, or rather triangulated categories. In work with Georg Hein (arXiv 0704.2512, 0901.1554), we have introduced a notion of “Postnikov stability” for general triangulated categories, the motivation being Falting’s observation that semistability of vector bundles on smooth, projective curves is characterised by the existence of orthogonal sheaves. Our main result is that for projective varieties, classical stability of sheaves can always be captured by an appropriate Postnikov stability.

Some applications of this theory: compactifications of classical moduli spaces using genuine complexes; a more conceptual look on questions around “preservation of stability”; answer to a conjecture of Friedman on stable sheaves on elliptic surfaces (the latter is Bernadara/Hein, 1002.4986).