Abstract:

Stability, first introduced by Mumford in the 1960's, is used as a tool to construct moduli space of sheaves on algebraic varieties. Motivated by homological mirror symmetry conjecture Bridgeland introduces stability conditions on triangulated categories which depends on the existence of Harder-Narasimhan (HN) filtration and central charges on the relevant K group of associated triangulated categories. The existence of Bridgeland stability is still an open problem in general varieties even in normal projective surfaces. Therefore, I introduce a notion of stability filtration in arbitrary categories which is equivalent to the existence of HN sequence on objects. Indeed, it is equivalent to existences of a zero morphism, a partial order on objects, and a collection of some universal sequences. In this talk, I would focus on normal projective varieties over a perfect field, and present the result of the existence of stability filtration (of degree 1) which is equivalent to Bridgeland stability on normal projective surfaces. As an application this result implies new effective restriction theorems of semistable sheaves on normal projective varieties over perfect fields. Moreover, our approach also gives new proofs of Hodge Index Theorem and Bogomolov Inequality.