Vector bundles and reflexive sheaves with extremal Chern characters

Moduli spaces of vector bundles, or more generally sheaves on algebraic varieties, are usually badly behaved. As soon as the dimension of the variety is at least three, they satisfy Murphy's Law in algebraic geometry, i.e., all types of singularities can occur on them. In this talk, I will introduce a class of stable reflexive sheaves in three-dimensional projective space, whose moduli spaces are smooth and irreducible, contrary to the general picture. These spaces are closely related to the numerical study of Chern characters of semistable sheaves. As a corollary one can obtain strict bounds on Chern characters of semistable sheaves of rank two on certain general type surfaces.