Stable reflexive sheaves and localization

Abstract

We study moduli spaces N of rank 2 stable reflexive sheaves on P3. Fixing Chern classes c1, c2, and summing over c3, we consider the generating function Zrefl(q) of Euler characteristics of such moduli spaces. The action of the torus T on P3 lifts to N and we classify all sheaves in NT. This leads to an explicit expression for Zrefl(q). Since c3 is bounded below and above, Zrefl(q) is a polynomial. We find a simple formula for its leading term when c1=−1. Next, we study moduli spaces of rank 2 stable torsion free sheaves on P3 and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of Zrefl(q) and Euler characteristics of Quot schemes of certain T -equivariant reflexive sheaves, which are studied elsewhere. Many techniques of this paper apply to any toric 3-fold. In general, Zrefl(q) depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of P2×P1.