Stable reflexive sheaves and localization

We study moduli spaces N of rank 2 stable reflexive sheaves on P-3. Fixing Chern classes c(1), c(2), and summing over c(3), we consider the generating function Z(refl)(q) of Euler characteristics of such moduli spaces. The action of the torus T on P-3 lifts to N and we classify all sheaves in N-T. This leads to an explicit expression for Z(refl)(q). Since c(3) is bounded below and above, Z(refl)(q) is a polynomial. We find a simple formula for its leading term when c(1) = -1. Next, we study moduli spaces of rank 2 stable torsion free sheaves on P-3 and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of Z(refl)(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, Z(refl)(q) depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of P-2 x P-1. Published by Elsevier B.V.