Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants

Abstract

We formulate conjectural blowup formulas for Segre and Verlinde numbers on moduli spaces of sheaves on projective surfaces S S with p g ( S ) > 0 p_g(S)>0 and b 1 ( S ) = 0 b_1(S)=0 . As application we give a give a conjectural formula for the Donaldson invariants of S S in arbitrary rank, generalizing the famous structure formula for Donaldson invariants of Kronheimer and Mrowka. Furthermore we give conjectural formulas for the K K -theoretic Donaldson invariants, as well as for some Donaldson invariants with fundamental matters.