Cosection localization and the Quot scheme QuotSl(E)

Abstract

Let E be a locally free sheaf of rank r on a smooth projective surface S . The Quot scheme Quot S l ( E ) of length l coherent sheaf quotients of E is a natural higher-rank generalization of the Hilbert scheme of l points of S . We study the virtual intersection theory of this scheme. If C ⊂ S is a smooth canonical curve, we use cosection localization to show that the virtual fundamental class of Quot S l ( E ) is ( − 1 ) l times the fundamental class of the smooth subscheme Quot C l ( E | C ) ⊂ Quot S l ( E ) . We then prove a structure theorem for virtual tautological integrals over Quot S l ( E ) . From this, we deduce, among other things, the equality of virtual Euler characteristics χ vir ( Quot S l ( E ) ) = χ vir ( Quot S l ( O ⊕ r ) ) .