Generating functions for 𝐾-theoretic Donaldson invariants and Le Potier’s strange duality

Abstract

For a projective algebraic surface X X with an ample line bundle H H , let M H X ( c ) M_H^X(c) be the moduli space H H -semistable sheaves E \mathcal {E} of class c c in the Grothendieck group K ( X ) K(X) . We write c = ( r , c 1 , c 2 ) c=(r,c_1,c_2) or c = ( r , c 1 , χ ) c=(r,c_1,\chi ) with r r the rank, c 1 , c 2 c_1,c_2 the Chern classes, and χ \chi the holomorphic Euler characteristic. We also write M H X ( 2 , c 1 , c 2 ) = M X X ( c 1 , d ) M_H^X(2,c_1,c_2)=M_X^X(c_1,d) , with d = 4 c 2 − c 1 2 d=4c_2-c_1^2 . The K K -theoretic Donaldson invariants are the holomorphic Euler characteristics χ ( M H X ( c 1 , d ) , μ ( L ) ) \chi (M_H^X(c_1,d),\mu (L)) , where μ ( L ) \mu (L) is the determinant line bundle associated to a line bundle on X X . More generally for suitable classes c ∗ ∈ K ( X ) c^*\in K(X) there is a determinant line bundle D c , c ∗ \mathcal {D}_{c,c^*} on M H X ( c ) M^X_H(c) . We first compute some generating functions for K K -theoretic Donaldson invariants on P 2 \mathbb {P}^2 and rational ruled surfaces, using the wallcrossing formula of [Pure Appl. Math. Q. 5 (2009), pp. 1029–1111]. Then we show that Le Potier’s strange duality conjecture relating H 0 ( M H X ( c ) , D c , c ∗ ) H^0(M^X_H(c),\mathcal {D}_{c,c^*}) and H 0 ( M H X ( c ∗ ) , D c ∗ , c ) H^0(M^X_H(c^*),\mathcal {D}_{c^*,c}) holds for the cases c = ( 2 , c 1 = 0 , c 2 > 2 ) c=(2,c_1=0,c_2>2) and c ∗ = ( 0 , L , χ = 0 ) c^{*}=(0,L,\chi =0) with L = − K X L=-K_X on P 2 \mathbb {P}^2 , and L = − K X L=-K_X or − K X + F -K_X+F on P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 and P 2 ^ \widehat {\mathbb {P}^2} with F F the fibre class of the ruling, and also the case c = ( 2 , H , c 2 ) c=(2,H,c_2) and c ∗ = ( 0 , 2 H , χ = − 1 ) c^*=(0,2H,\chi =-1) on P 2 \mathbb {P}^2 .