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

For a projective algebraic surface $X$ with an ample line bundle $H$, let $M_H^X(c)$ be the moduli space $H$-semistable sheaves $\mathcal {E}$ of class $c$ in the Grothendieck group $K(X)$. We write $c=(r,c_1,c_2)$ or $c=(r,c_1,\chi )$ with $r$ the rank, $c_1,c_2$ the Chern classes, and $\chi$ the holomorphic Euler characteristic. We also write $M_H^X(2,c_1,c_2)=M_X^X(c_1,d)$, with $d=4c_2-c_1^2$. The $K$-theoretic Donaldson invariants are the holomorphic Euler characteristics $\chi (M_H^X(c_1,d),\mu (L))$, where $\mu (L)$ is the determinant line bundle associated to a line bundle on $X$. More generally for suitable classes $c^*\in K(X)$ there is a determinant line bundle $\mathcal {D}_{c,c^*}$ on $M^X_H(c)$. We first compute some generating functions for $K$-theoretic Donaldson invariants on $\mathbb {P}^2$ 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 $H^0(M^X_H(c),\mathcal {D}_{c,c^*})$ and $H^0(M^X_H(c^*),\mathcal {D}_{c^*,c})$ holds for the cases $c=(2,c_1=0,c_2>2)$ and $c^{*}=(0,L,\chi =0)$ with $L=-K_X$ on $\mathbb {P}^2$, and $L=-K_X$ or $-K_X+F$ on $\mathbb {P}^1\times \mathbb {P}^1$ and $\widehat {\mathbb {P}^2}$ with $F$ the fibre class of the ruling, and also the case $c=(2,H,c_2)$ and $c^*=(0,2H,\chi =-1)$ on $\mathbb {P}^2$.