Higher direct images of the structure sheaf via the Hilbert-Chow morphism

Abstract

Let $X$ be a projective smooth surface over $\mathbb{C}$ with $H^2(\mathcal{O}_X)=0$. Let $M=M(L,\chi)$ be the moduli space of 1-dimensional semistable sheaves with determinant $\mathcal{O}_X(L)$ and Euler characteristic $\chi$. We have the Hilbert-Chow morphism $\pi:M\rightarrow |L|$. We give explicit forms of the higher direct images $R^i\pi_*\mathcal{O}_M$ under some mild conditions on $M$ and $|L|$. Our result shows that $R^i\pi_*\mathcal{O}_M$ are direct sums of line bundles. In particular, using our result one gets explicit formulas for the Euler characteristic of $\pi^*\mathcal{O}_{|L|}(m)$, which in $X=\mathbb{P}^2$ case was once conjectured by Chung-Moon.