The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface
We shall compute the Betti numbers of moduli spaces of stable sheaves of rank 2 on a ruled surface or \(\mathbb{P}^2\), under some conditions on polarizations and Chern classes. Our approach is the same as that of Harder and Narasimhan. That is, we shall consider the weighted numbers \(\sum_E {1 \over \# \Aut (E)}\) of torsion free sheaves defined over a finite field \(\mathbb{F}_q\) and by using Harder-Narasimhan filtration, we can compute the number of \(\mathbb{F}_q\)-rational points of moduli spaces. Then, by virtue of the Weil conjectures, we obtain the Betti numbers.
scientific article; zbMATH DE number 799558