Moduli of vector bundles and class numbers (Q807690)

Let \(M=M(c_ 1,c_ 2)\) be the moduli space of stable vector bundles of rank 2 over the complex projective space \({\mathbb{P}}^ 2\) with Chern classes \(c_ 1, c_ 2\) and let \(D=4c_ 2-c^ 2_ 1\). The author shows that the Euler characteristic \(\chi\) (M) of M with respect to cohomology with compact supports is equal to 3H(D) if \(c_ 1\) is odd, and to 3H(D)-(3/2)d(D/4) if \(c_ 1\) is even. (Here H is the Hurwitz function, H(D) being equal to the number of classes of integral binary quadratic forms of discriminant \(-D,\) counted with weights 2/\(| Aut(D)|\), and d(n) is the number of divisors of n.) The proof depends on the Lefschetz fixed point formula for the action of a torus and a description of equivariant bundles on a toric variety.