Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
The author computes Euler characteristics of moduli spaces \(\mathcal{M}\) of torsion free \(\mu-\)stable sheaves over a toric surface \(S\). The action of the torus \(T\) on \(S\) lifts to \(\mathcal{M}\) and the fixed point locus \(\mathcal{M}^T\), by the author's previous result [\textit{M. Kool}, Adv. Math. 227, No. 4, 1700--1755 (2011; Zbl 1227.14018)], consists of \(T-\)equivariant \(\mu-\)stable sheaves. Hence Klyachko's description of \(T-\)equivariant torsion free sheaves applies. The author follows Göttsche's strategy to reduce the computation of \(e(\mathcal{M}_S^H(r,c_1,c_2))\) to \(e(\mathcal{N}_S^H(r,c_1,c_2))\), where \(\mathcal{N}_S^H(r,c_1,c_2)\) is the moduli space of \(\mu-\)stable locally free sheaves with rank \(r\) and \(i\)-th Chern class \(c_i\). By \textit{A. A. Klyachko}'s result [Math. USSR, Izv. 35, No. 2, 337--375 (1990); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 53, No. 5, 1001--1039 (1989; Zbl 0706.14010)], the fixed point locus \(\mathcal{N}_S^H(r,c_1,c_2)^T\) can be stratified into finite strata, each of which can be realized as a good geometric \(\mathrm{SL}(r,\mathbb{C})-\)quotient of a locally closed subset \(\mathcal{D}_{v,\delta}^s\) of some flag variety, and has a very explicit combinatorial description. The main theorem of the paper expresses the generating function \(\displaystyle{\sum_{c_2}}e(\mathcal{M}_S^H(r,c_1,c_2))q^{c_2}\) in terms of \(e(\mathcal{D}_{v,\delta}^s/\mathrm{SL}(r,\mathbb{C}))\). Finally, the author applies the main theorem to the following examples: rank 1 case, rank 2 and 3 case on the projective plan, and rank 2 case on a Hirzebruch surface, also he explains the concordance between his results and the results in the literature.
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