Mathematics – Algebraic Geometry
Scientific paper
2009-06-18
Mathematics
Algebraic Geometry
41 pages, minor corrections made, minor change in exposition
Scientific paper
As an application of the combinatorial description of fixed point loci of moduli spaces of sheaves on toric varieties derived in \cite{Koo}, we study generating functions of Euler characteristics of moduli spaces of $\mu$-stable torsion free sheaves on nonsingular complete toric surfaces. We obtain a general expression for such generating functions in terms of Euler characteristics of moduli spaces of stable configurations of linear subspaces. The expression holds for any choice of $X$, ample divisor, rank and first Chern class. It can be further simplified in examples, which allows us to compute some new and known generating functions (due to G\"ottsche, Klyachko and Yoshioka). In general, these generating functions depend on choice of stability condition, enabling us to study wall-crossing phenomena and relate to work of G\"ottsche and Joyce. Much of our work is in the spirit of Klyachko \cite{Kly4} and based on theory developed in \cite{Koo}.
Kool Martijn
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