Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces

Abstract

Given a smooth toric variety $X$, the action of the torus $T$ lifts to the moduli space $\mathcal{M}$ of stable sheaves on $X$. Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus ${\mathcal{M}}^T$ can be given (as shown by earlier work of the author). In this paper, we apply this description to the case of torsion free sheaves on a smooth toric surface $S$. A general expression for the generating function of the Euler characteristics of such moduli spaces is obtained. The generating function is expressed in terms of Euler characteristics of certain moduli spaces of stable configurations of linear subspaces appearing in classical GIT. The expression holds for any choice of $S$, polarization, rank, and first Chern class. Specializing to various examples allows us to compute some new as well as known generating functions.

30 pages. Published version. Results unchanged. Completely rewritten in order to improve exposition and notation following the referee's suggestions

References in corpus (5)

• The Betti numbers of the moduli space of stable sheaves of rank 3 on P2
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• Poincare polynomial of moduli spaces of framed sheaves on (stacky) Hirzebruch surfaces
• BPS invariants of N=4 gauge theory on a surface
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• Gauge theories on compact toric surfaces, conformal field theories and equivariant Donaldson invariants
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• BPS invariants of N=4 gauge theory on a surface
• Vafa-Witten theory and iterated integrals of modular forms
• Crystal melting on toric surfaces
• Gauge theories on compact toric manifolds
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• Invariants of pure 2-dimensional sheaves inside threefolds and modular forms
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• Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P^2
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• Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity
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