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A theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface

Takuro Mochizuki
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Abstract

Let $X$ be a smooth polarized algebraic surface over the compex number field. We discuss the invariants obtained from the moduli stacks of semistable sheaves of arbitrary ranks on $X$. For that purpose, we construct the virtual fundamental classes of some moduli stacks, and we show the transition formula of the integrals over the moduli stacks of the $δ$-stable Bradlow pairs for the variation of the parameter $δ$. Then, we study the relation among the invariants. In the case $p_g>0$, we show that the invariants are independent of the choice of a polarization of $X$. We also show that the invariants can be reduced to the invariants obtained from the moduli of abelian pairs and the Hilbert schemes. In the case $p_g=0$, we obtain the weak wall crossing formula and the weak intersection rounding formula, which describes the dependence of the invariants on the polarization.

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Takuro Mochizuki. 2006-07-07. A theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface. https://arxiv.org/abs/math/0210211

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