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

2002/10/15 by Takuro Mochizuki, Mochizuki, Takuro
Mathematics · #14D20 #14J60 #14J80 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.math/0210211

openalex publication_date 2002/10/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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