Journal of the
Korean Mathematical Society JKMS

In Donaldson-Thomas theory, moduli spaces are locally the critical locus of a holomorphic function defined on a complex manifold. In this paper, we develop a theory of critical virtual manifolds which are the gluing of critical loci of holomorphic functions. We show that a critical virtual manifold $X$ admits a natural semi-perfect obstruction theory and a virtual fundamental class $[X]\virt$ whose degree $DT(X)=\deg [X]\virt$ is the Euler characteristic $\chi_\nu(X)$ weighted by the Behrend function $\nu$. We prove that when the critical virtual manifold is orientable, the local perverse sheaves of vanishing cycles glue to a perverse sheaf $P$ whose hypercohomology has Euler characteristic equal to the Donaldson-Thomas type invariant $DT(X)$. In the companion paper \cite{KLp2}, we proved that a moduli space $X$ of simple sheaves on a Calabi-Yau 3-fold $Y$ is a critical virtual manifold whose perverse sheaf categorifies the Donaldson-Thomas invariant of $Y$ and also gives us a mathematical theory of Gopakumar-Vafa invariants.

Keywords: Donaldson-Thomas invariant, critical virtual manifold, perverse sheaves, mixed Hodge module

MSC numbers: 14N35, 14C30, 32S35