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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.
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참고문헌 (26)
참고문헌 신청
K. Behrend / 2009 / Donaldson-Thomas type invariants via microlocal geometry / Ann. of Math. 170(3):1307~1338
K. Behrend / 1997 / The intrinsic normal cone / Invent. Math. 128(1):45~88
A. A. Beilinson / 1982 / Analysis and topology on singular spaces, I / Soc. Math. France :5~171
O. Ben-Bassat / 2015 / A 'Darboux theorem' for shifted symplectic structures on derived Artin stacks, with applications / Geom. Topol. 19(3):1287~1359
C. Brav / 2015 / Symmetries and stabilization for sheaves of vanishing cycles. With an appendix by Jorg Schurmann / J. Singul. 11:85~151
K. Behrend / 1997 / The intrinsic normal cone / Invent. Math. 128(1):45~88
A. A. Beilinson / 1982 / Analysis and topology on singular spaces, I / Soc. Math. France :5~171
O. Ben-Bassat / 2015 / A 'Darboux theorem' for shifted symplectic structures on derived Artin stacks, with applications / Geom. Topol. 19(3):1287~1359
C. Brav / 2015 / Symmetries and stabilization for sheaves of vanishing cycles. With an appendix by Jorg Schurmann / J. Singul. 11:85~151
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