Categorification of Donaldson-Thomas invariants via perverse ...

CATEGORIFICATION OF DONALDSON-THOMAS INVARIANTS 3which coincides with the 2-cocycle obstruction for gluing the determinant line bundlesof the tangent bundles of CS charts. Since the perfect obstruction theoryExt • π(E, E) for X is symmetric, the determinant of the tangent bundle is a squareroot of det Ext • π(E, E). Therefore the local perverse sheaves {P α} • glue to a globalperverse sheaf if and only if there is a square root of det Ext • π(E, E) in Pic(X). (SeeTheorem 3.14.)When X is the moduli scheme of one dimensional stable sheaves on Y , we showthat the torsion-free part of det Ext • π(E, E) has a square root by Grothendieck-Riemann-Roch. More generally, Z. Hua ([11]) proved that it is true for all sheaves.In §9, we simplify his proof and generalize his result to the case of perfect complexes.By taking a spectral cover using a torsion line bundle, we obtain a finite étale Galoiscover ρ : X † → X with a cyclic Galois group G and a perverse sheaf P • on X †which is locally the perverse sheaf of vanishing cycles of a local CS functional. (SeeTheorem 3.16.)The layout of this paper is as follows. In §2, we recall necessary facts about theperverse sheaves of vanishing cycles and their gluing. In §3, we collect the mainresults of this paper. In §4, we prove that there exists a structure which we callpreorientation data on X. In §5, we show that preorientation data induce CS datamentioned above. In §6, we show that CS data induce local perverse sheaves, gluingisomorphisms and the obstruction class for gluing. In §7, we prove an analogue ofTheorem 1.1 for mixed Hodge modules. In §8, we develop a theory of GV invariants.In §9, we discuss the existence of square root of det Ext • π(E, E).An incomplete version of this paper was posted in the arXiv on October 15,2012 (1210.3910). Some related results were independently obtained by C. Brav,V. Bussi, D. Dupont, D. Joyce and B. Szendroi in [5]. We are grateful to DominicJoyce for his comments and suggestions. We thank Martin Olsson for his comments,Zheng Hua for informing us of his paper [11] and Yan Soibelman for his comments.We also thank Takuro Mochizuki for answering questions on mixed Hodge modules.Notations. A complex analytic space is a local ringed space which is coveredby open sets, each of which is isomorphic to a ringed space defined by an idealof holomorphic functions on an analytic open subset of C n for some n > 0, andwhose transition maps preserve the sheaves of holomorphic functions. A complexanalytic variety is a reduced complex analytic space. We will denote the varietyunderlying a complex analytic space X by X. We will use smooth functions to meanC ∞ functions. In case the space is singular, with a stratification by smooth strata,smooth functions are continuous functions that are smooth along each stratum. Weuse analytic functions to mean continuous functions that locally have power seriesexpansions in the real and imaginary parts of coordinate variables. We will workwith analytic topology unless otherwise mentioned.2. Perverse sheaves of vanishing cyclesIn this section, we recall necessary facts about perverse sheaves of vanishingcycles. Let X be a complex analytic variety and Dc(X) b the bounded derivedcategory of constructible sheaves on X over Q. Perverse sheaves are sheaf complexeswhich behave like sheaves.