Dimensional reduction in cohomological Donaldson-Thomas theory

Abstract

For oriented −1-shifted symplectic derived Artin stacks, Ben-Bassat-Brav-Bussi-Joyce introduced certain perverse sheaves on them which can be regarded as sheaf theoretic categorifications of the Donaldson-Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the −1-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel-Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison.We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson-Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of Thom isomorphism theorem for dual obstruction cones, we propose a sheaf theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks. Contents 1. Introduction 1 2. Shifted symplectic structures and vanishing cycles 5 3. Dimensional reduction for schemes 16 4. Dimensional reduction for stacks 21 5. Applications 36 Appendix A. Remarks on the determinant functor 39 References 44. By taking the derived functor of f * we obtain Rf * . When f is a smooth morphism, f * is nothing but the restriction functor. In general f * is constructed by taking simplicial covers, but we use pullback functors only for smooth morphisms in this paper, so we do not need this general construction. To define Rf ! and f ! we use the Verdier duality functor. Arguing as in [LO08, §3], we can construct the dualizing complex ω X ∈ D (b) c (X lis−an , Q) and define the Verdier duality functor D X := RHom(−, ω X ) : D (−) c (X lis−an , Q) op → D (+) c (X lis−an , Q).

Purity and 2-Calabi-Yau categories

“…In general, the zeroth perverse cohomology at a point F should be obtained by taking symmetric convolution powers of BPS sheaves corresponding to classes in K 0 (A ) dividing that of F . Recently Tasuki Kinjo has extended the dimensional reduction isomorphism to general 2CY categories [Kin21]. With some substantial work, one should be able to use his version of the dimensional reduction isomorphism along with the arguments of [Dav20] to produce an alternative proof of Theorem 6.1.…”

“…This is an analogue of the "less" perverse filtration from[Dav20], which is denoted in [ibid] by L ≤• (hence our notation here). This is not to be confused with the perverse filtration on critical cohomology (that A also carries, using Kinjo's dimensional reduction theorem[Kin21]), which is quite different, and is denoted by P ≤• in the case of preprojective algebras considered in[Dav20].…”

Virtual classes via vanishing cycles

“…Detailed analysis on the canonical orientation for T * [−1]U as in [21] shows that the local system L is trivial. Therefore the first statement follows immediately.…”

“…Therefore we can define a natural perverse sheaf ϕ T * [−1]Y ∈ Perv( Y ). The following isomorphisms are proved in [21,Theorem 3.1]:…”

“…The isomorphism γ is constructed in[21, Theorem 3.1] and the commutativity of the diagram (5.2) follows from its proof. The isomorphism γ can be constructed using the Verdier self-duality of ϕ T * [−1]Y , and the commutativity of the diagram (5.3) follows from the Verdier dual of the diagram (5.2), the commutativity of the diagram (2.33), and the discussion after[21, Theorem 2.17]. Now consider the following composition(5.4) …”