The integrality conjecture and the cohomology of preprojective stacks

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Abstract / Description of output

We study the Borel-Moore homology of stacks of representations of preprojective algebras Π Q, via the study of the DT theory of the undeformed 3-Calabi-Yau completion Π Q [ x ]. Via a result on the supports of the BPS sheaves for Π Q[ x ] -mod Π Q[x], we prove purity of the BPS cohomology for the stack of Π Q [ x ] -modules and define BPS sheaves for stacks of Π Q-modules. These are mixed Hodge modules on the coarse moduli space of Π Q-modules that control the Borel-Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure and thus that the Borel-Moore homology of stacks of Π Q -modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of Π Q -modules. We use our results to prove positivity of a number of "restricted"Kac polynomials, determine the critical cohomology of Hilb n (A 3), and the Borel-Moore homology of genus one character stacks, as well as providing various applications to the cohomological Hall algebras associated to Borel-Moore homology of stacks of modules over preprojective algebras, including the PBW theorem, and torsion-freeness.

Original languageEnglish
Pages (from-to)105-154
JournalJournal für die reine und angewandte Mathematik
Volume2023
Issue number804
Early online date24 Oct 2023
DOIs
Publication statusPublished - 1 Nov 2023

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