# The integrality conjecture and the cohomology of preprojective stacks

@article{Davison2016TheIC, title={The integrality conjecture and the cohomology of preprojective stacks}, author={Ben Davison}, journal={arXiv: Algebraic Geometry}, year={2016} }

By importing the compactly supported cohomology of various stacks of representations of the preprojective algebra $\Pi_Q$, for $Q$ an arbitrary quiver, into the theory of cohomological Donaldson--Thomas invariants, we prove that this cohomology is pure. In addition, we prove a generalisation of Hausel's formula for the Betti numbers of Nakajima quiver varieties, a degeneration result for preprojective cohomological Hall algebras with extra equivariant parameters, and calculate the mixed Hodge…

## 29 Citations

On cohomological Hall algebras of quivers : Yangians

- Mathematics
- 2017

We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology…

Nonabelian Hodge theory for stacks and a stacky P=W conjecture

- Mathematics, Physics
- 2021

We introduce a version of the P=W conjecture relating the Borel– Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel–Moore homology of…

The cohomological Hall algebra of a surface and factorization cohomology

- Mathematics
- 2019

For a smooth quasi-projective surface S over complex numbers we consider the Borel-Moore homology of the stack of coherent sheaves on S with compact support and make this space into an associative…

Cohomological Donaldson-Thomas theory

- Mathematics
- 2015

This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau…

KAC POLYNOMIALS AND LIE ALGEBRAS ASSOCIATED TO QUIVERS AND CURVES

- MathematicsProceedings of the International Congress of Mathematicians (ICM 2018)
- 2019

A survey of the theory of Kac polynomials for quivers and for curves. In particular, we describe the representation-theoretic meaning of Kac polynomials in terms of Hall algebras, and the geometric…

Theta-stratifications, Theta-reductive stacks, and applications

- Mathematics
- 2016

These are expanded notes on a lecture of the same title at the 2015 AMS summer institute in algebraic geometry. We give an introduction and overview of the "beyond geometric invariant theory" program…

Rational Links and DT Invariants of Quivers

- Mathematics, PhysicsInternational Mathematics Research Notices
- 2019

We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson–Thomas invariants of…

K-theoretic Hall algebras of quivers with potential as Hopf algebras

- Mathematics
- 2021

Preprojective K-theoretic Hall algebras (KHAs), particular cases of KHAs of quivers with potential, are conjecturally positive halves of the Okounkov– Smirnov affine quantum algebras. It is thus…

Cohomological $\chi$-independence for Higgs bundles and Gopakumar-Vafa invariants

- Mathematics, Physics
- 2021

The aim of this paper is two-fold: Firstly, we prove Toda’s χ-independence conjecture for Gopakumar–Vafa invariants of arbitrary local curves. Secondly, following Davison’s work, we introduce the BPS…

Stabilization of Kac polynomials: some conjectures

- Mathematics
- 2020

We give some conjectures concerning the behaviour of Kac polynomials of quivers when increasing the number of arrows: they seem to converge in the ring of power series, with a linear rate of…

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