Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

Abstract

Abstract. The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

Positivity for quantum cluster algebras

“…After a reminder of some notation and constructions regarding quiver representations in §6.1, in §6.2 we prove a positivity result for the refined DT invariants of the category of right modules for the path algebra CQ of a quiver Q. This generalizes a positivity result in [MR19] which applied only to the case of acyclic quivers. The proof of this theorem is within the framework of cohomological Donaldson-Thomas theory.…”

“…On the other hand, since with a little work, the original positivity result of Meinhardt and Reineke [MR19], written in the language of refined DT theory, can be used to prove the pL preservation statement of Theorem 2.15, we present this version of the theory in §6.3. Since the pL property is stronger than positivity, this provides an alternative proof of the (strong) positivity of quantum theta functions.…”

“…We show that the positivity problem in the base cases involves the DT invariants of a quiver depending on n = ω(v 1 , v 2 ), and the parities of m 1 and m 2 . In the case in which m 1 and m 2 are both even, this turns out to be a problem that has already been solved: positivity amounts to proving positivity of the DT invariants for the n-Kronecker quiver, which has been established as a very special case of a positivity result of Meinhardt and Reineke regarding acyclic quivers [MR19] using refined DT theory.…”

Equivariant characteristic classes of external and symmetric products of varieties

“…As a concrete example, for Z pure dimensional with coefficients given by the intersection cohomology Hodge module IC H Z on Z, the corresponding Schur functor S µ of π n * IC H Z n is given by the twisted intersection cohomology Hodge module S µ (π n * IC H Z n ) = IC H Z (n) (V µ ) with twisted coefficients corresponding to the local system on the configuration space B(Z, n) of unordered n-tuples of distinct points in Z, induced from V µ by the group homomorphism π 1 (B(Z, n)) → Σ n (compare [28][p.293] and [30][Prop.3.5]). For Z projective and pure-dimensional, by taking the degrees in (19) for the present choice of coefficients IC H Z and representation V µ , we obtain the following identity for the χ y -polynomial of the twisted intersection cohomology:…”

An Introduction to (Motivic) Donaldson-Thomas Theory

“…Despite some computations of motivic or even numerical Donaldson-Thomas invariants for quivers with potential (see [2,6,4,27]), the true nature of Donaldson-Thomas invariants for quiver with potential remained mysterious for quite some time. A full understanding has been obtained recently and is the content of a series of papers [7,5,26,24].…”