Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems

TL;DR

This paper extends motivic Hirzebruch classes to naive Donaldson-Thomas type invariants, explores their categorification from a bivariant-theoretic perspective, and proposes open questions for future research.

Contribution

It introduces a new extension of motivic Hirzebruch classes to DT-type invariants and discusses their categorification, providing a novel framework for further exploration.

Findings

Extended motivic Hirzebruch classes to DT-type invariants

Proposed a categorification approach from bivariant theory

Posed open questions for future research

Abstract

Donaldson-Thomas invariant is expressed as the weighted Euler characteristic of the so-called Behrend (constructible) function. In \cite{Behrend} Behrend introduced a DT-type invariant for a morphism. Motivated by this invariant, we extend the motivic Hirzebruch class to naive Donaldson-Thomas type analogues. We also discuss a categorification of the DT-type invariant for a morphism from a bivariant-theoretic viewpoint, and we finally pose some related questions for further investigations.