The motivic Donaldson-Thomas invariants of (-2) curves

TL;DR

This paper computes motivic Donaldson-Thomas invariants for (-2)-curves from 3-fold flopping contractions, revealing nontrivial monodromy actions that differ from known invariants in similar geometric contexts.

Contribution

It introduces a method to calculate motivic Donaldson-Thomas invariants for (-2)-curves and uncovers nontrivial monodromy actions, expanding understanding of these invariants in algebraic geometry.

Findings

Motivic Donaldson-Thomas invariants for (-2)-curves are explicitly computed.

Monodromy actions on these invariants are shown to be nontrivial.

Results contrast with known invariants for small resolutions of Gorenstein singularities.

Abstract

In this paper we calculate the motivic Donaldson-Thomas invariants for (-2)-curves arising from 3-fold flopping contractions in the minimal model programme. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed previously by the authors we describe the monodromy on these invariants. In particular, in contrast to all existing known Donaldson-Thomas invariants for small resolutions of Gorenstein singularities these monodromy actions are nontrivial.