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

Abstract

We calculate the motivic Donaldson–Thomas invariants for $\left(-2\right)$-curves arising from $3$-fold flopping contractions in the minimal model program. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed earlier 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.