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

Davison, Ben; Meinhardt, Sven

doi:10.2140/ant.2017.11.1243

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

Ben Davison and Sven Meinhardt

Vol. 11 (2017), No. 6, 1243–1286

DOI: 10.2140/ant.2017.11.1243

Abstract

We calculate the motivic Donaldson–Thomas invariants for $(- 2)$ -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.

Keywords

Donaldson–Thomas theory, minus two curves, motivic invariants

Mathematical Subject Classification 2010

Primary: 14N35

Milestones

Received: 6 February 2016

Revised: 23 November 2016

Accepted: 1 February 2017

Published: 16 August 2017

Authors

Ben Davison
School of Mathematics and Statistics University of Glasgow 15 University Gardens Glasgow G128QW United Kingdom

Sven Meinhardt
School of Mathematics and Statistics University of Sheffield Hicks Building, Hounsfield Road Sheffield S37RH United Kingdom

Vol. 11, No. 6, 2017