Donaldson-Thomas theory: generalizations and related conjectures

Donaldson-Thomas theory: generalizations and related conjectures

Tue, 08/11/2011
15:45 		Vittoria Bussi (Oxford) Algebraic and Symplectic Geometry Seminar Add to calendar L3
Donaldson-Thomas theory: generalizations and related conjectures
Generalized Donaldson-Thomas invariants $ \bar{DT}^\alpha(\tau) $ defined by Joyce and Song are rational numbers which 'count' both $ \tau $-stable and $ \tau $-semistable coherent sheaves with Chern character $ \alpha $ on a Calabi-Yau 3-fold X, where $ \tau $ denotes Gieseker stability for some ample line bundle on X. The theory of Joyce and Song is valid only over the field $ \mathbb C $. We will extend it to algebraically closed fields $ \mathbb K $ of characteristic zero. We will describe the local structure of the moduli stack $ \mathfrak M $ of coherent sheaves on X, showing that an atlas for $ \mathfrak M $ may be written locally as the zero locus of an almost closed 1-form on an étale open subset of the tangent space of $ \mathfrak M $ at a point, and use this to deduce identities on the Behrend function $ \nu_{\mathfrak M} $ of $ \mathfrak M $. This also yields an extension of generalized Donaldson-Thomas theory to noncompact Calabi-Yau 3-folds. Finally, we will investigate how our argument might yield generalizations of the theory to a even wider context, for example the derived framework using Toen's theory and to motivic Donaldson-Thomas theory in the style of Kontsevich and Soibelman.