Orientation data for moduli spaces of coherent sheaves over CalabiYau 3folds
Abstract
Let $X$ be a compact CalabiYau 3fold, and write $\mathcal M,\bar{\mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$, analogues of canonical bundles. Orientation data on $\mathcal M,\bar{\mathcal M}$ is an isomorphism class of square root line bundles $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$, satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman arXiv:1006.270 in their theory of motivic DonaldsonThomas invariants, and is important in categorifying DonaldsonThomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact CalabiYau 3folds, and also for compactlysupported coherent sheaves and perfect complexes on noncompact CalabiYau 3folds $X$ with a spin smooth projective compactification $X\hookrightarrow Y$. This proves a longstanding conjecture in DonaldsonThomas theory. These are special cases of a more general result. Let $X$ be a spin smooth projective 3fold. Using the spin structure we construct line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal M}$. We define spin structures on $\mathcal M,\bar{\mathcal M}$ to be isomorphism classes of square roots $K_{\mathcal M}^{1/2},K_{\bar{\mathcal M}}^{1/2}$. We prove that natural spin structures exist on $\mathcal M,\bar{\mathcal M}$. They are equivalent to orientation data when $X$ is a CalabiYau 3fold with the trivial spin structure. We prove this using our previous paper arXiv:1908.03524, which constructs 'spin structures' (square roots of a certain complex line bundle $K_P\to\mathcal B_P$) on differentialgeometric moduli stacks $\mathcal B_P$ of connections on a principal U$(m)$bundle $P\to X$ over a compact spin 6manifold $X$.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:2001.00113
 Bibcode:
 2020arXiv200100113J
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 46 pages. (v2) final version, to appear in Advances in Mathematics