Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds


Joyce, D
Upmeier, M

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1 January 2020



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Let X be a compact Calabi-Yau 3-fold, and write M,M¯ for the moduli stacks of objects in coh(X),D^bcoh(X). There are natural line bundles K_M→M, K_M¯→M¯, analogues of canonical bundles. Orientation data on M,M¯ is an isomorphism class of square root line bundles K^{1/2}M,K^{1/2}_{M¯}, 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 Donaldson-Thomas invariants, and is important in categorifying Donaldson-Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds X with a spin smooth projective compactification X↪Y. This proves a long-standing conjecture in Donaldson-Thomas theory. These are special cases of a more general result. Let X be a spin smooth projective 3-fold. Using the spin structure we construct line bundles K_M→M, K_M¯→M¯. We define spin structures on M,M¯ to be isomorphism classes of square roots K^{1/2}M,K^{1/2}_{M¯}. We prove that natural spin structures exist on M,M¯. They are equivalent to orientation data when X is a Calabi-Yau 3-fold 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→B_P) on differential-geometric moduli stacks B_P of connections on a principal U(m)-bundle P→X over a compact spin 6-manifold X.

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