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Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds

Dennis Borisov and Dominic Joyce

Geometry & Topology 21 (2017) 3231–3311

Let (X,ωX) be a separated, 2–shifted symplectic derived –scheme, in the sense of Pantev, Toën, Vezzosi and Vaquié (2013), of complex virtual dimension vdimX = n , and Xan the underlying complex analytic topological space. We prove that Xan can be given the structure of a derived smooth manifold Xdm, of real virtual dimension vdimXdm = n. This Xdm is not canonical, but is independent of choices up to bordisms fixing the underlying topological space Xan. There is a one-to-one correspondence between orientations on (X,ωX) and orientations on Xdm.

Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented 2–shifted symplectic derived –schemes have virtual classes, in either homology or bordism. This is surprising, as conventional algebrogeometric virtual cycle methods fail in this case. Our virtual classes have half the expected dimension.

Now derived moduli schemes of coherent sheaves on a Calabi–Yau 4–fold are expected to be 2–shifted symplectic (this holds for stacks). We propose to use our virtual classes to define new Donaldson–Thomas style invariants “counting” (semi)stable coherent sheaves on Calabi–Yau 4–folds Y over , which should be unchanged under deformations of Y .

Calabi–Yau manifold, coherent sheaf, moduli space, virtual class, derived algebraic geometry
Mathematical Subject Classification 2010
Primary: 14A20
Secondary: 14N35, 14J35, 14F05, 55N22, 53D30
Received: 7 April 2015
Revised: 3 October 2016
Accepted: 23 November 2016
Published: 31 August 2017
Proposed: Simon Donaldson
Seconded: Richard Thomas, Jim Bryan
Dennis Borisov
Mathematisches Institut
Georg-August-Universität Göttingen
Dominic Joyce
The Mathematical Institute
University of Oxford
United Kingdom