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2017 Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds
Dennis Borisov, Dominic Joyce
Geom. Topol. 21(6): 3231-3311 (2017). DOI: 10.2140/gt.2017.21.3231


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.


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Dennis Borisov. Dominic Joyce. "Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds." Geom. Topol. 21 (6) 3231 - 3311, 2017.


Received: 7 April 2015; Revised: 3 October 2016; Accepted: 23 November 2016; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06779917
MathSciNet: MR3692967
Digital Object Identifier: 10.2140/gt.2017.21.3231

Primary: 14A20
Secondary: 14F05, 14J35, 14N35, 53D30, 55N22

Rights: Copyright © 2017 Mathematical Sciences Publishers


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Vol.21 • No. 6 • 2017
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