Geometry & Topology

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

Dennis Borisov and Dominic Joyce

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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 .

Article information

Geom. Topol., Volume 21, Number 6 (2017), 3231-3311.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14A20: Generalizations (algebraic spaces, stacks)
Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45] 14J35: $4$-folds 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 53D30: Symplectic structures of moduli spaces

Calabi–Yau manifold coherent sheaf moduli space virtual class derived algebraic geometry


Borisov, Dennis; Joyce, Dominic. Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds. Geom. Topol. 21 (2017), no. 6, 3231--3311. doi:10.2140/gt.2017.21.3231.

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