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

Borisov, Dennis; Joyce, Dominic

doi:10.2140/gt.2017.21.3231

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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

DOI: 10.2140/gt.2017.21.3231

Abstract

Let $(X, ω_{X}^{*})$ be a separated, $- 2$ –shifted symplectic derived $ℂ$ –scheme, in the sense of Pantev, Toën, Vezzosi and Vaquié (2013), of complex virtual dimension ${vdim}_{ℂ} X = n \in ℤ$ , and $X_{a n}$ the underlying complex analytic topological space. We prove that $X_{a n}$ can be given the structure of a derived smooth manifold $X_{d m}$ , of real virtual dimension ${vdim}_{ℝ} X_{d m} = n$ . This $X_{d m}$ is not canonical, but is independent of choices up to bordisms fixing the underlying topological space $X_{a n}$ . There is a one-to-one correspondence between orientations on $(X, ω_{X}^{*})$ and orientations on $X_{d m}$ .

Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented $- 2$ –shifted symplectic derived $ℂ$ –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 $ℂ$ , which should be unchanged under deformations of $Y$ .

Keywords

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

Mathematical Subject Classification 2010

Primary: 14A20

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

References

Publication

Received: 7 April 2015

Revised: 3 October 2016

Accepted: 23 November 2016

Published: 31 August 2017

Proposed: Simon Donaldson

Seconded: Richard Thomas, Jim Bryan

Authors

Dennis Borisov
Mathematisches Institut Georg-August-Universität Göttingen Göttingen Germany
https://sites.google.com/site/dennisborisov/
Dominic Joyce
The Mathematical Institute University of Oxford Oxford United Kingdom
http://people.maths.ox.ac.uk/~joyce/

Volume 21, issue 6 (2017)