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

Journal article

Let $({\bf X},\omega_{\bf X}^*)$ be a separated, $-2$-shifted symplectic derived $\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension ${\rm vdim}_{\mathbb C}{\bf }=n\in\mathbb Z$, and $X_{\rm an}$ the underlying complex analytic topological space. We prove that $X_{\rm an}$ can be given the structure of a derived mooth manifold ${\bf X}_{\rm dm}$, of real virtual dimension ${\rm vdim}_{\mathbb R}{\bf X}_{\rm dm}=n$. This ${\bf X}_{\rm dm}$ is not canonical, but is independent of choices up to bordisms fixing the underlying topological pace $X_{\rm an}$. There is a 1-1 correspondence between orientations on $({\bf X},\omega_{\bf X}^*)$ and orientations on ${\bf X}_{\rm dm}$.
Because compact, oriented derived manifolds have virtual classes, this means that proper, oriented $-2$-shifted symplectic derived $\mathbb C$-schemes have virtual classes, in either homology or bordism. This is surprising, as conventional algebro-geometric virtual cycle methods fail in this case. Our virtual classes have half the expected dimension, and from purely complex algebraic input, can yield a virtual class of odd real 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 $\mathbb C$, which should be unchanged under deformations of $Y$.

Authors: Borisov, D; Joyce, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Mathematical Sciences Publishers
• Journal: Geometry and Topology
• Volume: 21
• Issue: 6
• Pages: 3231–3311
• Publication date: 2017-08-31
• Acceptance date: 2016-11-24
• DOI: 10.2140/gt.2017.21.3231
• EISSN: 1364-0380
• ISSN: 1465-3060
• Keywords: Article; math.AG; math.DG