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

Abstract

Let $(\mathbf{X},{\omega }_{\mathbf{X}}^{\ast })$ 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 ${\mathrm{vdim}}_{\mathbb{C}}\mathbf{X}=n\in \mathbb{Z}$, and $X_{\mathrm{an}}$ the underlying complex analytic topological space. We prove that $X_{\mathrm{an}}$ can be given the structure of a derived smooth manifold ${\mathbf{X}}_{\mathrm{dm}}$, of real virtual dimension ${\mathrm{vdim}}_{\mathbb{R}}{\mathbf{X}}_{\mathrm{dm}}=n$. This ${\mathbf{X}}_{\mathrm{dm}}$ is not canonical, but is independent of choices up to bordisms fixing the underlying topological space $X_{\mathrm{an}}$. There is a 1-1 correspondence between orientations on $(\mathbf{X},{\omega }_{\mathbf{X}}^{\ast })$ and orientations on ${\mathbf{X}}_{\mathrm{dm}}$. 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 $Y$.

(v2) 69 pages. Final version, to appear in Geometry and Topology

References in corpus (1)

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