Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
The main result of this paper relates two classes of geometric objects: derived \(\mathbb{C}\) schemes \(\mathbf{X}\) of Toën-Vezzosi [\textit{T. Pantev} et al., Publ. Math., Inst. Hautes Étud. Sci. 117, 271--328 (2013; Zbl 1328.14027)] equipped with a \(-2\) shifted symplectic structure \(\omega^*_{\mathbf{X}}\) of [loc. cit.] (expected structures on 4D Calabi-Yau derived moduli \(\mathbb{C}\)-schemes); and derived smooth manifolds \(\mathbf{X}_{\text{dm}}\) of \textit{D. I. Spivak} [Duke Math. J. 153, No. 1, 55--128 (2010; Zbl 1420.57073)] and the first author with \textit{J. Noel} [``Simplicial approach to derived differential manifolds'', Preprint, \url{arXiv:1112.0033}] (the authors work with them in terms of Kuranishi atlases). Namely, the underlying complex analytic space \(X_{\text{an}}\) of \(\mathbf{X}\) can be given a structure of smooth derived manifold \(\mathbf{X}_{\text{dm}}\) of \(\text{vdim}_{\mathbb{R}}\mathbf{X}_{\text{dm}}=\text{vdim}_{\mathbb{C}}\mathbf{X}\). Although not canonical, \(\mathbf{X}_{\text{dm}}\) is independent of choices of derived bordisms fixing \(X_{\text{an}}\), and there is a \(1-1\) correspondence between orientations on \((\mathbf{X},\omega^*_{\mathbf{X}})\) and \(\mathbf{X}_{\text{dm}}\). A surprising consequence is that \(-2\) shifted symplectic derived \(\mathbb{C}\)-schemes \((\mathbf{X},\omega^*_{\mathbf{X}})\) have virtual classes, surprising because the standard Behrend-Fantechi construction [\textit{K. Behrend} and \textit{B. Fantechi}, Invent. Math. 128, No. 1, 45--88 (1997; Zbl 0909.14006)] of them only works for obstruction theories perfect in \([-1,0]\), not \([-2,0]\) as here. Moreover, the virtual classes have half the expected dimension, roughly reflecting the base of a ``Lagrangian fibration'' \(\mathbf{X}^{C^\infty}\to\mathbf{X}_{\text{dm}}\) with derived \( C^\infty\) scheme \(X^{C^\infty}\). The motivation for, and the intended application of, the results is to define new invariants, conjectured by \textit{S. K. Donaldson} and \textit{R. P. Thomas} [in: The geometric universe: science, geometry, and the work of Roger Penrose. Proceedings of the symposium on geometric issues in the foundations of science. Oxford: Oxford University Press. 31--47 (1998; Zbl 0926.58003)], that ``count'' semistable coherent sheaves on Calabi-Yau \(4\)-folds. They are complex analogs of the Donaldson invariants for \(4\)-manifolds. The results hint at a special geometry on Calabi-Yau \(4\)-folds like the one known for \(3\)-folds (mirror symmetry, etc.). An alternative approach to counting coherent sheaves on Calabi-Yau \(4\)-folds, based on gauge theory, was recently proposed by \textit{Y. Cao} and \textit{N. C. Leung} [Trans. Am. Math. Soc. 369, No. 9, 6631--6659 (2017; Zbl 1373.14053)] (following a Donaldson-Thomas' suggestion). But the derived approach pursued in this paper often delivers compactness of the moduli for free by working with coherent sheaves instead of vector bundles, and might be more effective.
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