Virtual classes via vanishing cycles

Abstract

We develop a new method to construct the virtual fundamental classes for quasi-smooth derived schemes using the perverse sheaves of vanishing cycles on their $-1$-shifted contangent spaces. It is based on the author's previous work that can be regarded as a version of the Thom isomorphism for $-1$-shifted cotangent spaces. We use the Fourier-Sato transform to prove that our classes coincide with the virtual fundamental classes introduced by the work of Behrend-Fantechi and Li-Tian, under the quasi-projectivity assumption. We also discuss an approach to construct DT4 virtual classes for $-2$-shifted symplectic derived schemes using the perverse sheaves of vanishing cycles.

arXiv comment45 pages. Version 2 added Section 1.4 and removed Section 6.2 in the previous version

References in corpus (12)

• The intrinsic normal cone
• Gromov-Witten invariants in algebraic geometry
• Cycle groups for Artin stacks
• A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications
• Symmetries and stabilization for sheaves of vanishing cycles
• Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds
• Riemann-Roch theorems and elliptic genus for virtually smooth Schemes
• Donaldson-Thomas theory for Calabi-Yau 4-folds
• Counting sheaves on Calabi-Yau 4-folds, I
• Perversely categorified Lagrangian correspondences
• Dimensional reduction in cohomological Donaldson-Thomas theory
• Natural Commuting of Vanishing Cycles and the Verdier Dual