We develop a new method to construct the virtual fundamental
classes for quasismooth derived schemes (and more generally, derived
-Artin
stacks) using the perverse sheaves of vanishing cycles on their
-shifted
cotangent spaces. It is based on the author’s previous work that
can be regarded as a version of the Thom isomorphism for
-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 schematic and quasiprojectivity
assumption. We also discuss an approach to construct DT4 virtual classes for
-shifted
symplectic derived schemes using the perverse sheaves of vanishing cycles.
Keywords
virtual fundamental classes, vanishing cycles, derived
algebraic geometry, shifted symplectic geometry,
cohomological Donaldson–Thomas theory
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