Virtual classes via vanishing cycles

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.