  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.
