||We discuss a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories but sufficiently flexible to admit global complex Kuranishi charts. As an application, we prove Borisov-Joyce's virtual cycle of a projective moduli space of stable sheaves on a Calabi-Yau 4-fold coincides with the algebraic cycle in homology after inverting 2 in the coefficients. In particular, when Borisov-Joyce's real virtual dimension is odd, their virtual cycle is torsion. This is a joint work with Richard P. Thomas.