Twisted cohomology on algebraic and analytic varieties

In this article, we study and review some aspects of twisted cohomologies on algebraic and analytic varieties. We compared such cohomologies in both the algebraic and analytic categories and defined two types of twisting parameters in the analytic setting, and also discussed algebraic twisting. Several computations are given. We have also reviewed some isomorphisms of such cohomologies for cohomologous twisting parameters. We discussed constraints on the algebraic varieties that should be assumed so that the algebraic de Rham cohomologies on the variety can be twisted.