Twisted cohomology on algebraic and analytic varieties

TL;DR

This paper reviews and compares twisted cohomologies on algebraic and analytic varieties, introduces twisting parameters, and discusses their properties and constraints for algebraic de Rham cohomologies.

Contribution

It provides a comparative analysis of algebraic and analytic twisted cohomologies, introduces new types of twisting parameters, and discusses conditions for their application.

Findings

Comparison of algebraic and analytic twisted cohomologies

Introduction of two types of twisting parameters in the analytic setting

Discussion of isomorphisms and constraints for twisted algebraic de Rham cohomologies

Abstract

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.