Twisted Poincaré lemma and twisted Čech–de Rham isomorphism in case dimension =1

Abstract

For a compact Riemann surface, $(n+1)$-tuple $x:=(x_0,\dots ,x_n)$ of points on it, and a holomorphic vector bundle with an integrable connection on the open Riemann surface $X_x$ deprived of $(n+1)$ points $x_0,\dots ,x_n$, let $\mathcal{L}$ be the local system of horizontal sections of the connection. In this article, we give a suitable covering of $X_x$ to calculate the Čech cohomology and describe the isomorphism between the cohomology and the twisted de Rham cohomology, which is the cohomology of the complex with the differentials given by the connection. This isomorphism is given by the integrations over Aomoto’s regularized paths, the so-called Euler type integrals.

For the family $\{X_x{\}}_x$ parametrized by $x$, we give a variant of the isomorphism.