Cohomology of flat vector bundles over tori (Q2782758)

Let \(E\) be a flat complex vector bundle over a torus \(\mathbb{T}^n= \mathbb{R}^n\). Taking a flat connection of \(E\), \(H^k(\mathbb{T}^n,E)\), the cohomology of \(k\)-forms on \(\mathbb{T}^n\) with values in \(E\) is defined.NEWLINENEWLINENEWLINEIn this paper, after remarking \(\dim H^0(\mathbb{T}^n, E)=1\) if and only if \(E\) is trivial, NEWLINE\[NEWLINE\dim H^k (\mathbb{T}^n,E) ={n\choose k}\dim H^0(\mathbb{T}^n,E),NEWLINE\]NEWLINE is shown (theorem). Since \(E\) is induced from a representation \(\rho\) of \(\mathbb{Z}^n\) in the structure group of \(E\), \(H^k(\mathbb{T}^n,E)\) is isomorphic to the cohomology of multiplicative forms on \(\mathbb{R}^n\) with respect to \(\rho\). Then, applying Hodge theory, the theorem is proved by counting multiplicative harmonic forms on \(\mathbb{R}^n\).