Cohomology of local systems on the complement of hyperplanes
Let \({\mathcal H}=\{H_ i\}_{i \in I}\) be a finite collection of hyperplanes in a complex projective space \(\mathbb{P}^ N\), \(U=\mathbb{P}^ N- \bigcup_{H \in {\mathcal H}}H\). Suppose that to each \(H \in {\mathcal H}\) a complex \(r \times r\) matrix \(P(H)\) is assigned, such that \(\sum_{H \in {\mathcal H}} P(H)=0\). This defines a connection \(\nabla=d+\omega\) on a trivial \(r\)-dimensional bundle \(E={\mathcal O}_ U \otimes_ \mathbb{C} \mathbb{C}^ r\), \({\mathcal O}_ U\) being the sheaf of holomorphic functions, and the closed \(\text{End} (E)\)-valued 1-form \(\omega\) is defined as \(\omega=\sum_{i \in I} \eta_{i,i_ 0} \otimes P(H_ i)\) where \(\eta_{ij}:=\text{dlog} (f_{ij})\), \(H_ i-H_ j=\text{div} (f_{ij})\). Suppose that \(\nabla\) is integrable, i.e. \(\omega \wedge \omega=0\). The sheaf of horizontal sections of \(\nabla\) is a complex local system \({\mathcal L}\) on \(U\). Let \(\Omega_ \nabla\) denote the subcomplex of the complex of global sections of the de Rham complex of \((E,\nabla)\) generated as a \(\mathbb{C}\)-vector space by all forms \(\eta_{i_ 1j_ 1} \wedge \cdots \wedge \eta_{i_ pj_ p} \otimes v\), \(v \in \mathbb{C}^ r\). It is complex of finite dimensional vector spaces. One has a natural map \[ H^*(\Omega_ \nabla) \to H^*(U,{\mathcal L}) \tag{1} \] The following result is proven. Theorem. Suppose that for any nonempty intersection \(L\) of hyperplanes from \({\mathcal H}\), none of the eigenvalues of the operator \(\sum_{H | L \subset H} P(H)\) is a positive integer. Then the map (1) is an isomorphism. This establishes a conjecture made by K. Aomoto. The result is a not difficult consequence of a local result by Deligne, Brieskorn theorem, and mixed Hodge theory.
scientific article; zbMATH DE number 220750