Iara Goncalves: Direct Images of Locally Constant Sheaves on Complements to Plane Line Arrangements

On the complement \(X= \mathbb C^2 - \bigcup_{i=1}^n L_i\) (where each \(L_i\) is a line passing through the origin) to a plane line arrangement \(\bigcup_{i=1}^n L_i \subset \mathbb C^2\), a locally constant sheaf of complex vector spaces is given by a multi-index \(\alpha \in \mathbb C^n\). Using the description of Mac-Pherson and Vilonen we obtain a criterion for the irreducibility of the direct image \({Rj_* \mathcal L_{\alpha}}\) as a perverse sheaf, where \(j: X \rightarrow \mathbb C^2\) is the canonical inclusion.