Hodge Number of Cohomology of Local Systems on the Complement of Hyperplanes in $\Bbb P^3$

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 2, Page 177--199.

Kawahara, Yukihito
Hodge Number of Cohomology of Local Systems on the Complement of Hyperplanes in $\Bbb P^3$
The cohomology of the local system on the complement of hyperplanes has a Hodge structure as the $Î±$-invariant cohomology of a Kummer covering ramified along their hyperplanes for a generic character $Î±$. In this paper we study the case of arrangements of hyperplanes in the three dimensional complex projective space. We construct a resolution for an arrangement of hyperplanes and compute its Chow group. By computing the first Chern class of logarithmic 1-forms, we can obtain the Euler characteristic and the Hodge numbers of cohomology of local systems using the intersection set of the arrangement of hyperplanes and binomial coefficients.