Logarithmic Connections and Mixed Hodge Structures

Abstract

In this chapter we prove a version of the Tangent Cone Theorem for smooth quasi-projective varieties. Then we discuss the mixed Hodge structure on the cohomology of the hyperplane complement $M({\mathscr {A}})$ and of the Milnor fiber F. We define the corresponding spectrum, and state the key results of N. Budur and M. Saito stating that this spectrum is determined by the intersection lattice and giving an explicit formula in the case of a line arrangement in $\mathbb {P}^2$. Next we discuss the polynomial count property of algebraic varieties Y defined over the rationals $\mathbb {Q}$. This property always holds when $Y=M({\mathscr {A}})$, while in the case when Y is the Milnor fiber F of such an arrangement, this property is related to the triviality of the monodromy action on $H^*(F)$. A discussion of Hodge–Deligne polynomials completes this chapter.