Hodge Numbers of Arbitrary Sections from Linear Sections

TL;DR

This paper develops recursive formulas to compute Hodge numbers of smooth complete intersections in a projective setting, extending classical Lefschetz properties and utilizing asymptotic mixed Hodge structures.

Contribution

It introduces new recursive methods for calculating Hodge numbers of arbitrary sections, weakening the Lefschetz hyperplane property in this context.

Findings

Recursive formulas for Hodge numbers in terms of degrees and linear sections

Weakening of the Lefschetz hyperplane property by one degree

Explicit calculations of Hodge numbers for three-dimensional base cases

Abstract

Let $Y$ be a projective submanifold of the total space of the inverse of a very ample line bundle $\pi:L^{-1}\rightarrow B$ over a projective manifold $B$. Any section of $L^{-1}\rightarrow B$ is isomorphic to $B$ and the Hodge numbers of any proper smooth multisection are determined by the degree $d$ of that multi-section as are the Hodge numbers of any smooth complete intersection of multi-sections of degrees $\left(d_{1},\ldots,d_{r}\right)$. In this paper recursive formulae are given for those Hodge numbers in terms of the integers $\left\{ d_{1},\ldots,d_{r}\right\} $ and the Hodge numbers of the linear sections. The recursion proceeds by induction on dimension and degree. Its proof relies on the theory of asymptotic mixed Hodge structures. An interesting corollary is that the Lefschetz hyperplane property is weakened by one degree in this setting. That is, relative vanishing does not reach the middle degree of the hyperplane section but only to degree one less than the middle degree. As an application, in an Appendix, we calculate closed formulae for all Hodge numbers of all smooth complete intersections for the case $\dim B=3$.