Hodge Numbers of Arbitrary Sections from Linear Sections

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 $\dim B=3$.