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Let $Y$ be a projective submanifold of the total space of the inverse of avery 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 Hodgenumbers of any proper smooth multisection are determined by the degree $d$ ofthat multi-section as are the Hodge numbers of any smooth complete intersectionof multi-sections of degrees $\left(d_{1},\ldots,d_{r}\right)$. In this paperrecursive 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 linearsections. The recursion proceeds by induction on dimension and degree. Itsproof relies on the theory of asymptotic mixed Hodge structures. An interestingcorollary is that the Lefschetz hyperplane property is weakened by one degreein this setting. That is, relative vanishing does not reach the middle degreeof 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 Hodgenumbers of all smooth complete intersections for the case $\dim B=3$.

Hodge Numbers of Arbitrary Sections from Linear Sections