The Hodge ring of Kaehler manifolds

TL;DR

This paper characterizes the structure of the Hodge ring for compact Kähler manifolds, revealing the relations among Hodge numbers and their invariance properties, and solving a classical problem of Hirzebruch.

Contribution

It determines the structure of the Hodge ring, clarifies relations among Hodge numbers, and identifies invariants, extending results to all Kähler manifolds and solving a classical problem.

Findings

No unexpected relations among Hodge numbers.

Hodge numbers of Kähler and projective varieties are essentially the same.

Complete solution to Hirzebruch's classical problem.

Abstract

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kaehler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kaehler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch's.