The Hodge ring of Kaehler manifolds

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.

Dedicated to the memory of F. Hirzebruch. To appear in Compositio Math

References in corpus (5)

• Topologically invariant Chern numbers of projective varieties
• Characteristic numbers of algebraic varieties
• Chern numbers and diffeomorphism types of projective varieties
• Dualization invariance and a new complex elliptic genus
• Pontryagin numbers and nonnegative curvature

Cited by in corpus (10)

• On the construction problem for Hodge numbers
• Kaehler structures on spin 6-manifolds
• The construction problem for Hodge numbers modulo an integer in positive characteristic
• The Hodge ring of varieties in positive characteristic
• Algebraic structures with unbounded Chern numbers
• Sasaki structures distinguished by their basic Hodge numbers
• Examples of diffeomorphic complete intersections with different Hodge numbers
• Updates on Hirzebruch's 1954 Problem List
• A characterization of the $\hat{A}$-genus as a linear combination of Pontrjagin numbers
• Homological congruence formulae for characteristic classes of singular varieties