7 research outputs found
The Hodge ring of Kaehler manifolds
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.Comment: Dedicated to the memory of F. Hirzebruch. To appear in Compositio
Mat
Kähler structures on spin 6-manifolds
We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a K\ue4hler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible K\ue4hler structures