Stefan Schreieder

On the construction problem for Hodge numbers

For any symmetric collection of natural numbers h^{p,q} with p+q=k, we construct a smooth complex projective variety whose weight k Hodge structure has these Hodge numbers; if k=2m is even, then we have to impose that h^{m,m} is bigger than some quadratic bound in m. Combining these results for different weights, we solve the construction problem for the truncated Hodge diamond under two additional assumptions. Our results lead to a complete classification of all nontrivial dominations among Hodge numbers of Kaehler manifolds.

DUALIZATION INVARIANCE AND A NEW COMPLEX ELLIPTIC GENUS

We define a new elliptic genus on the complex bordism ring. With co- efficients in Z(1=2), we prove that it induces an isomorphism of the complex bordism ring modulo the ideal which is generated by all differences P(E) − P(E ∗ ) of projective bundles and their duals onto a polynomial ring on 4 generators in degrees 2, 4, 6 and 8. As an alternative geometric description of , we prove that it is the universal genus which is multiplicative in projective bundles over Calabi-Yau 3-folds. With the help of the q-expansion of modular forms we will see that for a complex manifold M , the value (M ) is a holomorphic Euler characteristic. We also compare with Krichever-H¨ complex elliptic genus and see that their only common specializations are Ochanines elliptic genus and the y-genus.

The Hodge ring of Kähler manifolds

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kahler 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 Kahler 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.

THETA DIVISORS WITH CURVE SUMMANDS AND THE SCHOTTKY PROBLEM

We prove the following converse of Riemann’s Theorem: let

Decomposable theta divisors and generic vanishing

(A,\Theta )

Generic vanishing and minimal cohomology classes on abelian fivefolds

(A,Θ) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety

On the rationality problem for quadric bundles

\Theta =C+Y