48 research outputs found
Multiplicative sub-Hodge structures of conjugate varieties
For any subfield K of the complex numbers which is not contained in an
imaginary quadratic number field, we construct conjugate varieties whose
algebras of K-rational (p,p)-classes are not isomorphic. This compares to the
Hodge conjecture which predicts isomorphisms when K is contained in an
imaginary quadratic number field; additionally, it shows that the complex Hodge
structure on the complex cohomology algebra is not invariant under the
Aut(\C)-action on varieties. In our proofs, we find simply connected conjugate
varieties whose multilinear intersection forms on their second real cohomology
groups are not (weakly) isomorphic. Using these, we detect non-homeomorphic
conjugate varieties for any fundamental group and in any birational equivalence
class of dimension at least 10.Comment: 26 pages; final version, to appear in Forum of Mathematics, Sigm
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.Comment: 34 pages; final version, to appear in Geometry & Topolog
Theta divisors with curve summands and the Schottky problem
We prove the following converse of Riemann's Theorem: let (A,\Theta) be an
indecomposable principally polarized abelian variety whose theta divisor can be
written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C
is smooth, A is the Jacobian of C, and Y is a translate of W_{g-2}(C). As
applications, we determine all theta divisors that are dominated by a product
of curves and characterize Jacobians by the existence of a d-dimensional
subvariety with curve summand whose twisted ideal sheaf is a generic vanishing
sheaf.Comment: 23 pages; final version, to appear in Mathematische Annale
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
Algebraic structures with unbounded Chern numbers
We determine all Chern numbers of smooth complex projective varieties of
dimension at least four which are determined up to finite ambiguity by the
underlying smooth manifold. We also give an upper bound on the dimension of the
space of linear combinations of Chern numbers with that property and prove its
optimality in dimension four.Comment: 15 pages; final version, to appear in Journal of Topolog