Finiteness for self-dual classes in integral variations of Hodge structure

Infinitesimal variations of Hodge structure at infinity

By analyzing the local and infinitesimal behavior of degenerating polarized variations of Hodge structure the notion of infinitesimal variation of Hodge structure at infinity is introduced. It is shown that all such structures can be integrated to polarized variations of Hodge structure and that, conversely, all are limits of infinitesimal variations of Hodge structure at finite points. As an i...

Singularities of Variations of Mixed Hodge Structure

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over R, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, has a real limit at the puncture. In particul...

Monodromy of variations of Hodge structure

We give a survey of properties of the monodromy of local systems on quasi-projective varieties which underlie a variation of Hodge structure. In the last section a less widely known version of a NoetherLefschetz type theorem is discussed.

Variations of Hodge Structure on Curves

On integral Hodge classes on uniruled or Calabi-Yau threefolds

Let X be a smooth complex projective variety of dimension n. The Hodge conjecture is then true for rational Hodge classes of degree 2n−2, that is, degree 2n−2 rational cohomology classes of Hodge type (n − 1, n − 1) are algebraic, which means that they are the cohomology classes of algebraic cycles with Q-coefficients. Indeed, this follows from the hard Lefschetz theorem, which provides an isom...