On Hodge-Riemann Cohomology Classes

Notes on absolute Hodge classes

Absolute Hodge classes first appear in Deligne’s proof of the Weil conjectures for K3 surfaces in [14] and are explicitly introduced in [16]. The notion of absolute Hodge classes in the singular cohomology of a smooth projective variety stands between that of Hodge classes and classes of algebraic cycles. While it is not known whether absolute Hodge classes are algebraic, their definition is bo...

Hodge Theory and A∞ Structures on Cohomology

We use Hodge theory and a construction of Merkulov to construct A∞ structures on de Rham cohomology and Dolbeault cohomology. Hodge theory is a powerful tool in differential geometry. Classically, it can be used to identify the de Rham cohomology of a closed oriented Riemannian manifold with the space of harmonic forms on it as vector spaces. The wedge product on differential forms provides an ...

The Hodge filtration on nonabelian cohomology

Whereas usual Hodge theory concerns mainly the usual or abelian cohomology of an algebraic variety—or eventually the rational homotopy theory or nilpotent completion of π1 which are in some sense obtained by extensions—nonabelian Hodge theory concerns the cohomology of a variety with nonabelian coefficients. Because of the basic fact that homotopy groups in higher dimensions are abelian, and si...

Hodge structures on cohomology algebras and geometry

It is well-known (see eg [22]) that the topology of a compact Kähler manifold X is strongly restricted by Hodge theory. In fact, Hodge theory provides two sets of data on the cohomology of a compact Kähler manifold. The first data are the Hodge decompositions on the cohomology spaces H(X,C) (see (1.1) where V = H(X,Q)); they depend only on the complex structure. The second data, known as the Le...

Hodge loci and absolute Hodge classes