The mixed Hodge–Riemann bilinear relations for compact Kähler manifolds

Abstract

We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kähler manifolds in the mixed situation.

Equidistribution problems in complex dynamics of higher dimension

“…Recall that a direct consequence of the mixed Hodge-Riemann theorem applied to the graphs of f n , see e.g. [20,51], implies that, when f is a meromorphic map, the function p → log d p (f ) is concave. Equivalently, we have…”

Cauchy–Schwarz-type inequalities on Kähler manifolds-II

“…The mixed Hodge-Riemann bilinear relations. In this subsection we briefly recall the mixed Hodge-Riemann biliner relations established in [2] by Dinh and Nguyên.…”

“…Define the mixed Hodge-Riemann bilinear form Q Ω (·, ·) with respect to Ω on H p,q (M, C) by and Q(·, ·) respectively in [2]. We use the current symbols to avoid confusion as they stress the dependence on the choices of ω and Ω, whose advantage will be clear in the process of our proof in Theorem 1.3 in the next section.…”

Hodge-Index Type Inequalities, Hyperbolic Polynomials, and Complex Hessian Equations

“…Note that the Laplacian equation can be seen as a special case of the more general dd c -equations used in [DN06].…”

“…For example, it can be proved by reducing the global case to the local case by harmonic forms (see e.g. [Voi07]), or by using the deep relationship between polarized Hodge-Lefschetz modules and variations of Hodge structures (see [Cat08]), or by reducing to the local case by applying the L 2 -method to solve a dd c -equation (see [DN06]). Note that the later two proofs apply to the more general mixed situation.…”