Hodge--Riemann relations for $m$-positive and generalized $m$-positive forms

Abstract

We study Hodge--Riemann relations for ordinary, sequential, and weak $m$-positive real $(1,1)$-forms in complex dimension $d$. For ordinary $m$-positive forms, we prove the fully mixed relations in every permitted bidegree without assuming semipositivity, generalizing results by Xiao. As a generalization of ordinary $m$-positivity, sequential $m$-positivity is defined by the hyperbolicity cones of successive directional derivatives of the determinant. We formulate a generalized mixed Hodge--Riemann conjecture for this condition and prove several cases beyond the scalar inequalities supplied by hyperbolicity. We prove strict strong positivity for good prefixes with at most three repeated blocks after the first factor. We also consider weak $m$-positivity which corresponds to the pointwise subsolution condition for the complex $m$-Hessian equation. In this setting, we prove the Hodge--Riemann relations for $p\ne q$ and in several diagonal cases, and show that the associated power map induces a pointwise diffeomorphism onto the cone of positive $(d-1,d-1)$-forms. Using Dinh and Nguyên's $L^2$ globalization method, we pass from the pointwise relations to cohomology classes on compact Kähler manifolds. Finally, we propose conjectures on a numerical criterion for sequential $m$-positive classes and on the solvability of the generalized $m$-Hessian equation.

arXiv comment48 pages