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

Abstract

It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using Gårding's theory for hyperbolic polynomials, we obtain a mixed Hodge-index type theorem for classes of type (1, 1). The new feature is that this Hodge-index type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate and even negative along some directions.

Concave elliptic equations and generalized Khovanskii-Teissier inequalities

“…Corollary 2.16 was known to Tosatti-Weinkove [32] who observed that it followed from Dinew-Ko lodziej's work [11], and Demailly's technique [9]. It was also proved independently by Xiao [36], who also proves polarized versions of (2.5).…”

“…There has recently been some interest in generalizing the Khovanskii-Teissier inequalities using different concave functions. For some recent work in this direction, using the σ k operators, see [36,37] and the references therein.…”

Teissier problem for nef classes

“…The (4.1) is proved in [18, Theorem 1.6 and Example 2.11] (also see [3,17] for the special case that ω 1 = ... = ω m = ω X ) when each [α j ] ∈ Γ (indeed, this is implied by Remark 2.5(c)); then taking a limit gives (4.1). Comparing with (1.1), [α j ]'s and [α] in (4.1) are not necessarily nef, however we pay the price that the ω j 's in (4.1) have been assumed to be Kähler.…”

“…There are many remarkable further developments on Khovanskii-Teissier type inequalities (see e.g. [1,3,4,5,6,7,8,9,10,11,12,13,15,17] and references therein), among which we may recall the following one over an n-dimensional compact Kähler manifold (X, ω X ) as an example (see [4,5,7]), which will be closely related to our study here. In this note, when we write a class as [α], then α always is a smooth representative of [α].…”

“…[18, Remark 2.9] We remark some consequences of the Hodge index theorem, which should be well-known (see e.g. [5,17]).…”