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On the Lefschetz and Hodge-Riemann theorems. (English) Zbl 1302.32020

The main result of this article concerns generalizations of the following three theorems: the hard Lefschetz theorem, the Lefschetz decomposition theorem, and the Hodge-Riemann theorem.
Let us first review these three theorems. Let \((X, \omega)\) be a compact Kähler manifold of dimension \(n\), and \(p,q\) be non-negative integers with \(p+q\leq n\). Put \(\Omega:=\omega^{n-p-q}\). Then \(\Omega\) induces a Hermitian form \(Q_\Omega\) on \(H^{p, q}(X, \mathbb{C})\) by \[ Q_\Omega(\{\alpha\}, \{\beta\}):=i^{p-q}(-1)^{\frac{(p+q)(p+q-1)}{2}}\int_X \alpha\wedge\overline{\beta}\wedge\Omega \] for smooth closed \((p, q)\)-forms \(\alpha\) and \(\beta\). The Hodge-Riemann theorem says that the Hermitian form \(Q_\Omega\) is positive-definite on the subspace \(H^{p, q}(X, \mathbb{C})_{\mathrm{prim}}\subset H^{p, q}(X, \mathbb{C})\), which is defined as follows: \[ H^{p, q}(X, \mathbb{C})_{\mathrm{prim}}:=\Big\{\{\alpha\}\Big| \{\alpha\}\cup\{\Omega\}\cup\{\omega\}=0\Big\}. \] The hard Lefschetz theorem says that the map defined by \(\{\alpha\}\mapsto \{\alpha\}\wedge\{\Omega\}\) gives an isomorphism between \(H^{p, q} (X, \mathbb{C})\) and \(H^{n-p, n-q}(X, \mathbb{C})\). The Lefschetz decomposition theorem says that the cohomology group \(H^{p, q}(X, \mathbb{C})\) can be decomposed as follows: \[ H^{p, q}(X, \mathbb{C})=\{\omega\}\cup H^{p-1, q-1}(X, \mathbb{C})\oplus H^{p, q}(X, \mathbb{C})_{\mathrm{prim}}. \] This decomposition is orthogonal with respect to \(Q_\Omega\). It is known that these three theorems are not true if we replace \(\{\Omega\}\) with an arbitrary class in \(H^{n-p-q, n-p-q}(X, \mathbb{R})\), even when the class contains a strictly positive form ([B. Berndtsson and N. Sibony, Invent. Math. 147, No.2, 371–428 (2002; Zbl 1031.32005)], see also Remark 2.9 in the present article).
In this article, the authors give sufficient conditions on \(\{\Omega\}\) for which these three theorems hold. More precisely, they define the notion of a “Hodge-Riemann cone associated with \(X\)” and show that \(\{\Omega\}\) satisfies the above three theorems if \(\Omega\) takes values only in the Hodge-Riemann cone associated with \(X\). Roughly speaking, this sufficient condition means that we can deform \(\Omega\) continuously to \(\omega^{n-p-q}\) in a “nice way” at every point \(x\) of \(X\). This deformation need not depend continuously on \(x\) and a priori does not preserve the closedness nor the smoothness of the form.
As an application of this theorem and V. A. Timorin’s result [Funct. Anal. Appl. 32, No.4, 268–272 (1998); translation from Funkts. Anal. Prilozh. 32, No. 4, 63–68 (1998; Zbl 0948.32021)], it can be said that the cohomology class defined by the wedge product of \(n-p-q\) Kähler forms on \(X\) satisfies the above three theorems (this theorem was obtained by the authors in [Geom. Funct. Anal. 16, No. 4, 838–849 (2006; Zbl 1126.32018)]). In the last section, they study an explicit family of Hodge-Riemann forms in the context of compact symplectic Kähler manifolds.

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
58A14 Hodge theory in global analysis
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