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Mixed Hodge-Riemann bilinear relations in the linear situation. (English. Russian original) Zbl 0948.32021

Funct. Anal. Appl. 32, No. 4, 268-272 (1998); translation from Funkts. Anal. Prilozh. 32, No. 4, 63-68 (1998).
Introduction: The Aleksandrov-Fenchel inequalities in the theory of mixed volumes are a far-reaching generalization of the isoperimetric inequality. Even the linear variant of these inequalities, which relates the mixed discriminants of various collections of positive definite quadratic forms, is nontrivial. For instance, the old van der Waerden problem on permanents of doubly stochastic matrices was solved with the help of inequalities for mixed discriminants. It is known that there is a close connection between the Aleksandrov-Fenchel inequalities and the Hodge-Riemann relations in the cohomology ring of a compact Kähler manifold. More precisely, the Aleksandrov-Fenchel inequalities correspond to the Hodge-Riemann relations in the \((1,1)\)-cohomology. The theory of harmonic forms reduces the Hodge-Riemann relations to their linear variant.
In this paper, we generalize the linear variant of the Hodge-Riemann relations by constructing their mixed analog. For \((1,1)\)-forms, this analog is equivalent to the Aleksandrov inequalities for mixed discriminants.

MSC:

32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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