Hodge-index type inequalities, hyperbolic polynomials and complex Hessian equations

Jian Xiao

Hodge-index type inequalities, hyperbolic polynomials and complex Hessian equations

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arxiv 1810.04662 v1 pith:YV73Y5VC submitted 2018-10-10 math.AG math.CV

Jian Xiao This is my paper · ORCID

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keywords typehodge-indexpolynomialscomplexequationshessianhyperbolicinequalities

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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 G{\aa}rding'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.

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Complex Hessian equations with prescribed singularity on compact K\"ahler manifolds
math.CV 2019-09 conditional novelty 7.0 of 10

The total Hessian mass is monotone in singularity type, and Hessian equations H_m(u)=µ have unique solutions in prescribed singularity classes on compact Kähler manifolds.

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