Positivity conditions for Hermitian symmetric functions

Details Positivity conditions for Hermitian symmetric functions Positivity conditions for Hermitian symmetric functions

2003-06-13

arxiv.org/abs/math/0306220v1

Mathematics

Complex Variables

Dedicated to Yum-Yong Siu on the occasion of his sixtieth birthday

Scientific paper

We introduce a countable collection of positivity classes for Hermitian symmetric functions on a complex manifold, and establish their basic properties. We study a related notion of stability. The first main result shows that, if the underlying matrix of coefficients of an entire Hermitian symmetric function has at most k positive eigenvalues, then it can lie in the k-th positivity class only if it is a squared norm. We establish a similar result for Hermitian symmetric functions on the total space of a holomorphic line bundle. Finally we study the positivity classes for a natural one-parameter family of Hermitian metrics on a power of the universal bundle over complex projective space; we obtain sharp information about the parameter values in order to be in the k-th class. The paper closes with some additional information about the case when k is 2, where a nonlinear version of the Cauchy-Schwarz inequality arises.

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