Abstract
We consider positive Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds. In particular, we consider the tensor product of positive powers of a positive Hermitian algebraic function satisfying the strong global Cauchy–Schwarz condition on a holomorphic line bundle with another fixed positive Hermitian algebraic function on another holomorphic line bundle. Our main result is to give an effective estimate (in terms of certain geometric data) on the smallest power needed to be taken so that the resulting tensor product is a maximal sum of Hermitian squares, or equivalently, the induced Hermitian metric on the resulting line bundle is the pull-back (via some holomorphic map) of the standard Hermitian metric on the universal line bundle over some complex projective space. This result is an effective version of Catlin–D’Angelo’s Hermitian Positivstellensatz.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.