Effective Geometric Hermitian Positivstellensatz

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.