KASS seminar

Abstract: A graded norm on a section ring of a polarised projective manifold is called submultiplicative if the norm of products of holomorphic sections is no bigger than the products of norms. Such norms arise naturally in complex geometry and functional analysis. For example, in the former context, they appear in the study of holomorphic extension problems, submultiplicative filtrations (related to K-stability) and Narasimhan-Simha pseudonorms. In the latter context, they appear in the study of projective tensor norms on polynomial rings.

We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarising line bundle. We then derive several applications of this result to the aforementioned problems. The holomorphic extension theorem of Ohsawa and Takegoshi, semiclassical analysis in complex geometry and pluripotential theory play a prominent role in our work.