Siarhei Finski

 For a complex projective manifold polarised by an ample line bundle, we study the asymptotic properties of submultiplicative filtrations on the associated section ring and show that these are related to the geometry at infinity of the space of Kähler metrics on the manifold. The construction of geodesic rays from test configurations due to Phong and Sturm and the study of Mabuchi geometry on the space of positive metrics lies at the core of our approach.

Among others, we give an asymptotic formula for the dimension of the vector subspace of the space of holomorphic sections of high tensor powers of the polarising line bundle such that for any holomorphic section from our subspace the difference between the orders of annihilation along two fixed submanifolds lies in a given interval.