We propose a new construction which associates to any ample (or big) line bundle on a projective manifold a canonical growth condition (i.e., a choice of a plurisubharmonic (psh) function well defined up to a bounded term) on the tangent space of any given point . We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows one to recover all the infinitesimal Okounkov bodies of at . The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case, the growth condition is “equivalent” to the moment polytope. As in the toric case, the growth condition says a lot about the Kähler geometry of the manifold. We prove a theorem about Kähler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.
"Canonical growth conditions associated to ample line bundles." Duke Math. J. 167 (3) 449 - 495, 15 February 2018. https://doi.org/10.1215/00127094-2017-0031