Canonical growth conditions associated to ample line bundles

Abstract

We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e. a choice of a psh function well-defined up to a bounded term) on the tangent space $T_{p}X$ of any given point $p$. We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows you to recover all the infinitesimal Okounkov bodies of $L$ at $p$. 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.

34 pages. Added outline of proofs for the big case

References in corpus (4)

• Canonical growth conditions associated to ample line bundles
• Transcendental Morse Inequality and Generalized Okounkov Bodies
• Positivity of line bundles and Newton-Okounkov bodies
• Okounkov bodies and Seshadri constants

Cited by in corpus (2)

• Canonical growth conditions associated to ample line bundles
• A remark on the Alexandrov-Fenchel inequality