## Duke Mathematical Journal

### Canonical growth conditions associated to ample line bundles

David Witt Nyström

#### 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 plurisubharmonic (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 one 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.

#### Article information

Source
Duke Math. J., Volume 167, Number 3 (2018), 449-495.

Dates
Revised: 24 June 2017
First available in Project Euclid: 5 January 2018

https://projecteuclid.org/euclid.dmj/1515143006

Digital Object Identifier
doi:10.1215/00127094-2017-0031

Mathematical Reviews number (MathSciNet)
MR3761104

Zentralblatt MATH identifier
06848177

#### Citation

Witt Nyström, David. Canonical growth conditions associated to ample line bundles. Duke Math. J. 167 (2018), no. 3, 449--495. doi:10.1215/00127094-2017-0031. https://projecteuclid.org/euclid.dmj/1515143006

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