## Duke Mathematical Journal

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

### 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 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**

Received: 7 October 2015

Revised: 24 June 2017

First available in Project Euclid: 5 January 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00127094-2017-0031

**Mathematical Reviews number (MathSciNet)**

MR3761104

**Zentralblatt MATH identifier**

06848177

**Subjects**

Primary: 14C20: Divisors, linear systems, invertible sheaves

Secondary: 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture

**Keywords**

growth condition ample line bundle toric geometry Seshadri constant Okounkov body Kähler embeddings

#### 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