Duke Mathematical Journal

Canonical growth conditions associated to ample line bundles

David Witt Nyström

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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 TpX 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

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

Received: 7 October 2015
Revised: 24 June 2017
First available in Project Euclid: 5 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14C30: Transcendental methods, Hodge theory [See also 14D07, 32G20, 32J25, 32S35], Hodge conjecture

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


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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