15 February 2018 Canonical growth conditions associated to ample line bundles
David Witt Nyström
Duke Math. J. 167(3): 449-495 (15 February 2018). DOI: 10.1215/00127094-2017-0031


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.


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David Witt Nyström. "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


Received: 7 October 2015; Revised: 24 June 2017; Published: 15 February 2018
First available in Project Euclid: 5 January 2018

zbMATH: 06848177
MathSciNet: MR3761104
Digital Object Identifier: 10.1215/00127094-2017-0031

Primary: 14C20
Secondary: 14C30

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

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 3 • 15 February 2018
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