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Duality between the pseudoeffective and the movable cone on a projective manifold


Author: David Witt Nyström; with an appendix by Sébastien Boucksom
Journal: J. Amer. Math. Soc. 32 (2019), 675-689
MSC (2010): Primary 32L10, 32Q15, 32U05
DOI: https://doi.org/10.1090/jams/922
Published electronically: April 11, 2019
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove a conjecture of Boucksom-Demailly-Păun-Peternell,
namely that on a projective manifold $ X$ the cone of pseudoeffective classes in $ H^{1,1}_{\mathbb{R}}(X)$ is dual to the cone of movable classes in $ H^{n-1,n-1}_{\mathbb{R}}(X)$ via the Poincaré pairing. This is done by establishing a conjectured transcendental Morse inequality for the volume of the difference of two nef classes on a projective manifold. As a corollary the movable cone is seen to be equal to the closure of the cone of balanced metrics. In an appendix by Boucksom it is shown that the Morse inequality also implies that the volume function is differentiable on the big cone, and one also gets a characterization of the prime divisors in the non-Kähler locus of a big class via intersection numbers.


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Additional Information

David Witt Nyström
Affiliation: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
Email: wittnyst@chalmers.se, danspolitik@gmail.co

Sébastien Boucksom
Affiliation: CNRS–CMLS, École Polytechnique, F-91128 Palaiseau Cedex, France
Email: sebastien.boucksom@polytechnique.edu

DOI: https://doi.org/10.1090/jams/922
Received by editor(s): April 11, 2017
Received by editor(s) in revised form: December 6, 2018
Published electronically: April 11, 2019
Article copyright: © Copyright 2019 American Mathematical Society