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Intersection theory of nef b-divisor classes

Part of: Birational geometry Cycles and subschemes

Published online by Cambridge University Press:  05 September 2022

Nguyen-Bac Dang  and
Charles Favre
Nguyen-Bac Dang
Affiliation:
Institut de Mathématiques d'Orsay, Université Paris-Saclay, 307 Rue Michel Magat, 91400 Orsay, France nguyen-bac.dang@universite-paris-saclay.fr
Charles Favre
Affiliation:
Centre de Mathématiques Laurent Schwartz, Institut polytechnique de Paris, 91128 Palaiseau, France charles.favre@polytechnique.edu
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Abstract

We prove that any nef $b$-divisor class on a projective variety defined over an algebraically closed field of characteristic zero is a decreasing limit of nef Cartier classes. Building on this technical result, we construct an intersection theory of nef $b$-divisors, and prove several variants of the Hodge index theorem inspired by the work of Dinh and Sibony. We show that any big and basepoint-free curve class is a power of a nef $b$-divisor, and relate this statement to the Zariski decomposition of curves classes introduced by Lehmann and Xiao. Our construction allows us to relate various Banach spaces contained in the space of $b$-divisors which were defined in our previous work.

Keywords

b-divisorsbirational geometry

MSC classification

Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14E05: Rational and birational maps
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 7 , July 2022 , pp. 1563 - 1594
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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