Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Differentiability of volumes of divisors and a problem of Teissier


Authors: Sébastien Boucksom, Charles Favre and Mattias Jonsson
Journal: J. Algebraic Geom. 18 (2009), 279-308
DOI: https://doi.org/10.1090/S1056-3911-08-00490-6
Published electronically: April 23, 2008
MathSciNet review: 2475816
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Abstract | References | Additional Information

Abstract: We give an algebraic construction of the positive intersection products of pseudo-effective classes and use them to prove that the volume function on the Néron-Severi space of a projective variety is $ \mathcal{C}^1$-differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes. We then apply our differentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-Teissier inequalities for nef and big classes.


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Additional Information

Sébastien Boucksom
Affiliation: CNRS-Université Paris 7 Institut de Mathématiques, F-75251 Paris Cedex 05, France
Email: boucksom@math.jussieu.fr

Charles Favre
Affiliation: CNRS-Université Paris 7, Institut de Mathématiques, F-75251 Paris Cedex 05, France
Email: favre@math.jussieu.fr

Mattias Jonsson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043 USA and Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden
Email: mattiasj@umich.edu; mattiasj@kth.se

DOI: https://doi.org/10.1090/S1056-3911-08-00490-6
Received by editor(s): December 21, 2006
Received by editor(s) in revised form: April 16, 2007
Published electronically: April 23, 2008
Additional Notes: The second author was supported by the Japanese Society for the Promotion of Science. The third author was supported by NSF Grant No. DMS-0449465, the Swedish Science Council, and the Gustafsson Foundation.

American Mathematical Society