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19 April 2024 |
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Differentiability of non-archimedean volumes and non-archimedean Monge-Amp'ere equations (with an appendix by Robert Lazarsfeld) | | José Ignacio Burgos Gil
; Walter Gubler
; Philipp Jell
; Klaus Kuennemann
; Florent Martin
; | | Date: |
5 Aug 2016 | | Abstract: | Let $X$ be a normal projective variety over a complete discretely valued
field and $L$ an ample line bundle on $X$. We denote by $X^ extrm{an}$ the
analytification of $X$ in the sense of Berkovich and equip the analytification
$L^ extrm{an}$ of $L$ with a continuous metric $| |$. We study
non-archimedean volumes, a tool which allows us to control the asymptotic
growth of small sections of big powers of $L$. We prove that the
non-archimedean volume is differentiable at a continuous semipositive metric
and that the derivative is given by integration with respect to a
Monge-Amp’ere measure. Such a differentiability formula had been proposed by
M. Kontsevich and Y. Tschinkel. In residue characteristic zero, it implies an
orthogonality property for non-archimedean plurisubharmonic functions which
allows us to drop an algebraicity assumption in a theorem of S. Boucksom, C.
Favre and M. Jonsson about the solution to the non-archimedean Monge-Amp’{e}re
equation. The appendix by R. Lazarsfeld establishes the holomorphic Morse
inequalities in arbitrary characteristic. | | Source: | arXiv, 1608.1919 | | Services: | Forum | Review | PDF | Favorites | |
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