Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations

Let X be a normal projective variety over a complete discretely valued field and L a line bundle on X. We denote by X the analytification of X in the sense of Berkovich and equip the analytification L of L with a continuous metric k k. We study nonarchimedean 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-Ampère 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-Ampère equation. The appendix by R. Lazarsfeld establishes the holomorphic Morse inequalities in arbitrary characteristic.This work was supported by the collaborative research center SFB 1085 funded by the DeutscheForschungsgemeinschaft. J. I. Burgos was partially supported by MINECO research projects MTM2016-79400-P and by ICMAT Severo Ochoa project SEV-2015-0554. Robert Lazarsfeld was partially sup-ported by NSF grant DMS-143928