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^an the analytification of X in the sense of Berkovich and equip the analytification L^an 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.