Differentiability of non-archimedean volumes and non-archimedean Monge-Amp\`ere equations (with an appendix by Robert Lazarsfeld)

Let $X$ be a normal projective variety over a complete discretely valued field and $L$ a line bundle on $X$. We denote by $X^{\text{an}}$ the analytification of $X$ in the sense of Berkovich and equip the analytification $L^{\text{an}}$ of $L$ with a continuous metric $\mathrm{\parallel }\mathrm{\parallel }$. 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.