Distribution of logarithmic spectra of the equilibrium energy
Let \(X\) be a projective variety defined on a complete valued field, and let \(L\) be a big line bundle on \(X\). Suppose that this line bundle \(L\) is equipped with two continuous metrics. In the paper under review, the authors show that the distribution of the eigenvalues of the transition matrix between the \(L^2\)-norms on \(H^0(X,nL)\) with respect to these two metrics converges (in law) as \(n\to \infty\) to a Borel probability measure on \(\mathbb{R}\). This result is proved using the ideas from Arakelov geometry, and it can be viewed as a generalization of the existence of the energy at the equilibrium as a limit, or an extension of Berndtsson's results to the more general context of graded linear series and a more general class of line bundles.
scientific article