Growth of Balls of Holomorphic Sections and Energy at Equilibrium

Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and the weight φ of a continuous Hermitian metric e on L, we define the energy at equilibrium of (K,φ) as the Aubin-Mabuchi energy of the extremal psh weight associated to (K,φ). We prove the differentiability of the energy at equilibrium with respect to φ, and we show that this energy describes the asymptotic behaviour as k → ∞ of the volume of the sup-norm unit ball induced by (K, kφ) on the space of global holomorphic sections H(X, kL). As a consequence of these results, we recover and extend Rumely’s Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.