The volume of pseudoeffective line bundles and partial equilibrium

Journal article, 2024

Let (L, he-u) be a pseudoeffective line bundle on an n–dimensional compact Kähler manifold X. Let h0 (X, Lk ⊗(ku)) be the dimension of the space of sections s of Lk such that hk (s, s)e-ku is integrable. We show that the limit of k-n h0 (X, Lk ⊗J(ku)) exists, and equals the nonpluripolar volume of P[u]J, the J–model potential associated to u. We give applications of this result to Kähler quantization: fixing a Bernstein–Markov measure v, we show that the partial Bergman measures of u converge weakly to the nonpluripolar Monge–Ampère measure of P[u]J, the partial equilibrium.