The volume of pseudoeffective line bundles and partial equilibrium

Let (L, he−u) be a pseudoeffective line bundle on an n-dimensional compact Kähler manifold X. Let h0(X,Lk ⊗ I(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−nh0(X,Lk ⊗ I(ku)) exists, and equals the non-pluripolar volume of P [u]I , the I-model potential associated to u. We give applications of this result to Kähler quantization: fixing a Bernstein– Markov measure ν, we show that the partial Bergman measures of u converge weakly to the non-pluripolar Monge–Ampère measure of P [u]I , the partial equilibrium.

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