The volume of pseudoeffective line bundles and partial equilibrium

Abstract

Let $(L,h{e}^{-u})$ be a pseudoeffective line bundle on an $n$-dimensional compact Kähler manifold $X$. Let $h^0(X,L^k\otimes \mathcal{I}(ku))$ be the dimension of the space of sections $s$ of $L^k$ such that $h^k(s,s)e^{-ku}$ is integrable. We show that the limit of $k^{-n}{h}^0(X,L^k\otimes \mathcal{I}(ku))$ exists, and equals the non-pluripolar volume of $P[u{]}_{\mathcal{I}}$, the $\mathcal{I}$-model potential associated to $u$. We give applications of this result to Kähler quantization: fixing a Bernstein-Markov measure $\nu$, we show that the partial Bergman measures of $u$ converge weakly to the non-pluripolar Monge--Ampère measure of $P[u{]}_{\mathcal{I}}$, the partial equilibrium.

arXiv commentv3. addresses and references updated, typos fixed v.4 more typos fixed, to appear on Geometry & Topology

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