The volume of pseudoeffective line bundles and partial equilibrium

Darvas, Tamás¹ ; Xia, Mingchen²

¹ Department of Mathematics, University of Maryland, College Park, MD, United States
² Department of Mathematics, Chalmers Tekniska Högskola, Göteborg, Sweden, Institut de mathématiques de Jussieu, Sorbonne Université, Paris, France

Geometry & topology, Tome 28 (2024) no. 4, pp. 1957-1993.

Résumé

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 nonpluripolar 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 nonpluripolar Monge–Ampère measure of $P{[u]}_{\mathcal{ℐ}}$, the partial equilibrium.

Détail

DOI : 10.2140/gt.2024.28.1957

Keywords: Hermitian line bundle, volume, Bergman kernel, equilibrium

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