The volume of pseudoeffective line bundles and partial equilibrium

Darvas, Tamás; Xia, Mingchen

doi:10.2140/gt.2024.28.1957

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The volume of pseudoeffective line bundles and partial equilibrium

Tamás Darvas and Mingchen Xia

Geometry & Topology 28 (2024) 1957–1993

DOI: 10.2140/gt.2024.28.1957

Abstract

Let $(L, h e^{- u})$ be a pseudoeffective line bundle on an $n$ –dimensional compact Kähler manifold $X$ . Let $h^{0} (X, L^{k} \otimes ℐ (k u))$ be the dimension of the space of sections $s$ of $L^{k}$ such that $h^{k} (s, s) e^{- k u}$ is integrable. We show that the limit of $k^{- n} h^{0} (X, L^{k} \otimes ℐ (k u))$ exists, and equals the nonpluripolar 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 nonpluripolar Monge–Ampère measure of $P {[u]}_{ℐ}$ , the partial equilibrium.

Keywords

Hermitian line bundle, volume, Bergman kernel, equilibrium

Mathematical Subject Classification

Primary: 32W20

Secondary: 53C55

References

Publication

Received: 29 May 2022

Revised: 29 September 2022

Accepted: 16 November 2022

Published: 18 July 2024

Proposed: Gang Tian

Seconded: Leonid Polterovich, John Lott

Authors

Tamás Darvas
Department of Mathematics University of Maryland College Park, MD United States

Mingchen Xia
Department of Mathematics Chalmers Tekniska Högskola Göteborg Sweden	Institut de mathématiques de Jussieu Sorbonne Université Paris France

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Volume 28, issue 4 (2024)