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The volume of pseudoeffective line bundles and partial equilibrium

Tamás Darvas and Mingchen Xia

Geometry & Topology 28 (2024) 1957–1993
Abstract

Let (L,heu) 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)eku is integrable. We show that the limit of knh0(X,Lk (ku)) exists, and equals the nonpluripolar volume of P[u], the –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], 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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