The volume of pseudoeffective line bundles and partial equilibrium

2021/12/07 by Darvas, Tamás, Xia, Mingchen · 1 citation
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2112.03827

Abstract

Let (L,he^-u) be a pseudoeffective line bundle on an n-dimensional compact Kähler manifold X. Let h⁰(X,L^k⊗ \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^-nh⁰(X,L^k⊗ \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 ν, 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.

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• Relative non-pluripolar product of currents on compact Hermitian manifolds

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