On the volume of a line bundle. (English) Zbl 1101.14008

Let \(L\) be a holomorphic line bundle on a compact \(n\)-dimensional Kähler manifold \(X\). The volume \(v(L)\) of \(L\) is defined as \[ v(L):=\limsup_{k\to \infty} \frac{n!}{k^n}h^0(X,kL). \] The bundle \(L\) has maximal Kodaira-Iitaka dimension, i.e. \(L\) is big, if and only if \(v(L)>0\). This implies that \(v(mF)=m^nv(F)\) for every \(m\in\mathbb Z\) and allows the definition \(v(F)=m^{-n}v(mF)\) for a \(\mathbb Q\)-line bundle \(F\) when \(mF\) is a holomorphic line bundle. The volume of \(L\) depends only on the Chern class \(c_1(L)\) if \(L\) is numerically effective. In this case it follows from Demailly’s Morse inequalities and the asymptotic Riemann-Roch formula that \(v(L)=\int_Xc_1(L)^n\).
The author shows that it is possible in general to express \(v(L)\) in terms of \(c_1(L)\). By a theorem of T. Fujita [Kodai Math. J. 17, No. 1, 1–3 (1994; Zbl 0814.14006)] the volume of the pull back of a big line bundle \(L\) on a projective manifold with respect to a suitable modification can be arbitrarily close approximated by the volume of ample \(\mathbb{Q}\)-line bundles \(A\). The author generalizes this theorem to Kähler manifolds (Theorem 1.4). Using the Calabi-Yau methods to solve Monge-Ampère equations he shows that one can find a hermitian metric on \(A\) such that the product of the curvature eigenvalues equals \(v(A)\). Moreover this yields metrics on \(L\) converging to some singular metric whose product of the curvature eigenvalues equals \(v(L)\).
Combining this result with the holomorphic Morse equalities the author proves the following equality for \(v(L)\) when \(c_1(L)\) is a pseudoeffective class, i.e. it contains closed positive currents \(T\): \(v(L)=\max_T\int_X T^n_{ac}=:v(c_1(L))\), where \(T_{ac}\) is the absolutely continuous part of \(T\). The function \(v:H^{(1,1)}(X,\mathbb R)\to\mathbb R\), \(v(\alpha):=0\) if \(\alpha\in H^{(1,1)}(X,\mathbb R)\) does not contain a closed positive current, is continuous. This is a corollary of Theorem 4.7, which says that \(\alpha\in H^{(1,1)}(X,\mathbb R)\) is the Chern class of a big line bundle if and only if \(v(\alpha)>0\). The proof of this theorem uses methods developed by J. P. Demailly and M. Paun [Ann. Math. (2) 159, No. 3, 1247–1274 (2004; Zbl 1064.32019)].


14C20 Divisors, linear systems, invertible sheaves
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32J27 Compact Kähler manifolds: generalizations, classification
Full Text: DOI arXiv


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