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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)].

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