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Asymptotic invariants of line bundles. (English) Zbl 1139.14008
This paper is an appealing survey on recent directions in the study of asymptotic invariants of line bundles on a smooth projective variety \(X\). The volume of a line bundle \(L\) on \(X\), which is defined as \(\text{vol}_X(L) = \limsup_{m \to \infty} \frac{h^0(X,\mathcal O_X(mL))}{m^d/d !}\), where \(d=\dim(X)\), is the most well-known of such invariants, due to its connection with the Riemann–Roch problem. For an ample line bundle \(L\), \(\text{vol}_X(L)\) depends only on the numerical class of \(L\) and is computed by a function on the cone of nef numerical equivalence classes in the real Néron–Severi space, which is given by a polynomial and is log-concave. Focusing on the volume, the authors illustrate the philosophy that big line bundles display a surprising number of properties analogous to those of ample line bundles, from the point of view of asymptotic invariants.
The discussion includes the function on the real Néron–Severi space computing \(\text{vol}_X\) and its continuity, log-concavity and local polynomiality for \(d=2\) on the big cone, as well as the interpretation of the volume in terms of asymptotic moving self-intersection number (i. e., counted outside the base locus). The text is enriched by several illuminating examples. Then the authors discuss higher asymptotic cohomological functions, obtained by replacing \(h^0\) with \(h^i\) in the definition of \(\text{vol}_X(L)\), their behavior and their use to characterize ample classes. Finally, further invariants like asymptotic order of vanishing and restricted volumes are combined with the notions of stable and augmented base loci of a big divisor in order to state a result describing the irreducible components of the augmented base locus, proven by the authors in [Restricted volumes and base loci of linear series,arXiv:math/0607221v2, to appear in Am. J. Math.]

14C20 Divisors, linear systems, invertible sheaves
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