Restricted volumes and base loci of linear series. (English) Zbl 1179.14006

Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea

Am. J. Math. 131, No. 3, 607-651 (2009).

This paper is part of general program of the authors of using asymptotic invariants of divisors in order to get information about the geometry of linear series, base loci, and cones of divisors on a projective variety [Pure Appl. Math. Q. 1, No. 2, 379–403 (2005; Zbl 1139.14008); Ann. Inst. Fourier 56, No. 6, 1701–1734 (2006; Zbl 1127.14010)]. Many properties of ample line bundles on a smooth complex projective variety \(X\) extend to arbitrary divisors provided one works asymptotically.
In this paper, the authors focus on the restricted volume of a divisor \(D\) along a \(d\)-dimensional subvariety \(V\) defined as \[ \text{vol}_{X|V}(D):=\limsup _{m\rightarrow\infty} \frac{\dim_{\mathbb{C}}\text{{Im}} (\, H^0(X,\mathcal{O}_X(mD))\rightarrow H^0(V,\mathcal{O}_V(mD))\, )}{m^d/d!} . \] This is a generalization of the usual degree when \(D\) is ample. It is shown that \(vol_{X|V}(D)\) can be computed in terms of “moving” intersection numbers of divisors with \(V\), a result also attributed to Demailly and S. Takayama [Invent. Math. 165, No. 3, 551–587 (2006; Zbl 1108.14031)].
The main result is about the augmented base locus \(\mathbf{B}_+(D)\) of a divisor \(D\). This is a better-behaved cousin of the stable base locus \(\mathbf{B}(D)={\bigcap_m\bigcap_{D'\in |mD|}D'},\) and is defined as \(\mathbf{B}(D-A)\) for a sufficiently small \(\mathbb{Q}\)-ample divisor \(A\). By a previous result of the authors, \(\mathbf{B}_+(D)\) depends only on the numerical equivalence class of \(D\). Let \(\text{Big}^V(X)\) be the numerical equivalence classes of \(\mathbb{R}\)-divisors \(D\) such that \(V\) is not properly contained in any irreducible component of \(\mathbf{B}_+(D)\). The authors show that \(vol_{X|V}(D)\) defines a continuous, log concave, real function on \(\text{Big}^V(X)\). It is zero on \(D\) iff \(V\) is an irreducible component of \(\mathbf{B}_+(D)\).
Using these notions, the authors generalize a result of U. Angehrn and Y.-T. Siu [Invent. Math. 122, No. 2, 291–308 (1995; Zbl 0847.32035)] on effective base-point freeness of adjoint bundles, and strengthen a result of M. Nakamaye [Trans. Am. Math. Soc. 355, No. 2, 551–566 (2003; Zbl 1017.14017)] on moving Seshadri constants.

Reviewer: Nero Budur (Notre Dame)

MSC:

14C20	Divisors, linear systems, invertible sheaves
14J99	Surfaces and higher-dimensional varieties

Keywords:

divisor; restricted volume; base locus

Citations:

Zbl 1139.14008; Zbl 1127.14010; Zbl 1108.14031; Zbl 0847.32035; Zbl 1017.14017