Seshadri constants and a criterion for bigness of pseudo-effective line bundles (Q1566387)

Let \(X\) be a smooth complex projective variety. For a nef line bundle \(L\) on \(X\), Seshadri's constant of \(L\) at a point \(x\in X\) is defined by \(\varepsilon(L,x):= \inf_{x\in C}{L\cdot C\over \text{mult}_xC}\), where the infimum is taken over all integral curves \(C\) passing through \(x\) and \(\text{mult}_xC\) denotes the multiplicity of \(C\) at \(x\). Seshadri's constant \(\varepsilon(L,x)\) measures how positive \(L\) is locally near \(x\) and the global Seshadri constant \(\varepsilon(L):= \inf_{x\in X}\varepsilon(L,x)\) is positive if and only if the nef line bundle \(L\) is ample. This is the so-called Seshadri criterion of ampleness for nef line bundles [cf.: \textit{R. Hartshorne}, Ample subvarieties of algebraic varieties, Lect. Notes Math. 156 (1970; Zbl 0208.48901)]. The purpose of the paper under review is to generalize Seshadri's criterion of ampleness to a criterion of bigness for only pseudo-effective line bundles. Analytically, a line bundle \(L\) is pseudo-effective if it admits a singular Hermitian metric \(h\) with nonnegative curvature current and multplier ideal sheaf \(I(h)\). For such a datum \((L,h)\), the author defines an analogue \(\varepsilon((L,h),x)\) of Seshadri's local constant at \(x\in X\) and proves the following bigness criterion: A pseudo-effective line bundle \(L\) on \(X\) is big if and only if it admits such a singular Hermitian metric \(h\) with nonnegative curvature current that \(\varepsilon((L,h),x)\) is positive for some point \(x\in X\setminus\text{Sing}(h)\). Moreover, in that case, the volume \(v(L)\) of \(L\) is bounded from below by \(\varepsilon((L,h),x)^{\dim X}\). The proof is based on an appropriate intersection theory for pseudo-effective line bundles as originally proposed by \textit{H. Tsuji} [Numerical trivial fibrations, Preprint, \texttt{http://arxiv.org/math.AG/0001023}], on an approximation theorem for the modified local Seshadri constants as a local version of Fujita's volume approximation theorem [\textit{T. Fujita}, Kodai Math. J. 17, 1--3 (1994; Zbl 0814.14006)], on \textit{Y.-T. Siu's} results on multiplier ideal sheaves [Invent. Math. 134, No. 3, 661--673 (1998; Zbl 0955.32017)], and on \textit{A. M. Nadel's} cohomological vanishing theorem for pseudo-effective line bundles [Ann. Math. (2) 129, No. 1, 161--178 (1989; Zbl 0675.14018)]. The author gives several geometric applications of his bigness criterion for pseudo-effective line bundles on projective manifolds, among them a local freeness result for adjoint linear series associated with a big line bundle, which points in the direction of the famous Fujita conjecture [\textit{Y. Miyaoka} and \textit{T. Peternell}, Geometry of higher-dimensional algebraic varieties (DMV Seminar 26, Basel: Birkhäuser Verlag) (1997; Zbl 0865.14018)].