Strongly pseudo-effective and numerically flat reflexive sheaves
Let \(X\) be a compact Kähler manifold, let \(E\) be a holomorphic vector bundle over \(X\), and let \(\pi\colon\mathbb{P}(E)\rightarrow X\) be the projectivization of hyperplanes of \(E\). \(E\) is called nef if its hyperplane bundle \(\mathcal{O}_{\mathbb{P}(E)}(1)\) is nef. Analogously, \(E\) is called (weakly) pseudoeffective if \(\mathcal{O}_{\mathbb{P}(E)}(1)\) is pseudoeffective. If moreover the image of \(\mathcal{O}_{\mathbb{P}(E)}(1)\)'s nonnef locus under \(\pi\) is not equal to \(X\), then \(E\) is called strongly pseudoeffective. The main result of the present work is that a strongly pseudoeffective \(E\) with vanishing first Chern class is nef. To prove this, the author constructs Segre currents of singular Hermitian metrics with analytic singularities using techniques similar to [\textit{M. Andersson}, et al., Indiana Univ.\ Math.\ J., 68, 1217--1231 (2019; Zbl 1439.32079)]. Although such Segre currents were constructed in [\textit{R. Lärkäng}, et al., Adv. Math., 326, 465--489 (2018; Zbl 1391.32024); Indiana Univ.\ Math.\ J., 71, No. 1, 153--189 (2022; Zbl 1487.32105)] before, the approach of the present work allows to estimate the current's Lelong numbers (generalizing results in [\textit{J.-P.\ Demailly}, J. Differ.\ Geom., 37, 323--374 (1993; Zbl 0783.32013)]) which are essential for the proof of the main result. As an corollary of the main result, the author obtains that a strongly pseudoeffective reflexive sheaf over a compact Kähler manifold with vanishing first Chern class is locally free and nef. For projective \(X\), this has been proven before in [\textit{F. Campana} et al., Algebr.\ Geom. 8, No. 4, 430--464 (2021; Zbl 1483.14016)] and [\textit{G. Hosono} et al., J. Inst.\ Math.\ Jussieu (to appear)].
scientific article; zbMATH DE number 7493855
scientific article