Effectivity of semi-positive line bundles
type: Item
instance of: scholarly article
title: Effectivity of semi-positive line bundles
author: Fabrizio Anella
author: Daniel Huybrechts
published in: Milan Journal of Mathematics
publication date: 2022-12-02T00:00:00Z
review text: A compact simply connected Kähler manifold \(X\) is hyper-Kähler (HK) if \(X\) admits an everywhere non-degenerate holomorphic \(2\)-form (unique up to scaling). HK manifolds appear as one of the three building blocks of compact Kähler manifolds with holomorphically torsion canonical bundle as a consequence of the Beauville-Bogomolov decomposition theorem. A HK surface is a K3 surface. Up to deformation, in each even dimension (\(2n \geq 4\)), the only known examples are: Hilbert schemes of points on K3 surfaces, generalized Kummer varieties associated with an abelian surface that is a product of two elliptic curves and the two examples of O'Grady in dimensions 6 and 10. The following result was first proved in the non-algebraic setting by \textit{F. Campana} et al. [J. Differ. Geom. 85, No. 3, 397--424 (2010; 1232.53042)], and later extended to the algebraic setting by \textit{M. Verbitsky} [Geom. Funct. Anal. 19(2009), No. 5, 1481--1493 (2010; Zbl 1188.53046)]: Theorem 1. Let \(\mathcal{L} \to X\) be a semi-positive line bundle on a HK manifold. Then \(\mathcal{L}\) is \(\mathbb{Q}\)-effective in the sense that \(H^0(X, \mathcal{L}^{\otimes m}) \neq 0\) for some \(m >0\). Recall that a line bundle \(\mathcal{L} \to X\) is said to be semi-positive if it admits a smooth Hermitian metric whose curvature form is semi-positive (in the sense of \((1,1)\)-forms). Semi-positive line bundles are nef, but the converse is not true in general. The situation is expected to be better for HK manifolds (or, more generally, for Calabi-Yau manifolds): {Conjecture 2.} Let \(\mathcal{L} \to X\) be a nef line bundle on a HK manifold. Then \(\mathcal{L}\) is semi-ample. Let \(q\) denote the Beauville-Bogomolov quadratic pairing on \(H^2(X,\mathbb{Z})\). Given a nef line bundle \(\mathcal{L} \to X\) as in the above conjecture, either \(q(\mathcal{L})=0\) and \(q(\mathcal{L})>0\). If \(q(\mathcal{L}) > 0\), \(\mathcal{L}\) is nef and big, \(X\) is projective, and hence, \(\mathcal{L}\) i…
reviewed by: Kyle Broder
Mathematics Subject Classification ID: 32Q15
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32L05
zbMATH DE Number: 7626078
zbMATH Keywords: SYZ conjecture
zbMATH Keywords: hyper-Kähler manifolds
zbMATH Keywords: twists of vector bundles
MaRDI profile type: Publication
full work available at URL: s00032-022-00356-z
OpenAlex ID: W4293084311
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DOI: 10.1007/S00032-022-00356-Z
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description: scientific article; zbMATH DE number 7626078