Effectivity of semi-positive line bundles
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, \(\mathcal{L}\) is semi-ample by the base-point free theorem. Hence, only the \(q(\mathcal{L})=0\) case needs to be considered. If \(\mathcal{L}\) is a non-trivial semi-ample line bundle with \(q(\mathcal{L})=0\) on a HK manifold \(X\) of dimension \(2n\), the linear system \(| \mathcal{L}^{\otimes m} |\), for \(m>0\) large, defines a Lagrangian fibration \(X \to B\) over a normal base \(B\) of dimension \(n\). Since any HK manifold with \(b_2 \geq 5\) deforms to a HK manifold with a nef line bundle of square zero, Conjecture 2 would confirm the following HK SYZ conjecture: {Conjecture 3.} A HK manifold can be deformed to a HK manifold that admits a holomorphic Lagrangian fibration. This conjecture has been verified for the four aforementioned known deformation types of HK manifolds. The proof of Theorem 1 was obtained only for non-projective HK manifolds by Campana-Oguiso-Peternell. Verbitsky later showed how to combine the original approach with Boucksom's Zariski decomposition to also cover the algebraic case. The present article provides an alternative approach that reduces the projective case to the non-projective case via deformation. The techniques may be relevant to further progress on Conjecture 2.
scientific article; zbMATH DE number 7626078
scientific article