Holomorphic 2-Forms on Complex Threefolds

Holomorphic differential forms play an important role in the global study of complex projective or compact Kähler manifolds. Forms of degree 1 are well understood: there is a universal map, the Albanese map, to a complex torus and every 1-form is a pull-back. Forms of top degree also play a special role because they are sections in a line bundle, the canonical bundle. Forms of degree d with 2 ≤ d ≤ dimX − 1 are however much harder to study. Especially 2-forms are very interesting, e.g. in symplectic geometry, or by the fundamental theorem of Kodaira describing them as obstruction for Kähler manifolds to be projective. In this paper we want to study the ”first interesting case” of 2-forms on 3-folds. As general guideline we ask whether there is some kind of analogue of the Albanese for 2forms. This means that we should try to find some universal object from which all 2-forms arise. This is impossible to some extend because we can take products of surfaces with curves and then would have to consider 2-forms on surfaces which are ”wild”. So instead we ask whether every 2-forms is produced by some canonical procedure. In sect. 1, which is partially written as an extended introduction, this is explained in great detail. Cum grano salis, we hope that every 2-form is induced by a meromorphic map to a surface, to a torus or to a symplectic manifold. We will explain in sect.1, what we mean by ”induced” (not just pull-back, of course). In sections 3,4 and 5 we try to get a picture on 2-forms on 3-folds in terms of the Kodaira dimension. We describe this briefly a more detailed account on the results of this paper is given in sect. 2. Let ω be a 2-form on X. (a) If κ(X) = −∞ and if X is not simple (i.e. there is a compact subvariety of positive dimension through every point), then there is a meromorphic map onto a surface and ω is a pull-back. (b) If κ(X) = 0 and if X is projective, then after finite cover, étale in codimension 2, X is birationally to either a product of a K3-surface and an elliptic curve or a torus and ω is induced in an obvious way. The same holds also in the Kähler case (sect. 8) unless X is simple and not covered by torus, a case which is expected not to exist. (c) If 1 ≤ κ(X) ≤ 2, we consider the Iitaka reduction f : X −→ B and relate ω to f . We have to distinguish the cases that ω is vertical to f or not. In many cases we prove that ω is induced. (d) If X is of general type, we cannot say anything.