Compact Kähler 3-manifolds without non-trivial subvarieties

Jean-Pierre Demailly (1) , Misha Verbitsky (2) , Frederic Campana (3)

1 IF - Institut Fourier
2 Laboratory of algebraic geometry and its applications,Faculty of Mathematics, Higher School of Economics
3 IECL - Institut Élie Cartan de Lorraine

We prove that any compact Kähler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact Kähler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of Siu and of the second author on the Lelong numbers of closed positive (1,1)-currents, and with a version of the hard Lefschetz theorem for pseudo-effective line bundles, due to Takegoshi and Demailly-Peternell-Schneider. In a similar vein, we show that a normal compact and Kähler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple non-projective torus if it carries no effective divisor. This is a crucial step to complete the bimeromorphic classification of compact Kähler 3-folds

Mots clés: complex torus; complex 3-manifold; simple variety; algebraic dimension; compact kähler manifold; Albanese map; closed positive current; pseudo-effective line bundle; nef line bundle; multiplier ideal sheaf; hard Lefschetz theorem

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]

Jean-Pierre Demailly, Misha Verbitsky, Frederic Campana. Compact Kähler 3-manifolds without non-trivial subvarieties. Algebraic Geometry, 2014, 1 (2), pp.131-139. ⟨10.14231/AG-2014-007⟩. ⟨hal-00819044v2⟩