Compact Kahler 3-manifolds without nontrivial subvarieties

Algebraic Geometry. 2014. Vol. 2. P. 131–139.

Campana F., Demailly J., Вербицкий М. С.

We prove that any compact Kahler 3-dimensional manifold which has no nontrivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of so-called simple manifolds, central in the bimeromorphic classication of compact Kahler manifolds. The proof follows from the Brunella pseudo-eectivity theorem, combined with fundamental results of Siu and of the second author on the Le- long numbers of closed positive (1;1)-currents, and with a version of the hard Lefschetz theorem for pseudo-eective line bundles, due to Takegoshi and Demailly-Peternell- Schneider. In a similar vein, we show that a normal compact and Kahler 3-dimensional analytic space with terminal singularities and nef canonical bundle is a cyclic quotient of a simple nonprojective torus if it carries no eective divisor. This is a crucial step towards completing the bimeromorphic classication of compact Kahler threefolds.

Ключевые слова: Kähler manifold, complex torus, algebraic dimension, holomorphic foliations, кэлерово многообразие, алгебраическая размерность, голоморфное слоение, комплексный тор

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