Compact subvarieties with ample normal bundles, algebraicity, and cones of cycles

Abstract

The second part of the paper is concerned with projective manifolds X and curves C ⊂ X with ample normal bundles. In the "dual" situation of a hypersurface Y with ample normal bundle, the line bundle O X (Y ) is big and therefore in the interior of the pseudo-effective cone. Therefore we expect that the class [C] is in the interior of the Mori cone N E(X) of curves:1.3. Conjecture. Let X be a projective manifold, C ⊂ X a curve with ample normal bundle. Then [C] is in the interior of N E(X).Equivalently, if L is any nef line bundle, such that L · C = 0, then L ≡ 0. We prove 1.4. Theorem. Let X be a projective manifold, C ⊂ X a smooth curve with ample normal bundle and L a nef line bundle on X. If H 0 (X, mL) = 0 for some m > 0 and if L · C = 0, then L ≡ 0.

Submanifolds with ample normal bundle

“…Theorem A and Question 1.1 should be compared with a conjecture by Peternell [20] which predicts that field $$\mathbb {C}(X)$$ of meromorphic functions on a compact Kähler manifold $$X$$ admitting a subvariety $$Z$$ with ample normal bundle satisfies $$\operatorname{tr\, deg}_{\mathbb {C}}\mathbb {C}(X) \geqslant \dim Z + 1$$. We do not know if we can construct examples of pairs $$(X,Y)$$ with $$X$$ compact Kähler with arbitrary $$\operatorname{tr\, deg}_{\mathbb {C}}\mathbb {C}(X,Y)$$.…”