The ample-normal-bundle conjecture for compact Kähler submanifolds

Let $X$ be a compact Kähler manifold and let $Y\subset X$ be a positive-dimensional compact submanifold with ample normal bundle. Then

$$a(X)\geq \dim Y+1.$$

Ample-normal-bundle conjecture. The algebraic dimension of $X$ should be at least one greater than the dimension of any positive-dimensional compact submanifold with ample normal bundle. In dimension three this is presented as a direct consequence of a preceding theorem; the general statement is posed as a conjecture.