The algebraic-dimension conjecture for submanifolds with ample normal bundle

Let $X$ be a compact Kähler manifold containing a compact submanifold $Z$ of dimension $d\geq 1$ with ample normal bundle. The algebraic dimension $a(X)$ is the maximal number of algebraically independent meromorphic functions on $X$.

Algebraic-dimension conjecture. One has

$$a(X)\geq d+1.$$

The assertion is known when $Z$ is a hypersurface, and the paper proves it in several further cases, including when $Z$ moves in a covering family, when $X$ is hyperkähler with $a(X)\geq 1$, and when $Z$ is uniruled. The general higher-codimension case remains open.