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# “Special” manifolds: rational points and entire curves

## Frédéric Campana

We prove (joint with J. Winkelmann) for any rationally connected projective manifold $$X$$ analytic analogues of several conjectural properties in arithmetic geometry: the “Potential Density”, the Weak Approximation Property, and the Hilbert Property (in the form stated and conjectured by Corvaja-Zannier). In particular: given any countable set $$N$$ on any rationally connected $$X$$, there is a holomorphic map $$\mathbf C\to X$$ whose image contains $$N$$. The proof rests on deformation properties of rational curves.

More general conjectures claim similar (arithmetic and analytic) properties for the much larger class of “special manifolds”, defined as the ones having no “$$\Omega^\bullet$$-big” line subbundle. They lie on the geometric spectrum at the opposite side of manifolds of “general type”, for which Lang conjectured “Mordellicity”. We conjecture that ‘potential density’ is equivalent to “specialness” for any $$X$$ defined over a number field $$k$$.

These two conjectures combine to describe the structure of $$X(k)$$ for any $$X/k$$, by means of the “core map” $$c : X\to C$$, which “splits” $$X$$ into its “special” part (the fibres), and “general type” part (the orbifold base).