Is there a closed subscheme $D$ in $\mathbb P^2\_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map


$$ \phi: X(\mathbb C) \to \mathbb P^2(\mathbb C) - D(\mathbb C)$$


algebraizes? (That is, is there a morphism $f:X\to \mathbb P^2\_{\mathbb C}- D$ whose analytification is $\phi$?)



Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
On page 73 of Kobayashi's book Hyperbolic Complex spaces he shows that if D is a certain configuration of 6 lines in the plane then its complement is complete hyperbolic and hyperbolically embedded in the projective plane.By theorem 6.3.24 page 290 of the same book your map from X to the complement extends meromorphically to an algebraic compactification of X .The algebraicity should follow from Serre's GAGA .


