In a complex projective 3-space, we consider a domain with a projective line. If there is a compact non-singular quotient of the domain and the quotient manifold admits a non-constant meromorphic function, then the domain is dense in the projective 3-space and its complement is properly contained in a finite union of complex hypersurfaces and a set with Hausdorff dimension not more than two. Further, if the complement admits a certain fiber space structure, then it is either a disjoint union of two projective lines, a projective line, or an empty set.
"Compact Quotients of Large Domains in a Complex Projective 3-space." Tokyo J. Math. 29 (1) 209 - 232, June 2006. https://doi.org/10.3836/tjm/1166661875