Line bundles on complex tori and a conjecture of Kodaira
  A famous conjecture attributed to Kodaira asks whether any compact Kaehler manifold can be approximated by projective manifolds. We confirm this conjecture on projectivized direct sums of three line bundles on three-dimensional complex tori which appears rather surprising in view of expected dimensions of certain families of tori. We also discuss possible counter examples. 
Jean-Pierre Demailly, Thomas Eckl, Thomas Peternell