An equivariant compactification of a vector group V (i.e. a vector space with addition as the group law) is a smooth
projective variety X endowed with a generically transitive action of V. The arithmetics (mainly Mannin's conjecture) of such varieties has been studied by Chambert-Loir and Tschinkel in a series of interesting papers. I'll report a joint work with Jun-Muk Hwang which classifies such varieties of Picard number one over the field of complex numbers, under a mild condition.