Complete polynomial vector fields on the complex plane

In this outstanding work the author gives a full classification of complete polynomial vector fields in two complex variables. This classification is obtained in terms of three models up to polynomial automorphisms and under no hypothesis on the singular set of the vector field. The two main ingredients in the proof are the pioneering work of M. Suzuki [Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 517--546; MR0590938 (58 \#28702)] on holomorphic flows on Stein surfaces and the work of M. McQuillan [in European Congress of Mathematics, Vol. II (Barcelona, 2000), 47--53, Progr. Math., 202, Birkhäuser, Basel, 2001; MR1905350 (2003j:14048)] on the classification of holomorphic foliations on projective surfaces admitting a nonalgebraic entire leaf. The main difficulties arise from the fact that one needs to adapt the "projective classification" of McQuillan to the given "affine problem". This is done by passing to a suitable projective compactification of the complex affine plane. The notion of Kodaira dimension of a foliation on a projective surface as well as the classification of foliations according to the Kodaira dimension also play a fundamental role. The paper may be difficult to read for a nonexpert due to the depth of the results. It is truly an important contribution to the subject and highlights the interplay between different fields such as several complex variables, potential theory, holomorphic foliations and affine algebraic geometry.