  We prove that every trajectory of a polynomial vector field on the complex
projective plane accumulates to the singular locus of the vector field. This
statement represents a holomorphic version of the Poincare-Bendixson theorem
and solves the complex analytic counterpart of Hilbert's 16th problem. The main
result can be also reformulated as the nonexistence of "exceptional minimals"
of holomorphic foliations on $\pp^2$ and, in particular, implies the
nonexistence of real analytic Levi flat hypersurfaces in the complex projective
plane. Finally, we describe (in the first approximation) the way a minimal
complex trajectory approaches the singular locus of the vector field.
