S may be negative if S is a dicritical separatrix (here, N is the normal bundle of the foliation). Brunella in (1997), and reads: the number c 1( N ) - S. The obstruction in this case was given by M. Simple examples show that, when S is a dicritical separatrix of, the search for a positive solution to the problem is meaningless. Key words: holomorphic foliations, invariant varieties, polar classes, degrees. We consider the question of relating extrinsic geometric characters of a smooth irreducible complex projective variety, which is invariant by a one-dimensional holomorphic foliation on a complex projective space, to geometric objects associated to the foliation. Manuscript received on Jaccepted for publication on July 18, 2001. SOARES * * Member of Academia Brasileira de Ciências E-mail: de Matemática, ICEx, UFMG - 31270-901 Belo Horizonte, Brazil On the geometry of Poincaré's problem for one-dimensional projective foliations
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