By Marco Brunella

ISBN-10: 3319143093

ISBN-13: 9783319143095

ISBN-10: 3319143107

ISBN-13: 9783319143101

The textual content provides the birational type of holomorphic foliations of surfaces. It discusses at size the idea constructed by means of L.G. Mendes, M. McQuillan and the writer to check foliations of surfaces within the spirit of the class of complicated algebraic surfaces.

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Example text

J on Uj \ Ui for every i ¤ j . j where fgij g is a OX -cocycle defining NF . e. 1; 0/-forms gijij C ˇj ˇi can be thought as a cocycle of smooth sections of NF , because it vanishes identically on F . Now, the sheaf of smooth sections of NF is certainly fine and so it has trivial cohomology. j on Uj n Vj . ˇj C j/ on Uj is therefore a well-defined closed 2-form on X , which represents the first Chern class of NF according to one of the possible definitions of Chern class [21]. 2; 0/, and thus ˝ represents NF in de Rham sense (as an element of H 2 ) but not in Dolbeault sense (as an element of H 1;1 ).

It follows that the foliation around F is given by the Dulac normal form ! 2 i /w. By flipping multivalued functions w D cz exp zk the fibre, one finds also that can be changed to C n, n 2 Z, so that the monodromy and the multiplicity of the fibre give a complete description of F up to bimeromorphism. However, in the generic case the two saddle-nodes have no weak separatrix. The most classical example is Euler’s equation ! 0; 0/ has no weak separatrix. The reader may find in [27] examples where both saddlenodes have no weak separatrix, with any prescribed monodromy.

E. F ; L; p/ 2. It follows that such a line is in fact invariant by F . z; w/ @w , with A and B affine functions (or even linear, after a translation of the origin). F / 1 at 3 Some Examples 19 infinity). The line at infinity is invariant if and only if R Á 0. F / 2 then it may happen that the foliation has no invariant line. F0 /. Let p 2 CP 2 be the point which is blown-up, let E X be the exceptional divisor, and let L X be the (strict) transform of a line which does not pass through p. p/ be defined in the usual way.

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Birational Geometry of Foliations by Marco Brunella

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