## Birational Geometry of Foliations by Marco Brunella

By Marco Brunella

The textual content offers the birational type of holomorphic foliations of surfaces. It discusses at size the idea built by means of L.G. Mendes, M. McQuillan and the writer to review foliations of surfaces within the spirit of the category of complicated algebraic surfaces.

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Recall that Hirzebruch surfaces form a countable family fXk g, k 2 NC [f0g, with X0 D CP 1 CP 1 ; for k > 0, Xk has a unique structure of CP 1 -bundle, and a unique section C with negative self-intersection, more precisely C C D k. We only consider the case k > 0, leaving as an exercise for the reader the case k D 0. Then we have to prove that F coincides with the (unique) fibration. Xk ; Q/ the section C and a fibre F . Then we can write, at the cohomological level, NF D aF C bC with a; b 2 Q.

2 1i/2 @Vj ˇQj ^ d ˇQj , which is nothing but than Baum–Bott index. t u This theorem is in fact a particular case of much more general results [3, 45] for foliations in any dimension, but the underlying philosophy is always the same: Chern numbers of the normal bundle (or sheaf) of a foliation can be localized at singular points (or sets) of the foliation, thanks to Frobenius Theorem. F //. 2 Camacho–Sad Formula Let again F be a foliation on X , and suppose moreover that C X is an F invariant curve (not necessarily smooth, nor irreducible).

X; NF / > 0: Proof. X / > 0 2 . X / > 0 because X is rational). X; NF ˝ KX / D 0. 2, let R X be a smooth rational curve of positive selfintersection and not F -invariant, so that NF R > 0. R/ < 0. Hence NF ˝ KX has negative degree on also have KX R D R R R, and thus it cannot be effective being R R > 0. t u 40 3 Index Theorems P Let now kj D1 aj Dj be an effective divisor representing NF (Dj irreducible curves, aj positive integers). Remark that it is not trivial, for NF KX ¤ 0 (for instance). Let us prove that each Dj is smooth rational curve.