6533b861fe1ef96bd12c4299

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Subharmonic variation of the leafwise Poincar� metric

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Let X be a compact complex algebraic surface and let F be a holomorphic foliation, possibly with singularities, on X. On each leaf of F we put its Poincare metric (this will be defined below in more precise terms). We thus obtain a (singular) hermitian metric on the tangent bundle TF of F , and dually a (singular) hermitian metric on the canonical bundle KF = T ∗ F of F . The main aim of this paper is to prove that this metric on KF has positive curvature, in the sense of currents. Of course, the positivity of the curvature in the leaf direction is an immediate consequence of the definitions; the nontrivial fact is that the curvature is positive also in the directions transverse to the leaf. This last fact can be rephrased by saying that the Poincare metric on the leaves has a subharmonic variation. In order to give more precise statements, let us firstly set out the types of singularities that will be allowed. Concerning X, we shall not require that X be smooth, but (for a reason which will be clear later) we will allow Hirzebruch–Jung singularities: around such a singularity, X looks like a quotient of the ball B ⊂ C2 by a linear action of a finite cyclic group [BPV, pp. 80–84]. These are very mild singularities: they are rational and X is even projective [Art]; but we shall not need these facts. Concerning F , we shall require only that its singularities Sing(F ) are isolated and disjoint from Sing(X), in the sense that around a point in Sing(X) the foliation looks like a quotient of a regular foliation on B2. We shall set X ′ = X \ Sing(F ) X ′′ = X ′ \ Sing(X)