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Diophantine approximations and foliations. (English) Zbl 1006.32020

Author’s summary: The main result of this article is the following semi-effective version of the Green-Griffiths conjecture for surfaces.
Theorem 0.2. Let \(X/\mathbb{C}\) be a smooth projective surface of general type with \(s_2(X)>0\). Then there exists a proper subvariety \(Z\) in \(X\) such that if \(f:Y\to X\) is a holomorphic map from a finite ramified cover \(p:Y\to\mathbb{C}\) of \(\mathbb{C}\) such that \(f(Y)\not\subset Z\) after a possible étale base change \((f\to g:= h^* f)\) there exists a constant \(\alpha\) depending on \(g\) such that \[ h_{K_X} \bigl( g(r)\bigr) \leq\alpha.d \bigl(g(r)\bigr)+ o\biggl(h_H \bigl(g(r)\bigr) \biggr) \] for all \(r\in\mathbb{R}^+\) outside a set a finite Lebesgue measure. Here \[ h_D\bigl(f(r) \bigr):=\int^r_0 {dt\over t}\int_{p^{-1} \bigl(|z|\leq r\bigr)}f^* \bigl(c_1(D)\bigr) \] where \(c_1(D)\) is the Chern form of the metricised Cartier divisor \(D\) on \(X\), \(H\) is an ample line bundle on \(X\) and \(d(f(r))\) is the ramification number of \(f\) minus the ramification number of \(p\) on \(p^{-1} (|z|\leq r)\).
The first main ingredient of the proof is the following “tautological” inequality result.
Theorem A. Let \(X\) be a smooth projective variety, \(H\) an ample line bundle on \(X\) and \(f:Y\to X\) a holomorphic map from a finite ramified cover \(p:Y\to \mathbb{C}\) of the complex line. Let \(f':Y\to \mathbb{P}(\Omega^1_X)\) the derivative of \(f\) and denote by \(\mathbb{O} (1)\) the tautological line bundle on \(\mathbb{P}(\Omega^1_X)\). Then \[ h_{\mathbb{O}(1)} \bigl(f'(r) \bigr)\leq d\bigl(f(r) \bigr)+O\Bigl(\log r+\log\biggl[ h_H\bigl(f(r) \bigr)\biggr] \Bigr) \] for all \(r\in\mathbb{R}^+\) outside a set of finite Lebesgue measure.
The second main ingredient is the
Theorem B. Let \(X\) be a surface of general type and \(f:Y\to X\) is a holomorphic map from a finite ramified cover \(p:Y\to\mathbb{C}\) of \(\mathbb{C}\). Assume that \(f(Y)\) is a Zariski dense leaf of a foliation on \(X\). Then after a possible base change \((f\to g:=h^*f)\) there exists a constant \(\alpha\) depending on \(g\) such that \[ h_{K_X}\bigl( g(r) \bigr)\leq \alpha.d \bigl(f(r)\bigr) +o\biggl( h_H\bigl(g(r) \bigr)\biggr) \] for all \(r\in\mathbb{R}^+\) outside a set of finite Lebesgue measure.
To treat the case where \(f'\) has a Zariski dense image in \(\mathbb{P}(\Omega^1_X)\), using results of Bogomolov and Miyaoka, Theorem A is enough. When \(f'(Y)\) lies in a divisor \(D\subset \mathbb{P}(\Omega^1_X)\), lifting now \(f\) to a map from \(Y\) to a desingularization \(\widetilde D\) of \(D\), the image of \(f\) is now a leaf of a foliation of \(\widetilde D\) and Theorem B allows to conclude.

MSC:

32H25 Picard-type theorems and generalizations for several complex variables
32S65 Singularities of holomorphic vector fields and foliations
14J29 Surfaces of general type
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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