## 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

### Keywords:

second main theorem; Green-Griffiths conjecture; foliation
Full Text:

### References:

 [1] E. Arrondo, I. Sols, B. Speiser,Global Moduli For Contacts, MSRI preprint (1992). · Zbl 0930.14035 [2] P. Baum, R. Bott, Singularities Of Holomorphic Foliations,J. Differential Geometry 7 (1972), 279–342. · Zbl 0268.57011 [3] F. Bogomolov, Families Of Curves On Surfaces Of General Type,Soviet Math. Dokl. 18 (1977), 1294–1297. · Zbl 0415.14013 [4] F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties,Math. USSR Izv. 13 (1978), 499–555. · Zbl 0439.14002 [5] E. Bombieri, The Mordell Conjecture Revisited,Ann. Scuola Norm. Sup. Pisa Cl. Sci (4)17 (1990), no 4, 615–640. Errata-corrigé: The Mordell conjecture revisited,Ann. Scuola Norm. Sup. Pisa Cl. Sci (4)18 (1991), no 3, 473. · Zbl 0722.14010 [6] A. Connes,Non Commutative Geometry, Academic Press, 1994. [7] D. Cerveau, J.-F. Mattei, Formes intégrables holomorphes singulières,Astérisque vol. 97 (1982). · Zbl 0545.32006 [8] M. Deschamps, Courbes de genres géométrique borné sur une surface de type général (d’après F. A. Bogomolov),Séminaire Bourbaki 519 (1977/1978). [9] J. P. Demailly, Monge-Ampère operators, Lelong numbers, and intersection theory,Complex Analysis and Geometry, V. Ancona and A. Silva Eds, Plenum, New York, London (1993). · Zbl 0792.32006 [10] J. P. Demailly,Algebraic Criteria for Kobayashi Hyperbolic Varieties and Jet Differentials, Lecture notes of a course given at AMS summer research Institute, Santa Cruz (1995). [11] J. P. Demailly, Regularization of closed positive currents and intersection theory,JAG 1 (1992), 361–409. · Zbl 0777.32016 [12] T. Ekedahl, Foliations and Inseparable Morphisms, Algebraic Geometry, Bowdoin,Proc. Symp. Pure Math. 46, S. Bloch ed., AMS, Providence (1987), 139–149. · Zbl 0659.14018 [13] G. Faltings, Diophantine Approximation on abelian Varieties,Ann. Math. 133 (1991), 549–576. · Zbl 0734.14007 [14] W. Fulton,Intersection Theory, New York, Springer Verlag, Berlin, Heidelberg (1984). · Zbl 0541.14005 [15] T. Fujita, On the Zariski problem,Proc. Japan Acad. 55 (1979), 106–110. · Zbl 0444.14026 [16] M. Green, P. Griffiths, Two applications of algebraic geometry to entire holomorphic mappings,The Chern Symposium, New York, Springer Verlag, Berlin, Heidelberg, (1979), 41–74. [17] R. Hartshorne,Ample subvarieties of algebraic varieties, LNM156 (1970). · Zbl 0208.48901 [18] J. P. Jouanolou, Hypersurface solutions d’une équation de Pfaff analytique,Math. Ann. 232 (1978), 239–245. · Zbl 0363.34002 [19] S. Lang,Number Theory III, Encyclopedia of Mathematical Sciences, volume 60, New York, Springer Verlag, Berlin, Heidelberg, 1991. [20] S. Lang,Introduction to complex hyperbolic spaces, New York, Springer Verlag, Berlin, Heidelberg (1987). · Zbl 0628.32001 [21] S. Lang, W. Cherry,Topics In Nevanlinna Theory, LNM1433 (1990). · Zbl 0709.30030 [22] S. Lu, S. T. Yau, Holomorphic curves in surfaces of general type,Proc. Nat. Acad. Sci. USA 87 (1990), 80–82. · Zbl 0702.32015 [23] M. McQuillan, A new proof of the Bloch conjecture,JAG 5 (1996), 107–117. · Zbl 0862.14027 [24] M. McQuillan,A Dynamical Counterpart To Faltings’ Diophantine Approximation On Abelian Varieties, IHES preprint, 1996. [25] M. McQuillan,La mappa di Faltings e la construzione grafica di Macpherson, lectures given at the Università di Napoli, ”Federico II”, to appear. [26] Y. Miyaoka, Algebraic Surfaces With Positive Index,Classification of algebraic and analytic manifolds (Kata Symposium Proc. 1982),Progress in Math. 39, Birkhauser, Boston Basel Stuttgart (1983), 281–301. [27] Y. Miyaoka, Deformations of Morphisms Along a Foliation,Algebraic Geometry, Bowdoin, Proc. Symp. Pure Math.46, S. Bloch ed., AMS, Providence (1987), 245–268. · Zbl 0659.14008 [28] M. Nakamaye, conversation at tea, MSRI, 1992. [29] J. Noguchi, On the value distribution of meromorphic mappings of covering spaces over C m into algebraic varieties,J. Math. Soc. Japan 37 (1985), 295–313. · Zbl 0566.32019 [30] M. Reid, Bogomolov’s Theoremc 1 2 4c 2,Proc. Intern. Colloq. Algebraic Geometry (Kyoto, 1977), 623–642. [31] A. Seidenberg, Reduction of singularities of the differential equation Ady=Bdx, American Journ. Math. 89 (1967), 248–269. · Zbl 0159.33303 [32] F. Sakai, Weil Divisors on normal surfaces,Duke Math. J. 51 (1984), 877–887. · Zbl 0602.14006 [33] N. Shepherd-Baron, Miyaoka’s Theorems, Flips And Abundance For Algebraic Threefolds (János Kollár ed.),Asterisque vol.211 (1992), 103–114. [34] Y. T. Siu, Analyticity of sets associated to Lelong numbers and the extension of closed positive currents,Invent. Math. (1974), 53–156. · Zbl 0289.32003 [35] C. Soulé,Lectures On Arakelov Geometry, Cambridge Studies in Advanced Mathematics33, Cambridge, 1992. [36] E. Ullmo,Positivité et discrétion des points algébriques des courbes, to appear. [37] P. Vojta,Diophantine Approximations and Value Distribution Theory, LNM1239 (1987). · Zbl 0609.14011 [38] P. Vojta, On Algebraic Points On Curves,Comp. Math. 78 (1991), 29–36. · Zbl 0731.14015 [39] P. Vojta, Siegel’s Theorem In The Compact Case,Ann. Math. 133 (1991), 509–548. · Zbl 0774.14019 [40] P. Vojta, Integral points on subvarieties of semi-abelian varieties I, to appear. · Zbl 1011.11040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.