##
**Generalizations of Siegel’s and Picard’s theorems.**
*(English)*
Zbl 1250.11067

This paper presents new conjectures and new results on the distribution of integral points and of entire curves on quasi-projective algebraic varieties.

Let \(X\) be a smooth projective variety defined over a number field. Let \(D\subset X\) be a divisor on \(X\), with normal crossing singularities (if any). Given a ring of \(S\)-integers over which the variety and the divisor can be defined, one is interested on the set of \(S\)-integral points of the open variety \(X\setminus D\).

The celebrated Vojta’s conjecture (Conjecture 3.4.3 in [P. Vojta, Diophantine approximations and value distribution theory. Lecture Notes in Mathematics, 1239. Berlin etc.: Springer-Verlag (1987; Zbl 0609.14011)]) asserts the degeneracy of the set of \(S\)-integral points of \(X\setminus D\) under a suitable condition on \(D\) and the canonical line bundle of \(X\).

In the paper under review, Levin considers conjectures of similar type, but resting on conditions on the number of components of \(D\) and their dimension, defined as follows:

Given a divisor \(D\), its dimension is the well-defined integer \(\kappa(D)\) with the property that there exists positive constants \(c_1,c_2\) such that for all sufficiently divisible \(n>0\), \[ c_1n^{\kappa(D)}\leq h^0(nD)\leq c_2 n^{\kappa(D)}. \]

The first Diophantine conjecture of Levin reads as follows:

Conjecture. Let \(D=D_1+\ldots+D_r\) be a divisor on \(X\) such that the intersection of any \(m+1\)distinct \(D_i\) is empty. Suppose that \(\kappa(D_i)\geq \kappa_0>0\) for all \(i=i,\ldots,r\). If \(r>m+m/\kappa_0\), then the set of \(S\)-integral points is degenerate. Under the same condition, no entire map \({\mathbb C}\to X\setminus D\) has Zariski-dense image.

(Of course, in the second statement one considers varieties defined over the complex number field).

In the case of curves, the conditions are satisfied whenever \(r\geq 3\), thus recovering Siegel’s theorem on integral points and Picard theorem on entire functions, hence the title of the article.

A first result of the paper is the complete solution of the above conjecture for surfaces. A particular case of Theorem 6.2A, states, in the above notation, that

Theorem. Suppose \(D=D_1+\ldots+D_r\) is a divisor on a surface \(X\), such that no three distinct \(D_i\) meet. If \(\kappa(D_i)>0\) and \(r\geq 5\), or \(\kappa(D_i)=2\) and \(r\geq 4\) then the integral points are not Zariski-dense. Moreover, in the case \(\kappa(D_i)=2, r\geq 4\), the positive dimensional part of the Zariski-closure of the set of integral point is contained in a curve independent of the ring of \(S\)-integers. If moreover the components \(D_i\) are ample divisors and \(r\geq 5\), then the set of integral points is finite.

A completely analogous result holds in the complex-analytic settings.

The example of three lines in general position on the plane shows that the inequalities on \(r\) are sharp when \(\kappa(D_i)\) is assumed to be maximal. When \(\kappa(D_i)=1\), the examples of four divisors of type \((1,0)\) and \((0,1)\) on a smooth quadric prove that again the inequality is sharp.

In higher dimension, the main result is Theorem 9.11A:

Theorem. Let \(D=D_1+\ldots+D_r\) be a divisor on a projective algebraic variety \(X\) of dimension \(q\), such that no \(m+1\) distinct \(D_i\) intersect. If \(\kappa(D_i)=q\) for all \(i\) and \(r>2mq\) there exists a closed proper subvariety of \(X\), independent of the ring of \(S\)-integers, containing all but finitely many integral points of \(X\setminus D\). If moreover all the \(D_i\) are ample, then all sets of integral points are finite.

Again, the analogue holds for entire curves on \(X\setminus D\). In arbitrary dimension, the result is probably not optimal; the author conjectures that \(r>2m\) should suffice. For instance, in dimension three it is conjecture that removing five ample divisors in general position leads to degeneracy of integral points. A recent result of P. Autissier [“Sur la non-densité des points entiers”, Duke Math. J. 158, No. 1, 13–27 (2011; Zbl 1217.14020)], following previous work of the author, U. Zannier and the reviewer [“Integral points on threefolds and other varieties”, Tohoku Math. J. (2) 61, No. 4, 589–601 (2009; Zbl 1250.11066)] states that the removal of six ample divisors suffices.

In another direction, Levin considers the problem of describing the distribution of integral points of bounded degree. The above theorem generalizes as follows:

Theorem. Under the above notation, if \(r>2d^2mq\), then no set of integral points of degree \(d\) is Zariski-dense.

In the penultimate paragraph of this long paper, Levin discusses the problem of degeneracy of integral points on the complement of an irreducible divisor on the plane. Generalizing results of G. Faltings [“A new application of Diophantine approximations”, Cambridge: Cambridge University Press. 231–246 (2002; Zbl 1080.14025)] and U. Zannier [“On the integral points on the complement of ramification-divisors”, J. Inst. Math. Jussieu 4, No. 2, 317–330 (2005; Zbl 1089.11021)] he proves the degeneracy of integral points on the complement of a ’sufficiently singular’ curve.

Finally, in the last paragraph he discusses the relation between his results and his conjectures to the celebrated conjectures of Vojta, formulated in [Zbl 0609.14011].

The proofs of Levin’s theorems are based on the Subspace Theorem of Schmidt and Schlickewei, following the pattern introduced by U. Zannier and the reviewer in [“A subspace theorem approach to integral points on curves”, C. R., Math., Acad. Sci. Paris 334, No. 4, 267–271 (2002; Zbl 1012.11051)] and [“On integral points on surfaces”, Ann. Math. (2) 160, No. 2, 705–726 (2004; Zbl 1146.11035)]. In particular, the Main Theorem of this second paper by Corvaja-Zannier contains the crucial ingredient to obtain Levin’s result on surfaces; the author combines it with a new result from intersection theory (his Lemma 9.7), valid in arbitrary dimension.

Let \(X\) be a smooth projective variety defined over a number field. Let \(D\subset X\) be a divisor on \(X\), with normal crossing singularities (if any). Given a ring of \(S\)-integers over which the variety and the divisor can be defined, one is interested on the set of \(S\)-integral points of the open variety \(X\setminus D\).

The celebrated Vojta’s conjecture (Conjecture 3.4.3 in [P. Vojta, Diophantine approximations and value distribution theory. Lecture Notes in Mathematics, 1239. Berlin etc.: Springer-Verlag (1987; Zbl 0609.14011)]) asserts the degeneracy of the set of \(S\)-integral points of \(X\setminus D\) under a suitable condition on \(D\) and the canonical line bundle of \(X\).

In the paper under review, Levin considers conjectures of similar type, but resting on conditions on the number of components of \(D\) and their dimension, defined as follows:

Given a divisor \(D\), its dimension is the well-defined integer \(\kappa(D)\) with the property that there exists positive constants \(c_1,c_2\) such that for all sufficiently divisible \(n>0\), \[ c_1n^{\kappa(D)}\leq h^0(nD)\leq c_2 n^{\kappa(D)}. \]

The first Diophantine conjecture of Levin reads as follows:

Conjecture. Let \(D=D_1+\ldots+D_r\) be a divisor on \(X\) such that the intersection of any \(m+1\)distinct \(D_i\) is empty. Suppose that \(\kappa(D_i)\geq \kappa_0>0\) for all \(i=i,\ldots,r\). If \(r>m+m/\kappa_0\), then the set of \(S\)-integral points is degenerate. Under the same condition, no entire map \({\mathbb C}\to X\setminus D\) has Zariski-dense image.

(Of course, in the second statement one considers varieties defined over the complex number field).

In the case of curves, the conditions are satisfied whenever \(r\geq 3\), thus recovering Siegel’s theorem on integral points and Picard theorem on entire functions, hence the title of the article.

A first result of the paper is the complete solution of the above conjecture for surfaces. A particular case of Theorem 6.2A, states, in the above notation, that

Theorem. Suppose \(D=D_1+\ldots+D_r\) is a divisor on a surface \(X\), such that no three distinct \(D_i\) meet. If \(\kappa(D_i)>0\) and \(r\geq 5\), or \(\kappa(D_i)=2\) and \(r\geq 4\) then the integral points are not Zariski-dense. Moreover, in the case \(\kappa(D_i)=2, r\geq 4\), the positive dimensional part of the Zariski-closure of the set of integral point is contained in a curve independent of the ring of \(S\)-integers. If moreover the components \(D_i\) are ample divisors and \(r\geq 5\), then the set of integral points is finite.

A completely analogous result holds in the complex-analytic settings.

The example of three lines in general position on the plane shows that the inequalities on \(r\) are sharp when \(\kappa(D_i)\) is assumed to be maximal. When \(\kappa(D_i)=1\), the examples of four divisors of type \((1,0)\) and \((0,1)\) on a smooth quadric prove that again the inequality is sharp.

In higher dimension, the main result is Theorem 9.11A:

Theorem. Let \(D=D_1+\ldots+D_r\) be a divisor on a projective algebraic variety \(X\) of dimension \(q\), such that no \(m+1\) distinct \(D_i\) intersect. If \(\kappa(D_i)=q\) for all \(i\) and \(r>2mq\) there exists a closed proper subvariety of \(X\), independent of the ring of \(S\)-integers, containing all but finitely many integral points of \(X\setminus D\). If moreover all the \(D_i\) are ample, then all sets of integral points are finite.

Again, the analogue holds for entire curves on \(X\setminus D\). In arbitrary dimension, the result is probably not optimal; the author conjectures that \(r>2m\) should suffice. For instance, in dimension three it is conjecture that removing five ample divisors in general position leads to degeneracy of integral points. A recent result of P. Autissier [“Sur la non-densité des points entiers”, Duke Math. J. 158, No. 1, 13–27 (2011; Zbl 1217.14020)], following previous work of the author, U. Zannier and the reviewer [“Integral points on threefolds and other varieties”, Tohoku Math. J. (2) 61, No. 4, 589–601 (2009; Zbl 1250.11066)] states that the removal of six ample divisors suffices.

In another direction, Levin considers the problem of describing the distribution of integral points of bounded degree. The above theorem generalizes as follows:

Theorem. Under the above notation, if \(r>2d^2mq\), then no set of integral points of degree \(d\) is Zariski-dense.

In the penultimate paragraph of this long paper, Levin discusses the problem of degeneracy of integral points on the complement of an irreducible divisor on the plane. Generalizing results of G. Faltings [“A new application of Diophantine approximations”, Cambridge: Cambridge University Press. 231–246 (2002; Zbl 1080.14025)] and U. Zannier [“On the integral points on the complement of ramification-divisors”, J. Inst. Math. Jussieu 4, No. 2, 317–330 (2005; Zbl 1089.11021)] he proves the degeneracy of integral points on the complement of a ’sufficiently singular’ curve.

Finally, in the last paragraph he discusses the relation between his results and his conjectures to the celebrated conjectures of Vojta, formulated in [Zbl 0609.14011].

The proofs of Levin’s theorems are based on the Subspace Theorem of Schmidt and Schlickewei, following the pattern introduced by U. Zannier and the reviewer in [“A subspace theorem approach to integral points on curves”, C. R., Math., Acad. Sci. Paris 334, No. 4, 267–271 (2002; Zbl 1012.11051)] and [“On integral points on surfaces”, Ann. Math. (2) 160, No. 2, 705–726 (2004; Zbl 1146.11035)]. In particular, the Main Theorem of this second paper by Corvaja-Zannier contains the crucial ingredient to obtain Levin’s result on surfaces; the author combines it with a new result from intersection theory (his Lemma 9.7), valid in arbitrary dimension.

Reviewer: Pietro Corvaja (Udine)

### MSC:

11G35 | Varieties over global fields |

32H30 | Value distribution theory in higher dimensions |

11J97 | Number-theoretic analogues of methods in Nevanlinna theory (work of Vojta et al.) |

14G25 | Global ground fields in algebraic geometry |

### Keywords:

integral points; holomorphic curves; Schmidt’s subspace theorem; second main theorem; hyperbolicity### Citations:

Zbl 0609.14011; Zbl 1217.14020; Zbl 1080.14025; Zbl 1089.11021; Zbl 1012.11051; Zbl 1146.11035; Zbl 1250.11066### References:

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