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Paul Vojta

Publications and source records attributed to Paul Vojta.

14 recordsLinked to original sources
arXiv2025

The non-Archimedean Green--Griffiths--Lang--Vojta conjecture for commutative algebraic groups with unipotent rank 1

Let $k$ be algebraically closed field of characteristic zero, let $G$ be a commutative algebraic group over $k$ such that the linear part of $G$ is isomorphic to $\mathbb{G}_a$, and let $X$ be a closed subvariety of $G$. We show that the Kawamata locus of $X$ is equal to a Lang-like exceptional locus of $X$, and furthermore, we identify a condition on $X$ that implies that these loci are proper subschemes of $X$. We also prove the strong form of the non-Archimedean Green--Griffiths--Lang--Vojta conjecture for closed subvarieties of commutative algebraic groups where the linear part is isomorphic to $\mathbb{G}_a \times \mathbb{G}_m^t$.

math.AG↗
arXiv2020

Birational Nevanlinna constants, beta constants, and diophantine approximation to closed subschemes

In an earlier paper (joint with Min Ru), we proved a result on diophantine approximation to Cartier divisors, extending a 2011 result of P. Autissier. This was recently extended to certain closed subschemes (in place of divisors) by Ru and Wang. In this paper we extend this result to a broader class of closed subschemes. We also show that some notions of $β(\mathscr L,D)$ coincide, and that they can all be evaluated as limits.

math.NT↗
arXiv2020

An Evertse-Ferretti Nevanlinna constant and its consequences

In this paper, we introduce the notion of an Evertse-Ferretti Nevanlinna constant and compare it with the birational Nevanlinna constant introduced by the authors in a recent joint paper. We then use it to recover several previously known results. This includes a 1999 example of Faltings from his Baker's Garden article.

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arXiv2016

A birational Nevanlinna constant and its consequences

The purpose of this paper is to modify the notion of the Nevanlinna constant $\operatorname{Nev}(D)$, recently introduced by the first author, for an effective Cartier divisor on a projective variety $X$. The modified notion is called the birational Nevanlinna constant and is denoted by $\operatorname{Nev}_{\text{bir}}(D)$. By computing $\operatorname{Nev}_{\text{bir}}(D)$ using the filtration constructed by Autissier in 2011, we establish a general result (see the General Theorem in the Introduction), in both the arithmetic and complex cases, which extends to general divisors the 2008 results of Evertse and Ferretti and the 2009 results of the first author. The notion $\operatorname{Nev}_{\text{bir}}(D)$ is originally defined in terms of Weil functions for use in applications, and it is proved later in this paper that it can be defined in terms of local effectivity of Cartier divisors after taking a proper birational lifting. In the last two sections, we use the notion $\operatorname{Nev}_{\text{bir}}(D)$ to recover the proof of an example of Faltings from his 2002 Baker's Garden article.

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arXiv2013

Jets via Hasse-Schmidt Derivations

This note is intended to provide a general reference for jet spaces and jet differentials, valid in maximal generality (at the level of EGA). The approach is rather concrete, using Hasse-Schmidt (divided) higher differentials. Discussion of projectivized jet spaces (as in Green and Griffiths (1980)) is included.

math.AG↗
arXiv2009

Transplanting Faltings' garden

In his contribution to the Baker's Garden book, Faltings gives a family of examples of irreducible divisors $D$ on $\Bbb P^2$ for which $\Bbb P^2\setminus D$ has only finitely many integral points over any given localization of a number ring away from finitely many places. He also notes that neither $\Bbb P^2\setminus D$ nor the étale covers used in his proof embed into semiabelian varieties, so his examples do not easily reduce to known results about such subvarieties. In this note, we show how Faltings' results follow directly from a theorem of Evertse and Ferretti; hence these examples can be explained by noting that if one pulls back to a cover of $\Bbb P^2$ étale outside of $D$ and then adds components to the pull-back of $D$ then one can embed the complement into a semiabelian variety and obtain useful diophantine approximation results for the original divisor $D$.

math.NT↗
arXiv2008

On the Nochka-Chen-Ru-Wong Proof of Cartan's Conjecture

In 1982-83, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in $\Bbb P^n$ relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subseqently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides a mild simplification of the proof of the construction of the weights, by construing the ``Nochka diagram'' of Ru and Wong as a convex hull.

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arXiv2007

Multiplier ideal sheaves, Nevanlinna theory, and diophantine approximation

This note states a conjecture for Nevanlinna theory or diophantine approximation, with a sheaf of ideals in place of the normal crossings divisor. This is done by using a correction term involving a multiplier ideal sheaf. This new conjecture trivially implies earlier conjectures in Nevanlinna theory or diophantine approximation, and in fact is equivalent to these conjectures. Although it does not provide anything new, it may be a more convenient formulation for some applications.

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arXiv2007

On McQuillan's "tautological inequality" and the Weyl-Ahlfors theory of associated curves

In 1941, L. Ahlfors gave another proof of a 1933 theorem of H. Cartan on approximation to hyperplanes of holomorphic curves in P^n. Ahlfors' proof built on earlier work of H. and J. Weyl (1938), and proved Cartan's theorem by studying the associated curves of the holomorphic curve. This work has subsequently been reworked by H.-H. Wu in 1970, using differential geometry, M. Cowen and P. A. Griffiths in 1976, further emphasizing curvature, and by Y.-T. Siu in 1987 and 1990, emphasizing meromorphic connections. This paper gives another variation of the proof, motivated by successive minima as in the proof of Schmidt's Subspace Theorem, and using McQuillan's "tautological inequality." In this proof, essentially all of the analysis is encapsulated within a modified McQuillan-like inequality, so that most of the proof primarily uses methods of algebraic geometry, in particular flag varieties. A diophantine conjecture based on McQuillan's inequality is also posed.

math.CV↗
arXiv2007

Nagata's embedding theorem

In 1962-63, M. Nagata showed that an abstract variety could be embedded into a complete variety. Later, P. Deligne translated Nagata's proof into the language of schemes, but did not publish his notes. This paper, which is to appear as an appendix in a forthcoming book, gives an elaboration of Deligne's notes. It also contains some complementary results on extending divisors and vector sheaves to suitable completions.

math.AG↗
arXiv1999

On the abc conjecture and diophantine approximation by rational points

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split function field case of characteristic zero, and in the case of holomorphic curves (Nevanlinna theory). The proof of the latter involves a geometric generalization of the classical lemma on the logarithmic derivative, due to McQuillan. This lemma may be of independent interest.

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arXiv1998

Integral points on subvarieties of semiabelian varieties, II

This paper proves a finiteness result for families of integral points on a semiabelian variety minus a divisor, generalizing the corresponding result of Faltings for abelian varieties. Combined with the main theorem of the first part of this paper, this gives a finiteness statement for integral points on a closed subvariety of a semiabelian variety, minus a divisor. In addition, the last two sections generalize some standard results on closed subvarieties of semiabelian varieties to the context of closed subvarieties minus divisors.

math.NT↗
arXiv1998

A more general abc conjecture

This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new conjecture is implied by the earlier conjecture. As with most of the author's conjectures, this new conjecture stems from analogies with Nevanlinna theory; in this case it corresponds to a Second Main Theorem in Nevanlinna theory with truncated counting functions. The original abc conjecture of Masser and Oesterlé corresponds to the Second Main Theorem with truncated counting functions on P^1 for the divisor [0]+[1]+[\infty].

math.NT↗