Algebraic Hyperbolicity Research Articles

Boundedness in families with applications to arithmetic hyperbolicity

AbstractMotivated by conjectures of Demailly, Green–Griffiths, Lang and Vojta, we show that several notions related to hyperbolicity behave similarly in families. We apply our results to show the persistence of arithmetic hyperbolicity along field extensions for projective normal surfaces with non‐zero irregularity. These results rely on the mild boundedness of semi‐abelian varieties. We also introduce and study the notion of pseudo‐algebraic hyperbolicity which extends Demailly's notion of algebraic hyperbolicity for projective schemes.

Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures

In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.

Algebraic hyperbolicity of very general surfaces

Recently, Haase and Ilten initiated the study of classifying algebraically hyperbolic surfaces in toric threefolds. We complete this classification for $${\mathbb{P}^1} \times {\mathbb{P}^1} \times {\mathbb{P}^1},\,\,{\mathbb{P}^2} \times {\mathbb{P}^1},\,\,{\mathbb{F}_e} \times {\mathbb{P}^1}$$ and the blowup of ℙ3 at a point, augmenting our earlier work on ℙ3. Most importantly, we treat the boundary cases which are the hardest cases from the point of view of positivity. In the process, we codify several different techniques for proving algebraic hyperbolicity, allowing us to prove similar results for hypersurfaces in any threefold admitting a group action with dense orbit.

Algebraic hyperbolicity for surfaces in smooth projective toric threefolds with Picard rank 2 and 3

Algebraic hyperbolicity serves as a bridge between differential geometry and algebraic geometry. Generally, it is difficult to show that a given projective variety is algebraically hyperbolic. However, it was established recently that a very general surface of degree at least five in projective space is algebraically hyperbolic. We are interested in generalizing the study of surfaces in projective space to surfaces in smooth projective toric threefolds with Picard rank 2 or 3. Following Kleinschmidt and Batyrev, we explore the combinatorial description of smooth projective toric threefolds with Picard rank 2 and 3. We then use Haase and Ilten’s method of finding algebraically hyperbolic surfaces in toric threefolds. As a result, we determine many algebraically hyperbolic surfaces in each of these varieties.

Algebraic intermediate hyperbolicities

We extend Lang's conjectures to the setting of intermediate hyperbolicity and prove two new results motivated by these conjectures. More precisely, we first extend the notion of algebraic hyperbolicity (originally introduced by Demailly) to the setting of intermediate hyperbolicity and show that this property holds if the appropriate exterior power of the cotangent bundle is ample. Then, we prove that this intermediate algebraic hyperbolicity implies the finiteness of the group of birational automorphisms and of the set of surjective maps from a given projective variety. Our work answers the algebraic analogue of a question of Kobayashi on analytic hyperbolicity.

Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P[Formula: see text]un and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596].

Algebraic hyperbolicity for surfaces in toric threefolds

Adapting focal loci techniques used by Chiantini and Lopez, we provide lower bounds on the genera of curves contained in very general surfaces in Gorenstein toric threefolds. We illustrate the utility of these bounds by obtaining results on algebraic hyperbolicity of very general surfaces in toric threefolds.

Demailly\u2019s notion of algebraic hyperbolicity: geometricity, boundedness, moduli of maps

Demailly’s conjecture, which is a consequence of the Green–Griffiths–Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly’s conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly’s definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that {{,mathrm{Aut},}}(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore “weaker” notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green–Griffiths–Lang conjecture on hyperbolic projective varieties.

Algebraic hyperbolicity of the very general quintic surface in [formula omitted

We prove that a curve of degree dk on a very general surface of degree d≥5 in P3 has geometric genus at least dk(d−5)+k2+1. This gives a substantial improvement on the celebrated genus bounds of Geng Xu. As a corollary, we deduce the algebraic hyperbolicity of a very general quintic surface in P3, resolving a long-standing conjecture of Demailly. This completely determines which very general hypersurfaces in P3 are algebraically hyperbolic.

Algebraically hyperbolic manifolds have finite automorphism groups

A projective manifold [Formula: see text] is algebraically hyperbolic if there exists a positive constant [Formula: see text] such that the degree of any curve of genus [Formula: see text] on [Formula: see text] is bounded from above by [Formula: see text]. A classical result is that Kobayashi hyperbolicity implies algebraic hyperbolicity. It is known that Kobayashi hyperbolic manifolds have finite automorphism groups. Here, we prove that, more generally, algebraically hyperbolic projective manifolds have finite automorphism groups.

Asymmetric tensor representations in micropolar continuum mechanics theories

Получены новые представления трехмерного асимметричного тензора напряжений и соответствующие им формы дифференциальных уравнений равновесия. Асимметричные теории механики деформируемого твердого тела по прежнему привлекают пристальное внимание в связи с необходимостью математического моделирования механического поведения современных материалов (например, ауксетиков с помощью теорий гемитропной микрополярной упругости). Исследование ограничивается только такими асимметричными тензорами второго ранга, для которых удается сохранить понятие о вещественных собственных значениях, но отказаться от взаимной ортогональности направлений главного триэдра. Обсуждается точная алгебраическая формулировка указанных условий асимметричности. В статье обобщаются тензорные представления симметричного тензора напряжений, основанные на естественном репере асимптотических направлений. Полученные результаты являются ярким свидетельством в пользу алгебраической «гиперболичности» симметричных и асимметричных тензоров второго ранга в трехмерном пространстве.

Algebraic hyperbolicity of ramified covers of $\mathbb{G}^2_m$ (and integrap points on affine subsets of $\mathbb{P}_2$)

Let $X$ be a smooth affine surface, $X \to \mathbb{G}^2_m$ be a finite morphism. We study the affine curves on $X$, with bounded genus and number of points at infinity, obtaining bounds for their degree in terms of Euler characteristic. A typical example where these bounds hold is represented by the complement of a three-component curve in the projective plane, of total degree at least 4. The corresponding results may be interpreted as bounding the height of integral points on $X$ over a function field. In the language of Diophantine Equations, our results may be rephrased in terms of bounding the height of the solutions of $f(u, v, y) = 0$, with $u, v, y$ over a function field, $u, v$ $S$-units. It turns out that all of this contain some cases of a strong version of a conjecture of Vojta over function fields in the split case. Moreover, our method would apply also to the nonsplit case. We remark that special cases of our results in the holomorphic context were studied by M. Green already in the seventies, and recently in greater generality by Noguchi, Winkelmann, and Yamanoi; however, the algebraic context was left open and seems not to fall in the existing techniques.

On the logarithmic Kobayashi conjecture

We study the hyperbolicity of the log variety (ℙn,X), where X is a very general hypersurface of degree d ≧ 2n + 1 (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of (ℙn, X) is of log-general type, give a new proof of the algebraic hyperbolicity of (ℙn, X), and exclude the existence of maximal rank families of entire curves in the complement of the universal degree d hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.

Algebraic hyperbolicity of generic high degree hypersurfaces

Algebraic hyperbolicity of generic high degree hypersurfaces