Hyperbolic Varieties Research Articles

Hyperbolic secant varieties of M-curves

Abstract We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1.

Urata's theorem in the logarithmic case and applications to integral points

Urata showed that a pointed compact hyperbolic variety admits only finitely many maps from a pointed curve. We extend Urata's theorem to the setting of (not necessarily compact) hyperbolically embeddable varieties. As an application, we show that a hyperbolically embeddable variety over a number field K $K$ with only finitely many O L , T $\mathcal {O}_{L,T}$ -points for any number field L / K $L/K$ and any finite set of finite places T $T$ of L $L$ has, in fact, only finitely many points in any given Z $\mathbb {Z}$ -finitely generated integral domain of characteristic zero. We use this latter result in combination with Green's criterion for hyperbolic embeddability to obtain novel finiteness results for integral points on symmetric self-products of smooth affine curves and on complements of large divisors in projective varieties. Finally, we use a partial converse to Green's criterion to further study hyperbolic embeddability (or its failure) in the case of symmetric self-products of curves. As a by-product of our results, we obtain the first example of a smooth affine Brody-hyperbolic threefold over C $\mathbb {C}$ which is not hyperbolically embeddable.

Non-archimedean hyperbolicity and applications

Abstract Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K-analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.

On deformations of hyperbolic varieties

In this paper we study flat deformations of real subschemes of $\mathbb{P}^n$, hyperbolic with respect to a fixed linear subspace, i.e. admitting a finite surjective and real fibered linear projection. We show that the subset of the corresponding Hilbert scheme consisting of such subschemes is closed and connected in the classical topology. Every smooth variety in this set lies in the interior of this set. Furthermore, we provide sufficient conditions for a hyperbolic subscheme to admit a flat deformation to a smooth hyperbolic subscheme. This leads to new examples of smooth hyperbolic varieties.

Positively hyperbolic varieties, tropicalization, and positroids

A variety of codimension c in complex affine space is positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension c. Positively hyperbolic hypersurfaces are defined by stable polynomials. We characterize these varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. Positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type A hyperplane arrangement defined by xi=xj, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of results of Choe–Oxley–Sokal–Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.

Arithmetic hyperbolicity: automorphisms and persistence

We show that if the automorphism group of a projective variety is torsion, then it is finite. Motivated by Lang’s conjecture on rational points of hyperbolic varieties, we use this to prove that a projective variety with only finitely many rational points has only finitely many automorphisms. Moreover, we investigate to what extent finiteness of S-integral points on a variety over a number field persists over finitely generated fields. To this end, we introduce the class of mildly bounded varieties and prove a general criterion for proving this persistence.

Algebraicity of analytic maps to a hyperbolic variety

AbstractLet X be a complex algebraic variety. We say that X is Borel hyperbolic if, for every finite type reduced scheme S over the complex numbers, every holomorphic map from S to X is algebraic. We use a transcendental specialization technique to prove that X is Borel hyperbolic if and only if, for every smooth affine complex algebraic curve C, every holomorphic map from C to X is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.

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.

The Chow Form of a Reciprocal Linear Space

A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.

Real fibered morphisms and Ulrich sheaves

In this paper, we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces inPd\mathbb {P}^dand other hyperbolic varieties. We show that real fibered morphisms are intimately connected to Ulrich sheaves admitting positive definite symmetric bilinear forms.

Semi-inverted linear spaces and an analogue of the broken circuit complex

The image of a linear space under inversion of some coordinates is an affine variety whose structure is governed by an underlying hyperplane arrangement. In this paper, we generalize work by Proudfoot and Speyer to show that circuit polynomials form a universal Groebner basis for the ideal of polynomials vanishing on this variety. The proof relies on degenerations to the Stanley-Reisner ideal of a simplicial complex determined by the underlying matroid. If the linear space is real, then the semi-inverted linear space is also an example of a hyperbolic variety, meaning that all of its intersection points with a large family of linear spaces are real.

On the connection of the solvability of a nonlinear evolution equation of the fourth order and elliptic integrals

The purpose of the present paper is to cover two intentions: Firstly we introduce a new computational algebraic procedure that can be applied to derive classes of solutions of nonlinear partial differential equations especially of higher order important in scientific and technical applications. The crucial step needs an auxiliary variable satisfying some ordinary differential equations of the first order containing sine, cosine and their hyperbolic varieties introducing for the first time connected by a special functionally dependence. Stated in most general form the solution manifold of these ordinary differential equations admits elliptic integrals and functions, respectively, as analytical solutions. Secondly the validity and reliability of the method is tested by its application to a rarely studied nonlinear evolution equation of the fourth order and leads to new classes of solutions. Nevertheless it should be emphasised that this technique does not need the solution of complicated nonlinear ordinary differential equations as caused the case of similarity reduction techniques. Further the algorithm works efficiently, is clearly structured and can be used in applications independently of the order of the equation. For computational purposes the method is appropriate to be written in any computer language. Therefore the given novel algebraic approach is suitable for wider classes of nonlinear partial differential equations in order to augment the solution manifold by a straightforward alternative approach.

On non-linear evolution equations of higher order – the introduction and application of a novel computational approach

The purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied to derive class of solutions of non-linear partial differential equations (nPDE) especially of higher order. The crucial step needs an auxiliary variable satisfying some ordinary differential equations (ODE) of first order containing sine, cosine and their hyperbolic varieties introducing to the first time. General transformations are given to determine class of solutions explicitly. The validity and reliability of the method is tested by its application to some important non-linear evolution equations leading to new class of solutions with physical significance. Nevertheless it should be emphasised that this techniques do not need the solution of complicate nODEs as in the case of similarity reduction. Further, the algorithm works efficiently, is clear structured and can be used in any applications independent of the order of the nPDE. For computational purposes the method is appropriate to rewrite it in any computer languages. Therefore, the given novel algebraic approach is suitable for a wider class of nPDE in order to augment the solution manifold by a straightforward alternative approach.