scholarly journals Algebraically hyperbolic manifolds have finite automorphism groups

2019 ◽  
Vol 22 (02) ◽  
pp. 1950003
Author(s):  
Fedor A. Bogomolov ◽  
Ljudmila Kamenova ◽  
Misha Verbitsky
Keyword(s):  
Positive Constant ◽  
Classical Result ◽  
Automorphism Groups ◽  
Hyperbolic Manifolds ◽  
Projective Manifold ◽  
Kobayashi Hyperbolicity ◽  
Genus Formula ◽  
Projective Manifolds

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.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals ON PROJECTIVE MANIFOLDS SWEPT OUT BY CUBIC VARIETIES

2012 ◽  
Vol 23 (07) ◽  
pp. 1250058 ◽  
Author(s):  
KIWAMU WATANABE
Keyword(s):  
High Dimensional ◽  
Projective Manifold ◽  
Projective Bundle ◽  
Projective Manifolds ◽  
Cubic Hypersurfaces

We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold ℙ1 × ℙ2. In the course of the proof, we classify projective manifolds of dimension five swept out by planes.

scholarly journals A note on Lang's conjecture for quotients of bounded domains

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Sébastien Boucksom ◽  
Simone Diverio
Keyword(s):  
Plurisubharmonic Function ◽  
General Type ◽  
Universal Cover ◽  
Projective Manifold ◽  
Bounded Domains ◽  
Final Version ◽  
Lang's Conjecture ◽  
Projective Manifolds ◽  
Lang’S Conjecture ◽  
Complex Projective

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains. Comment: 10 pages, no figures, comments are welcome. v3: following suggestions made by the referee, the exposition has been improved all along the paper, we added a variant of Theorem A which includes manifolds whose universal cover admits a bounded psh function which is strictly psh just at one point, and we added a section of examples. Final version, to appear on \'Epijournal G\'eom. Alg\'ebrique

REAL PROJECTIVE MANIFOLDS DEVELOPING INTO AN AFFINE SPACE

1993 ◽  
Vol 04 (02) ◽  
pp. 179-191 ◽  
Author(s):  
YOUNKI CHAE ◽  
SUHYOUNG CHOI ◽  
CHAN-YOUNG PARK
Keyword(s):  
Projective Space ◽  
Affine Space ◽  
Dimensional Subspace ◽  
Affine Group ◽  
Projective Manifold ◽  
Hilbert Metric ◽  
Projective Manifolds

Suppose that an n-dimensional closed real projective manifold M, n ≥ 2, develops into an affine space RPn − RPn − 1 for an (n − 1)-dimensional subspace RPn − 1 of the projective space RPn. Then either M is convex or affine or M admits a flat foliation [Formula: see text] with a transverse invariant Hilbert metric. Further, if the codimension of [Formula: see text] is n − 1, then M is convex. We prove this statement by a use of a variation of Carrière's discompacté, a measure of non-compactedness of an affine group acting on an affine space.

scholarly journals Automorphism Groups of Wreath Product Digraphs

10.37236/106 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Edward Dobson ◽  
Joy Morris
Keyword(s):  
Wreath Product ◽  
Directed Graph ◽  
Classical Result ◽  
Automorphism Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Necessary And Sufficient ◽  
Transitive Graphs

We generalize a classical result of Sabidussi that was improved by Hemminger, to the case of directed color graphs. The original results give a necessary and sufficient condition on two graphs, $C$ and $D$, for the automorphsim group of the wreath product of the graphs, ${\rm Aut}(C\wr D)$ to be the wreath product of the automorphism groups ${\rm Aut}(C)\wr {\rm Aut}(D)$. Their characterization generalizes directly to the case of color graphs, but we show that there are additional exceptional cases in which either $C$ or $D$ is an infinite directed graph. Also, we determine what ${\rm Aut}(C \wr D)$ is if ${\rm Aut}(C \wr D) \neq {\rm Aut} (C) \wr {\rm Aut} (D)$, and in particular, show that in this case there exist vertex-transitive graphs $C'$ and $D'$ such that $C' \wr D' = C \wr D$ and ${\rm Aut} (C\wr D) = {\rm Aut} (C') \wr {\rm Aut}(D')$.

Cusps, triangle groups and hyperbolic 3-folds

1993 ◽  
Vol 55 (2) ◽  
pp. 149-182 ◽  
Author(s):  
Marston D. E. Conder ◽  
Gaven J. Martin
Keyword(s):  
Kleinian Group ◽  
Fundamental Domain ◽  
Small Volume ◽  
Automorphism Groups ◽  
Integral Formula ◽  
Hyperbolic Manifolds ◽  
Triangle Groups ◽  
Totally Geodesic ◽  
Volume Limit ◽  
Geodesic Surfaces

AbstractWe provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

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