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 Deformations of the tangent bundle of projective manifolds

2020 ◽  
Vol 31 (11) ◽  
pp. 2050087
Author(s):  
Thomas Peternell
Keyword(s):  
Projective Space ◽  
Tangent Bundle ◽  
Complex Projective Space ◽  
Projective Manifold ◽  
First Order ◽  
Projective Manifolds ◽  
Complex Projective

We investigate when the tangent bundle of a projective manifold has a nontrivial first-order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.

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 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.

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

On geometry of orbits of adapted projective frame space

2019 ◽  
pp. 88-98
Author(s):  
A. Kuleshov
Keyword(s):  
Normal Subgroup ◽  
Projective Space ◽  
Tangent Space ◽  
Affine Group ◽  
Current Paper ◽  
Previous Issue ◽  
Distinguished Point

The current paper continues consideration of geometry of projective frame orbits started in the author’s article in the previous issue. The ndimensional projective space with a distinguished point (the center) is considered. The action of matrix affine group of order n on the adapted projective frame manifold is given. It is shown that the linear frames, i. e., bases of the tangent space, can be identified with the orbits of adapted projective frames under the action of some normal subgroup of this group. Two adapted frames are said to be equivalent if they belong to the same orbit. The strict perspectivity relation between two adapted frames is introduced. The proofs of the theorem on the Desargues hyperplane and of the criterion of equivalence are simplified. According to this criterion, two adapted frames in strict perspective are equivalent if and only if the Desargues hyperplane generated by these frames is passing through the center.

Local Metrics for Rigid Body Displacements

2004 ◽  
Vol 126 (5) ◽  
pp. 805-812 ◽  
Author(s):  
Johannes K. Eberharter ◽  
Bahram Ravani
Keyword(s):  
Rigid Body ◽  
Projective Space ◽  
Affine Space ◽  
Three Dimensional ◽  
Real Space ◽  
Elegant Method ◽  
Rigid Body Displacement ◽  
Definition Of ◽  
Three Space ◽  
Motion Design

One hundred years ago, Eduard Study introduced a very elegant method to describe a rigid body displacement in three-space. He mapped each position of a rigid body onto a point on a quadric, now called the Study quadric. This quadric is a six-dimensional rational hyper-surface, embedded in a seven-dimensional projective real space, called Study’s soma space. More than half a century later Ravani and Roth reconfigured Study’s soma space into a three-dimensional dual projective space, and defined a geometric metric for rigid body displacements. Here, approximately 20 years later, we again use Study’s quadric and define a new metric for rigid body displacements based on an optimized local mapping of the quadric. The local mappings of the quadric are achieved using stereographic projections, resulting in an affine space where the Euclidean definition of a metric can be used for rigid body displacements and techniques from design of curves and surfaces can be directly utilized for motion design. The results are illustrated by examples.

On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa

1992 ◽  
Vol 44 (1) ◽  
pp. 206-214
Author(s):  
Jarosław A. Wiśniewski
Keyword(s):  
Line Bundle ◽  
Projective Space ◽  
Betti Numbers ◽  
Ample Line Bundle ◽  
Double Cover ◽  
Topological Properties ◽  
Surjective Morphism ◽  
Projective Manifolds ◽  
Dimensional Projective Space ◽  
Complex Projective

AbstractLet π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

Finding conjugate stabilizer subgroups in $\PSL$ and related groups

2010 ◽  
Vol 10 (3&4) ◽  
pp. 282-291
Author(s):  
DA. Denney ◽  
C. Moore ◽  
A. Russell
Keyword(s):  
Projective Space ◽  
Finite Groups ◽  
Quantum Algorithms ◽  
Affine Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Hidden Subgroup Problem ◽  
Subgroup Problem ◽  
Groups Of Lie Type ◽  
Positive Results

We reduce a case of the hidden subgroup problem (HSP) in $\SL$, $\PSL$, and $\PGL$, three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group $\AGL$. These groups act on projective space in an ``almost'' 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as $\PSL$.

scholarly journals Schematic homotopy types and non-abelian Hodge theory

2008 ◽  
Vol 144 (3) ◽  
pp. 582-632 ◽  
Author(s):  
L. Katzarkov ◽  
T. Pantev ◽  
B. Toën
Keyword(s):  
Homotopy Theory ◽  
Main Tool ◽  
Hodge Decomposition ◽  
Homotopy Groups ◽  
Projective Manifold ◽  
Hodge Structures ◽  
Homotopy Types ◽  
Projective Manifolds ◽  
Simply Connected ◽  
Complex Projective

AbstractWe use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto (X\otimes \mathbb {C})^{\mathrm {sch}}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a Hodge decomposition on $(X\otimes \mathbb {C})^{\mathrm {sch}}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb {C}^{\times \delta }$ on the object $(X\otimes \mathbb {C})^{\mathrm {sch}}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group, and, in the simply connected case, the Hodge decomposition on the complexified homotopy groups. We show that our Hodge decomposition satisfies a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As applications we construct new examples of homotopy types which are not realizable as complex projective manifolds and we prove a formality theorem for the schematization of a complex projective manifold.

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