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Holomorphic Cartan geometries and rational curves

Indranil Biswas, Benjamin McKay (2016)

Complex Manifolds

We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.

Homogénéité locale pour les métriques riemanniennes holomorphes en dimension 3

Sorin Dumitrescu (2007)

Annales de l’institut Fourier

Une métrique riemannienne holomorphe sur une variété complexe M est une section holomorphe q du fibré S 2 ( T * M ) des formes quadratiques complexes sur l’espace tangent holomorphe à M telle que, en tout point m de M , la forme quadratique complexe q ( m ) est non dégénérée (de rang maximal, égal à la dimension complexe de M ). Il s’agit de l’analogue, dans le contexte holomorphe, d’une métrique riemannienne (réelle). Contrairement au cas réel, l’existence d’une telle métrique sur une variété complexe compacte n’est...

Intrinsic geometric on the class of probability densities and exponential families.

Henryk Gzyl, Lázaro Recht (2007)

Publicacions Matemàtiques

We present a way of thinking of exponential farnilies as geodesic surfaces in the class of positive functions considered as a (multiplicative) sub-group G+ of the group G of all invertible elements in the algebra A of all complex bounded functions defined on a measurable space. For that we have to study a natural geometry on that algebra. The class D of densities with respect to a given rneasure will happen to be representatives of equivalence classes defining a projective space in A. The natural...

Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties

Giovanni Gaiffi, Michele Grassi (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove that one can obtain natural bundles of Lie algebras on rank two s -Kähler manifolds, whose fibres are isomorphic respectively to so ( s + 1 , s + 1 ) , su ( s + 1 , s + 1 ) and sl ( 2 s + 2 , ) . These bundles have natural flat connections, whose flat global sections generalize the Lefschetz operators of Kähler geometry and act naturally on cohomology. As a first application, we build an irreducible representation of a rational form of su ( s + 1 , s + 1 ) on (rational) Hodge classes of Abelian varieties with rational period matrix.

Métriques riemanniennes holomorphes en petite dimension

Sorin Dumitrescu (2001)

Annales de l’institut Fourier

Nous étudions les métriques riemanniennes holomorphes sur les variétés complexes compactes de dimension 3 . Nous montrons que, contrairement au cas réel, une métrique riemannienne holomorphe possède un “grand” pseudo-groupe d’isométries locales. Ceci implique qu’une telle métrique n’existe pas sur les variétés complexes compactes simplement connexes de dimension 3 .

Properties of a hypothetical exotic complex structure on P 3

J. R. Brown (2007)

Mathematica Bohemica

We consider almost-complex structures on P 3 whose total Chern classes differ from that of the standard (integrable) almost-complex structure. E. Thomas established the existence of many such structures. We show that if there exists an “exotic” integrable almost-complex structures, then the resulting complex manifold would have specific Hodge numbers which do not vanish. We also give a necessary condition for the nondegeneration of the Frölicher spectral sequence at the second level.

Propriétés globales de l'espace de twisteurs

Paolo De Bartolomeis, Luca Migliorini, Antonella Nannicini (1991)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We study global properties of the twistor space over an even dimensional conformally flat manifold, proving that the twistor space is Kähler if and only if the manifold is conformally equivalent to the standard 2 n -dimensional sphere ( n > 2 ).

Quaternionic and para-quaternionic CR structure on (4n+3)-dimensional manifolds

Dmitri Alekseevsky, Yoshinobu Kamishima (2004)

Open Mathematics

We define notion of a quaternionic and para-quaternionic CR structure on a (4n+3)-dimensional manifold M as a triple (ω1,ω2,ω3) of 1-forms such that the corresponding 2-forms satisfy some algebraic relations. We associate with such a structure an Einstein metric on M and establish relations between quaternionic CR structures, contact pseudo-metric 3-structures and pseudo-Sasakian 3-structures. Homogeneous examples of (para)-quaternionic CR manifolds are given and a reduction construction of non...

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