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Holomorphic Cartan geometries and rational curves

100%
Biswas I., McKay B.
Complex Manifolds
|
2016
|
tom 3
|
nr 1
EN
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.
2
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On geometry of curves of flags of constant type

100%
Doubrov B., Zelenko I.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1836-1871
EN
We develop an algebraic version of Cartan’s method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of GL(W). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure linear algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.
3
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The gap phenomenon in the dimension study of finite type systems

100%
Kruglikov B.
Open Mathematics
|
2012
|
tom 10
|
nr 5
1605-1618
EN
Several examples of gaps (lacunes) between dimensions of maximal and submaximal symmetric models are considered, which include investigation of number of independent linear and quadratic integrals of metrics and counting the symmetries of geometric structures and differential equations. A general result clarifying this effect in the case when the structure is associated to a vector distribution, is proposed.
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