Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle (original) (raw)
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The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We construct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to general parabolic geometries are discussed.
3 Geometry of Curves in Generalized Flag Varieties
2016
The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.
Cotangent Bundle to the Flag Variety - II
2016
Let P be a parabolic subgroup in SLn(C). We show that there is a SLn(C)-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety(associated to the Kac-Moody group ŜLn(C)) which is a natural compactification of the cotangent bundle to SLn(C)/P . As a consequence, we recover the Springer resolution for any orbit closure inside the variety of nilpotent matrices.
The Student Mathematical Library, 2003
Let G be a complex reductive linear algebraic group and G0 G a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that G0 \ L = L0 is a real form of L. Using the minimal globalization Vmin of a finite length admissible representation for L0 , one can define a homogeneous analytic vector bundle on the G0 orbit S of P in the generalized flag manifold Y = G=P. Let A(P;Vmin) denote the corresponding sheaf of polarized sections. In this article we analyze the G0 representations obtained on the compactly supported sheaf cohomology groups H p c (S; A(P;Vmin)).
Cotangent Bundle to the Flag Variety–I
Transformation Groups, 2017
We show that there is a SLn-stable closed subset of an affine Schubert variety in the infinite dimensional Flag variety (associated to the Kac-Moody group SLn) which is a natural compactification of the cotangent bundle to the finite-dimensional Flag variety SLn/B.
On geometry of curves of flags of constant type
2011
We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space WWW with respect to the action of a subgroup GGG of the GL(W)GL(W)GL(W). Under some natural assumptions on the subgroup GGG and on the flags, one can pass from the filtered objects
Transformation Groups, 2007
We define the odd symplectic grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group and generalizing the usual symplectic grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic grassmannians, and we prove a Borel-Weil type theorem for the odd symplectic group.
2021
In this paper we address Fano manifolds with positive higher Chern characters. They are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank 1 have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank 1 having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete interse...