Rational Homogeneous Spaces Research Articles

Smooth projective horospherical varieties with nef tangent bundles

We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.

Fano Varieties in Mori Fibre Spaces

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension three and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.

On the derived category of the Cayley plane II

We find a full strongly exceptional collection for the Cayley plane OP2, the simplest rational homogeneous space of the exceptional group E6. This collection, closely related to the one given by the second author in [J. Algebra, 330:177-187, 2011], consists of 27 vector bundles which are homogeneous for the group E6, and is a Lefschetz collection with respect to the minimal equivariant embedding of OP2.

Fano manifolds of Calabi–Yau Hodge type

We introduce and we study a class of odd dimensional compact complex manifolds whose Hodge structure in middle dimension looks like that of a Calabi–Yau threefold. We construct several series of interesting examples from rational homogeneous spaces with special properties.

Formal properties in small codimension

In this note we extend connectedness results to formal properties of inverse images under proper maps of Schubert varieties and of the diagonal in products of projective rational homogeneous spaces.

Seshadri constants and degrees of defining polynomials

In this paper, we study a relation between Seshadri constants and degrees of defining polynomials. In particular, we compute the Seshadri constants on Fano varieties obtained as complete intersections in rational homogeneous spaces of Picard number one.

Chern class formulas for $G_{2}$ Schubert loci

We define degeneracy loci for vector bundles with structure group $G_2$ and give formulas for their cohomology (or Chow) classes in terms of the Chern classes of the bundles involved. When the base is a point, such formulas are part of the theory for rational homogeneous spaces developed by Bernstein–Gelfand–Gelfand and Demazure. This has been extended to the setting of general algebraic geometry by Giambelli–Thom–Porteous, Kempf–Laksov, and Fulton in classical types; the present work carries out the analogous program in type $G_2$. We include explicit descriptions of the $G_2$ flag variety and its Schubert varieties, and several computations, including one that answers a question of W. Graham. In appendices, we collect some facts from representation theory and compute the Chow rings of quadric bundles, correcting an error in a paper by Edidin and Graham.

Classification of compact complex homogeneous manifolds with pseudo-kählerian structures

In this paper, we apply a modification theorem for a compact homogeneous solvmanifold to compact complex homogeneous manifolds with pseudo-kählerian structures. We are able to classify these manifolds as certain products of projective rational homogeneous spaces, tori, simple and double reduced primitive pseudo-kähler spaces.

Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of \cite{cmp2} and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in \cite{seidel}. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar \cite{Magyar}. It also uses Peterson's unpublished result \cite{Peterson} --- recently proved by Lam and Shimozono in \cite{Lam-Shi} --- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.

Holomorphic maps from rational homogeneous spaces onto projective manifolds

In [Math. Ann. 142, 453-468], Remmert and Van de Ven conjectured that if X X is the image of a surjective holomorphic map from P n \mathbb {P}^n , then X X is biholomorphic to P n \mathbb {P}^n . This conjecture was proved by Lazarsfeld [Lect. Notes Math. 1092 (1984), 29-61] using Mori’s proof of Hartshorne’s conjecture [Ann. Math. 110 (1979), 593-606]. Then Lazarsfeld raised a more general problem, which was completely answered in the positive by Hwang and Mok. Theorem 1 ([Invent. math. 136 (1999), 209-231] and [Asian J. Math. 8 (2004), 51-63]). Let S = G / P S=G/P be a rational homogeneous manifold of Picard number 1 1 . For any surjective holomorphic map f : S → X f:S\to X to a projective manifold X X , either X X is a projective space, or f f is a biholomorphism. The aim of this article is to give a generalization of Theorem 1. We will show that modulo canonical projections, Theorem 1 is true when G G is simple without the assumption on Picard number. We need to find a dominating and generically unsplit family of rational curves which are of positive degree with respect to a given nef line bundle on X X . Such a family may not exist in general, but we will prove its existence under a certain assumption which is applicable in our situation.

Homogeneous bundles and the first eigenvalue of symmetric spaces

In this note we prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kähler metric on a compact Hermitian symmetric spaces of ABCD–type.

Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation

Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation

BIRATIONALITY OF THE TANGENT MAP FOR MINIMAL RATIONAL CURVES

For a uniruled projective manifold, we prove that a general rational curve of minimal degree through a general point is uniquely determined by its tangent vector. As applications, among other things we give a new proof, using no Lie theory, of our earlier result that a holomorphic map from a rational homogeneous space of Picard number 1 onto a projective manifold different from the projective space must be a biholomorphic map.

Deformation rigidity of the rational homogeneous space associated to a long simple root

As a continuation of our previous works we study the conjecture on the rigidity under Kähler deformation of the complex structure of rational homogeneous spaces G/ P of Picard number 1, confirming its validity whenever G/ P is associated to a long simple root. For these rational homogeneous spaces the minimal G-invariant holomorphic distribution D is spanned by varieties of minimal rational tangents, and, excepting the symmetric and the contact cases, the complex structure of G/ P is completely determined by the nilpotent symbol algebra of the weak derived differential system of D. The problem is reduced, in a sense, to the invariance of this nilpotent symbol algebra under Kähler deformation. In our earlier works in relation to the question of the integrability of distributions spanned by varieties of minimal rational tangents we have established identities on Lie brackets using integral surfaces arising from pencils of rational curves. In the case on hand, at a point o∈ G/ P we prove that the nilpotent symbol algebra at o is nothing other than the universal Lie algebra generated by D o subject to these identities on Lie brackets, by verifying that they correspond to finiteness condition in the Serre presentation of the simple Lie algebra G.

Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds

Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds