Complex Flag Manifolds Research Articles

On a class of symmetric CR manifolds

We study and classify a large class of minimal orbits in complex flag manifolds for the holomorphic action of a real Lie group. These orbits are all symmetric CR spaces for the restriction of a suitable class of Hermitian invariant metrics on the ambient flag manifold. As a particular case we obtain that the standard compact homogeneous CR manifolds associated with semisimple Levi–Tanaka algebras are symmetric CR-spaces.

Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds

It is known [M4] that Kℂ-orbits S and Gℝ-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. It is possible to replace Kℂ by some conjugate xKℂx−1 so that the correspondence is preserved. We investigate the sets C(S) of such x for various orbits S and their relations with each other. We prove that for classical groups the intersection C = ∩S C(S) equals D0Z where D0 = D0/Kℂ is the universal domain in Gℂ/Kℂ introduced in [AG] and Z is the center of G. As a corollary we prove that D0 is Stein for classical groups. Moreover we conjecture that C(S)0 = D0 for generic S where C(S)0 is the connected component of C(S) containing the identity.

Schubert varieties and cycle spaces

Let $G_0$ be a real semisimple Lie group. It acts naturally on every complex flag manifold $Z=G/Q$ of its complexification. Given an Iwasawa decomposition $G_0 = K_0 A_0 N_0$, a $G_0$-orbit $γ⊂Z$, and the dual $K$-orbit $κ⊂Z$, Schubert varieties are studied and a theory of Schubert slices for arbitrary $G_0$-orbits is developed. For this, certain geometric properties of dual pairs $(γ,κ)$ are underlined. Canonical complex analytic slices contained in a given $G_0$-orbit γ which are transversal to the dual $K_0$-orbit γ ∩ κ are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space $Ω_W$($D$) is a Stein domain that contains the universally defined Iwasawa domain $Ω_I$. This is one of the main ingredients in the proof that $Ω_W(D)=Ω_{AG}$ for all but a few Hermitian exceptions. In the Hermitian case, $Ω_W(D)$ is concretely described in terms of the associated bounded symmetric domain.

Families of (1, 2)‐symplectic metrics on full flag manifolds

We obtain new families of (1, 2)‐symplectic invariant metrics on the full complex flag manifolds F(n). For n ≥ 5, we characterize n − 3 different n‐dimensional families of (1, 2)‐symplectic invariant metrics on F(n). Each of these families corresponds to a different class of nonintegrable invariant almost complex structures on F(n).

On index number and topology of flag manifolds

The k-number of a complex flag manifold and the index number of a real flag manifold is known to be equal to the sum of the Z 2 -Betti numbers of the manifold. We give an interpretation and alternative proofs of these results in the framework of symplectic topology and Morse theory

The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces

We compute the center and nilpotency of the graded Lie algebra π ∗(Ω Baut 1(X))⊗ Q for a large class of formal spaces X. The latter calculation determines the rational homotopical nilpotency of the space of self-equivalences aut 1( X) for these X. Our results apply, in particular, when X is a complex or symplectic flag manifold.

Real groups transitive on complex flag manifolds

Let Z = G / Q Z = G/Q be a complex flag manifold. The compact real form G u G_u of G G is transitive on Z Z . If G 0 G_0 is a noncompact real form, such transitivity is rare but occasionally happens. Here we work out a complete list of Lie subgroups of G G transitive on Z Z and pick out the cases that are noncompact real forms of G G .

Linear cycle spaces in flag domains

Let Z = G/Q, a complex flag manifold, where G is a complex semisimple Lie group and Q is a parabolic subgroup. Fix a real form \(G_0 \subset G\) and consider the linear cycle spaces \(M_D\), spaces of maximal compact linear subvarieties of open orbits \(D = G_0(z) \subset Z\). In general \(M_D\) is a Stein manifold. Here the exact structure of \(M_D\) is worked out when \(G_0\) is a classical group that corresponds to a bounded symmetric domain B. In that case \(M_D\) is biholomorphic to B if a certain double fibration is holomorphic, is biholomorphic to \(B \times \overline{B}\) otherwise. There are also a number of structural results that do not require \(G_0\) to be classical.

$(1,2)$-symplectic structures on flag manifolds

By using moving frames and directred digraphs, we study invariant (1,2)-symplectic structures on complex flag manifolds. Let $F$ be a flag manifold with height $k-1$. We show that there is a $k$-dimensional family of invariant (1,2)-symplectic metrics of any parabolic structure on $F$. We also prove any invariant almost complex structure $J$ on $F$ with height 4 admits an invariant (1,2)-symplectic metric if and only if $J$ is parabolic or integrable.

Hermitian structures and equi-harmonic tori on non-symmetric complex flag manifolds

Hermitian structures and equi-harmonic tori on non-symmetric complex flag manifolds

Systems of Roots and Topology of Complex Flag Manifolds

In this paper we study the topology of a complex homogeneous space M = G/H of complex dimension n, with non vanishing Euler characteristic and G of type A, D, E by means of a topological invariant φ 2, which is related to the Poincare polynomial of M. We introduce the function Q =φ 2/n and we examine how it varies as one passes from a principal orbit of the adjoint representation of a compact Lie group G to a more singular one. Moreover, it is proved that if M is a principal orbit G/T then Q depends only on the Weyl group of G.

Cayley transforms and orbit structure in complex flag manifolds

LetZ=G/Q be a complex flag manifold andG 0 a real form ofG. Suppose thatG 0 is the analytic automorphism group of an irreducible bounded symmetric domain and that some openG 0-orbit onZ is a semisimple symmetric space. Then theG 0-orbit structure ofZ is described explicitly by the partial Cayley transforms of a certain hermitian symmetric sub-flagF⊃Z. This extends the results and simplifies the proof for the classical orbit structure description of [10] and [11], which applies whenF=Z.

SOME PROPERTIES OF THE FLAG MANIFOLDS


 
 
 The aim of the present paper is to prove that the real, complex and quaternionic flag manifolds are $\Sigma$-space. 
 
 

On the syzygies of flag manifolds

We show that on a complex flag manifold, a very ample line bundle which is a p p -th power has property N p N_p in the sense of Green and Lazarsfeld. This is a partial answer to a problem raised by Fulton.

Compression semigroups of open orbits in complex manifolds

of the open C-orbit. Such semigroups play a central role in the theory holomorphic extensions of uni- tary representations of the group G (eft [HO0], [FHO], [Ols], IN5], IN6], [NT], [St]). More concretely we are dealing with the following two classes of homogeneous spaces, namely with complex flag manifolds and with certain embeddings of complex coadjoint orbits into complex homogeneous spaces. The main results for complex flag manifolds are fairly easy to describe. Since everything decomposes nicely according to the decomposition of Gc into simple factors, we may assume that Gc is simple. In this case three mutually exclusive possibilities occur (cf. Proposition II.3, Theorem III.14): (1)

Polar actions on Hilbert space

An isometricH-action on a Riemannian manifoldX is calledpolar if there exists a closed submanifoldS ofX that meets everyH-orbit and always meets orbits orthogonally (S is called a section). LetG be a compact Lie group equipped with a biinvariant metric,H a closed subgroup ofG ×G, and letH act onG isometrically by (h1,h2) ·x = h1xh2−1 · LetP(G, H) denote the group ofH1-pathsg: [0, 1] →G such that (g(0),g (1)) ∈H, and letP(G, H) act on the Hilbert spaceV = H0([0, 1], g) isometrically byg * u = gug−1 −g′g−1. We prove that if the action ofH onG is polar with a flat section then the action ofP(G, H) onV is polar. Principal orbits of polar actions onV are isoparametric submanifolds ofV and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples ofP(G, H)-actions for suitable choice ofH andG.

Exhaustion Functions and Cohomology Vanishing Theorems for Open Orbits on Complex Flag Manifolds

Abstract. Let G 0 bearealsemisimpleLiegroup,let R beaparabolicsubgroupofthecomplexification G of G 0 , let D beanopen G 0 -orbitinthe complex flag manifold X = G/R , andlet Y be a maximal compactlinearsubvarietyof D . First,anexplicitparabolicsubgroup Q ⊂ R ⊂ G isconstructedsothattheopen G 0 -orbitson W = G/Q aremeasurableandonesuchorbit D = G 0 ( w ) ⊂ W mapsonto D withaffinefibre. Second,itisshownthat D is( s +1)-completeinthesenseofAndreottiandGrauert, s =dim C Y ;thuscohomologies H q ( D ; F )=0for q>s whenever F→D isacoherentanalyticsheaf. Thiswasknown[7]forthecaseofmeasurableopenorbits,andtheproofusesthatresulton D . Third,itisshownthatthespace M D ofcompactlinearsubvarietiesof D isaSteinmanifold. Forthat,astrictlyplurisubharmonicexhaustionfunctionisconstructedasintheargument[9]forthecaseofmeasurableopenorbits. 1. BackgroundandstatementofresultsLet G 0 beaconnectedreductiverealLiegroup,g 0 itsrealLiealgebra,andg =g 0 ⊗ R C thecomplexification.Asusual,Int(g)denotesthecomplexconnectedsemisimpleLiegroupofallinnerautomorphismsofg,consistingoftheAd(

The Stein Condition for Cycle Spaces of Open Orbits on Complex Flag Manifolds

The G-orbit structure of X is well understood (see [13]). There are only finitely many orbits, in particular there are open orbits. If x E X, let Px be the corresponding parabolic subalgebra of g; that is, if x = gP, then Px = Ad(g)p. Let ( denote the complex conjugation of gc over g. Then Px n px contains a Cartan subalgebra of g of the form [ = io OR C, where ho is a Cartan subalgebra of go. Let A = A(g, [) denote the root system. Fix

Some results on the real k-theory of certain homogeneous spaces.

Let Fr(n) be the incomplete complex flag manifold of length r in Cn. We make a start on the complete determination of the torsion part of the group KO-i(Fr(n)) giving results here when r = 2, 3.

Tournaments, flags, and harmonic maps

Twistor space constructions have shown that under quite general hypotheses, harmonic maps of Riemann surfaces into Riemannian manifolds can be represented by holomorphic curves with respect to almost complex structures which are often non-integrable, see for example, [-19, 17, 5]. For maps into complex Grassmannians, the almost complex structures are invariant ones on flag manifolds, and can be identified with "tournaments", i.e. digraphs in which any two nodes are joined by exactly one oriented edge. Elementary properties of tournaments are summarised in Sect. I, the basic references are Moon [14] and Reid and Beineke [18]. Emphasis is given to certain types of tournament which are interpreted in Sect. 2 in terms of the homogeneous structure of complex flag manifolds. Holomorphic curves in such flag manifolds are described in Sect. 3 by collections of subbundles (of a trivial bundle over a Riemann surface) whose second fundamental forms define a tournament. This approach relies on techniques of Burstall and Wood [6] and gives rise naturally to harmonic maps into Grassmannians. When the Riemann surface has genus zero, a result from algebraic geometry, namely the Birkhoff-Grothendieck decomposition theorem for holomorphic vector bundles, fits into the scheme and allows one to reconstruct a tournament from a harmonic map into a Grassmannian. In Sect. 4 we use such techniques to show that harmonic maps of tl2P 1 into a complex Grassmannian can be built up successively from "holomorphic data". This last result extends results of Burstall and Wood [6] and is closely related to recent work of Uhlenbeck [21], and Wolfson [23].