Irreducible Hermitian Symmetric Space Research Articles

Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Pseudoeffective thresholds and cohomology of twisted symmetric tensor fields on irreducible Hermitian symmetric spaces

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over R, C or H. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

Homotopy commutativity in Hermitian symmetric spaces

AbstractGanea proved that the loop space of $\mathbb{C} P^n$ is homotopy commutative if and only if $n=3$ . We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C} P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency class of the loop spaces of generalized flag manifolds $G/T$ for a maximal torus T of a compact, connected Lie group G.

Stability of the tangent bundles of complete intersections and effective restriction

For n≥3 and r≥1, let M be an (n+r)-dimensional irreducible Hermitian symmetric space of compact type and let 𝒪 M (1) be the ample generator of pic(M). Let Y=H 1 ∩⋯∩H r be a smooth complete intersection of dimension n, where H i ∈|𝒪 M (d i )| with d i ≥2. We prove a vanishing theorem for twisted holomorphic forms on Y. As an application, we show that the tangent bundle T Y of Y is stable. Moreover, if X is a smooth hypersurface of degree d in Y such that the restriction pic(Y)→pic(X) is surjective, we establish some effective results for d to guarantee the stability of the restriction T Y | X . In particular, if Y is a general hypersurface in ℙ n+1 and X is a general smooth divisor in Y, we show that T Y | X is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.

Multicomponent Fokas–Lenells equations on Hermitian symmetric spaces

Multi-component integrable generalizations of the Fokas–Lenells equation, associated with each irreducible Hermitian symmetric space are formulated. Description of the underlying structures associated to the integrability, such as the Lax representation and the bi-Hamiltonian formulation of the equations is provided. Two reductions are considered as well, one of which leads to a nonlocal integrable model. Examples with Hermitian symmetric spaces of all classical series of types A.III, BD.I, C.I and D.III are presented in details, as well as possibilities for further reductions in a general form.

Homogeneous Higgs and co-Higgs bundles on Hermitian symmetric spaces

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

Volume-preserving mappings between Hermitian symmetric spaces of compact type

In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian symmetric space of compact type into its Cartesian products.

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X0, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X0, X) of the subdiagram type whenever X0 is nonlinear. It remains unsolved whether rigidity holds when (X0, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X0, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure $$\overline{\omega} : \mathscr{C}(S) \rightarrow S$$ on a uniruled projective manifold $$(X,\,{\cal K})$$ equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve l emanating from a general point x ∈ S, there exists an immersed neighborhood Nl of l which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ⊂ X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ⩾ 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X0 into X which induces an isomorphism on second homology groups. By studying ℂ*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X0 ⊂ X. Under the additional hypothesis that all holomorphic sections in Γ(X0, Tx∣x0) lift to global holomorphic vector fields on X, we prove that the admissible pair (X0, X) is rigid. As examples we check that (X0, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ≅ ℂ2n, n ⩾ 3, and when X0 ⊂ X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

Real hypersurfaces with isometric Reeb flow in Kähler manifolds

We investigate the structure of real hypersurfaces with isometric Reeb flow in Kähler manifolds. As an application we classify real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type.

Stability of Restrictions of the Cotangent Bundle of Irreducible Hermitian Symmetric Spaces of Compact Type

It is known that the cotangent bundle \Omega_Y of an irreducible Hermitian symmetric space Y of compact type is stable. We show that if X \subset Y is a subvariety whose structure sheaf has a short split resolution and such that the restriction map Pic (Y) \to Pic (X) is surjective, then, apart from a few exceptions, the restriction \Omega_{Y|X} is a stable bundle. We then address some cases where the Picard group increases by restriction.

Complex Riemannian foliations of open Kähler manifolds

Classification results for complex Riemannian foliations are obtained. For open subsets of irreducible Hermitian symmetric spaces of compact type, where one has explicit control over the curvature tensor, we completely classify such foliations by studying the infinitesimal model associated with the canonical connection. We also establish results for symmetric spaces of noncompact type and a general rigidity result for any irreducible Kähler manifold.

The symplectic area of a geodesic triangle in a Hermitian symmetric space of compact type

Let M be an irreducible Hermitian symmetric space of compact type, and let $$\omega $$ be its Kahler form. For a triplet $$(p_1,p_2,p_3)$$ of points in M we study conditions under which a geodesic triangle $$\mathcal T(p_1,p_2,p_3)$$ with vertices $$p_1,p_2,p_3$$ can be unambiguously defined. We consider the integral $$A(p_1,p_2,p_3)=\int _\Sigma \omega $$ , where $$\Sigma $$ is a surface filling the triangle $$\mathcal T(p_1,p_2,p_3)$$ and discuss the dependence of $$A(p_1,p_2,p_3)$$ on the surface $$\Sigma $$ . Under mild conditions on the three points, we prove an explicit formula for $$A(p_1,p_2,p_3)$$ analogous to the known formula for the symplectic area of a geodesic triangle in a non-compact Hermitian symmetric space.

The adapted hyper-Kähler structure on the crown domain

Let $$\Xi $$ be the crown domain associated with a non-compact irreducible Hermitian symmetric space G / K. We give an explicit description of the unique G-invariant adapted hyper-Kahler structure on $$\Xi $$ , i.e., compatible with the adapted complex structure $$J_\mathrm{ad}$$ and with the G-invariant Kahler structure of G / K. We also compute invariant potentials of the involved Kahler metrics and the associated moment maps.

Hermitian symmetric space, flat bundle and holomorphicity criterion

Let K\\G be an irreducible Hermitian symmetric space of noncompact type and Γ⊂G a closed torsionfree discrete subgroup. Let X be a compact Kähler manifold and ρ:π1(X,x0)⟶Γ a homomorphism such that the adjoint action of ρ(π1(X,x0)) on the Lie algebra Lie(G) is completely reducible. A theorem of Corlette associates to ρ a harmonic map H:X⟶K\\G/Γ. We give a criterion for this harmonic map H to be holomorphic. We also give a criterion for it to be anti-holomorphic.

Invariant envelopes of holomorphy in the complexification of a Hermitian symmetric space

In this paper we investigate invariant domains in $\, \Xi^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,\Xi^+$, recently introduced by Kr\otz and Opdam, contains the crown domain $\,\Xi\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,\Xi$. We prove that the envelope of holomorphy of an invariant domain in $\,\Xi^+$, which is contained neither in $\,\Xi\,$ nor in $\,S^+$, is univalent and coincides with $\,\Xi^+$. This fact, together with known results concerning $\,\Xi\,$ and $\,S^+$, proves the univalence of the envelope of holomorphy of an arbitrary invariant domain in $\,\Xi^+\,$ and completes the classification of invariant Stein domains therein.

The index of symmetry of a flag manifold

We study the index of symmetry of a compact generalized flag manifold M = G/H endowed with an invariant Kähler structure. When the group G is simple we show that the leaves of symmetry are irreducible Hermitian symmetric spaces which depend only on the invariant complex structure and we estimate their dimension

MOMENT MAPS AND ISOPARAMETRIC HYPERSURFACES IN SPHERES — HERMITIAN CASES

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. We prove that the Cartan-M\"unzner polynomials of these hypersurfaces can be written as squared-norms of the moment maps for some Hamiltonian actions. The proof is based on the structure theory of symmetric spaces.

Deforming symplectomorphism of certain irreducible Hermitian symmetric spaces of compact type by mean curvature flow

In this paper, we generalize Medos-Wang's arguments and results on the mean curvature flow deformations of symplectomorphisms of $\CP^n$ in \cite{MeWa} to complex Grassmann manifold $G(n, n+m;\C)$ and compact totally geodesic K\ahler-Einstein submanifolds of $G(n, 2n;\C)$ such as irreducible Hermitian symmetric spaces $SO(2n)/U(n)$ and $Sp(n)/U(n)$ (in the terminology of \cite[p. 518]{He}). We also give an abstract result and discuss the case of complex tori.

The intersection of two real forms in Hermitian symmetric spaces of compact type II

We show that the intersections of two real forms, certain totally geodesic Lagrangian submanifolds, in Hermitian symmetric spaces of compact type are antipodal sets. The intersection number of two real forms is invariant under the replacement of the two real forms by congruent ones. If two real forms are congruent, then their intersection is a great antipodal set of them. It implies that any real form in Hermitian symmetric spaces of compact type is a globally tight Lagrangian submanifold. Moreover we describe the intersection of two real forms in the irreducible Hermitian symmetric spaces of compact type.