Compact Hermitian Symmetric Spaces Research Articles

Birational Transformations on Irreducible Compact Hermitian Symmetric Spaces

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

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.

Weighted Bergman kernels for nearly holomorphic functions on bounded symmetric domains

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

On gap rigidity problems for compact Hermitian symmetric spaces

We prove a gap rigidity theorem for diagonal curves in maximal polyspheres of irreducible compact Hermitian symmetric spaces of tube type, which is a dual analog to a theorem obtained by Mok. Motivated by the proof we show that the Chow space of certain totally geodesic submanifolds is affine algebraic, which gives a weaker version of gap rigidity for a class of higher dimensional submanifolds.

Non-split supermanifolds associated with the cotangent bundle

Here, I study the problem of classification of non-split supermanifolds having as retract the split supermanifold $(M,\Omega)$, where $\Omega$ is the sheaf of holomorphic forms on a given complex manifold $M$ of dimension $> 1$. I propose a general construction associating with any $d$-closed $(1,1)$-form $\omega$ on $M$ a supermanifold with retract $(M,\Omega)$ which is non-split whenever the Dolbeault class of $\omega$ is non-zero. In particular, this gives a non-empty family of non-split supermanifolds for any flag manifold $M\ne \mathbb{CP}^1$. In the case where $M$ is an irreducible compact Hermitian symmetric space, I get a complete classification of non-split supermanifolds with retract $(M,\Omega)$. For each of these supermanifolds, the 0- and 1-cohomology with values in the tangent sheaf are calculated. As an example, I study the $\Pi$-symmetric super-Grassmannians introduced by Yu. Manin.

Nijenhuis tensor and invariant polynomials

We discuss the diagonalization problem of the Nijenhuis tensor in a class of Poisson-Nijenhuis structures defined on compact hermitian symmetric spaces. We study its action on the ring of invariant polynomials of a Thimm chain of subalgebras. The existence of ϕ-minimal representations defines a suitable basis of invariant polynomials that completely solves the diagonalization problem. We prove that such representations exist in the classical cases AIII, BDI, DIII and CI, and do not exist in the exceptional cases EIII and EVII. We discuss a second general construction that in these two cases computes partially the spectrum and hints at a different behavior with respect to the classical cases.

Symplectic geometry of Cartan–Hartogs domains

This paper studies the geometry of Cartan–Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan–Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan–Hartogs domains and their dual. Further, we compute their symplectic capacity and show that a Cartan–Hartogs domain admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

Complexified Hermitian Symmetric Spaces, Hyperkähler Structures, and Real Group Actions

There is a known hyperkähler structure on any complexified Hermitian symmetric space G/K, whose construction relies on identifying G/K with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space Gu/K0. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on G/K; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on \(T^{*}\left (G_{u}/K_{0}\right )\). We highlight the intermediate Kähler structures, which share a holomorphic action of G related to the one on G/K, but moment geometry related to that of \(T^{*}\left (G_{u}/K_{0}\right )\). As an application, for the real form G0 ⊂ G corresponding to G0/K0, the Hermitian symmetric space of noncompact type, we give a strategy for study of the action on G/K using the moment-critical subsets for the intermediate structures. We give explicit computations for SL(2).

Principal series of Hermitian Lie groups induced from Heisenberg parabolic subgroups

Let G be an irreducible Hermitian Lie group and D=G/K its bounded symmetric domain in Cd of rank r. Each γ of the Harish-Chandra strongly orthogonal roots {γ1,⋯,γr} defines a Heisenberg parabolic subgroup P=MAN of G. We study the principal series representations IndPG(1⊗eν⊗1) of G induced from P. These representations can be realized as the L2-space on the minimal K-orbit S=Ke=K/L of a root vector e of γ in Cd, and S is a circle bundle over a compact Hermitian symmetric space K/L0 of K of rank one or two. We find the complementary series, reduction points, and unitary sub-quotients in this family of representations.

Solitons for the Schrödinger flows on Hermitian symmetric spaces

We use loop group factorizations to construct Darboux transformations, Permutability formulas, Scaling Transformations, and explicit soliton solutions of the [Formula: see text]-d Schrödinger flows on compact Hermitian symmetric spaces.

Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow

Using a stability criterion due to Kröncke, we show, providing $${n\ne 2k}$$ , the Kähler–Einstein metric on the Grassmannian $$Gr_{k}(\mathbb {C}^{n})$$ of complex k-planes in an n-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on $$\mathbb {CP}^{n}$$ for $$n>1$$ . The key to the proof is using the description of Grassmannians as certain coadjoint orbits of SU(n). We are also able to prove that Kröncke’s method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.

Complete integrability from Poisson–Nijenhuis structures on compact hermitian symmetric spaces

We study a class of Poisson-Nijenhuis systems defined on compact hermitian symmetric spaces, where the Nijenhuis tensor is defined as the composition of Kirillov-Konstant-Souriau symplectic form with the so called Bruhat-Poisson structure. We determine its spectrum. In the case of Grassmannians the eigenvalues are the Gelfand-Tsetlin variables. We introduce the abelian algebra of collective hamiltonians defined by a chain of nested subalgebras and prove complete integrability. By construction, these models are integrable with respect to both Poisson structures. The eigenvalues of the Nijenhuis tensor are a choice of action variables. Our proof relies on an explicit formula for the contravariant connection defined on vector bundles that are Poisson with respect to the Bruhat-Poisson structure.

Cotangent bundles over all the Hermitian symmetric spaces

We construct the N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. In order to construct them we use the projective superspace formalism which is an N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the N =1 superfields, once the Kähler potentials of the base manifolds are obtained. Starting with N = 1 supersymmetric Kähler nonlinear sigma models on the Hermitian symmetric spaces, we extend them into the N = 2 supersymmetric models by using the projective superspace formalism and derive the general formula for the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. We apply to the formula for the non-compact Hermitian symmetric space E7/E6 × U(1)1.

CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS

Abstract. A Schubert variety in a rational homogeneous variety G/P isdefined by the closure of an orbit of a Borel subgroup Bof G. In general,Schubert varieties are singular, and it is an old problem to determinewhich Schubert varieties are smooth. In this paper, we classify all smoothSchubert varieties in the symplectic Grassmannians. 1. IntroductionA rational homogeneous manifold S= G/Pis a projective manifold, wherea connected complex semisimple group Gacts transitively. Under the actionof a Borel subgroup Bof G, Shas finitely many orbits. The closure of a B-orbit in Sis called a Schubert variety of S. In general, Schubert varieties aresingular, and it is an old problem to determine which Schubert varieties aresmooth. Lakshmibai-Weyman and Brion-Polo have studied the singular lociof Schubert varieties of S, when Sis a compact Hermitian symmetric space([9] and [2]). In particular, they showed that in this case any smooth Schubertvariety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S. For example, a Schubert variety of theGrassmannian Gr(k,V) of k-subspaces in a vector space V is smooth if andonly if it is a linearly embedded sub-Grassmannian.More generally, when Sis associated to a long simple root, we have:Theorem 1.1 (Proposition 3.7 of Hong-Mok [4]). Let S= G/P be a rationalhomogeneous manifold associated to a long simple root. Then any smooth Schu-bert variety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S.On the other hand, when Sis associated to a short simple root, there is asmooth Schubert variety that is not homogeneous. Let V be a vector spacewith a symplectic form ω, i.e., a nondegenerate skew-symmetric bilinear form.

Supersymmetry and cotangent bundle over non-compact exceptional Hermitian symmetric space

Abstract We construct $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the non-compact exceptional Hermitian symmetric spaces $$ \mathrm{\mathcal{M}}={E}_{6\left(-14\right)}/\mathrm{SO}(10)\times \mathrm{U}(1) $$ ℳ = E 6 − 14 / SO 10 × U 1 and E 7(−25) /E 6 × U(1). In order to construct them we use the projective superspace formalism which is an $$ \mathcal{N}=2 $$ N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the $$ \mathcal{N}=1 $$ N = 1 superfields, once the Kähler potentials of the base manifolds are obtained. We derive the $$ \mathcal{N}=1 $$ N = 1 supersymmetric nonlinear sigma models on the Kähler manifolds $$ \mathrm{\mathcal{M}} $$ ℳ . Then we extend them into the $$ \mathcal{N}=2 $$ N = 2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism. The resultant models are the $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the Hermitian symmetric spaces $$ \mathrm{\mathcal{M}} $$ ℳ . In this work we complete constructing the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces.

ON DIFFERENTIAL INVARIANTS OF HYPERPLANE SYSTEMS ON NONDEGENERATE EQUIVARIANT EMBEDDINGS OF HOMOGENEOUS SPACES

Given a complex submanifold M of the projective space P(T), the hyperplane system R on M characterizes the projective embedding of M into P(T) in the following sense: for any two nondegenerate com- plex submanifolds MP(T) and M ' � P(T ' ), there is a projective linear transformation that sends an open subset of M onto an open subset of M ' if and only if (M,R) is locally equivalent to (M ' ,R ' ). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of lin- ear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane sys- tems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.

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.

Multiplicative integrable models from Poisson–Nijenhuis structures

We discuss the role of Poisson-Nijenhuis geometry in the definition of multiplicative integrable models on symplectic groupoids. These are integrable models that are compatible with the groupoid structure in such a way that the set of contour levels of the hamiltonians in involution inherits a topological groupoid structure. We show that every maximal rank PN structure defines such a model. We consider the examples defined on compact hermitian symmetric spaces and studied in [arXiv:1503.07339].

On the Dolbeault-Dirac operator of quantized symmetric spaces

The Dolbeault complex of a quantized compact Hermitian symmetric space is expressed in terms of the Koszul complex of a braided symmetric algebra of Berenstein and Zwicknagl. This defines a spectral triple quantizing the Dolbeault‐Dirac operator associated to the canonical spin c structure.