[Paper Review] Holomorphic maps between closed SU(l,m)-orbits in Grassmannian

Sung‐Yeon Kim|arXiv (Cornell University)|Sep 6, 2018

Holomorphic and Operator Theory20 references3 citations

TL;DR

This paper studies holomorphic maps between closed $SU(\ell,m)$-orbits in Grassmannians, showing that under a signature difference condition on the Levi forms, such maps factor into a standard embedding and a holomorphic map into a sub-Grassmannian. The key result is that under additional dimension constraints, the map extends as a totally geodesic embedding of the ambient Grassmannian.

ABSTRACT

Orbits of $SU(\ell, m)$ in a Grassmannian manifold have homogeneous CR structures. In this paper, we study germs of smooth CR mappings sending a closed orbit of $SU(\ell,m)$ into a closed orbit of $SU(\ell',m')$ in Grassmannian manifolds. We show that if the signature difference of the Levi forms of two orbits is not too large, then the mapping can be factored into a simple form and one of the factors extends to a totally geodesic embedding of the ambient Grassmannian into another Grassmannian with respect to the standard metric. As an application, we give a sufficient condition for a smooth CR mapping sending a closed orbit of $SU(\ell,m)$ into a closed orbit of $SU(\ell',m')$ in Grassmannian manifolds to extend as a totally geodesic embedding of the Grassmannian into another Grassmannian.

Motivation & Objective

• To generalize rigidity results for holomorphic maps between minimal $SU(\ell,m)$-orbits in Grassmannians when maximal complex submanifolds differ between source and target.
• To investigate the propagation of CR rigidity along chains of maximal complex submanifolds when the signature difference of Levi forms is small.
• To establish conditions under which a smooth CR embedding between such orbits extends as a totally geodesic embedding of the ambient Grassmannian.
• To extend the lifting technique of Ng [Ng12] beyond the case of identical maximal complex submanifolds by introducing a factorization into standard and sub-Grassmannian components.

Proposed method

• Use the method of moving frames to analyze the CR structure of $S_{q,p}^\ell$, the minimal $SU(\ell,m)$-orbit in $Gr(q,p)$.
• Define universal spaces of $n$-confined subgrassmannians over $S_{q,p}^\ell$ to parameterize maximal complex submanifolds and their relations.
• Establish that if $q > 1$ and $|\ell' - q'| < 2(\ell - q)$, then the germ of a smooth transversal CR embedding factors into a standard embedding into $Gr(q',L)$ and a holomorphic map into $Gr(q',N)$ with $\dim N \leq \ell' - \ell + q$.
• Apply the result of Mok [M08] on characteristic bundles and use lifting techniques to show that the map preserves bundle structures.
• Use orthogonal decomposition with respect to the Hermitian form $\langle \cdot, \cdot \rangle_{\ell',m'}$ to analyze tangent and normal spaces at points.
• Prove that under the dimension condition $q' \times (\ell' - \ell + q - q') < q \times (\ell - q)$, the sub-Grassmannian component becomes constant, implying global extension.

Experimental results

Research questions

• RQ1 Under what conditions does a germ of a smooth CR embedding between $SU(\ell,m)$-orbits in Grassmannians extend to a totally geodesic embedding of the ambient Grassmannian?
• RQ2 How does CR rigidity propagate along chains of maximal complex submanifolds when the source and target orbits have different maximal complex submanifolds?
• RQ3 What is the structure of holomorphic maps between $SU(\ell,m)$-orbits when the signature difference of their Levi forms is bounded?
• RQ4 Can the lifting technique used in Ng [Ng12] be generalized to cases where the maximal complex submanifolds of source and target differ?
• RQ5 When does a holomorphic map into a $SU(\ell',m')$-orbit factor into a standard embedding and a constant map into a sub-Grassmannian?

Key findings

• If $q > 1$ and $|\ell' - q'| < 2(\ell - q)$, then any germ of a smooth transversal CR embedding $f: S_{q,p}^\ell \to S_{q',p'}^{\ell'}$ factors as $z \mapsto (f_1(z), f_2(z))$, where $f_1$ is a standard embedding into $Gr(q',L)$ and $f_2$ maps into $Gr(q',N)$ with $\dim N \leq \ell' - \ell + q$.
• Under the additional condition $q' \times (\ell' - \ell + q - q') < q \times (\ell - q)$, the map $f_2$ becomes constant, so the entire map $f$ extends as a standard embedding of $Gr(q,p)$ into $Gr(q',p')$.
• The factorization result generalizes Ng's [Ng12] result, which required $q = q'$ and $\ell = \ell'$, to the case of different maximal complex submanifolds.
• The proof relies on the method of moving frames and the orthogonal decomposition of the tangent space with respect to the $SU(\ell',m')$-invariant Hermitian form.
• The image of the embedding lies in a totally geodesic sub-Grassmannian, and the structure of the normal space ensures that the map preserves the characteristic bundle structure.
• The result confirms that under the stated signature and dimension constraints, holomorphic maps between such orbits are rigid and extend globally as standard embeddings.