Injectivity of the Double Fibration Transform for Cycle Spaces of Flag Domains

TL;DR

This paper proves the injectivity of the double fibration transform for cycle spaces of flag domains using Schubert intersection theory, under conditions of sufficient negativity of the vector bundle.

Contribution

It establishes the injectivity of the double fibration transform for certain holomorphic vector bundles on cycle spaces of flag domains, extending previous results with new geometric techniques.

Findings

The transform is injective when the vector bundle is sufficiently negative.

Schubert intersection theory is effective in analyzing the transform's properties.

The results apply to complex flag manifolds with real group actions.

Abstract

The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb E \to D$. The topic here is the double fibration transform ${\cal P}: H^q(D;{\cal O}(\mathbb E)) \to H^0({\cal M}_D;{\cal O}(\mathbb E'))$ where $q$ is given by the geometry of $D$, ${\cal M}_D$ is the cycle space of $D$, and $\mathbb E' \to {\cal M}_D$ is a certain naturally derived holomorphic vector bundle. Schubert intersection theory is used to show that ${\cal P}$ is injective whenever $\mathbb E$ is sufficiently negative.