Existence of nearly holomorphic sections on compact Hermitian symmetric spaces

TL;DR

This paper proves the existence of non-trivial nearly holomorphic sections on compact Hermitian symmetric spaces, extending harmonic analysis tools and the understanding of $L^2$-decomposition of sections.

Contribution

It establishes the global existence of nearly holomorphic sections on compact Hermitian symmetric spaces, building upon previous local and $L^2$-decomposition results.

Findings

Existence of non-trivial nearly holomorphic sections on $X=U/K$.

Extension of local nearly holomorphic sections to global sections.

Enhanced understanding of $L^2$-decomposition in harmonic analysis.

Abstract

Let $X=U/K$ be a compact Hermitian symmetric space, and let $\sE$ be a $U$-homogeneous Hermitian vector bundle on $X$. In a previous paper, we showed that the space of nearly holomorphic sections is well-adapted for harmonic analysis in $L^2(X,\sE)$ provided that non-trivial nearly holomorphic sections do exist. Here we investigate the problem of extending local nearly holomorphic sections to global ones and prove the existence of non-trivial nearly holomorphic sections. This extends the results on the $U$-type decomposition of $L^2(X,\sE)$ from our previous paper.