Topological property of the holonomy displacement on the principal $U(n)$-bundle over $D_{n,m},$ related to complex surfaces

Abstract

Consider $D_{n,m} = U(n,m)/\left(U(n) \times U(m)\right)$, the dual of the the Grassmannian manifold and the principal $U(n)$ bundle over $D_{n,m},$ $U(n)\rightarrow U(n,m)/U(m) \stackrel{\pi} \rightarrow D_{n,m}$. Given a nontrivial $X \in M_{m \times n}(\mathbb{C}),$ consider a two dimensional subspace $\mathfrak{m}' \subset \mathfrak{m} \subset \mathfrak{u}(n,m), $ induced by $X, iX \in M_{m \times n}(\mathbb{C}),$ and a complete oriented surface $S,$ related to $(X,g) \in M_{m \times n}(\mathbb{C}) \times U(n,m), $ in the base space $D_{n,m}$ with a complex structure from $\mathfrak{m}'.$ Let $c$ be a smooth, simple, closed, orientation-preserving curve on $S$ parametrized by $0\leq t\leq 1$, and $\hat{c}$ its horizontal lift on the bundle $U(n) \ra U(n,m)/U(m) \stackrel{\pi}\ra D_{n,m} $. Then the holonomy displacement is given by the right action of $e^\Psi$ for some $ \Psi \in \text{Span}_{\bbr}\{i(X^*X)^k\}_{k=1}^{q} \subset \mathfrak{u}(n), \: q=\text{rk}X, $ such that $$ \hat{c}(1) = \hat{c}(0) \cdot e^\Psi \text{24pt and 12pt } \text{Tr}(\Psi)= 2i \, \text{Area}(c), $$ where $\text{Area}(c)$ is the area of the region on the surface $S$ surrounded by $c,$ obtained from a special 2-form $\omega_{(X,g)}$ on $S,$ called an area form $\omega_{(X,g)}$ related to $(X,g)$ on $S.$