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 $D_{n,m},$ $U(n)\to U(n,m)/U(m)\stackrel{\pi }{\to }{D}_{n,m}$. Given a nontrivial $X\in M_{m\times n}(\mathbb{C}),$ consider a two dimensional subspace ${\mathfrak{m}}^{\prime }\subset \mathfrak{m}\subset \mathfrak{u}(n,m),$ 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 ${\mathfrak{m}}^{\prime }.$ Let $c$ be a smooth, simple, closed, orientation-preserving curve on $S$ parametrized by $0\le t\le 1$, and $\hat{c}$ its horizontal lift on the bundle $U(n) \ra U(n,m)/U(m) \stackrelπ\ra D_{n,m} $. Then the holonomy displacement is given by the right action of $e^{\Psi }$ for some $ Ψ\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 $S.$