The topological matter of holonomy displacement on the principal U(n)-bundle over Dn,m, related to complex surfaces

Consider U(n)→U(n,m)∕U(m)→πDn,m, where Dn,m=U(n,m)∕U(n)×U(m). Given a nontrivial X∈Mm×n(C) and g∈U(n,m), consider a complete oriented surface S=S(X,g) with a complex structure in Dn,m and a “new” area form ω(X,g) on the surface S. Let c:[0,1]→S be a smooth, simple, closed, orientation-preserving curve and cˆ:[0,1]→U(n,m)∕U(m) its horizontal lift. Then the holonomy displacement is given by the right action of eΨ for some Ψ∈SpanR{i(X∗X)k}k=1p⊂u(n),p=the number of distinct positiveeigenvalues ofX∗X, such that cˆ(1)=cˆ(0)⋅eΨandTr(Ψ)=2iArea(c),where Area(c) is the “area,” produced by ω(X,g), of the region on the surface S, surrounded by c. And Ψ can be represented as the solution of a system of first order ordinary linear differential equations.