The topological matter of holonomy displacement on the principal [formula omitted]-bundle over [formula omitted] related to complex surfaces.

Consider U ( n ) → U ( n , m ) ∕ U ( m ) → π D n , m , where D n , m = U ( n , m ) ∕ U ( n ) × U ( m ) . Given a nontrivial X ∈ M m × n ( C ) and g ∈ U ( n , m ) , consider a complete oriented surface S = S ( X , g ) with a complex structure in D n , 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 c ˆ : [ 0 , 1 ] → U ( n , m ) ∕ U ( m ) its horizontal lift. Then the holonomy displacement is given by the right action of e Ψ for some Ψ ∈ Span R { i ( X ∗ X ) k } k = 1 p ⊂ u ( n ) , p = the number of distinct positive eigenvalues of X ∗ X , such that c ˆ ( 1 ) = c ˆ ( 0 ) ⋅ e Ψ and Tr ( Ψ ) = 2 i Area ( 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. [ABSTRACT FROM AUTHOR]