math.DG — Differential Geometry

Collision Geometry and Holonomy of Finite Blaschke Model Spaces

Contributed by Zhenyu Sun

We investigate the rank-$n$ Hermitian vector bundle of finite Blaschke model spaces over the coefficient domain of monic polynomials whose zeros lie entirely in the open unit disk. By introducing a global anti-holomorphic rational frame, we show that both the bundle and its orthogonal-projection connection extend real-analytically across all internal root-collision strata. At the maximally coincident divisor, we compute the full curvature tensor explicitly; the successive differences of its matrix coefficients yield the standard matrix units, which implies that both the restricted and full holonomy groups coincide with the full unitary group $U(n)$. Furthermore, the induced metric on the determinant line bundle is shown to agree with the diagonal restriction of the Szegő kernel of the symmetrized polydisc, and its power-sum Kähler potential endows the coefficient domain with a complete Kähler metric. By contrast, once the Clark phase circle is adjoined to the base, the Clark spectral lines together with the canonical model-space conjugation single out a unique compatible unitary connection. This real-spectral connection is flat, extends real-analytically across internal collisions, and exhibits a finite signed cyclic holonomy group. We also derive a weighted residue formula for the complex-bilinear pairing induced by the canonical conjugation, and analyze explicit low-dimensional subfamilies that illustrate the sharp geometric contrast between the curved projection transport of the moving model space and the flat parallel transport of its real spectral lines.