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Collision Geometry and Holonomy of Finite Blaschke Model Spaces

A mix of human-written and AI-generated textHuman understanding: all partsmath.DG — Differential Geometrymath.CV — Complex Variablesmath.FA — Functional Analysismath.SP — Spectral Theory

Contributed by Zhenyu Sun ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We investigate the rank-nn 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)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.

Provenance statement

The manuscript was manually polished by the author. OpenAI Codex assisted with a subsequent mathematical audit, localized corrections, and submission preparation. AI-assisted analytic reviews were supplemented by finite symbolic and numerical consistency checks using Python, SymPy, and NumPy. These computations check explicit formulas and do not replace the proofs in the manuscript.

References

  1. Ambrose, Warren; Singer, Isadore M. A theorem on holonomy. Transactions of the American Mathematical Society, vol. 75, no. 3, pp. 428–443. 1953DOI
  2. Clark, Douglas N. One dimensional perturbations of restricted shifts. Journal d'Analyse Mathématique, vol. 25, pp. 169–191. 1972DOI
  3. Cowen, Michael J.; Douglas, Ronald G. Complex geometry and operator theory. Acta Mathematica, vol. 141, pp. 187–261. 1978DOI
  4. Garcia, Stephan Ramon; Putinar, Mihai. Complex symmetric operators and applications. Transactions of the American Mathematical Society, vol. 358, no. 3, pp. 1285–1315. 2006DOI
  5. Garcia, Stephan Ramon; Mashreghi, Javad; Ross, William T. Introduction to Model Spaces and Their Operators. Cambridge University Press, no. 148. 2016DOI
  6. Kato, Tosio. On the adiabatic theorem of quantum mechanics. Journal of the Physical Society of Japan, vol. 5, no. 6, pp. 435–439. 1950DOI
  7. Liaw, Constanze; Treil, Sergei. Clark model in the general situation. Journal d'Analyse Mathématique, vol. 130, no. 1, pp. 287–328. 2016DOI
  8. Misra, Gadadhar; Shyam Roy, Subrata; Zhang, Genkai. Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc. Proceedings of the American Mathematical Society, vol. 141, no. 7, pp. 2361–2370. 2013DOI
  9. O'Loughlin, Ryan. Symmetric matrix representations of truncated Toeplitz operators on finite dimensional spaces. Linear Algebra and its Applications, vol. 666, pp. 11–28. 2023DOI
  10. Sarason, Donald. Algebraic properties of truncated Toeplitz operators. Operators and Matrices, vol. 1, no. 4, pp. 491–526. 2007DOI
Show all 11 referencesShow fewer references
  1. Tanaka, Atushi; Miyamoto, Manabu. Quasienergy anholonomy and its application to adiabatic quantum state manipulation. Physical Review Letters, vol. 98, pp. 160407. 2007DOI

Version history

  1. v1Submitted by Zhenyu SunInitial depositCurrentOct 29, 2026