2009/11/19 by Alexander Zuevsky, Zuevsky, A.
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.0911.3908
openalex publication_date 2009/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the holomorphic unramified mapping of two arbitrary finite bordered Riemann surfaces. Extending the map to the doubles X1 and X2 of Riemann surfaces we define the vector bundle on the second double as a direct image of the vector bundle on first double. %% We choose line bundles of half-order differentials Δ1 and Δ2 so that the vector bundle VX2χ_\ad ⊗ Δ2 on X2 would be the direct image of the vector bundle VX1χ_\ao ⊗ Δ1. We then show that the Hardy spaces H2, J1(p) (S1,Vχ_\ao ⊗ Δ1) and H2,J2(p) (S2,Vχ_\ad ⊗ Δ2) are isometrically isomorphic. Proving that we construct an explicit isometric isomorphism and a matrix representation χ_\ad of the fundamental group \piod(X2, p0) given a matrix representation χ_\ao of the fundamental group \piod(X1, p'0). %% On the basis of the results of \citevin and Theorem \reftheorem1 proven in the present work we then conjecture that there exists a covariant functor from the category \cal RH of finite bordered surfaces with vector bundle and signature matrices to the category of Kre\uın spaces and isomorphisms which are ramified covering of Riemann surfaces.