2023/04/24 by Indranil Biswas, Biswas, Indranil
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2304.12086
openalex publication_date 2023/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A moduli space \mathcal N of stable parabolic vector bundles, of rank r and parabolic degree zero, on a n-pointed curve has two naturally occurring holomorphic T^*\mathcal N--torsors over it. One of them is given by the moduli space of pairs of the form (E_*, D), where E_* ∈ \mathcal N and D is a connection on E_*. This T^*\mathcal N--torsor has a C^∞ section that sends any E_* ∈ \mathcal N to the unique connection on it with unitary monodromy. The other T^*\mathcal N--torsor is given by the sheaf of holomorphic connections on a theta line bundle over \mathcal N. This T^*\mathcal N--torsor also has a C^∞ section given by the Hermitian structure on the theta bundle constructed by Quillen. We prove that these two T^*\mathcal N--torsors are isomorphic by a canonical holomorphic map. This holomorphic isomorphism interchanges the above two C^∞ sections.