2021/01/31 by Indranil Biswas, Biswas, Indranil, Jacques Hurtubise +1 · 1 citation
Mathematics · #14D21 #14H60 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2102.00624
openalex publication_date 2021/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the symplectic geometric and differential geometric aspects of\nthe moduli space of connections on a compact Riemann surface X. Fix a theta\ncharacteristic K1/2X on X; it defines a theta divisor on the moduli\nspace mathcal M of stable vector bundles on X of rank r degree zero.\nGiven a vector bundle E \∈ mathcal M lying outside the theta divisor, we\nconstruct a natural holomorphic connection on E that depends holomorphically\non E. Using this holomorphic connection, we construct a canonical holomorphic\nisomorphism between the following two: beginenumerate item the moduli space\n mathcal C of pairs (E, D), where E\∈ mathcal M and D is a\nholomorphic connection on E, and\n item the space rm Conn(\Θ) given by the sheaf of holomorphic\nconnections on the line bundle on mathcal M associated to the theta divisor.\n endenumerate The above isomorphism between mathcal C and rm\nConn(\Θ) is symplectic structure preserving, and it moves holomorphically\nas X runs over a holomorphic family of Riemann surfaces.\n