Topology of moduli of parabolic connections with fixed determinant

2023/11/22 by Nilkantha Das, Das, Nilkantha, Sumit Roy +1
Mathematics · #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2311.13477

Abstract

Let X be a compact Riemann surface of genus g ≥ 2 and D⊂ X be a fixed finite subset. Let ξ be a line bundle of degree d over X. Let M(α, r, ξ) (respectively, M_conn(α, r, ξ)) denote the moduli space of stable parabolic bundles (respectively, parabolic connections) of rank r (≥ 2), determinant ξ and full flag generic rational parabolic weight type α. We show that π_k(M_conn(α, r, ξ)) ≅ π_k(M(α, r, ξ)) for k ≤2(r-1)(g-1)-1. As a consequence, we deduce that the moduli space M_conn(α, r, ξ) is simply connected. We also show that the Hodge structures on the torsion-free parts of both the cohomologies H^k(M_conn(α, r, ξ),ℤ) and H^k(M(α, r, ξ),ℤ) are isomorphic for all k≤ 2(r-1)(g-1)+1.

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