Moduli spaces of connections and Higgs bundles and Spectral curves

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K13427
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Kobe University
• Principal Investigator: Saito Masa-Hiko 神戸大学, 数理・データサイエンスセンター, 教授 (80183044)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 放物接続 / ヒッグス束 / 見かけの特異点 / モジュライ空間 / ラグランジアン束 / 幾何学的ラングランズ対応 / スペクトル曲線 / BNR理論

Outline of Final Research Achievements

We constructed moduli spaces of stable parabolic connections with regular singular points of fixed spectral type as nonsingular symplectic algebraic manifolds with M. Inaba and we also obtained the dimension formula of moduli spaces by multiplicities of eigenvalues of residue matrix at each singular point. We also showed the Painleve property of the corresponding iso-monodromic deformation equation.
With S. Szabo, we developed apparent singularity theory for singular connections and Higgs bundle and developed a method to describe the detailed structure of the moduli spaces. In collaboration with A. Komyo, we give a detailed description of the moduli space of connections of rank 2 and with 5 regular singular points on the projection line. We are studying to establish the geometric Langlands correspondence as Fourier-Mukai transformation using detailed description of this moduli space.

Free Research Field

数学