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On certain Tannakian categories of integrable connections over Kähler manifolds

Part of: Modules, bimodules and ideals Global differential geometry Cycles and subschemes Abelian varieties and schemes

Published online by Cambridge University Press:  21 April 2021

Indranil Biswas ,
João Pedro dos Santos ,
Sorin Dumitrescu  and
Sebastian Heller
Indranil Biswas*
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai400005, India
João Pedro dos Santos
Affiliation:
Sorbonne Université, Institut de Mathématiques de Jussieu–Paris Rive Gauche, 4 place Jussieu, Case 247, 75252Paris Cedex 5, France e-mail: [email protected]
Sorin Dumitrescu
Affiliation:
Université Côte d’Azur, CNRS, LJAD, France e-mail: [email protected]
Sebastian Heller
Affiliation:
Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, 30167Hannover, Germany e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

Given a compact Kähler manifold X, it is shown that pairs of the form $(E,\, D)$ , where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on E, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.

Keywords

Integrable holomorphic connectionHiggs bundleneutral Tannakian categorycomplex torusTorelli theorem

MSC classification

Primary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein-Yang-Mills)
Secondary: 14C34: Torelli problem 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality 14K20: Analytic theory; abelian integrals and differentials
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 1034 - 1061
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first- and third-named authors were partially supported by the French government through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR2152IDEX201. The first-named author is partially supported by a J. C. Bose Fellowship, and school of mathematics, TIFR, is supported by 12-R&D-TFR-5.01-0500. The fourth-named author is supported by the DFG grant HE 6829/3-1 of the DFG priority program SPP 2026 Geometry at Infinity.

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