Criterion for existence of a logarithmic connection on a principal bundle over a smooth complex projective variety
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Criterion for existence of a logarithmic connection on a principal bundle over a smooth complex projective variety Sudarshan Gurjar1 · Arjun Paul1 Received: 15 February 2020 / Accepted: 16 June 2020 © Springer Nature B.V. 2020
Abstract Let X be a connected smooth complex projective variety of dimension n ≥ 1 . Let D be a simple normal crossing divisor on X. Let G be a connected complex Lie group, and EG a holomorphic principal G-bundle on X. In this article, we give criterion for existence of a logarithmic connection on EG singular along D. Keywords Logarithmic connection · Residue · Vector bundle · Principal bundle Mathematics Subject Classification 14J60 · 53C07 · 32L10
1 Introduction A theorem of Weil [12] says that a holomorphic vector bundle E on a smooth complex projective curve X admits a holomorphic connection if and only if each indecomposable holomorphic direct summand of E has degree 0; see [2]. For connected reductive linear algebraic group G over ℂ , this result of Weil and Atiyah is generalized to the case of holomorphic principal G-bundles on a smooth complex projective curve in [1]. It follows from [2, Theorem 4, p. 192] that, not every holomorphic bundle on a compact Kähler manifold can admit a holomorphic connection. Therefore, one can ask for criterion for a holomorphic bundle on X to admit a meromorphic connection. The simplest case of meromorphic connection is logarithmic connection. So it natural to ask when a given holomorphic bundle on X admits a logarithmic connection singular along a given divisor with prescribed residues. When X is a smooth complex projective curve, in [3], a necessary and sufficient criterion for a vector bundle on X to admit a logarithmic connection singular along a given reduced effective divisor D on X with prescribed rigid residues along D is given. This result is further generalized to the case of holomorphic principal G-bundles over smooth * Arjun Paul [email protected] Sudarshan Gurjar [email protected] 1
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, Maharashtra 400076, India
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Annals of Global Analysis and Geometry
complex projective curve in [4] when G is a connected reductive linear algebraic group over ℂ . When X is a smooth complex projective variety of dimension more than one, no such criterion for existence of logarithmic connection on a holomorphic bundle on X with prescribed residues along a given reduced effective divisor is known to the best of our knowledge. In this article, we attempt to study this problem.
1.1 Outline of the paper Unless otherwise specified, X is a connected smooth complex projective variety of dimension at least one, and D a reduced effective divisor on X. We denote by G a connected affine algebraic group over ℂ . In Sect. 2, we recall definitions of simple normal crossing divisor, logarithmic connection on holomorphic vector bundles on X, and their residues along a simple normal crossing divisor on X. In Sect. 3, we extend this notion of logarithmic
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