Characteristic forms for holomorphic families of local systems

2021/07/22 by Biswas, Indranil, Looijenga, Eduard
#14J42 #53B10 #53C07 #57R20 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2107.10482

Abstract

Let f:X→ S be a proper holomorphic submersion of complex manifolds and G a complex reductive linear algebraic group with Lie algebra \mathfrakg. Assume also given a holomorphic principal G-bundle P over X which is endowed with a holomorphic connection ∇ relative to f that is flat (this to be thought of as a holomorphic family of compact complex manifolds endowed with a holomorphic principal G-bundle with flat connection). We show that a refinement of the Chern-Weil homomorphism yields a graded algebra homomorphism ℂ[\mathfrakg]^G→ \bigoplus_{n≥ 0} H⁰(S, Ωⁿ_S,cl⊗ Rⁿf_*ℂ), where Ωⁿ_S,cl stands for the sheaf of closed holomorphic n-forms on S. If the fibers of f are compact Riemann surfaces and we take as our invariant the Killing form, then we recover Goldman's closed holomorphic 2-form on the base S.

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