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Schematic Harder–Narasimhan stratification for families of principal bundles in higher dimensions

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Abstract

Let G be a connected split reductive group over a field k of characteristic zero. Let \(X\rightarrow S\) be a smooth projective morphism of k-schemes, with geometrically connected fibers. We formulate a natural definition of a relative canonical reduction, under which principal G-bundles of any given Harder–Narasimhan type \(\tau \) on fibers of X / S form an Artin algebraic stack \(Bun_{X/S}^{\tau }(G)\) over S, and as \(\tau \) varies, these stacks define a stratification of the stack \(Bun_{X/S}(G)\) by locally closed substacks. This result extends to principal bundles in higher dimensions the earlier such result for principal bundles on families of curves. The result is new even for vector bundles, that is, for \(G = GL_{n,k}\).

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Acknowledgements

We thank the Center for Quantum Geometry of Moduli Spaces for their hospitality, supported by a Center of Excellence grant from the Danish National Research Foundation (DNRF95) and by a Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Union Framework Programme (FP7/2007–2013) under Grant agreement no 612534, project MODULI-Indo European Collaboration on Moduli Spaces. Sudarshan Gurjar would also like to thank the ICTP and the TIFR for support during part of the work. We thank Najmuddin Fakhruddin for producing the Example 2.6 of a line bundle defined on a relatively big open subscheme which does not admit a global prolongation.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute for Technology, Bombay, Powai, Mumbai, 400 076, India

    Sudarshan Gurjar

  2. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India

    Nitin Nitsure

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  1. Sudarshan Gurjar
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  2. Nitin Nitsure
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Correspondence to Sudarshan Gurjar.

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Gurjar, S., Nitsure, N. Schematic Harder–Narasimhan stratification for families of principal bundles in higher dimensions. Math. Z. 289, 1121–1142 (2018). https://doi.org/10.1007/s00209-017-1990-0

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  • DOI: https://doi.org/10.1007/s00209-017-1990-0

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