Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Parallel transport for vector bundles on $ p$-adic varieties


Authors: Christopher Deninger and Annette Werner
Journal: J. Algebraic Geom.
DOI: https://doi.org/10.1090/jag/747
Published electronically: September 26, 2019
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Abstract: We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a $ p$-adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous $ p$-adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a $ p$-adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings' $ p$-adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.


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Christopher Deninger
Affiliation: Department of Mathematics, University of Münster, Einsteinstr. 62, 48149 Münster, Germany
Email: c.deninger@uni-muenster.de

Annette Werner
Affiliation: Institute of Mathematics, Goethe University Frankfurt, Robert-Mayer-Str. 6-8, 60325 Frankfurt am, Germany
Email: werner@math.uni-frankfurt.de

DOI: https://doi.org/10.1090/jag/747
Received by editor(s): August 3, 2017
Received by editor(s) in revised form: May 20, 2019, and May 27, 2019
Published electronically: September 26, 2019
Article copyright: © Copyright 2019 University Press, Inc.