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arXiv · 2002.06629

Regular-singular connections on relative complex schemes

Phùng Hô Hai·João Pedro dos Santos
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Abstract

Deligne's celebrated "Riemann--Hilbert correspondence" relates representations of the fundamental group of a smooth complex algebraic variety and regular-singular integrable connections. In this work, we show how to arrive at a similar statement in the case of a smooth scheme $X$ over the spectrum of a ring $R=\mathbb C[[t_1,\ldots, t_r]]/I$. On one side of the correspondence we have representations on $R$-modules of the fundamental group of the special fibre, and on the other we have certain integrable $R$-connections admitting logarithmic models. The correspondence is then applied to give explicit examples of differential Galois groups of $\mathbb C[[t]]$--connections.

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Phùng Hô Hai, João Pedro dos Santos. 2020-02-16. Regular-singular connections on relative complex schemes. https://doi.org/10.2422/2036-2145.202006_010

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