Good formal structures for flat meromorphic connections, II: Excellent schemes

Kedlaya, Kiran

doi:10.1090/S0894-0347-2010-00681-9

Good formal structures for flat meromorphic connections, II: Excellent schemes
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by Kiran S. Kedlaya

J. Amer. Math. Soc. 24 (2011), 183-229

DOI: https://doi.org/10.1090/S0894-0347-2010-00681-9

Published electronically: September 17, 2010

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Abstract:

Given a flat meromorphic connection on an excellent scheme over a field of characteristic zero, we prove existence of good formal structures after blowing up; this extends a theorem of Mochizuki for algebraic varieties. The argument combines a numerical criterion for good formal structures from a previous paper, with an analysis based on the geometry of an associated valuation space (Riemann-Zariski space). We obtain a similar result over the formal completion of an excellent scheme along a closed subscheme. If we replace the excellent scheme by a complex analytic variety, we obtain a similar but weaker result in which the blowup can only be constructed in a suitably small neighborhood of a prescribed point.

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