Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base


Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A. We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.

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Correspondence to Jari Taskinen.

Additional information

Jari Taskinen was supported in part by the Academy of Finland and the Väisälä Foundation. Kari Vilonen was supported in part by NSF Grants DMS-1402928 & DMS-1069316, the Academy of Finland, the ARC Grant DP150103525, the Humboldt Foundation, and the Simons Foundation.

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Taskinen, J., Vilonen, K. Cartan Theorems for Stein Manifolds Over a Discrete Valuation Base. J Geom Anal 29, 577–615 (2019).

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  • Cartan theorem A and B
  • Stein manifold
  • Coherent module
  • Codimension-three conjecture
  • Discrete valuation ring
  • Banach algebra
  • Inductive limit

Mathematics Subject Classification

  • Primary 32C35
  • Secondary 32W05
  • 46A13
  • 46H99