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Regular logarithmic connections.

Piotr Achinger1

  • 1Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw, Poland.

Mathematische Annalen
|April 1, 2025
PubMed
Summary

We introduce regular integrable connections on log schemes, establishing an equivalence with connections on their analytifications. This work extends Deligne

Area of Science:

  • Algebraic Geometry
  • Number Theory

Background:

  • The study of integrable connections is fundamental in algebraic geometry and number theory.
  • Extending existing theories requires new frameworks for logarithmic structures.

Purpose of the Study:

  • To introduce and define regular integrable connections on smooth log schemes.
  • To establish an equivalence between categories of connections on log schemes and their analytifications.

Main Methods:

  • Development of the theory of regular integrable connections on smooth log schemes.
  • Construction of an equivalence using canonical extensions and good compactifications of log schemes.
  • Leveraging recent work by Włodarczyk on log scheme compactifications.

Main Results:

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  • An equivalence is established between regular integrable connections on log schemes and integrable connections on their analytifications.
  • This equivalence is compatible with de Rham cohomology.
  • The results extend Deligne's work for trivial log structures.

Conclusions:

  • The established equivalence provides a new perspective on regular integrable connections.
  • The findings offer a topological description of regular connections using constructible sheaves on the Kato-Nakayama space.
  • This research bridges concepts in algebraic geometry, number theory, and sheaf theory.