#### DOCUMENTA MATHEMATICA,
Extra Volume: Kazuya Kato's Fiftieth Birthday (2003), 655-724

** Arthur Ogus **
On the Logarithmic
Riemann-Hilbert Correspondence

We construct a classification of coherent sheaves with an integrable log
connection, or, more precisely, sheaves with an integrable connection
on a smooth log analytic space $X$ over $\bc$. We do this in three contexts:
sheaves and connections which are equivariant with respect to a torus action,
germs of holomorphic connections, and finally global log analytic spaces.
In each case, we construct an equivalence between the relevant category
and a suitable combinatorial or topological category. In the equivariant
case, the objects of the target category are graded modules endowed with
a group action. We then show that every germ of a holomorphic connection
has a canonical equivariant model. Global connections are classified by
locally constant sheaves of modules over a (varying) sheaf of graded rings
on the topological space $X_{log}$. Each of these equivalences is compatible
with tensor product and cohomology.

2000 Mathematics Subject Classification:

Keywords and Phrases: De Rham cohomology, Log scheme

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