log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over \(\mathbb{C}\)
For schemes over the complex numbers, there are two comparison theorems between topological and algebraic cohomology. Firstly the étale cohomology of a constructible (torsion) sheaf is equal to its topological cohomology. Secondly for manifolds the algebraic de Rham topology coincides with the topological complex cohomology. Also for complex manifolds locally constant systems correspond to vector bundles with integrable connections (``Riemann-Hilbert correspondence''). In the paper under review these are generalised to log-schemes and log-analytic spaces. The log-étale cohomology has been studied by one of the authors, and its log-analytic analogue uses a ``real blow-up'' of the underlying analytic space. The comparison for constructible sheaves holds in general, for the other two results one has to restrict the singularities of the spaces (fine and log smooth is sufficient, but the results are more general). Also the Riemann-Hilbert correspondence needs some hypothesis on the bundles (``unipotent monodromy at the boundary''). The strategy of the proof consists of reduction to the classical case via local computations.
scientific article; zbMATH DE number 1367783
scientific article