## Logarithmic De Rham, Infinitesimal and Betti Cohomologies

J. Math. Sci. Univ. Tokyo
Vol. 13 (2006), No. 2, Page 205--257.

Chiarellotto, Bruno; Fornasiero, Marianna
Logarithmic De Rham, Infinitesimal and Betti Cohomologies
Given a log scheme $Y$ over $\mathbb C$, Kato and Nakayama \cite{Nakayama} were able to associate a topological space $Y^{an}_{log}$. We will use the log infinitesimal site $Y^{log}_{inf}$ and its structural sheaf $\mathcal O_{Y^{log}_{inf}}$; we will prove that $H^{^.}(Y^{log}_{inf}, \mathcal O_{Y^{log}_{inf}}) \cong H^{^.}(Y^{an}_{log}, \mathbb C)$. The isomorphism will be obtained using log De Rham cohomological spaces $H^{^.}_{DR,log}(Y/\mathbb C)$ along the lines of \cite{Shiho2}. These results generalize the (ideally) log smooth case of \cite{Nakayama}.