The concept of *logrithmic connection* is that of *connection on a bundle* in the context of logarithmic geometry. Hence a logarithmic connection is a connection (principal connection or connection on a vector bundle etc.) on some space $X$ whose local connection differential 1-form $A$ is allowed to have logarithmic singularities at a prescribed subspace $D \hookrightarrow X$, hence is allowed to be a differential form with logarithmic singularities.

- Indranil Biswas, V.Munoz,
*Moduli spaces of connections on a Riemann surface*(pdf)

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