I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by $$ '\Omega^\bullet_{X^\infty} \langle Y\rangle = \Omega^\bullet_{X} \langle Y\rangle\otimes_{\Omega^\bullet_{X}} ('\Omega^\bullet_{X^\infty}), $$ where $X$ is a real or complex smooth proper analytic space, $Y\subset X$ is a normal crossing divisor, $X^\infty$ is $X$ as a $C^\infty$-manifold, $\Omega^\bullet_{X} \langle Y\rangle$ is the complex of sheaves of holomorphic differentials with logarithmic singularities, and $'\Omega^\bullet_{X^\infty}$ is the complex of sheaves of currents on $X^\infty$. It seems from this definition that the space of currents with log poles is larger than that of the usual currents.
However, Remark 1.8 c) says $'\Omega_{X^\infty}^{p,q}\langle Y\rangle$ is a quotient of $'\Omega_{X^\infty}^{p,q}$. The reference there (J. R. King, "Log complexes of currents and functorial properties of the Abel-Jacobi map") seems to use an apparently different definition. Which definition is correct, or how can I make sense of this situation?
