manifold

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1] is both a smooth manifold with boundary and a complex manifold, it is not a complex manifold with boundary.
We first state the usual definition of a complex manifold with boundary.
Recall that a homology d-manifold with boundary is said to be r-stacked if it has no interior faces of dimension [less than or equal to] d - r - 1 and that a homology manifold without boundary is said to be r-stacked if it is the boundary of an r- stacked homology manifold with boundary.