I'd say it's closer to an oriented manifold with corners (corners happening where the divisor is singular), or even that times a coefficient. In [these papers][1] Khesin, Rosly, and later Thomas build a homology theory based on this analogy.


  [1]: https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=rosly&terms-0-field=author&terms-1-operator=AND&terms-1-term=polar+homology&terms-1-field=all&classification-mathematics=y&classification-physics_archives=all&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first