Some people say that a Fano manifold with anticanonical divisor is an analogue of a manifold with boundary. Where does this intuition come from?

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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 Khesin, Rosly, and later Thomas build a homology theory based on this analogy.

analogous, not equal, to. $\endgroup$ – Allen Knutson Oct 16 '16 at 17:50