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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$\begingroup$ Simplest example of a Fano: $\mathbb{P}^1$, and $-K_{\mathbb{P}^1} \cong 2p$. Not really an example of manifold with boundary. $\endgroup$– EnricoCommented Oct 12, 2016 at 11:22
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1$\begingroup$ In a sense, a complex manifold with a divisor is an analogue (or a "complexification") of a real manifold with boundary. $\endgroup$– Alex DegtyarevCommented Oct 12, 2016 at 11:48
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1$\begingroup$ @Enrico Consider (the closure of) one connected component of $\mathbb{RP}^1 \setminus \{0,\infty\}$. That's the manifold with boundary that $\mathbb P^1$ is supposed to be analogous, not equal, to. $\endgroup$– Allen KnutsonCommented Oct 16, 2016 at 17:50
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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.
answered Oct 12, 2016 at 18:18