# Analogy of a Fano manifold with anticanonical divisor

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

• Simplest example of a Fano: $\mathbb{P}^1$, and $-K_{\mathbb{P}^1} \cong 2p$. Not really an example of manifold with boundary. – Enrico Oct 12 '16 at 11:22
• In a sense, a complex manifold with a divisor is an analogue (or a "complexification") of a real manifold with boundary. – Alex Degtyarev Oct 12 '16 at 11:48
• @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. – Allen Knutson Oct 16 '16 at 17:50