# Do non-compact Fano manifolds exist?

Suppose $$(M,g, \omega)$$ is a Kähler manifold with $$\text{Ric}(g) = g$$, i.e., $$M$$ is a Fano manifold. Is $$M$$ necessarily compact? If not, perhaps complete and Fano implies compact? I'd like to build a catalog of examples of Fano manifolds, so any input is appreciated.

• For the first question, taking any compact Fano and deleting a point doesn't change the condition on the Ricci tensor. For the second, in what sense do you mean "complete"? – Pop Feb 25 at 22:25
• @Pop Complete in the sense of metric completeness. – A Bit Too Curious Feb 25 at 22:26
• Thank you. I asked because in algebraic geometry "complete" has an alternative meaning, essentially the same as "compact". – Pop Feb 25 at 23:09
• @Pop Complete is a dangerous word, I should've been more specific. At least complete is not as bad as "regular" which is so well-defined that we have 20 different inequivalent definitions for it :) – A Bit Too Curious Feb 25 at 23:21