A complex manifold $M$ is said to be Fano if the Chern curvature $2$form is a positive definite $(1,1)$form. What happens if the Chern curvature $2$form is a negative definite $(1,1)$form? What are such manifolds called, and do they differ in any meaningful way from Fano manifolds?
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3$\begingroup$ Usually they are called general type and complex hyperbolic manifolds are examples. They behave vastly differently from Fano varieties. For example Fano varieties have Kodaira dimension $\infty$ whereas varieties of general type has maximal Kodaira dimension. $\endgroup$– GTACommented Sep 4, 2019 at 1:34
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Such manifold are called of general type (at least in the projective case). If you look at compact Riemann surfaces, then the only Fano variety is $\mathbb{P}^1$ while curves of general type have bigger moduli spaces. Their geometry, and even topology, is really different.