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?



Negative Definite Fano Manifolds
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.


