Suppose that $X$ is a complex, rational, $\mathbb{Q}$-Gorenstein $3$-fold with at most terminal singularities, and $H\_{2}(X,\mathbb{R})=\mathbb{R}$. Is $X$ Fano?


By Kleiman ampleness criterion this is equivalent to $(-K\_{X})^3>0$, but the standard argument from the smooth case of using Kodaira dimension doesn't quite work.



Singular $3$-fold with $b_{2}=1$
I guess this just follows from exercise 4.1.9 here <https://www.dpmms.cam.ac.uk/~cb496/birgeom.pdf>.


