A Fano variety over $\mathbb{C}$ with Gorenstein singularity is called weak Fano if the anti-canonical divisor is nef and big. 


Are there finite families of weak Fano 4-folds with canonical Gorenstein singularities? Moreover, in what sense a set of Fano varieties is called "in the same family"?


Any comment on finiteness of Fano varieties are very welcome!



Are weak Fano 4-folds with canonical Gorenstein singularities bounded?
If you further assume $X$ only has canonical singularity, and $-K\_X$ is ample, then for any dimension, this is proved in *ACC for log canonical thresholds* Corollary 1.8. 


If you only assume $-K\_X$ is big and nef (but still assume $X$ has canonical singularity), from the result above, I believe a standard argument using the finiteness of models (se e.g. [BCHM]) should yield it then.


If you only assume Gorenstein, I doubt it. Can you just take the cone over elliptic curves but with degree higher and higher embedding? I think this give you infinitely many Gorenstein log canonical surfaces with $-K\_X$ ample and can't sit in finitely many families. 


