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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!

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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.

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  • $\begingroup$ Thank you so much! I think I have canonical singularity. May I ask one more question? In what sense, do you call such varieties sit in a family? In the sense of the same Hodge number...? $\endgroup$ – Li Yutong Jan 24 '14 at 1:44
  • $\begingroup$ We say varieties sit in a family, if there is a flat family over a (irreducible or connected) variety of finite type, such that these varieties appear as fibers of this flat family. $\endgroup$ – CYXU Jan 30 '14 at 19:16

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