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We establish sharp upper bounds for the anticanonical volume of $\mathbb{Q}$-factorial Gorenstein Fano fourfolds with Picard number one. For terminal singularities, the bound is $(-K_X)^4 \le 648$, achieved by $\mathbb{P}(1,1,1,1,2)$, while for canonical singularities, the bound is $(-K_X)^4 \le 7332$, which refines to $1024$ under specific Weil polarization conditions. These results provide crucial constraints on the geometry of these higher-dimensional varieties.
The anticanonical volume of Gorenstein Fano fourfolds is bounded, with $\mathbb{P}(1,1,1,1,2)$ realizing the sharpest bound for terminal singularities.
Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold of Picard number one. We prove that, if $X$ has terminal singularities, then $(-K_X)^4\le648$, with equality precisely for $\mathbb{P}(1,1,1,1,2)$. If the singularities are canonical, we obtain $(-K_X)^4\le7332$. Under the additional assumption that a primitive Weil polarization is Cartier outside finitely many points, the latter bound improves to the sharp bound $1024$; if its non-Cartier locus has dimension at most one, we obtain $6084$.