Author

Pinxian Bie

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Preprint Oct 2026

Anticanonical Volumes of Terminal and Canonical Gorenstein Fano Fourfolds

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...

Pinxian Bie, Zheng-Jie Yu · 0 citations

Preprint Oct 2026

Sharp Polynomial Upper Bounds for Anticanonical Volumes

For every positive integer $n$, we prove that there is a constant $C_n$, depending only on $n$, such that $Vol(-K_X) \leq C_n \epsilon^{-(2^n-n-1)}$ whenever $X$ admits an $\epsilon$-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In par...

Pinxian Bie, Pei-En Du, Zheng-Jie Yu · 0 citations