Sharp anticanonical volume bounds for Gorenstein Fano fourfold of Picard number one

Abstract

Let $X$ be a complex $\mathbb{Q}$-factorial Gorenstein Fano fourfold with $\rho (X)=1$. If $X$ has canonical singularities, we prove that $(-K_X{)}^4\le 3528$, with equality if and only if $(X\simeq \mathbb{P}(1,1,12,28,42)$. In fact, $(-K_X{)}^4\ge 3130$ already implies this isomorphism. If $X$ has terminal singularities, we prove the sharp bound $(-K_X{)}^4\le 648$, with equality if and only if $X\simeq \mathbb{P}(1,1,1,1,2)$. In the canonical case, we also obtain the sharp bound $(-K_X{)}^4\le 1024$ when a positive generator of $Cl(X)/\mathrm{tors}$ is Cartier outside a finite set. The proofs use jet estimates involving all Harder--Narasimhan slopes of the tangent sheaf, the geometry of algebraic foliations, and the integrality of intersection numbers on a general leaf. The equality case in the canonical bound is determined by two computations of divisorial section rings.

arXiv comment37 pages; v2; we also obtained the sharp bound in the canonical case