Sharp anticanonical volume bounds for Gorenstein Fano fourfold of Picard number one
Abstract
Let X be a complex ℚ-factorial Gorenstein Fano fourfold with ρ(X)=1. If X has canonical singularities, we prove that (−KX)4≤3528, with equality if and only if (X≃ ℙ(1,1,12,28,42). In fact, (−KX)4≥3130 already implies this isomorphism. If X has terminal singularities, we prove the sharp bound (−KX)4≤648, with equality if and only if X≃ ℙ(1,1,1,1,2). In the canonical case, we also obtain the sharp bound (−KX)4≤1024 when a positive generator of Cl(X)/ 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.
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BibTeX
@misc{bie2026sharp,
title = {{Sharp anticanonical volume bounds for Gorenstein Fano fourfold of Picard number one}},
author = {Pinxian Bie and Zhengjie Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09419},
eprint = {2610.09419},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09419},
}
RIS
TY - UNPB
TI - Sharp anticanonical volume bounds for Gorenstein Fano fourfold of Picard number one
AU - Pinxian Bie
AU - Zhengjie Yu
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.09419
UR - https://arxiv.org/abs/2610.09419
AB - Let X be a complex ℚ-factorial Gorenstein Fano fourfold with ρ(X)=1. If X has canonical singularities, we prove that (−K_X)^4≤3528, with equality if and only if (X≃ ℙ(1,1,12,28,42). In fact, (−K_X)^4≥3130 already implies this isomorphism. If X has terminal singularities, we prove the sharp bound (−K_X)^4≤648, with equality if and only if X≃ ℙ(1,1,1,1,2). In the canonical case, we also obtain the sharp bound (−K_X)^4≤1024 when a positive generator of Cl(X)/ 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.
ER -