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Sharp Polynomial Upper Bounds for Anticanonical Volumes
Abstract
For every positive integer n, we prove that there is a constant Cn, depending only on n, such that Vol(−KX) ≤ Cn ε−(2n−n−1) whenever X admits an ε-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In particular, the optimal exponent for fourfolds is eleven. We also prove an anticanonical interpolation theorem with optimal exponent 2n−1. The proof combines signed discrepancy estimates under projection, finite morphisms to projective space, and the canonical bundle formula. A reduction to a base bounded independently of ε, followed by a volume estimate along a flag, yields the sharper volume exponent.
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BibTeX
@misc{bie2026sharp,
title = {{Sharp Polynomial Upper Bounds for Anticanonical Volumes}},
author = {Pinxian Bie and Peien Du and Zhengjie Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10537},
eprint = {2610.10537},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10537},
}
RIS
TY - UNPB
TI - Sharp Polynomial Upper Bounds for Anticanonical Volumes
AU - Pinxian Bie
AU - Peien Du
AU - Zhengjie Yu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10537
UR - https://arxiv.org/abs/2610.10537
AB - For every positive integer n, we prove that there is a constant C_n, depending only on n, such that Vol(−K_X) ≤ C_n ε^−(2^n−n−1) whenever X admits an ε-lc log Fano boundary. The exponent is optimal in every dimension at least two, even for toric Fano varieties of Picard number one. In particular, the optimal exponent for fourfolds is eleven. We also prove an anticanonical interpolation theorem with optimal exponent 2^n−1. The proof combines signed discrepancy estimates under projection, finite morphisms to projective space, and the canonical bundle formula. A reduction to a base bounded independently of ε, followed by a volume estimate along a flag, yields the sharper volume exponent.
ER -