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Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces

  • Pinxian Bie
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Abstract

For a complex ε-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least ε2/100 and that every integral nef and big Weil divisor has volume at least ε7/1728. Both exponents are optimal. For ε-log canonical log Calabi--Yau surface pairs of arbitrary Picard number, we prove a lower bound of order ε9/(1+log2(1/ε)), without a bigness assumption or further restrictions on the boundary coefficients. We also prove τ(X,H)≤18ε−3 for the pseudo-effective threshold on every projective ε-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order ε−11/2√(1+log2(1/ε)) in the log Calabi--Yau setting.

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Cite

BibTeXRIS
BibTeX
@misc{bie2026sharp,
  title = {{Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces}},
  author = {Pinxian Bie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23763},
  eprint = {2609.23763},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23763},
}
RIS
TY  - UNPB
TI  - Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces
AU  - Pinxian Bie
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.23763
UR  - https://arxiv.org/abs/2609.23763
AB  - For a complex ε-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least ε^2/100 and that every integral nef and big Weil divisor has volume at least ε^7/1728. Both exponents are optimal. For ε-log canonical log Calabi--Yau surface pairs of arbitrary Picard number, we prove a lower bound of order ε^9/(1+log_2(1/ε)), without a bigness assumption or further restrictions on the boundary coefficients. We also prove τ(X,H)≤18ε^−3 for the pseudo-effective threshold on every projective ε-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order ε^−11/2√(1+log_2(1/ε)) in the log Calabi--Yau setting.
ER  -