对数 Calabi-Yau 曲面的多项式体积界与有效双有理性
Sharp Volume Bounds in Picard Rank One and Polynomial Bounds for Log Calabi-Yau Surfaces
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中文总结 AI 辅助
本文证明了对数 Calabi-Yau 曲面上积分 nef 且 big Weil 除子体积的多项式下界，并推导出有效双有理性阶，无需系数大性假设。
中文摘要 AI 辅助
我们为复射影曲面上的积分 nef 且 big 的 Weil 除子提供了一个一致的多项式下界，该曲面允许一个 $\varepsilon$-对数典范的对数 Calabi-Yau 边界。更精确地，该体积被一个绝对正常数乘以 $\varepsilon^9/(1+\log(1/\varepsilon))$ 所下界。不需要 $B$ 的系数的大性或假设。我们还推导出对所有足够大的倍数（包括伴随系统和积分伪有效加法）具有阶 $\varepsilon^{-11/2}\sqrt{1+\log(1/\varepsilon)}$ 的有效双有理性。
英文摘要
For a complex $\varepsilon$-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least $\varepsilon^2/100$ and that every integral nef and big Weil divisor has volume at least $\varepsilon^7/1728$. Both exponents are optimal. For $\varepsilon$-log canonical log Calabi--Yau surface pairs of arbitrary Picard number, we prove a lower bound of order $\varepsilon^9/(1+\log_2(1/\varepsilon))$, without a bigness assumption or further restrictions on the boundary coefficients. We also prove $τ(X,H)\le18\varepsilon^{-3}$ for the pseudo-effective threshold on every projective $\varepsilon$-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order $\varepsilon^{-11/2}\sqrt{1+\log_2(1/\varepsilon)}$ in the log Calabi--Yau setting.
发表机构
- Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。
