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

Abstract

For a complex $\epsilon$-log canonical Fano surface of Picard number one, we prove that the anticanonical volume is at least ${\epsilon }^2/100$ and that every integral nef and big Weil divisor has volume at least ${\epsilon }^7/1728$. Both exponents are optimal. For $\epsilon$-log canonical log Calabi--Yau surface pairs of arbitrary Picard number, we prove a lower bound of order ${\epsilon }^9/(1+{\log }_2(1/\epsilon ))$, without a bigness assumption or further restrictions on the boundary coefficients. We also prove $\tau (X,H)\le 18{\epsilon }^{-3}$ for the pseudo-effective threshold on every projective $\epsilon$-log canonical surface, with optimal exponent three. A volume-to-birationality estimate gives a sufficient bound of order ${\epsilon }^{-11/2}\sqrt{1+{\log }_2(1/\epsilon )}$ in the log Calabi--Yau setting.

arXiv comment40 pages, v2; title changed; major revision: added several sharp order results in Picard rank one and a sharp order pseudo-effective threshold