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arXiv · 2609.23763

Polynomial Volume Bounds and Effective Birationality for Log Calabi-Yau Surfaces

Pinxian Bie
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Abstract

We provide a uniform polynomial lower bound for the volume of an integral nef and big Weil divisor on a complex projective surface admitting an $\varepsilon$-log canonical log Calabi--Yau boundary. More precisely, the volume is bounded below by an absolute positive constant times $\varepsilon^9/(1+\log(1/\varepsilon))$. No bigness or assumptions on the coefficients of $B$ are required. We also derive effective birationality of order $\varepsilon^{-11/2}\sqrt{1+\log(1/\varepsilon)}$ for all sufficiently large multiples, including adjoint systems and integral pseudo-effective additions.

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Pinxian Bie. 2026-09-20. Polynomial Volume Bounds and Effective Birationality for Log Calabi-Yau Surfaces. https://arxiv.org/abs/2609.23763

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