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

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 $\epsilon$-log canonical log Calabi–Yau boundary. More precisely, the volume is bounded below by an absolute positive constant times ${\epsilon }^9\mathrm{/}(1+\log (1\mathrm{/}\epsilon ))$. No bigness or assumptions on the coefficients of $B$ are required. We also derive effective birationality of order ${\epsilon }^{-11\mathrm{/}2}\sqrt{1+\log (1\mathrm{/}\epsilon )}$ for all sufficiently large multiples, including adjoint systems and integral pseudo-effective additions.