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

Source 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.