Numerical inequalities for quasi-projective surfaces

Abstract

Let $V$ be a smooth quasi-projective complex surface with compactification $(X,D)$ and set ${\overline{P}}_1(V):=h^0(X,K_X+D)$, $\overline{q}(V):=h^0(X,\hat{I}{\text{c◯}}_X^1(\log D))$. We prove that ${\overline{P}}_1(V)\ge \overline{q}(V)-1$ if $V$ has maximal Albanese dimension and ${\overline{P}}_1(V)\ge \frac{1}{6}(\overline{q}(V)-5)$ otherwise. Both bounds are sharp.

12 pages, comments are welcome