Numerical inequalities for quasi-projective surfaces

TL;DR

This paper establishes sharp numerical inequalities relating the logarithmic plurigenera and irregularity of smooth quasi-projective surfaces, depending on their Albanese dimension.

Contribution

It provides new sharp bounds connecting key invariants of quasi-projective surfaces based on their Albanese dimension.

Findings

Proves that ar P_1(V)  ar q(V) - 1 for surfaces with maximal Albanese dimension.

Establishes that ar P_1(V)  rac{1}{6} (bar q(V) - 5) otherwise.

Both bounds are shown to be sharp.

Abstract

Let $V$ be a smooth quasi-projective complex surface with compactification $(X,D)$ and set $\overline P_1(V):=h^0(X,K_X+D)$, $\overline q(V):=h^0(X,\Omega^1_X(\log D))$. We prove that $\overline P_1(V)\ge \overline q(V)-1$ if $V$ has maximal Albanese dimension and $\overline P_1(V)\ge\frac 16( \overline q(V)-5)$ otherwise. Both bounds are sharp.