Severi type inequalities for irregular surfaces with ample canonical class

Abstract

Let $S$ be a smooth minimal complex projective surface of maximal Albanese dimension. Under the assumption that the canonical class of $S$ is ample and $q(S):=h^0({\Omega }_S^1)\ge 5$, we show $K_S^2\ge 4\chi (S)+\frac{10}{3}q(S)-8,$ thus improving the well-known Severi inequality $K_S^2\ge 4\chi (S)$.

We also give stronger inequalities under extra assumptions on the Albanese map or on the canonical map of $S$.