Severi type inequalities for irregular surfaces with ample canonical class

TL;DR

This paper establishes improved inequalities relating the invariants of irregular surfaces with ample canonical class, specifically for surfaces with irregularity at least 5, refining classical Severi inequalities.

Contribution

The paper derives new lower bounds for K^2 in terms of  and q(S) for irregular surfaces with ample canonical class, extending Severi inequalities.

Findings

Proves K^2  4 + (10/3)q(S) - 8 for q(S)  5.

Provides stronger inequalities under additional assumptions on the Albanese map.

Improves classical Severi inequality for a class of irregular surfaces.

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 the irregularity of S, q(S), is greater or equal to 5 we show that K^2>= 4\chi(S)+(10/3)q(S)-8, thus improving the well known Severi inequality K^2>=4\chi(S). We also give stronger inequalities under extra assumptions on the Albanese map or on the canonical map of S.