On the Severi type inequalities for irregular surfaces

Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $qgeq 2$. We show that $K_X^2geq 4chi(mathcal O_X)+4(q-2)$ if $K_X^2<frac92chi(mathcal O_X)$, and also obtain the characterization of the equality. As a consequence, we prove a conjecture of Manetti on the geography of irregular surfaces if $K_X^2geq 36(q-2)$ or $chi(mathcal O_X)geq 8(q-2)$, and we also prove a conjecture that surfaces of general type and maximal Albanese dimension with $K_X^2=4chi(mathcal O_X)$ are exactly the resolution of double covers of abelian surfaces branched over ample divisors with at worst simple singularities.