On the BMY inequality on surfaces

TL;DR

This paper investigates the relationship between the ordinarity of surfaces of general type and the failure of the BMY inequality in positive characteristic, establishing a new inequality with a vanishing correction term for ordinary surfaces.

Contribution

It introduces a new inequality relating Chern numbers for ordinary surfaces with semistable fibrations, extending the understanding of BMY inequality in positive characteristic.

Findings

Established an inequality relating c_1^2 and c_2 for ordinary surfaces.

Identified a correction term that vanishes for ordinary fibrations.

Connected ordinarity with the validity of the BMY inequality in positive characteristic.

Abstract

In this paper, we are concerned with the relation between the ordinarity of surfaces of general type and the failure of the BMY inequality in positive characteristic. We consider semistable fibrations $\pi:S \longrightarrow C$ where $S$ is a smooth projective surface and $C$ is a smooth projective curve. Using the exact sequence relating the locally exact differential forms on $S$, $C$, and $S/C$, we prove an inequality relating $c_1^2$ and $c_2$ for ordinary surfaces which admit generically ordinary semistable fibrations. This inequality differs from the BMY inequality by a correcting term which vanishes if the fibration is ordinary.