Non-holomorphic Lefschetz fibrations with $(-1)$-sections

TL;DR

This paper constructs two types of non-holomorphic Lefschetz fibrations over the sphere with (-1)-sections, providing new examples that challenge the slope inequality and complex structure conditions, thus expanding understanding of Lefschetz fibrations.

Contribution

It introduces explicit constructions of non-holomorphic Lefschetz fibrations with (-1)-sections, including examples that violate the slope inequality and cannot admit complex structures.

Findings

One type has a simply-connected total space and violates the slope inequality.

The other type's total space cannot admit any complex structure.

Provides an alternative proof for the existence of non-holomorphic Lefschetz pencils.

Abstract

We construct two types of non-holomorphic Lefschetz fibrations over $S^2$ with $(-1)$-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simply-connected total space, and the other has a total space that cannot admit any complex structure in the first place. These give an alternative existence proof for non-holomorphic Lefschetz pencils without Donaldson's theorem.