Nonholomorphic Lefschetz fibrations with $(-1)$-sections

Noriyuki Hamada, Ryoma Kobayashi and Naoyuki Monden

Vol. 298 (2019), No. 2, 375–398

DOI: 10.2140/pjm.2019.298.375

Abstract

We construct two types of nonholomorphic Lefschetz fibrations over $S^2$ with $\left(-1\right)$-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 nonholomorphic Lefschetz pencils without Donaldson’s theorem.

Keywords

Lefschetz fibrations, $(-1)$-sections, slope inequality, complex structure

Mathematical Subject Classification 2010

Primary: 14D06, 55R55, 57R15, 57R20

Milestones

Received: 25 December 2016

Revised: 12 April 2018

Accepted: 6 June 2018

Published: 8 March 2019