Vol. 298, No. 2, 2019

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

Noriyuki Hamada, Ryoma Kobayashi and Naoyuki Monden

Vol. 298 (2019), No. 2, 375–398
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
Revised: 12 April 2018
Accepted: 6 June 2018
Published: 8 March 2019
Authors
 Noriyuki Hamada Department of Mathematics and Statistics University of Massachusetts Amherst, MA United States Ryoma Kobayashi Department of General Education Ishikawa National College of Technology Tsubata Japan Naoyuki Monden Department of Mathematics, Faculty of Science Okayama University Okayama Japan