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December 2020 Genus 2 Lefschetz fibrations with b 2 + = 1 and c 1 2 = 1 , 2
Anar Akhmedov, Naoyuki Monden
Kyoto J. Math. 60(4): 1419-1451 (December 2020). DOI: 10.1215/21562261-2019-0067

Abstract

In this article, we construct a family of genus 2 Lefschetz fibrations f n : X θ n S 2 with e ( X θ n ) = 11 , b 2 + ( X θ n ) = 1 , and c 1 2 ( X θ n ) = 1 by applying a single lantern substitution to the twisted fiber sums of Matsumoto’s genus 2 Lefschetz fibration over S 2 . Moreover, we compute the fundamental group of X θ n and show that it is isomorphic to the trivial group if n = 3 or 1 , Z if n = 2 , and Z | n + 2 | for all integers n 3 , 2 , 1 . Also, we prove that our fibrations admit 2 section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension κ = 2 . In addition, using techniques developed over the past decade with other authors, we also construct the genus 2 Lefschetz fibrations over S 2 with c 1 2 = 1 , 2 and χ = 1 via the fiber sums of Matsumoto’s and Xiao’s genus 2 Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small 4 -manifolds with b 2 + = 1 and b 2 + = 3 .

Citation

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Anar Akhmedov. Naoyuki Monden. "Genus 2 Lefschetz fibrations with b 2 + = 1 and c 1 2 = 1 , 2 ." Kyoto J. Math. 60 (4) 1419 - 1451, December 2020. https://doi.org/10.1215/21562261-2019-0067

Information

Received: 19 October 2015; Revised: 16 February 2018; Accepted: 20 December 2018; Published: December 2020
First available in Project Euclid: 7 October 2020

MathSciNet: MR4175814
Digital Object Identifier: 10.1215/21562261-2019-0067

Subjects:
Primary: 57R55
Secondary: 57R17

Rights: Copyright © 2020 Kyoto University

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Vol.60 • No. 4 • December 2020
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