Abstract
We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces \({\mathbb {CP}}^{2}\# p\, \overline{\mathbb {CP}}^{2}\) for \(p=7, 8, 9\), and to \(3 {\mathbb {CP}}^{2}\#q\, \overline{\mathbb {CP}}^{2}\) for \(q =12, \ldots ,19\). Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with \(b^+ \le 3\), with one possible exception when \(b^+=3\). Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of \({\mathbb {CP}}^{2}\# 4 \overline{\mathbb {CP}}^{2}\) and \(3 {\mathbb {CP}}^{2}\# 6 \overline{\mathbb {CP}}^{2}\) from small Lefschetz fibrations over surfaces of higher genera.
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Notes
Another example with \(c_1^2=1\), \(\chi _h=1\) and \(\pi _1= {\mathbb {Z}}_3\) is given in a recent preprint of Akhmedov and Monden using the lantern substitution [2]. Akhmedov pointed out to us that they have now updated their arxiv paper to include some nonexplicit examples with \(\pi _1 = {\mathbb {Z}}\) and \({\mathbb {Z}}\oplus {\mathbb {Z}}_m\).
One can certainly determine these commutators on the nose, but there will be no need for our calculations here.
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Acknowledgments
The main results of this article were presented at the Great Lakes Geometry Conference in Ann Arbor and the Fukaya Categories of Lefschetz Fibrations Workshop at MIT back in March 2015; we would like to thank the organizers for motivating discussions. We also thank Hisaaki Endo and Tom Mark for their comments. The first author was partially supported by the NSF Grant DMS-1510395 and the Simons Foundation Grant 317732.
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Baykur, R.İ., Korkmaz, M. Small Lefschetz fibrations and exotic 4-manifolds. Math. Ann. 367, 1333–1361 (2017). https://doi.org/10.1007/s00208-016-1466-2
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DOI: https://doi.org/10.1007/s00208-016-1466-2