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Unchaining surgery, branched covers, and pencils on elliptic surfaces

Terry Fuller

Algebraic & Geometric Topology 23 (2023) 2867–2893
Abstract

R İnanç Baykur, Kenta Hayano, and Naoyuki Monden used a technique called unchaining to construct a family of simply connected symplectic 4–manifolds Xg(i) for all g 3 and 0 i g 1 (Geom. Topol. 20 (2016) 2335–2395). Among this family, the manifolds Xg(g 2) are shown to be symplectic Calabi–Yau 4–manifolds. They also showed that each Xg(i) # [t]2¯ admits a pair of inequivalent genus g Lefschetz pencils. We show how to describe every Xg(i) as a 2–fold branched cover of a rational surface, and use this to prove that each Xg(i) is diffeomorphic to the elliptic surface E(g i). This has several notable consequences: each symplectic Calabi–Yau they construct is diffeomorphic to K3; for each n 3 and g n, the elliptic surface E(n) admits a genus g Lefschetz pencil; and for each n 3 and g n, the once blown up elliptic surface E(n) # [t]2¯ admits a pair of inequivalent genus g Lefschetz pencils.

Keywords
symplectic $4$–manifolds, Lefschetz fibration, Lefschetz pencil
Mathematical Subject Classification
Primary: 57K40, 57K43
References
Publication
Received: 23 August 2021
Revised: 30 November 2021
Accepted: 19 December 2021
Published: 7 September 2023
Authors
Terry Fuller
Department of Mathematics
California State University, Northridge
Northridge, CA
United States

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