Unchaining surgery, branched covers, and pencils on elliptic surfaces

TL;DR

This paper proves that an infinite family of symplectic manifolds are diffeomorphic to elliptic surfaces, revealing new Lefschetz pencil structures and confirming the diffeomorphism of certain Calabi-Yau 4-manifolds to the K3 surface.

Contribution

It establishes diffeomorphisms between a family of symplectic manifolds and elliptic surfaces, and constructs new Lefschetz pencils on these surfaces, including for Calabi-Yau 4-manifolds.

Findings

Symplectic manifolds are diffeomorphic to elliptic surfaces.

Calabi-Yau 4-manifolds are diffeomorphic to the K3 surface.

Existence of genus g Lefschetz pencils on elliptic surfaces E(n).

Abstract

We show that every member of an infinite family of symplectic manifolds constructed by R. Inanc Baykur, Kenta Hayano, and Naoyuki Monden (arXiv:1903:02906) is diffeomorphic to an elliptic surface. As a result: (1) the symplectic Calabi-Yau 4-manifolds among their family are diffeomorphic to the standard K3 surface; (2) each elliptic surface E(n) admits a genus g Lefschetz pencil, for all g greater than or equal to n; and (3) each elliptic surface E(n) blown up once admits a pair of inequivalent genus g Lefschetz pencils, for all g greater than or equal to n.