Unchaining surgery and topology of symplectic 4-manifolds

TL;DR

This paper introduces the unchaining surgery operation in symplectic 4-manifolds, enabling new constructions of symplectic Calabi-Yau surfaces and resolving longstanding conjectures about Lefschetz fibrations.

Contribution

It presents a novel symplectic surgery called unchaining, which reduces Betti number and Kodaira dimension, and applies it to construct exotic manifolds and solve conjectures.

Findings

Unchaining reduces second Betti number and symplectic Kodaira dimension.

Constructs symplectic Calabi-Yau surfaces from various complex surfaces.

Provides criteria for spin structures from monodromy.

Abstract

We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex surfaces of general type, as well as from rational and ruled surfaces via the natural inverse of this operation. Combining the unchaining surgery with others, which all correspond to certain monodromy substitutions for Lefschetz pencils, we provide further applications, such as a complete resolution of a conjecture of Stipsicz on the existence of exceptional sections in Lefschetz fibrations, new constructions of exotic symplectic 4-manifolds, and inequivalent pencils of the same genera and the same number of base points on families of symplectic 4-manifolds. Meanwhile, we give a handy criterion for determining from the monodromy of a pencil whether its total space is spin or not.