Unchaining surgery and topology of symplectic 4-manifolds

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 and completely resolve a conjecture of Stipsicz on the existence of exceptional sections in Lefschetz fibrations. Combining the unchaining surgery with others, which all correspond to certain monodromy substitutions for Lefschetz pencils, we provide further applications, such as 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 present a handy criterion for determining from the monodromy of a pencil whether its total space is spin or not.