Unique fiber sum decomposability of genus 2 Lefschetz fibrations

Abstract

By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over S2, we show that K3#2CP‾2 can be rationally blown down along seven disjoint copies of the configuration C2. By counting the number of triangles in the line arrangement of the branched locus for corresponding hyperelliptic Lefschetz fibration, we show the maximum number of possible lantern relation substitutions on the global monodromy of K3#2CP‾2 is seven. We compute the Seiberg–Witten invariant of the resulting symplectic 4-manifolds and show that they are symplectically minimal. We also investigate how these exotic smooth 4-manifolds constructed via lantern relation substitution method are fiber sum decomposable. Furthermore by considering all the possible decompositions for each of our decomposable exotic examples, we will find out that there is a uniquely decomposing genus 2 Lefschetz fibration which is not a self-sum of the same fibration up to diffeomorphism on the indecomposable summands.