Holomorphic Curves in Blown Up Open Books

We use contact fiber sums of open book decompositions to define an infinite hierarchy of filling obstructions for contact 3-manifolds, called planar k-torsion for nonnegative integers k, all of which cause the contact invariant in Embedded Contact Homology to vanish. Planar 0-torsion is equivalent to overtwistedness, while every contact manifold with Giroux torsion also has planar 1-torsion, and we give examples of contact manifolds that have planar k-torsion for any $k\ge 2$ but no Giroux torsion, leading to many new examples of nonfillable contact manifolds. We show also that the complement of the binding of a supporting open book never has planar torsion. The technical basis of these results is an existence and uniqueness theorem for J-holomorphic curves with positive ends approaching the (possibly blown up) binding of an ensemble of open book decompositions.