We study neighborhoods of configurations of symplectic surfaces in symplectic
4–manifolds. We show that suitably “positive” configurations have neighborhoods
with concave boundaries and we explicitly describe open book decompositions of the
boundaries supporting the associated negative contact structures. This is used to
prove symplectic nonfillability for certain contact 3–manifolds and thus nonpositivity
for certain mapping classes on surfaces with boundary. Similarly, we show that
certain pairs of contact 3–manifolds cannot appear as the disconnected convex
boundary of any connected symplectic 4–manifold. Our result also has the potential
to produce obstructions to embedding specific symplectic configurations in closed
symplectic 4–manifolds and to generate new symplectic surgeries. From a
purely topological perspective, the techniques in this paper show how to
construct a natural open book decomposition on the boundary of any plumbed
4–manifold.
Keywords
symplectic, contact, concave, open book, plumbing, fillable
