Description
Abstract: We consider in this talk the following natural question in contact
topology: how many different ways can a given contact manifold arise
as the convex boundary of a compact symplectic manifold (i.e. a
symplectic filling)? One way to attack this question is by considering
certain geometric decompositions that reduce the dimension of the
problem, e.g. a Lefschetz fibration on a symplectic 4-manifold
presents it as a 2-parameter family of symplectically embedded surfaces,
which can then be turned into J-holomorphic curves. The natural
structure arising on the boundary of a 4-manifold with a Lefschetz
fibration is called a spinal open book. In a joint paper with Sam
Lisi and Jeremy Van Horn-Morris, we proved that for "most" contact
3-manifolds admitting spinal open books with genus zero pages, all
possible symplectic fillings come from Lefschetz fibrations whose
fibers are J-holomorphic curves, including finitely many nodal curves
(Lefschetz singular fibers). In this talk, I will sketch the geometric
construction and analytical machinery that leads to this result,
give a few examples of contact manifolds whose fillings can be
classified as a corollary, and also discuss the following caveat:
for more complicated spinal open books, the holomorphic curves one
obtains on the filling can also include finitely many so-called
"exotic fibers", a new type of degeneration that cannot be described
in the language of Lefschetz fibrations.
For Zoom access please contact Andras Stipsicz (stipsicz.andras[a]renyi.hu).