Joint IAS/Princeton University Symplectic Geometry Seminar
Symplectomorphisms mirror to birational transformations of the complex plane
We construct a non-finite type four-dimensional Liouville manifold $M$ and describe a correspondence between certain birational transformations of the complex plane preserving a standard holomorphic volume form and symplectomorphisms of $M$. This manifold $M$ is universal in the sense it admits every Liouville four-manifold mirror to a log Calabi-Yau surface as a subdomain; our construction recovers a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the symplectomorphism group of its mirror by restriction to these subdomains. This is joint work in progress with Paul Hacking and Ailsa Keating.