Symplectic cohomology with support and displaceability inside convex symplectic manifolds

Let W be a complete finite type Liouville manifold. One can associate to each closed subset K of W that is conical at infinity an invariant SH_W(K). I will first explain the construction of SH_W(K) and note how it recovers known invariants through special choices of K. Then, I will prove a big fiber theorem for certain contact manifolds using a descent property. I will end by briefly discussing an extension to negative line bundles. This is joint work with Yuhan Sun and Igor Uljarevic.