Symplectic geometry of conical symplectic resolutions

Ever since Kronheimer's celebrated hyperkähler construction of gravitational instantons thirty years ago, many similar constructions have appeared in the mathematical literature, eventually producing a vast class of holomorphic symplectic manifolds nowadays called Conical Symplectic Resolutions (CSRs). So far, they have been considered in representation theory, algebraic and differential geometry, and also in theoretical physics as they arise from certain supersymmetric gauge theories. However, much of their symplectic geometry is unexplored. This thesis aims to make the first steps in this direction. There are two essentially different natural real symplectic structures on CSRs; for this reason, this thesis splits into two major parts. The first part views CSRs as Liouville real symplectic manifolds (the symplectic form is exact). We investigate the presence of smooth closed exact Lagrangian submanifolds. The main theorem is that for any CSR there is a non-empty collection of non-isotopic such Lagrangians which arise from contracting C*-actions that act on the holomorphic symplectic structure by weight one. Lagrangians obtained from this method will be called minimal components. In particular, we use these to obtain (non-zero) lower bounds for the rank of symplectic cohomology. We will investigate two large subfamilies of CSRs where we are able to count minimal components and describe them. The family of Nakajima Quiver Varieties, in particular, we study in detail those of Dynkin type A; and the family of resolutions of Slodowy varieties that arise from the representation theory of semisimple Lie algebras and involve Springer theory. In the latter, we also obtain some further families of Lagrangians using Springer theoretic methods and certain crystal operators. The second part studies CSRs with respect to non-exact symplectic structures arising from S1-invariant Kähler structures on CSRs. We construct a family of symplectic cohomologies for them, labelled by different contracting C*-actions. Although we prove that these vanish, this vanishing result allows us to obtain an action-dependent filtration by ideals on the ordinary cohomology of a CSR. Using Morse-Bott-Floer spectral sequences, we give many examples that explicitly describe these filtrations, in particular, we show an example where distinct filtrations arise from different C*-actions.