Abstract

Monotone symplectic (aka symplectic Fano) manifolds are pretty rare in the universe of all symplectic manifolds, in much the same way that Fano varieties or Ricci-positive manifolds are rare. Positivity usually has strong implications for the underlying topology and one wonders if the same is true here. However, the twistor space of a hyperbolic 2n-manifold M (n bigger than or equal to 3) was observed to be a monotone symplectic manifold by Fine and Panov in 2009 and these examples counter many of one’s expectations of what a symplectic Fano manifold ought to look like. We explore the symplectic topology of these spaces (for the simplest case n=3) further by computing their quantum cohomology ring and the self-Floer cohomology of certain natural (equally unexpected) monotone Lagrangian submanifolds (Reznikov Lagrangians) associated to totally geodesic n-dimensional submanifolds of M. We will see evidence that there might be (yet more unusual) Lagrangians hiding in these spaces that we haven’t yet observed.