Research interest

My research interest relies in the Gromov-Witten theory and the related areas, with focus on the Hamiltonian Gromov-Witten invariants. As the name suggests, these quantities are symplectic invariants associated to Hamiltonian group actions on symplectic manifolds. Their definition present many similarities with that of the usual Gromov-Witten invariants, and in fact the two notions are conjecturally related. However, this issue is still to be clarified yet.

   Shortly, the GW-theory deals with spaces of pseudo-holomorphic curves in symplectic manifolds, that is with spaces of maps with source a Riemann surface and the target a symplectic manifold. These maps should obey in addition a partial differential equation saying that the map is pseudo-holomorphic.

   In the HGW-theory the picture is similar, with the change that instead of considering maps whose sources are curves, one is interested in equivariant maps defined on the total space of principal bundles over Riemann surfaces with target a symplectic manifold, and which satisfy the so-called vortex equations. The equivariance property refers to the given group action on the manifold.

  When the target manifold is a complex projective variety, the connection between the gauge theoretical setup of the HGW-theory and the algebraic-geometric one is achieved through a Kobayashi-Hitchin type correspondence. This result transforms the initial gauge theoretical problem into an algebraic-geometric one, which belongs to the area of the geometric invariant theory.