RESUMO: A driving (meta)question in symplectic geometry is to understand the extent to which compact symplectic manifolds differ from complex projective (or Kähler) smooth varieties. This problem can be refined by imposing further conditions: for instance, we may ask how compact monotone symplectic manifolds differ from smooth Fano varieties. While it is known that there are examples of the former that cannot be Kähler and, hence, Fano (due to Reznikov, in real dimension 12), only recently have people started asking this question in the presence of a Hamiltonian torus action. Motivated by very recent results by Lindsay and Panov in dimension 6 in the presence of a Hamiltonian S 1 -action, the aim of this talk is to prove that monotone compact symplectic 2n-dimensional manifolds endowed with a Hamiltonian T n−1 -action (for any n) enjoy some topological properties that their Fano counterparts do. This is joint work with Silvia Sabatini..