On a symplectic generalization of a Hirzebruch problem

Motivated by a problem of Hirzebruch, we study closed, symplectic, 8-dimensional manifolds having a Hamiltonian torus action with isolated fixed points and second Betti number equal to 1. Such manifolds are automatically positive monotone. Our main result concerns those endowed with a Hamiltonian T²-action and fourth Betti number equal to 2. We classify their isotropy data, (equivariant) cohomology rings and (equivariant) Chern classes, and prove that they agree with those of certain explicit Fano 4-folds with torus actions. Moreover, under more general assumptions, we prove several finiteness results concerning Betti and Chern numbers of positive monotone, 8-dimensional symplectic manifolds with a Hamiltonian torus action.