Positive monotone symplectic manifolds are the symplectic analogues of Fano varieties, namely they are compact symplectic manifolds for which the first Chern class equals the cohomology class of the symplectic form. In dimension 6, if the positive monotone symplectic manifold is acted on by a circle in a Hamiltonian way, a conjecture of Fine and Panov asserts that it is diffeomorphic to a Fano variety. In this talk I will report on recent classification results of positive monotone symplectic manifolds endowed with some special Hamiltonian actions of a torus, showing some evidence that they are indeed (homotopy equivalent/homeomorphic/diffeomorphic to) Fano varieties.