A compact symplectic manifold (M, ω) is called positive monotone if its first Chern class is a positive multiple of [ω] in H^2_{dR}(M). A Fano variety is a smooth complex variety that admits a holomorphic embedding into CP^N (for some N). Such a variety can be endowed with a symplectic form such that it is a positive monotone symplectic manifold. For this reason, Fano varieties are considered the algebraic counterparts of positive monotone symplectic manifolds. A general outstanding issue in symplectic geometry is the question of whenever a positive monotone symplectic manifold is diffeomorphic to a Fano variety. In low dimensions, namely two and four, it is proven by Gromov, Taubes, McDuff, Ohat-Ono that any positive monotone symplectic manifold is diffeomorphic to a Fano variety. Analogous results are not known in higher dimensions.

In this talk I will explain what is known about the difference between Fano varieties and positive monotone symplectic manifolds endowed with a Hamiltonian torus action. In particular, I will represent new results for the case that the complexity of the action is one, i.e., the dimension of the torus is equal to (1/2)dim(M)−1. This is joint work with Liat Kessler, Silvia Sabatini and Daniele Sepe.