A compact symplectic manifold (M, ω) is called positive monotone if its first Chern class is a positive multiple of [ω] in H2 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 monotone symplectic manifolds. A general outstanding issue in symplectic geometry is the question of whenever a positive monotone Fano manifold is diffeomorphic to a Fano variety. In low dimensions, namely two and four, it is proven by McDuff-Gromov-Taubes that any positive monotone symplectic manifold is diffeomorphic to a Fano variety. In higher dimensions analogues results are not known. In this talk I will explain what is known about the difference between Fano varieties and positive monotone symplectic manifolds endowed with 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 dim(M)/2−1. This is joint work with Liat Kessler, Silvia Sabatini and Daniele Sepe.
