A compact symplectic manifold $(M, \omega)$ is called positive monotone if its first Chern class is a positive multiple of $[\omega]$ in the second de Rham group $H^2(M)$. A Fano variety is a smooth complex variety that admits a holomorphic embedding into $\mathbb{C} P^N$ for some $N$. Such a variety can be endowed with a symplectic form such that it becomes a positive monotone symplectic manifold. For this reason, positive monotone symplectic manifolds are considered the symplectic counterparts of smooth Fano varieties.

In the field of symplectic geometry, a general outstanding issue is understanding in what context positive monotone symplectic manifolds differ from Fano varieties. In low dimensions, namely two and four, it has been proven by Gromov, Taubes, McDuff, and Ohat-Ono that any positive monotone symplectic manifold is symplectomorphic to a Fano variety. Starting from dimension twelve, work by Fine and Panov provides examples of positive monotone symplectic manifolds that are not even homotopy equivalent to a Fano variety.

In this talk, I will explain what is known about the differences between Fano varieties and positive monotone symplectic manifolds endowed with a Hamiltonian action of a compact torus $T$. In particular, I will present new results for the case where the complexity of the action is one, i.e., $\frac{1}{2}\dim(M)-\dim(T)=1$.

This talk is based on joint work with Liat Kessler, Silvia Sabatini, and Daniele Sepe.