A Hamiltonian T-space is a symplectic manifold (M, ω) endowed with an effective and symplectic action of a compact torus T that admits a moment map. The complexity of this space is the non-negativeinteger 1/2dim(M) − dim(T). Karshon has classified compact Hamiltonian T-spaces of complexity one in dimension four by their so-called decorated graphs. We extend the definition of decorated graphs to compact Hamiltonian T-spaces of complexity one in any dimension and prove that these graphs determine the T-equivariant homeomorphism type of the one-skeletons, as well as the integer equivariant and ordinary cohomology rings of their underlying spaces. If the underlying manifold is of dimension six and simply connected, we show that the decorated graph also determines the diffeomorphism type.

Furthermore, building on results by Godinho and Sabatini, we classify the decorated graphs of compact six-dimensional Hamiltonian T-spaces of complexity one with isolated fixed points, whose underlying symplectic manifold is monotone. A direct consequence of this classification is that the decorated graph of such a space is isomorphic to that of a smooth Fano threefold endowed with a holomorphic complexity one action.

These two steps lead to the conclusion that the underlying manifold of a six-dimensional, compact, and monotone Hamiltonian T-space of complexity one with a discrete fixed point set is diffeomorphic to a smooth Fano threefold.

The talk is based on joint work with Liat Kessler.