Hofer-Zehnder capacity of toric varieties

Question

Consider the complex algebraic torus $(\mathbb{C}^*)^n$ where $\mathbb{C}^*$ stands for $\mathbb{C} \setminus \{0\}$. Fix a finite set of lattice points $A = \{\alpha_0, \ldots, \alpha_r\} \subset \mathbb{Z}^n$ and define the map $$\phi_A: (\mathbb{C}^*)^n \to \mathbb{C}\mathbb{P}^r,$$ by $x \mapsto (x^{\alpha_0}:\cdots:x^{\alpha_r})$. Let us assume that $\phi_A$ is an embedding and equip the torus $(\mathbb{C}^*)^n$ with the Kahler form $\omega_A = \phi_A^*(\omega_{FS})$ which is the pull-back, via $\phi_A$, of the standard Kahler form (the Fubini-Study form) on the projective space $\mathbb{C}\mathbb{P}^r$.

We regard $((\mathbb{C}^*)^n, \omega_A)$ as a symplectic manifold. The compact torus $(S^1)^n$ acts on $(\mathbb{C}^*)^n$ in a Hamiltonian fashion and the image of its moment map is the interior of the convex hull of the finite set $A$.

Two important notions of capacity for a symplectic manifold are the Gromov width and the Hofer-Zehnder capacity. One can give estimates for the Gromov width of $((\mathbb{C}^*)^n, \omega_A)$ in terms of the moment polytope, i.e. the convex hull of $A$.

My question is: are there any estimates known for the Hofer-Zehnder capacity of $((\mathbb{C}^*)^n, \omega_A)$ in terms of the finite set $A$ or its convex hull?

Any statement in this regard would be interesting.

Answer 1

There is a paper Symplectic capacities from Hamiltonian circle actions by Taekgyu Hwang and Dong Youp Suh, see http://arxiv.org/abs/1305.2989 which in particular gives an answer to your question in a very special case.

Namely, Theorem 1.1 implies that if your toric manifold $(X,\omega)$ is Fano and there is one-parameter subgroup $S^1\subset (S^1)^n$ acting semifreely, then that Hofer-Zehnder capacity of $(X,\omega)$ equals $H_{max}-H_{min}$, where $H_{max}$ and $H_{min}$ are the maximal and the minimal values of the Hamiltonian of the semifree circle action.