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Suppose I am given a symplectic toric manifold $(M,\omega,\psi)$ which is also monotone, hence the symplectic form can be rescaled so that $c_1=[\omega]$. Then the moment map can be taken so that its image is a reflexive polytope $\Delta$. Being reflexive implies that there is only one integral point in the interior, namely the origin.

What is the "symplectic meaning" of the fact that there is only one integral point in the interior?

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    $\begingroup$ Related to your question: Monotone toric symplectic manifolds have the unique toric structure (McDuff 2011) every smooth integral polytope is a facet of a monotone polytope, See the paper of Haase, Christian; Melnikov, Ilarion V. The reflexive dimension of a lattice polytope. Ann. Comb. 10 (2006), no. 2, 211–217. $\endgroup$
    – user21574
    Commented Oct 18, 2017 at 5:15

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The geometric interpretation is quite simple: there is a unique torus fiber $L\subset M$ of the moment map $\mu:M\rightarrow\Delta_M$ which is monotone, and this fiber lies over the unique integral point. So this Lagrangian $L$ (equipped with its Spin structure) actually defines an object of the monotone Fukaya category $\mathcal{F}(M)$.

There should be infinitely many monotone Lagrangian tori in $M$ which are mutually not Hamiltonian isotopic (in the two-dimensional case, this is proved by Vianna for general del Pezzo surfaces). However, $L$ is the only one which appears as a fiber of the standard toric fibration.

It can also be shown that $L$ is non-displaceable, and in fact admits non-trivial Floer cohomology $\mathit{HF}^\ast(L,L)\neq0$, so it defines a non-trivial object in $\mathcal{F}(M)$. This is in fact the Lagrangian we use to recover the whole $\mathcal{F}(M)$ (defined over a field $\mathbb{K}$ with $\mathrm{char}(\mathbb{K})\neq2$). One can actually show that by taking iterated mapping cones in terms of $L$ (equipped with suitable local systems) and splitting off direct summands, one can obtain every oriented monotone Lagrangian submanifold in $M$ with non-trivial Floer cohomology. This is the work of Abouzaid-Fukaya-Oh-Ohta-Ono.

In conclusion, you can interpret the existence of a unique integral point as revealing the fact that all the geometric information about (oriented, monotone) Lagrangian submanifolds in $M$ is contained in a unique Lagrangian submanifold $L$. As a simple corollary, we have the following:

For every oriented monotone Lagrangian submanifold $K\subset M$ with $\mathit{HF}^\ast(K,K)\neq0$, we have $K\cap L\neq\emptyset$.

Addendum Let me remark that this pretty simple geometric picture is a consequence of both the torus symmetry and monotonicity. If monotonicity is not assumed, every blow-up (even a very small toric blow-up) on $M$ will create an additional non-displaceable Lagrangian torus, which has been proved recently by Woodward.

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