Almost Toric Symplectic Four-Manifolds

Question

Let $(M,\omega)$ be a closed, symplectic four-manifold admitting an almost toric fibration, in the sense of Symington and Leung (e.g. http://arxiv.org/pdf/math/0210033.pdf). That is, there is a Lagrangian torus fibration $$\pi\,:\,(M,\omega)\rightarrow B$$ such that the fibers attain at worst nodal or elliptic singularities. Then, $B$ has the natural structure of an integral affine-linear surface and is the natural generalization of the moment polygon. As far as I can tell, Symington proves that given an appropriate base $B$, there is a unique symplectic manifold $(M,\omega)$ admitting a unique almost toric fibration over $B$.

My question is the following; it may be very stupid, since I don't know too much symplectic geometry. Let $(M,\omega)$ be a symplectic four-manifold that admits some almost toric fibration. Is there a unique base $B$ over which $(M,\omega)$ admits an almost toric fibration? (Emphasis on uniqueness of the base!)

In case this turns out to be false in general, I'm specifically interested in the case where $M$ is a rational algebraic surface.

Answer 1

This is not true for toric fibrations. Namely there exist symplectic 4 manifolds with inequivalent Hamiltonian 2-torus actions: see Maximal tori in the symplectomorphism groups of Hirzebruch surfaces by Yael Karshon. These manifolds are not very exotic.

Answer 2

The answer is trivially no, even in the topological sense. For example, you consider $\mathbb{CP}^2$, the toric fibration $f_1:\mathbb{CP}^2\rightarrow B_1$ is of course almost-toric, since only elliptic singularities are involved. Notice that in this case, $B_1$ is an triangle with its symplectic affine structure. In particular, $B_1$ is an integral affine manifold with corners. On the other hand, by the technique of "nodal trade" in the paper you referred to, $\mathbb{CP}^2$ admits another almost toric fiberation $f_2:\mathbb{CP}^2\rightarrow B_2$ whose base $B_2$ is diffeomorphic to the a disc with boundary. $f_2$ has 3 nodal fibers in the interior of $B_1$ whose topologies are given by $T^2$ with a cycle in $H^1(T^2,\mathbb{Z})$ collapsed to a point. In this case, $B_2$ is a singular integral affine manifold with boundary. In a similar way you can produce examples of almost-toric symplectic 4-manifolds which admits different almost-toric Lagrangian fibrations with different bases. For example, you can apply the nodal trade technique to an arbitrary del-Pezzo surface $X$. You can also consider non-compact examples, although this is not treated in the paper you referred to. For example, when $X=\mathcal{O}(-2)\rightarrow\mathbb{P}^1$. This is a non-compact toric variety, so there is a toric fibration whose base $B_1$ is a noncompact polytope $P\subset\mathbb{R}^2$ with two corners. On the other hand, you can pull the two corners into the interior of $P$ to get another almost-toric fibration $X\rightarrow B_2$. In this case, the new base $B_2$ is the half plane $\mathbb{R}\times\mathbb{R}_{\geq0}$, and there are two nodal fibers. So $B_2$ is a non-compact singular affine manifold with boundary. This example is first considered by Mark Gross in his paper "Examples of special Lagrangan fibrations".