Fibrations with torus symmetry. [04JP]

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Fibrations with torus symmetry.

Let $(X,\omega)$ be a symplectic $2n$ -manifold and let $\mu:(X,\omega)\rightarrow\mathfrak{t}^{\ast}$ be the moment map of a Hamiltonian $T^{k}$ -action. Let $t\in\mu(X)$ and let $\pi_{t}:\mu^{-1}(t)\rightarrow X_{t}$ be the projection modulo the $T^{k}$ action. When $t$ is a regular value of $\mu$ , $X_{t}$ is a smooth manifold and the symplectic form $\omega$ descends to a symplectic form $\omega_{t}$ on $X_{t}$ . When $t$ is a critical value of $\mu$ , $X_{t}$ may be a singular space and $\omega_{t}$ will be only defined on the smooth part of $X_{t}$ . The space $(X_{t},\omega_{t})$ is the Marsden-Weinstein reduced space at $t$ .

Remark 5.1.

We shall denote by

\omega_{\mathbb{C}^{m}}=\frac{i}{2}\sum_{k}dz_{k}\wedge d\overline{z}_{k}

the standard symplectic structure on $\mathbb{C}^{m}$ and $\omega_{0}$ will denote the reduced symplectic form of the reduced space $(X_{t},\omega_{t})$ at time $t=0$ .

Goldstein [5] and Gross [6] used reduced spaces to construct $T^{k}$ -invariant (special) Lagrangian fibrations. The following is a particular case of [6]Thm. 1.2:

Proposition 5.2.

Let $T^{k}$ act effectively on $X$ , $k\leq n-1$ . Suppose that there is a continuous map $G:X\rightarrow M$ to an $(n-k)$ -dimensional manifold $M$ such that $G(T\cdot x)=G(x)$ for all $T\in T^{k}$ . Suppose that for $t$ in a dense subset of $\mu(X)$ the induced maps $G_{t}:X_{t}\rightarrow M$ have fibres that are Lagrangian with respect to $\omega_{t}$ . Then $f:X\rightarrow\mu(X)\times M$ given by:

f=(\mu,G)

(23)

defines a $T^{k}$ -invariant Lagrangian fibration.

When the $T^{k}$ -action has fixed points, the construction of Proposition 5.2 will produce fibrations with interesting singular fibres. We will give some explicit examples shortly.

Remark 5.3.

In the extremal case when $k=n-1$ , constructing Lagrangian fibrations using Proposition 5.2 is very easy. In this situation, the reduced spaces $X_{t}$ are two dimensional and every map $G_{t}:X_{t}\rightarrow\mathbb{R}$ with $1$ -dimensional level sets defines a Lagrangian fibration on $X_{t}$ . In particular, any $T^{n-1}$ -invariant continuous map $G:X\rightarrow\mathbb{R}$ which, on each $X_{t}$ , descends to a map $G_{t}$ with $1$ -dimensional level sets can be used to construct Lagrangian fibrations. We will make much use of this fact later on.