# Another way to get toric symplectic manifolds

The above construction relies on the short exact sequence of tori

$0\rightarrow N=\mathbb {T} ^{d-n}\rightarrow \mathbb {T} ^{d}\rightarrow \mathbb {T} ^{n}\rightarrow 0$ which via the splitting defines an action of $\mathbb {T} ^{n}$ on the symplectic quotient $\mu ^{-1}(0)/N$ where $\mu$ is the moment map.

Even without a polytope, we can construct symplectic toric varieties in a similar fashion by starting with a torus homomorphism as above.

Similarly to before, consider a torus $(S^{1})^{m}$ with a Hamiltonian action on $\mathbb {C} ^{m}$ . Now let $r , let $T=(S^{1})^{r}$ be a torus of lower dimension and consider an injective group morphism

$T\hookrightarrow (S^{1})^{m}.$ Since we have a canonical identification
$\mathbb {Z} ^{m}\rightarrow \operatorname {Hom} ,\quad u\mapsto \lambda ^{u}{\text{ defined by }}\lambda ^{u}(t)=(t^{u_{1}},\dots ,t^{u_{m}}),$ we conclude that the above injection can be represented by an $m\times r$ matrix $M$ with integer entries, corresponding to the $r$ maps $S^{1}\rightarrow (S^{1})^{m}$ . The injection requirement says that $M$ has full rank $r$ .

Via $T\hookrightarrow (S^{1})^{m}\curvearrowright \mathbb {C} ^{d}$ , this defines a Hamiltonian action of $T$ on $\mathbb {C} ^{d}$ . The moment map of this action decomposes as

$\mu _{T}:X{\xrightarrow {\mu }}{\mathfrak {s}}^{\ast }{\xrightarrow {M^{T}}}{\mathfrak {t}}^{\ast }$ where ${\mathfrak {s}}:=Lie((S^{1})^{m})\cong (\mathbb {R} ^{m})^{\ast }$ and ${\mathfrak {t}}:=Lie(T)\cong (\mathbb {R} ^{r})^{\ast }$ are the corresponding Lie-algebras, $\mu$ is the moment map of the action of $(S^{1})^{m}$ , and $M^{T}$ denotes the transpose of $M$ , which represents the projection that is dual to $M:\mathbb {R} ^{r}\hookrightarrow \mathbb {R} ^{d}$ .

If now $w\in t^{\ast }$ is a regular value of the moment map, we can form the symplectic quotient $X=\mu _{T}^{-1}(w)/T$ of real dimension $2(m-r)$ . This is in fact a toric symplectic manifold: Analoguous to the construction in section 2, we have an exact sequence of tori

$0\rightarrow T\rightarrow (S^{1})^{m}\rightarrow (S^{1})^{m-r}\rightarrow 0$ Since the $(S^{1})^{m}$ -action on $X$ is trivial on the image of $T$ , we get a torus action of $(S^{1})^{m-r}$ .