# Another way to get toric symplectic manifolds

The above construction relies on the short exact sequence of tori

${\displaystyle 0\rightarrow N=\mathbb {T} ^{d-n}\rightarrow \mathbb {T} ^{d}\rightarrow \mathbb {T} ^{n}\rightarrow 0}$
which via the splitting defines an action of ${\displaystyle \mathbb {T} ^{n}}$ on the symplectic quotient ${\displaystyle \mu ^{-1}(0)/N}$ where ${\displaystyle \mu }$ 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 ${\displaystyle (S^{1})^{m}}$ with a Hamiltonian action on ${\displaystyle \mathbb {C} ^{m}}$. Now let ${\displaystyle r, let ${\displaystyle T=(S^{1})^{r}}$ be a torus of lower dimension and consider an injective group morphism

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

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

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

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

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