My question is about the "algebraic moment map", as discussed by Frank Sottile in the final section of this paper, or by Bill Fulton in his Introduction to Toric Varieties, where he referes to it as the "moment map". Note that, as I shall recall below, it is not the same as the usual moment map of symplectic geometry.
Let me set up some notation in order to define this map. Begin with $\mathcal A=\{m_1,\dots,m_r\}$, a collection of points in $\mathbb Z^n$, with convex hull $\Delta$.
The quickest way to define the associated toric variety $Y$ is to say that it is the closure (in the Zariski or analytic topology) of the points $[t^{m_1},\dots,t^{m_r}]$ inside the projective space $\mathbb P^{r-1}$, as $t$ runs through $\mathbb C^n$.
Write $x_1,\dots,x_r$ for the corresponding coordinate functions on $\mathbb P^{r-1}$ (defined, of course, only up to overall rescaling).
The usual moment map from $Y$ to $\mathbb R^n$ is defined by:
$$y\mapsto \frac 1 {\sum_{i=1}^r |x_i(y)|^2} \sum_{i=1}^r |x_i(y)|^2 m_i$$
The algebraic moment map is defined by:
$$y \mapsto \frac 1 {\sum_{i=1}^r |x_i(y)|} \sum_{i=1}^r |x_i(y)| m_i$$
It is also natural to restrict to the non-negative part of $Y$, which is defined by taking the closure (in the analytic topology) of the points $[t^{m_1},\dots,t^{m_r}]$ with $t$ in $\mathbb R^n_{>0}$. There, we don't need to take norms anymore; the algebraic moment map is simply given by:
$$y \mapsto \frac 1 {\sum_{i=1}^r x_i(y)} \sum_{i=1}^r x_i(y) m_i$$
Both the usual moment map and the algebraic moment map have the property that their image is $\Delta$, and in fact the algebraic moment map gives a homeomorphism from the non-negative part of $Y$ to $\Delta$.
The usual moment map is, of course, very important for symplectic geometry. What about this algebraic moment map? Does it interact with any symplectic geometry notions? Is it in any sense a moment map?
My reason for asking this question, in case it helps explain what kind of answer I am looking for, is that the algebraic moment map appears in this paper by Arkani-Hamed, Bai, and Lam (in section 7.3.2) and I was hoping that a bit more context might help me understand better what is going on there.
