Lemma
Let \(V\) be a toric variety and \(T_V\) its torus. Let \(T\) be some other torus and endow \(\mathcal X = [V/T_V]\) with the trivial \(T\) action. Then
\begin{align*} [V/T_V]^T = \coprod_{\chi\in \Hom(T, T_V)} [V^\chi/T_V] \end{align*}where \(V^\chi\subset V\) is the fixed locus of \(V\) under the \(T\)-action given by \(\chi:T\to T_V\). Each component \([V^\chi/T_V]\) is an open and closed substack of \([V/T_V]^T\).
Proof
Collect Definitions. Technically a “trivial action” is a “weakly trivial action” [Rom22, Sec. 4.5] but \(\mathcal X^T\) is defined by a 2-universal property of the \(T\)-stack \((\mathcal X, \mu)\) where \(\mu\) is the action map on \(\mathcal X\), and so \(T\)-isomorphic actions have equivalent fixed stacks. This means we can take \(\mu = \pr_2\), the projection \(T\times \mathcal X\to \mathcal X\).
Objects of \(\mathcal X^T\) over \(S\) are pairs \((x, \{\alpha_g\})\) with \(x\in \mathcal X(S)\) and isomorphisms \(\alpha_g:g\cdot x\xrightarrow{\sim} x\) satisfying the cocycle condition and functoriality in \(g\in T(S')\), \(S'\to S\).
In this case, the trivial action means the cocycle condition specializes to a homomorphism; we can view the collection \(\{\alpha_g\}_{g\in T(S)}\) as a group homomorphism \(\alpha:T(S)\to \underline{\Aut}_S(x)\). The fiber of \(\eta:\mathcal X^T\to \mathcal X\) over \(S\) is the sheaf \(\underline{\Hom}(T_S, \underline{\Aut}_S(x))\).
Finally, a morphism \(\phi:(x,\alpha) \to (x', \alpha')\) in \(\mathcal X^T\) is a morphism \(\phi:x\to x'\) in \(\mathcal X\) with \(\phi\circ \alpha_g = \alpha'_g \circ \phi\), this is a condition on \(\phi\), not extra data – so there can only be FEWER morphisms in \(\mathcal X^T\) than in \(\mathcal X\).
Torsor Automorphisms An object in \(\mathcal X(S) = [V/T_V](S)\) is a \(T_V\)-torsor \(P\to S\) with a \(T_V\)-equivariant map \(f:P\to V\). An automorphism of \(x = (P, f)\) is a \(T_V\)-equivariant map \(\phi:P\to P\) so that
commutes. For each \(h\in T_V(S)\), the morphism \(\phi_h(p) = h\cdot p\) is a \(T_V\)-equivariant automorphism of the space \(P\). This actually requires the commutativity of \(T_V\), because equivariance demands that
\begin{align*} \phi_h(h'\cdot p) = h\cdot (h'\cdot p) = (h\cdot h') \cdot p = (h' \cdot h) = h'\phi_h(g) \end{align*}which holds if and only if \(h\cdot h' = h' \cdot h\) (for noncommutative groups, \(\phi_h\) is only a torsor automorphism for central \(h\)).
And actually, all the automorphisms of \(P\) itself are of this form. If \(\phi:P\to P\) is any other \(T_V\)-equivariant automorphism of \(P\), then choose a local trivialization of \(P\) over \(S'\) – this is equivalent to chooseing a section \(s:S' \to P\times_S S' = P_{S'}\). Since \(\phi(s)\) and \(s\) are two points of \(P_{S'}\) over the same base, there is some \(h_s\) such that \(h_s\cdot s = \phi(s)\). But now if we choose any other point \(p\in P_{S'}\), we can find a unique \(g\in T_V(S')\) such that \(p = g\cdot s\), and then
\begin{align*} \phi(p) = \phi(g\cdot s) = g\cdot \phi(s) = g\cdot (h_s\cdot s) = h_s\cdot (g\cdot s) = h_s\cdot p \end{align*}so \(\phi\) is merely multiplication by \(h_s\in T_V(S')\) over \(S'\). Descent then shows it is the same \(h_s\) on every section, hence \(\phi = h_s\cdot (-) \). To upgrade \(\phi_h\) to a \(T_V\)-torsor automorphism of \(P\), we need \(f = f\circ \phi_h\), or in other words \(f(p) = f(hp) = h\cdot f(p)\) for all \(p\in P\). Thus
\begin{align*} \underline{\Aut}_S((P, f)) &= \{h\in T_{V,S} ~ \mid ~ f(p) = h\cdot f(p) \text{ for all } p\in P\} \\& = \Stab_{T_V}(f(P)) \\ &\subseteq T_{V,S}. \end{align*}What this means for us is that if we have a point \((x,\alpha) \in \mathcal X^T(S)\), then \(x = (P, f)\) is some torsor over \(S\) and \(\alpha:T_S\to \underline{\Aut}_S(x)\) is a homomorphism. Composing \(\alpha\) with the inclusion \(\underline{\Aut}_S(x)\hookrightarrow T_{V,S}\) yields a homomorphism \(\chi_S:T_S\to T_{V,S}\). Thus, every point \((x,\alpha) \in \mathcal X^T(S)\) has a “type” \(\chi_S\).
Rigidity of tori gives disjoint union of stacks The sheaf \(\underline{\Hom}_{S-gp}(T_S, T_{V,S})\) is the constant sheaf \(\Hom(T, T_V)\). This means that the map \(\chi_S:T_S\to T_{V,S}\) is locally constant over \(S\), and on a connected component of \(S\) it is a fixed \(\chi\in \Hom(T, T_V)\). Define \(\mathcal X^T_\chi\) to be the full substack of objects whose type is constant and equal to \(\chi\).
Note that there are no morphisms between objects with different types. If \(\phi:(x,\alpha)\to (x',\alpha')\) is a morphism in \(\mathcal X^T\) between objects of types \(\chi\) and \(\chi'\) respectively, then \(\phi:P\to P'\) is a torsor isomorphism and we get \(\phi\circ \phi_{\chi(g)} = \phi_{\chi'(g)}\circ \phi\). For \(p\in P\), equivariance gives \(\phi(\chi(g)\cdot p) = \chi(g)\cdot \phi(p)\), and then the compatibility condition gives
\begin{align*} \phi(\chi(g)\cdot p) = \chi'(g)\cdot \phi(p), \end{align*}so \(\chi(g)\cdot \phi(p) = \chi'(g)\cdot \phi(p)\) in \(P'\) and freeness of the action gives \(\chi(g) = \chi'(g)\). Hence
\begin{align*} \chi^T = \coprod_{\chi} \mathcal X^T_{\chi} \end{align*}where each \(\mathcal X^T_{\chi}\) is open and closed.
Structure of \(\mathcal X^T_\chi\) Fix a \(\chi\in \Hom(T, T_V)\) now and consider a point \((x, \alpha) \in \mathcal X^T_\chi\). By identifying \(\underline{\Aut}(x)\) with its image in \(\Hom(T, T_V)\) we get \(\alpha = \chi\), and then it remains to decide what kind of point \(x\) might be. We need that \(\phi_\chi:P\to P\) is an automorphism of \(x = (P, f)\), which means \(\phi_\chi\) must preserve \(f\). If \(x\) is in the open orbit of \(T_V\) then \(\underline{\Aut}(x) = 1\) and only \(\chi = 0\) is admissable. With \(\chi\) fixed, we need
\begin{align*} \chi(g)\cdot f(p) = f(p)\quad \text{for all ~}p\in P(S'), g\in T(S'). \end{align*}This means the image of \(f\) must land inside \(V^\chi\subseteq V\), the fixed locus of \(V\) under the \(T\)-action given by \(\chi:T\to T_V\). That is, \(x = (P,f)\) may as well be a \(T_V\)-torsor over \(V^\chi\). Thus,
\begin{align*} \mathcal X^T_\chi = [V^\chi/T_V]. \end{align*}