Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and residue field. Let $S=\mathrm{Spec}R$ and $S_n=\mathrm{Spec} R/(\pi)^{n+1}$. For any $S$-scheme $V$ we denote the scheme $V\times_S S_n$ as $V_n$.
In particular, $A_K$ has semistable reduction, $A_K$ extends to a semiabelian scheme $G$ over $S=\mathrm{Spec} R$, and the corresponding Raynaud extension is just a torus $T$ over $S$, analytically $A_K=T_K/Y_K$, and we have a pairing $Y_K\times X_K\rightarrow \mathbb{G}_{m,K}$, where $Y_K$ is the periods lattice and $X$ is the character group of $T$. Moreover, we assume both $X$ and $Y$ (the character group of the Raynaud extension associated to the dual abelian variety $A_K^{\vee}$ which gives $Y_K$) are free $\mathbb{Z}$-modules.
Starting from this, we can construction lots of projective models of $A_K$ by using Mumford's construction, see Mumford compositio 1972, Faltings-Chai's book, the 1999 Tohoku paper of Aleexev and Nakamura. Given a nice (being nice see [Mum1973]section 6) $Y_K$-invariant polyhedral decomposition $\Sigma$ of the affine space $E=\mathrm{Hom}(X,\mathbb{Q})$, we can construct a projective model $P_{\Sigma}$ of $A_K$ and cut out the semiabelian scheme $G$ from $P_{\Sigma}$. Over $S_n$ we have group action $G_n\times_{S_n} P_{\Sigma,n}\rightarrow P_{\Sigma,n}$.
Now the question is: could we have an action of $G$ on $P_{\Sigma}$ which extends the group operation on $G$ and the actions on the infinitesimal fibres? It seems to me in Aleexev's 2002 Annals paper, the answer is yes, at least for certain canonical models obtained from a given theta divisor of some ample line bundle (since $A_K$ extends to a stable semiabelic scheme over $S$ once you are given a ample line bundle on $A_K$, and being a stable semiabelic scheme implies you have an action on the model, see Theorem 5.7.1 case 2 in that paper), but I couldn't find a concrete argument for this in the paper.
Could anyone help me point out where the hidden argument is? Or give me some other reference. I know in dimension $1$ case, the paper by Deligne and Rapoport describes this action in order to define generalised elliptic curves via a different approach (using $\mathrm{Pic}^0$), but seems not easy to generalise to hight dimensional case.
