In characteristic zero we have the following decomposition theorem for polarized abelian varieties: it gives an isomorphism between a PPAV and a product of PPAV's of lower dimension and is valid (as far as I have been able to find) in characteristic 0. I would like to know whether or not this holds in characteristic $p>0$. Below is the statement as it can be found in [Lange-Birkenhake, Theorem 4.3.1].
Let $(X,L)$ be a principally polarized complex abelian variety and decompose the linear system $|L|$ as $$|L|=|M|+F_1+\cdots+F_r$$ where $M$ is the moving part and $F_1+\cdots+F_r$ is the decomposition of the fixed part into irreducible components. Denote $N_{\ell}=O_X(F_{\ell})$.
Denote by $p_M:M\rightarrow X_M=X/K(M)_0$ and $p_{N_{\ell}}:N \rightarrow X_{N_{\ell}}=N/K(N_{\ell})_0$ the canonical projections, where for a pair $(X,L\in Pic(X))$, if $\phi_L:X\rightarrow \hat{X}$ denotes the map $x\mapsto \phi_L(x)=t_x^{\ast}L\otimes L^{-1}$, we denote by $K(X)_0$ the connected component of $\ker\phi_L$ containing $0$.
There are positive line bundles $\bar{M}\in Pic(X_M)$ and $\bar{N}_{\ell}\in Pic(X_{N_{\ell}})$ such that $M=p_M^{\ast}\bar{M}$ and $N_{\ell}=p_{N_{\ell}}^{\ast}\bar{N}_{\ell}$ and the pairs $(X_M,\bar{M})$ and $(X_{N_{\ell}},\bar{N}_{\ell})$ are polarized abelian varieties. \medskip
Consider the product $X_M\times X_{N_1} \times \cdots \times X_{N_r}$ and denote be $q_M$ and $q_{N_{\ell}}$ the corresponding projections. Then we have:
Decomposition Theorem. The map $$(p_M,p_{N_1},\ldots,p_{N_r}):X\longrightarrow X_M\times X_{N_1} \times \cdots \times X_{N_r}$$ is an isomorphism of polarized abelian varieties $$(X,L)\simeq (X_M\times X_{N_1} \times \cdots \times X_{N_r}, q_M\bar{M}\otimes q_{N_1}^{\ast}\bar{N}_1 \otimes \cdots \otimes q_{N_r}^{\ast}\bar{N}_r)$$
Thanks in advance for any insights.
