A result of Serre-Tate states that we can canonically lift an ordinary abelian variety over a perfect field $k$ of positive characteristic to an abelian scheme over the ring of Witt vectors of $k$ and that we can also lift an endomorphism of the variety uniquely to an endomorphism of the canonical lifting.

In the case where $k$ is finite, is the canonical lift a variety? Do we know its dimension? I'd be glad to read your answers and the references to those answers as well.