When is an isogeny $F\colon A \rightarrow A$ a closed map?

Question

Let $A$ be an abelian variety and $F\colon A \rightarrow A$ an isogeny. Is it a closed map?

Answer 1

It follows from the following Lemma: Let $\phi:X \to Y$ and $\psi: Y \to Z$ be morphisms of varieties (so in particular separated), if $\psi \circ \phi$ is proper then $\phi$ is proper.

This implies that any morphism of varieties between proper varieties must itself be proper, hence closed.