Let $A$ be an abelian manifold. Let $T$ be an abelian manifold and complex torus. Suppose $A\to T$ is surjective holomorphic homomorphism of abelian manifold.
$\textbf{Q:}$ Do I need $T$ to be an abelian variety to start with? What is the meaning of quotient torus in $A\to T$ sense here? Since $A$ is isogenous to $T\times T'$ where $T'$ is another abelian manifold, do I call $T$ quotient? Since isogeny says there is still a finite kernel, I cannot really say $\frac{A}{T'}\cong T$ in any sense.(Normally I would expect quotient as normal quotient in group.) What is the meaning of the quotient torus?