# Difference between stabilizer and automorphism group

Let $$X$$ be a smooth closed subvariety of a complex abelian variety $$A$$. Assume $$X$$ is of general type and of codimension one with $$\omega_X$$ ample.

Often, people speak about the stabilizer $$\mathrm{Stab}_A(X)$$ of $$X$$ in $$A$$. This is the group of $$a$$ in $$A$$ such that $$X+a = X$$.

What is the relation of $$\mathrm{Stab}_A(X)$$ to $$\mathrm{Aut}(X)$$?

They are both finite. Are they equal? If the stabilizer is trivial, does that imply $$\mathrm{Aut}(X)$$ is trivial? What about vice versa?