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Over an algebraically closed field $k$, an ample line bundle $L$ on an abelian variety $A$ defines a homomorphismwhere $A^{\vee }$ is the dual abelian variety and $t_x$ is translation by $x$. This isogeny is a polarization of an abelian variety. Over a general field, a polarization is an isogeny that has this form after extending to an algebraic closure; an inducing ample line bundle need not descend to the original field.^[ref]

A principal polarization is an isomorphism $A\to A^{\vee }$. An elliptic curve with its degree-one origin divisor gives a basic example. A polarization supplies positivity data relating the geometry of line bundles to the algebra of isogenies; twisting $L$ by a degree-zero line bundle does not change ${\lambda }_L$.

A principal polarization of an abelian variety $A$ is a polarization of an abelian variety $\lambda :A\to A^{\vee }$ that is an isomorphism. Equivalently, its isogeny has degree one. It identifies $A$ with its dual abelian variety through positivity supplied by an ample line bundle; an arbitrary isomorphism with the dual need not be a polarization of an abelian variety.^[ref]

For an elliptic curve $E$ with origin $0$, the degree-one line bundle ${\mathcal{O}}_E(0)$ gives a principal polarization. Replacing it by ${\mathcal{O}}_E(n\cdot 0)$, for positive $n$, multiplies the corresponding map to the dual by $n$ and gives degree $n^2$. Thus positivity alone does not make a polarization of an abelian variety principal: the size of its kernel matters.