= Polarization of an abelian variety
{title2=$\lambda_L:A\to A^\vee$}
Over an algebraically closed <field> $k$, an ample <line bundle> $L$ on an <abelian variety> $A$ defines a <homomorphism>
$$
\lambda_L:A\longrightarrow A^\vee,\qquad x\longmapsto t_x^*L\otimes L^{-1},
$$
where $A^\vee$ 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>.<Abelian Varieties by J. S. Milne (course notes)>[Chapter I, section 11, Polarizations and Invertible Sheaves]{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 $\lambda_L$.
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