= Principal polarization
{title2=$\lambda:A\xrightarrow{\sim}A^\vee$}
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>.<Abelian Varieties by J. S. Milne (course notes)>[Chapter I, section 11, definition of a principal polarization]{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 $\mathcal O_E(n\cdot0)$, 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.
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