A note on dual abelian varieties

Abstract

For any non-principal polarisation D, we explicitly construct D-polarised abelian variety A, such that its dual abelian variety is not (abstractly) isomorphic to A. For $\dim \left(A\right)>3$ the construction includes examples with submaximal Picard number equal to ${(\dim (A)-1)}^2+1$. As a corollary, we show that a very general non-principally polarised abelian variety is not isomorphic to its dual. Moreover, we show an example of an abelian variety that is isomorphic to its dual, yet it does not admit a principal polarisation.