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Abelian varieties are projective algebraic varieties endowed with an Abelian group structure. Over the complex numbers, they can be described as quotients of a vector space by a lattice of full rank. They are analogs in higher dimensions of elliptic curves, and play an important role in algebraic geometry and number theory.
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non principally polarized complex abelian varieties
Here is another construction, followed by some comments on how to solve the existence problem in general.
If $A$ is a $g$-dimensional principally polarized abelian variety over $\mathbf{C}$ with $ …