Remark 39.9.12. Let $k$ be a field. There are $2 \times 4 \times 2 = 16$ equivalent definitions of abelian varieties. Let

• projective, proper,

• geometrically irreducible, irreducible, geometrically connected, connected,

• smooth, geometrically reduced

be three sets of properties, pick one from each of them, and let $A$ be a group scheme over $k$ with the chosen properties over $k$. Then $A$ is an abelian variety. If we pick the options “proper, geometrically irreducible, geometrically reduced”, then we recover Definition 39.9.1 (use Varieties, Lemma 33.9.2). The weakest possible options would be “proper, connected, and geometrically reduced”, see for example Morphisms, Lemma 29.43.5 and Varieties, Lemma 33.25.4. So say $A$ is a proper, connected, and geometrically reduced group scheme over $k$. Then $A$ is geometrically irreducible by Lemmas 39.7.10 and 39.7.4 and hence an abelian variety. Finally, if $A/k$ is an abelian variety, then it is projective and smooth over $k$ (Proposition 39.9.11), whence satisfies the strongest possible options "projective, geometrically irreducible, smooth".

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