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6.1. Abelian varieties 1616.1 Abelian varieties6.1.1 Definition and first propertiesDefinition 6.1.1 (Abelian variety). An abelian variety is a connected and complete algebraicgroup variety.For example, elliptic curves are abelian varieties of dimension 1. We usually denote by g thedimension of an abelian variety.The rigidity lemma [204, Proposition I.1.1], [132, Lemma A.7.1.1] implies that every morphismbetween abelian varieties is the composite of a homomorphism and a translation and that thegroup law on an abelian variety is commutative.Proposition 6.1.2 (Commutativity [213, II.4], [204, Corollary I.1.4], [132, Corollary A.7.1.3]).Let A be an abelian variety. Then its group law is commutative.It is a consequence of the more difficult theorem of the cube [213, II.6], [204, Theorem I.5.1],[132, Corollary A.7.2.2] that every abelian variety is projective.Theorem 6.1.3 (Projectivity [213, II.6], [204, Theorem I.6.4], [132, Corollary A.7.2.1]). Let Abe an abelian variety. Then A is projective.Hence, we could have just defined abelian varieties as projective varieties with a commutativegroup law. Nonetheless, this is not the historical approach.As was the case for elliptic curves, complex abelian varieties are also isomorphic to complextori. Indeed, over the complex numbers, an abelian variety A is a compact connected complex Liegroup and is equipped with the so-called exponential map exp from the tangent space V ≃ C 2gat the zero element to itself [168, 15.4]:exp : V → A .A fundamental fact is that this map is surjective and its kernel is a lattice Λ in V .Theorem 6.1.4 ([213, I.1], [204, Proposition I.2.1]). Let A be a complex abelian variety and Vthe tangent space at 0. Then the exponential map is a surjective homomorphism of complex Liegroups with kernel Λ a lattice in V .HenceA ≃ V/Λis a complex torus.As soon as g > 1, the converse of this theorem is however not true anymore. All complex toriare indeed not projective anymore [65, Exercice III.3], [242].6.1.2 Theta functions and Riemann formsTo characterize the complex tori which are projective, and so are actually complex abelian varietiesa powerful tool is theta functions.Definition 6.1.5 (Theta function [213, I.3], [65, Définition 1.1]). Let V be a vector space and Λa lattice in V . A theta function θ associated with Λ is a non-zero function such that there existlinear forms a λ and constants b λ for every λ ∈ Λ verifyingθ(z + λ) = e 2iπ(a λ(z)+b λ ) θ(z)for every z ∈ V . The theta function is said to be of type (a λ , b λ ) λ∈Λ .