if the corresponding matrix constructed by adjoining the complex conjugate matrix to , so is nonsingular. This guarantees the column vectors of span a lattice in hence must be linearly independent vectors over . Example For a two-dimensional complex torus, it has a period matrix of the form for example, the matrix forms a period matrix since the associated period matrix has determinant 4. Normalized period matrix For any complex torus of dimension it has a period matrix of the form where is the identity matrix and where . We can get this from taking a change of basis of the vector space giving a block matrix of the form above. The condition for follows from looking at the corresponding -matrix since this must be a non-singular matrix. This is because if we calculate the determinant of the block matrix, this is simply which gives the implication. Example For example, we can write a normalized period matrix for a 2-dimensional complex torus as one such example is the normalized period matrix since the determinant of is nonzero, equal to . Period matrices of Abelian varieties To get a period matrix which gives a projective complex manifold, hence an algebraic variety, the period matrix needs to further satisfy the Riemann bilinear relations. Homomorphisms of complex tori If we have complex tori and of dimensions then a homomorphismpg 11 of complex tori is a function such that the group structure is preserved. This has a number of consequences, such as every homomorphism induces a map of their covering spaces which is compatible with their covering maps. Furthermore, because induces a group homomorphism, it must restrict to a morphism of the lattices In particular, there are injections and which are called the analytic and rational representations of the space of homomorphisms. These are useful to determining some information about the endomorphism ring which has rational dimension . Holomorphic maps of complex tori The class of homomorphic maps between complex tori have a very simple structure. Of course, every homomorphism induces a holomorphic map, but every holomorphic map is the composition of a special kind of holomorphic map with a homomorphism. For an element we define the translation map sending Then, if is a holomorphic map between complex tori , there is a unique homomorphism such that showing the holomorphic maps are not much larger than the set of homomorphisms of complex tori. Isogenies