Complex torus
In mathematics, a complex torus is a particular kind of complex manifold "M" whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number "N" circles). Here "N" must be the even number 2"n", where "n" is the complex dimension of "M". All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to C considered as real vector space; then the quotient group formula_1 is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For "n  = 1 this is the classical period lattice construction of elliptic curves. For n" > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when "n" > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for "n" = 1. Computer algebra can handle cases for  small "n" reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space. Definition. One way to define complex tori is as a compact connected complex Lie group formula_2. These are Lie groups where the structure maps are holomorphic maps of complex manifolds. It turns out that all such compact connected Lie groups are commutative, and are isomorphic to a quotient of their Lie algebra formula_3 whose covering map is the exponential map of a Lie algebra to its associated Lie group. The kernel of this map is a lattice formula_4 and  formula_5. Conversely, given a complex vector space formula_6 and a lattice formula_7 of maximal rank, the quotient complex manifold formula_8 has a complex Lie group structure, and is also compact and connected. This implies the two definitions for complex tori are equivalent. Period matrix of a complex torus. One way to describe a complex torus is by using a formula_9 matrix formula_10 whose columns correspond to a basis formula_11 of the lattice formula_12 expanded out using a basis formula_13 of formula_6. That is, we writeformula_15soformula_16We can then write the torus formula_17 asformula_18If we go in the reverse direction by selecting  a matrix formula_19, it corresponds to a period matrix if and only if the corresponding matrix formula_20 constructed by adjoining the complex conjugate matrix formula_21 to formula_10, soformula_23is nonsingular. This guarantees the column vectors of formula_10 span a lattice in formula_25 hence must be linearly independent vectors over formula_26. Example. For a two-dimensional complex torus, it has a period matrix of the formformula_27for example, the matrixformula_28forms a period matrix since the associated period matrix has determinant 4. Normalized period matrix. For any complex torus formula_17 of dimension formula_30 it has a period matrix formula_10 of the formformula_32where formula_33 is  the identity matrix and formula_34 where formula_35. We can get this from taking a change of basis of the vector space formula_6 giving a block matrix of the form above. The condition for formula_35 follows from looking at the corresponding formula_38-matrixformula_39since this must be a non-singular matrix. This is because if we calculate the determinant of the block matrix, this is simplyformula_40which gives the implication. Example. For example, we can write a normalized period matrix for a 2-dimensional complex torus asformula_41one such example is the normalized period matrixformula_42since the determinant of formula_43 is nonzero, equal to formula_44 . 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 formula_17 and formula_46 of dimensions formula_47 then a homomorphism of complex tori is a functionformula_48such that the group structure is preserved. This has a number of consequences, such as every homomorphism induces a map of their covering spacesformula_49which is compatible with their covering maps. Furthermore, because formula_50 induces a group homomorphism, it must restrict to a morphism of the latticesformula_51In  particular, there are injectionsformula_52 and formula_53which are called the analytic and rational representations of the space of homomorphisms. These are useful to determining some information about the endomorphism ring formula_54 which has rational dimension formula_55. 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 formula_56 we define the translation mapformula_57 sending formula_58Then, if formula_59 is a holomorphic map between complex tori formula_60, there  is a unique homomorphism formula_61 such thatformula_62showing the holomorphic maps are not much larger than the set of homomorphisms of complex tori. Isogenies. One distinct class of homomorphisms of complex tori are called isogenies. These are endomorphisms of complex tori with a non-zero kernel. For example, if we let formula_63 be an integer, then there is an associated mapformula_64 sending formula_65which has kernelformula_66isomorphic to formula_67. Isomorphic complex tori. There is an isomorphism of complex structures on the real vector space formula_68 and the setformula_69and isomorphic tori can be given by a change of basis of their lattices, hence a  matrix in formula_70. This gives the set of isomorphism classes of complex tori of dimension formula_30, formula_72, as the Double coset spaceformula_73Note that as a real manifold, this has dimensionformula_74this is important when considering the dimensions of moduli of Abelian varieties, which shows there are far more complex tori than Abelian varieties. Line bundles and automorphic forms. For complex manifolds formula_75, in particular complex tori, there is a construction relating the holomorphic line bundles formula_76 whose pullback formula_77 are trivial using the group cohomology of formula_78. Fortunately for complex tori, every complex line bundle formula_79 becomes trivial since formula_80. Factors  of automorphy. Starting from the first group cohomology groupformula_81we recall how its elements can be represented. Since formula_78 acts on formula_83 there is an induced action on all of its sheaves, hence onformula_84The formula_78-action can then be repsented as a holomorphic map formula_86. This map satisfies the cocycle condition ifformula_87for every formula_88 and formula_89. The abelian group of 1-cocycles formula_90 is called the group of factors of automorphy. Note that such functions formula_91 are also just called factors. On complex tori. For complex tori, these functions formula_91 are given by functionsformula_93which follow the cocycle condition. These are automorphic  functions, more precisely, the automorphic functions used in the transformation laws for theta functions. Also, any such map can be written asformula_94forformula_95which is useful for computing invariants related to the associated line bundle. Line bundles from factors of automorphy. Given a factor of automorphy formula_91 we can define a line bundle on formula_75 as follows: the trivial line bundle formula_98 has a formula_78-action given byformula_100for the factor formula_91. Since this action is free and properly discontinuous, the quotient bundleformula_102is a complex manifold. Furthermore, the projection formula_103 induced from the covering projection formula_104. This gives a mapformula_105which induces an isomorphismformula_106giving  the desired result. For complex tori. In the case of complex tori, we have formula_107 hence there is an isomorphismformula_108representing line bundles on complex tori as 1-cocyles in the associated group cohomology. It is typical to write down the group formula_78 as the lattice formula_12 defining formula_75, henceformula_112contains the isomorphism classes of line bundles on formula_75. First chern class of line bundles on complex tori. From the exponential exact sequenceformula_114the connecting morphismformula_115is the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out there is an isomorphism between  formula_116 and the module of alternating forms on the lattice formula_12, formula_118. Therefore, formula_119 can be considered as an alternating formula_120-valued 2-form formula_121 on formula_12. If formula_123 has factor of automorphy formula_124 then the alternating form can be expressed asformula_125for formula_126 and formula_127. Example. For a normalized period matrixformula_128expanded using the standard basis of formula_129 we have the column vectors defining the lattice formula_130. Then, any alternating form formula_121 on formula_12 is of the formformula_133where a number of compatibility conditions must be satisfied. Sections of line bundles and theta functions. For a line bundle formula_123 given by a  factor of automorphy formula_135, so formula_136 and formula_137, there is an associated sheaf of sections formula_138 whereformula_139with formula_140 open. Then, evaluated on global sections, this is the set of holomorphic functions formula_141 such thatformula_142which are exactly the theta functions on the plane. Conversely, this process can be done backwards where the automorphic factor in the theta function is in fact the factor of automorphy defining a line bundle on a complex torus. Hermitian forms and the Appell-Humbert theorem. For the alternating formula_120-valued 2-form formula_121 associated to the line bundle formula_76, it can be extended to be formula_26 -valued. Then, it turns out any formula_26-valued alternating form formula_148 satisfying the following conditions is the extension of some first Chern class formula_119 of a line bundle formula_76. Moreover, there is an associated Hermitian form formula_154 satisfying for any formula_157. Neron-Severi group. For a complex torus formula_17 we can define the Neron-Serveri group formula_159 as the group of Hermitian forms formula_160 on formula_6 withformula_162Equivalently, it is the image of the homomorphismformula_115from the first Chern class. We can also identify it with the group of alternating real-valued alternating forms formula_164 on formula_6 such that formula_149. Example of  a Hermitian form on an elliptic curve. For an elliptic curve formula_167 given by the lattice formula_168 where formula_169 we can find the integral form formula_170 by looking at a generic alternating matrix and finding the correct compatibility conditions for it to behave as expected. If we use the standard basis formula_171 of formula_172as a real vector space (so formula_173), then we can write out an alternating matrix formula_174 and calculate the associated products on the vectors associated to formula_175. These are formula_176 Then, taking the inner products (with the standard inner product) of these vectors with the vectors formula_175  we get formula_178 so if formula_179, then formula_180 We can then directly verify formula_181, which holds for the matrix above. For a fixed formula_182, we will write the integral form as formula_183. Then, there is an associated Hermitian formformula_184given byformula_185 where formula_186 Semi-character pairs for Hermitian forms. For a Hermitian form formula_160 a semi-character is a mapformula_188such thatformula_189hence the map formula_190 behaves like a character twisted by the Hermitian form. Note that if formula_160 is the zero element in formula_159, so it corresponds to the trivial line bundle formula_193, then the associated semi-characters are the group of  characters on formula_12. It will turn out this corresponds to the group formula_195 of degree formula_196 line bundles on formula_75, or equivalently, its dual torus, which can be seen by computing the group of charactersformula_198whose elements can be factored as mapsformula_199showing a character is of the formformula_200for some fixed dual lattice vector formula_201. This gives the isomorphismformula_202of the set of characters with a real torus. The set of all pairs of semi-characters and their associated Hermitian form formula_203, or semi-character pairs, forms a group formula_204 whereformula_205This group structure comes from applying the previous commutation law for semi-characters  to the new semicharacter formula_206:formula_207It turns out this group surjects onto formula_159 and has kernel formula_209, giving a short exact sequenceformula_210This surjection can be constructed through associating to every semi-character pair a line bundle formula_211. Semi-character pairs and line bundles. For a semi-character pair formula_212 we can construct a 1-cocycle formula_213 on formula_12 as a mapformula_215defined asformula_216The cocycle relationformula_217can be easily verified by direct computation. Hence the cocycle determines a line bundleformula_218where the formula_12-action on formula_220 is given byformula_221Note this action can be used to show the sections of the line bundle formula_211 are  given by the theta functions with factor of automorphy formula_213. Sometimes, this is called the canonical factor of automorphy for formula_123. Note that because every line bundle formula_76 has an associated Hermitian form formula_160, and a semi-character can be constructed using the factor of automorphy for formula_123, we get a surjectionformula_228Moreover, this is a group homomorphism with a trivial kernel. These facts can all be summarized in the following commutative diagramformula_229where the vertical arrows are isomorphisms, or equality. This diagram is typically called the Appell-Humbert theorem. Dual complex torus. As mentioned before, a character on the lattice can  be expressed as a functionformula_200for some fixed dual vector formula_231. If we want to put a complex structure on the real torus of all characters, we need to start with a complex vector space which formula_232 embeds into. It turns out that the complex vector spaceformula_233of complex antilinear maps, is isomorphic to the real dual vector space formula_234, which is part of the factorization for writing down characters. Furthermore, there is an associated latticeformula_235called the dual lattice of formula_12. Then, we can form the dual complex torusformula_237which has the special property that that dual of the dual complex torus is  the original complex torus. Moreover, from the discussion above, we can identify the dual complex torus with the Picard group of formula_75formula_239by sending an anti-linear dual vector formula_240 toformula_241giving the mapformula_242which factors through the dual complex torus. There are other constructions of the dual complex torus using techniques from the theory of Abelian varieties. Essentially, taking a line bundle formula_123 over a complex torus (or Abelian variety) formula_75, there is a closed subset formula_245 of formula_75 defined as the points of formula_247 where their translations are invariant, i.e.formula_248Then, the dual complex torus can be constructed asformula_249presenting it as an  isogeny. It can be shown that defining formula_250 this way satisfied the universal properties of formula_195, hence is in fact the dual complex torus (or Abelian variety). Poincare bundle. From the construction of the dual complex torus, it is suggested there should exist a line bundle formula_252 over the product of the torus formula_75 and its dual which can be used to present all isomorphism classes of degree 0 line bundles on formula_75. We can encode this behavior with the following two properties where the first is the property discussed above, and the second acts as a normalization property. We  can construct formula_252 using the following hermitian formformula_260and the semi-characterformula_261for formula_160. Showing this data constructs a line bundle with the desired properties follows from looking at the associated canonical factor of formula_212, and observing its behavior at various restrictions.