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Complex torus

A complex torus is a compact complex manifold of dimension g defined as the quotient \mathbb{C}^g / \Lambda, where \Lambda \subset \mathbb{C}^g is a lattice, meaning a discrete additive subgroup isomorphic to \mathbb{Z}^{2g}.^[1]^[2] This construction inherits from \mathbb{C}^g the structure of an abelian complex Lie group, with the group operation given by addition modulo \Lambda.^[3]^[4] Complex tori generalize elliptic curves, which correspond precisely to the one-dimensional case (g=1), where every such torus admits a projective embedding as a smooth cubic curve in \mathbb{P}^2.^[5]^[6] In higher dimensions, not all complex tori are projective; those that are form the class of abelian varieties over \mathbb{C}, characterized by the existence of an ample line bundle.^[2]^[7] The Jacobians of compact Riemann surfaces of genus g are principally polarized abelian varieties, a special class of complex tori, linked to the cohomology of the surfaces via the Abel-Jacobi map.^[2] These objects are central to several fields: in algebraic geometry, they parametrize families of abelian varieties via the moduli space \mathcal{A}_g; in number theory, they underpin the study of complex multiplication, where endomorphisms by rings of algebraic integers yield special loci with arithmetic significance; and in complex analysis, they serve as domains for theta functions and elliptic functions in higher dimensions.^[6]^[5] Their cohomology rings and Hodge structures further connect them to topological invariants and mirror symmetry conjectures.^[1]^[2]

Fundamentals

Definition

A complex torus of dimension g is defined as the quotient space \mathbb{C}^g / \Lambda, where \Lambda \subset \mathbb{C}^g is a lattice, i.e., a discrete additive subgroup isomorphic to \mathbb{Z}^{2g} as an abelian group. The natural projection \mathbb{C}^g \to \mathbb{C}^g / \Lambda is a local biholomorphism, making the quotient a compact complex manifold of dimension g that inherits a complex structure from \mathbb{C}^g. Moreover, it forms an abelian complex Lie group under the operation of addition modulo \Lambda. Lattices admit a basis \{ \omega_1, \dots, \omega_g, \eta_1, \dots, \eta_g \}, and two lattices yield isomorphic tori if and only if their bases are related by a complex linear transformation with determinant in \mathbb{Z}^\times = \{\pm 1\}.^[4]

Examples

A fundamental example of a complex torus arises in one dimension, where it coincides with an elliptic curve defined as the quotient space \mathbb{C} / (\mathbb{Z} + \tau \mathbb{Z}) for \tau \in \mathbb{C} with \operatorname{Im}(\tau) > 0. Such tori are classified up to isomorphism over \mathbb{C} by the j-invariant, a holomorphic function on the upper half-plane that serves as a complete modulus for elliptic curves.^[8] A simple explicit construction is the square torus, obtained by taking \tau = i, so the lattice is \mathbb{Z} + i\mathbb{Z}, the ring of Gaussian integers.^[5] This yields an elliptic curve with j-invariant $1728$. In higher dimensions, complex tori include products of lower-dimensional ones; for instance, a two-dimensional complex torus can be the product of two elliptic curves, \left( \mathbb{C} / (\mathbb{Z} + \tau_1 \mathbb{Z}) \right) \times \left( \mathbb{C} / (\mathbb{Z} + \tau_2 \mathbb{Z}) \right).^[8] To build intuition, consider the analogy with the real torus, which is topologically the quotient \mathbb{R}^2 / \mathbb{Z}^2 \cong S^1 \times S^1; while sharing a lattice quotient structure, the real case lacks the compatible complex structure that endows the complex torus with its holomorphic properties.^[9]

Representation

Period Matrix

A complex torus of dimension n can be represented using a period matrix derived from a suitable basis of its defining lattice. Let T = \mathbb{C}^n / \Lambda, where \Lambda \subset \mathbb{C}^n is a lattice, meaning a discrete subgroup isomorphic to \mathbb{Z}^{2n} that spans \mathbb{C}^n over \mathbb{R}. Choose a \mathbb{Z}-basis \{\omega_1, \dots, \omega_{2n}\} for \Lambda. Fixing the standard \mathbb{C}-basis for \mathbb{C}^n, the period matrix \Omega is the n \times 2n complex matrix whose columns are the coordinate vectors of \omega_1, \dots, \omega_n in the first block and \omega_{n+1}, \dots, \omega_{2n} in the second block, so \Omega = [\omega_1 \ \cdots \ \omega_n \ | \ \omega_{n+1} \ \cdots \ \omega_{2n}]. Different choices of \mathbb{Z}-basis for \Lambda yield equivalent representations of the torus up to isomorphism. Specifically, if \{\omega'_j\} is another \mathbb{Z}-basis, then there exists a unimodular integer matrix U \in \mathrm{GL}(2n, \mathbb{Z}) such that the corresponding period matrix is \Omega' = \Omega U. In block form, this action can be expressed as \Omega' = \Omega (A \ | \ B), where A, B \in M_n(\mathbb{Z}) are such that the block matrix has determinant \pm 1. This equivalence captures the freedom in basis selection while preserving the lattice structure. For the case of dimension n=1, an elliptic curve T = \mathbb{C} / \Lambda admits a period matrix \Omega = [1 \ \tau], where \tau \in \mathbb{C} with \operatorname{Im}(\tau) > 0 parametrizes the lattice \Lambda = \mathbb{Z} + \tau \mathbb{Z}. This form arises by choosing the basis \omega_1 = 1, \omega_2 = \tau, and ensures the fundamental domain is a parallelogram in the upper half-plane.

Normalization

To normalize the period matrix of a principally polarized complex torus (i.e., an abelian variety) \mathbb{C}^n / \Lambda, where \Lambda \subset \mathbb{C}^n is a lattice of rank $2n, one equips the underlying real vector space \mathbb{R}^{2n} (identifying \mathbb{C}^n \cong \mathbb{R}^{2n}) with the standard symplectic structure given by the alternating bilinear form defined by the matrix J = \begin{pmatrix} 0_n & I_n \\ -I_n & 0_n \end{pmatrix}. This form captures the intersection pairing on the homology group H_1(\mathbb{C}^n / \Lambda, \mathbb{Z}) \cong \Lambda, enabling the selection of a symplectic basis \{\omega_1, \dots, \omega_n, \eta_1, \dots, \eta_n\} for \Lambda such that the integrals of a basis of holomorphic differentials over these cycles yield the normalized period matrix \Omega = \begin{pmatrix} I_n & Z \end{pmatrix}, where Z = X + iY with X real symmetric, Y positive definite, and the Riemann bilinear relations ensuring \operatorname{Im}(Z) > 0.^[10]^[11] The set of all such Z \in M_n(\mathbb{C}) that are symmetric with positive definite imaginary part forms the Siegel upper half-space \mathcal{H}_n = \{ Z \in M_n(\mathbb{C}) \mid Z = Z^t, \operatorname{Im}(Z) > 0 \}, which parameterizes normalized period matrices of principally polarized abelian varieties up to isomorphism. The moduli space of such abelian varieties has complex dimension n(n+1)/2.^[10]^[11]^[12] In contrast, the moduli space of all complex tori (including non-projective ones) has complex dimension n^2, parametrized by non-symmetric Z with \operatorname{Im}(Z) > 0, up to the action of \mathrm{GL}(n, \mathbb{Z}) \times \mathrm{GL}(n, \mathbb{Z}). For n=1, this reduces to the classical upper half-plane \mathcal{H}_1 = \{ \tau \in \mathbb{C} \mid \operatorname{Im}(\tau) > 0 \}, where elliptic curves \mathbb{C}/\langle 1, \tau \rangle are classified up to isomorphism via the modular group action.^[10] Two normalized period matrices Z, Z' \in \mathcal{H}_n correspond to isomorphic principally polarized abelian varieties if and only if there exists M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in [\operatorname{Sp}(2n, \mathbb{Z})](/page/Sp)—the modular group preserving the symplectic form J, i.e., M J M^t = J—such that Z' = (A Z + B)(C Z + D)^{-1}, the standard fractional linear transformation action on the Siegel upper half-space.^[10]^[11] This action ensures that the normalization via symplectic bases provides a complete set of invariants for the moduli space of principally polarized abelian varieties of dimension n.^[12]

Relation to Abelian Varieties

A complex torus is an abelian variety precisely when it admits a projective embedding into complex projective space, which occurs if and only if it admits a principal polarization, corresponding to a period matrix that can be normalized such that the upper block consists of the identity matrix and the lower block is symmetric with positive definite imaginary part.^[13] This normalization ensures the existence of a positive-definite Hermitian form on the underlying complex vector space that is compatible with the lattice, allowing the torus to be realized as the complex points of an algebraic abelian variety.^[14] In higher dimensions, only a measure-zero subset of complex tori admit such a principal polarization; the Siegel upper half-space parametrizes these principally polarized abelian varieties, while the full moduli space of complex tori is larger.^[8] Central to this algebraic structure is the notion of principal polarization, which equips the torus with an ample line bundle whose first Chern class induces the standard alternating symplectic form on the first homology group H_1(T, \mathbb{Z}).^[14] This symplectic form arises from the imaginary part of a nondegenerate Riemann form on the complex vector space, ensuring that the polarization is of type (1, \dots, 1) and provides an isomorphism between the torus and its dual.^[13] Such a principal polarization guarantees the existence of sufficiently many global sections to embed the torus projectively, distinguishing algebraic tori from their non-projective counterparts.^[8] Jacobian varieties of compact Riemann surfaces exemplify complex tori that are abelian varieties, constructed as quotients \mathbb{C}^g / \Lambda where the period matrix derives from integrals of holomorphic differentials over a basis of homology cycles, yielding a specific Riemann form from the cup product on cohomology.^[8] These period matrices automatically satisfy the normalization due to the geometric origin of the symplectic structure on H_1(C, \mathbb{Z}), and the canonical theta divisor induces a principal polarization on the Jacobian.^[8] For genus g=1, every elliptic curve is a Jacobian, but in higher genus, Torelli's theorem asserts that the polarized Jacobian uniquely determines the curve.^[8] In contrast, non-algebraic complex tori arise when no ample line bundle exists, such as for generic period matrices in dimension g \geq 2, where the Siegel modular variety parametrizes only the algebraic ones, leaving most tori non-projective and thus outside the category of abelian varieties.^[13] For instance, a torus with a period matrix whose imaginary part lacks a compatible positive-definite Hermitian form with integer Riemann form admits no non-constant meromorphic functions and cannot be embedded projectively.^[8] These examples highlight the analytic flexibility of complex tori versus the rigid algebraic constraints of abelian varieties.^[14]

Morphisms

Holomorphic Maps

A holomorphic map f: \mathbb{C}^n / \Lambda \to \mathbb{C}^m / \Gamma between complex tori is defined as a holomorphic function satisfying the quotient structure. Such a map lifts uniquely (up to deck transformations) to a holomorphic function \phi: \mathbb{C}^n \to \mathbb{C}^m that is equivariant with respect to the lattices, meaning \phi(z + \lambda) - \phi(z) \in \Gamma for all z \in \mathbb{C}^n and \lambda \in \Lambda, or equivalently, \phi(\Lambda) \subset \Gamma. The map f is then given by f() = [\phi(z)], where [ \cdot ] denotes the class in the quotient. The lift \phi must be affine linear due to the periodicity imposed by the lattices. Specifically, \phi(z) = A z + b, where A is an m \times n complex matrix and b \in \mathbb{C}^m, with the compatibility condition A \Lambda \subset \Gamma. This ensures the map descends well to the quotients, as \phi(z + \lambda) = A(z + \lambda) + b = A z + b + A \lambda, and A \lambda \in \Gamma identifies the classes. Constant maps correspond to A = 0, while non-constant maps have A \neq 0.^[4] This affine form induces a group homomorphism on the fundamental groups of the tori, which are isomorphic to the lattices \pi_1(\mathbb{C}^n / \Lambda) \cong \Lambda and \pi_1(\mathbb{C}^m / \Gamma) \cong \Gamma. The induced map \Lambda \to \Gamma is precisely the linear part \lambda \mapsto A \lambda, reflecting the lattice compatibility. In the special case of one-dimensional tori (elliptic curves), the matrix A reduces to a scalar m \in \mathbb{C} with m \Lambda \subset \Gamma. The image of f is a translate of a subtorus determined by the image of the linear map A, and when the rank of A equals the dimension of the target torus m, the map is surjective. The fibers are then cosets of the kernel of the induced lattice homomorphism. If this kernel is finite, the map has finite fibers, yielding a finite-sheeted covering structure. This property holds when the source and target have compatible dimensions (e.g., n \geq m) and the map is non-degenerate.^[4]

Isogenies

In the category of complex tori, an isogeny is defined as a surjective holomorphic map f: T \to T' between complex tori such that the kernel \ker(f) is a finite subgroup of T.^[15] This finite kernel ensures that f induces a finite covering map away from the origin, and the image is the entire torus T' due to surjectivity.^[15] Isogenies form a fundamental class of morphisms, generalizing endomorphisms with nontrivial structure while preserving the analytic properties of the tori involved. For polarized complex tori (abelian varieties), isogenies interact with the polarization; see the section on relation to abelian varieties.^[16] The degree of an isogeny f, denoted \deg(f), is the cardinality of its kernel, |\ker(f)|.^[15] This degree is multiplicative under composition: if f: T \to T' and g: T' \to T'' are isogenies, then g \circ f is an isogeny with \deg(g \circ f) = \deg(g) \cdot \deg(f).^[16] On the level of homology, an isogeny f induces a map f_*: H_1(T, \mathbb{Z}) \to H_1(T', \mathbb{Z}), and for polarized tori equipped with a Riemann form, there exists a dual induced map on the homology of the dual tori that is adjoint to f_* with respect to the Riemann form, a positive definite Hermitian form compatible with the lattice structure.^[15] A representative example arises in dimension one, where complex tori are elliptic curves. The multiplication-by-m map : E \to E for m \in \mathbb{Z} \setminus \{0\} is an isogeny with kernel isomorphic to (\mathbb{Z}/m\mathbb{Z})^2, hence degree m^2.^[15] This kernel consists of the m-torsion points, and the map factors through the quotient by this finite subgroup, illustrating how isogenies capture lattice inclusions in the universal cover \mathbb{C}.^[15]

Isomorphisms

Two complex tori of dimension n, given as \mathbb{C}^n / \Lambda and \mathbb{C}^n / \Gamma, are isomorphic as complex manifolds if and only if there exists a \mathbb{C}-linear isomorphism \phi: \mathbb{C}^n \to \mathbb{C}^n such that \phi(\Lambda) = \Gamma. This condition equates the isomorphism of the tori to the existence of a lattice isomorphism compatible with the complex structure.^[17] The period matrix of a complex torus encodes the choice of basis for the lattice. For normalized period matrices of the form \Pi = (I_n \mid \Omega), where \Omega \in M_n(\mathbb{C}) with \operatorname{Im} \Omega positive definite, two such matrices \Pi and \Pi' define isomorphic tori if and only if there exist A \in \mathrm{GL}(n, \mathbb{C}) and M \in \mathrm{GL}(2n, \mathbb{Z}) such that \Pi' = A \Pi M. This equivalence arises from changing the basis of the universal cover via A and the lattice basis via M.^[18] The moduli space of isomorphism classes of complex tori of dimension n is the space of n \times 2n period matrices of full rank, modulo the action of \mathrm{GL}(n, \mathbb{C}) on the left and \mathrm{GL}(2n, \mathbb{Z}) on the right; this space has complex dimension n^2. For the subclass of principally polarized complex tori (abelian varieties), the moduli space is the Siegel modular variety \mathcal{A}_n = \mathfrak{H}_n / \mathrm{Sp}(2n, \mathbb{Z}), which has dimension n(n+1)/2 and classifies them up to isomorphism preserving the polarization.^[19] In the special case of dimension n=1, corresponding to elliptic curves, two such tori are isomorphic if and only if their j-invariants coincide, providing a complete discrete invariant for the isomorphism class.^[20]

Line Bundles

Factors of Automorphy

In the context of complex tori, factors of automorphy serve as cocycle data that describe holomorphic line bundles via their action on the universal cover. For a complex torus X = \mathbb{C}^n / \Lambda, where \Lambda is a lattice in \mathbb{C}^n, a factor of automorphy is a holomorphic function \chi: \Lambda \times \mathbb{C}^n \to \mathbb{C}^* satisfying the cocycle condition

\chi(\lambda_1 + \lambda_2, z) = \chi(\lambda_1, z + \lambda_2) \chi(\lambda_2, z)

for all \lambda_1, \lambda_2 \in \Lambda and z \in \mathbb{C}^n.^[21]^[22] This condition ensures that \chi defines a consistent twisting of the trivial bundle on the universal cover \mathbb{C}^n \to X.^[22] Such factors provide a trivialization of the transition functions for line bundles on X. Specifically, given a factor of automorphy \chi, the associated line bundle L_\chi is obtained by quotienting the trivial bundle on the cover by the equivalence relation (z, \xi) \sim (z + \lambda, \chi(\lambda, z) \xi) for \lambda \in \Lambda and \xi \in \mathbb{C}. This construction trivializes L_\chi over the universal cover while accounting for the deck transformations induced by the lattice.^[22] An illustrative example arises in the case of an elliptic curve, the one-dimensional complex torus X = \mathbb{C} / (\mathbb{Z} + \mathbb{Z}\tau) with \operatorname{Im} \tau > 0. Here, a factor of automorphy can take the form \chi(m + n\tau, z) = \exp(2\pi i (n a z + b(m,n))) for integers m, n and parameters a \in \mathbb{R}, where b(m,n) is chosen (e.g., bilinear in m,n) to satisfy the cocycle condition.^[23] This yields line bundles of various degrees, depending on the choices of a and b. Factors of automorphy corresponding to trivial line bundles satisfy the coboundary condition: there exists a holomorphic function g: \mathbb{C}^n \to \mathbb{C}^* such that \chi(\lambda, z) = g(z + \lambda) g(z)^{-1} for all \lambda \in \Lambda and z \in \mathbb{C}^n. In this case, the associated bundle L_\chi is isomorphic to the trivial bundle on X.^[22] The space of such coboundaries forms the 1-coboundaries in the cohomology group H^1(\Lambda, \mathcal{O}^*(\mathbb{C}^n)), classifying trivial automorphic bundles.^[21]

Construction and Chern Classes

The line bundle L_\chi on a complex torus X = \mathbb{C}^n / \Lambda, where \Lambda is a lattice in \mathbb{C}^n, is constructed via descent from the trivial holomorphic line bundle on \mathbb{C}^n. Specifically, consider the trivial bundle \mathbb{C}^n \times \mathbb{C}, and define an equivalence relation (z, v) \sim (z + \lambda, v \cdot \chi(\lambda, z)) for \lambda \in \Lambda, where \chi: \Lambda \times \mathbb{C}^n \to \mathbb{C}^\times is a factor of automorphy satisfying the cocycle condition \chi(\lambda + \mu, z) = \chi(\lambda, z + \mu) \chi(\mu, z) and holomorphy in the second variable. The quotient by this relation yields L_\chi, a holomorphic line bundle on X.^[24] The first Chern class c_1(L_\chi) resides in the cohomology group H^2(X, \mathbb{Z}), which is isomorphic to \wedge^2 H^1(X, \mathbb{Z})^* \cong \mathrm{Alt}^2(\Lambda, \mathbb{Z}), the space of alternating \mathbb{Z}-bilinear forms on \Lambda. By the Appell-Humbert theorem, factors of automorphy are parametrized by Appell-Humbert data consisting of a Hermitian form H on \mathbb{C}^n with \mathrm{Im}\, H(\Lambda, \Lambda) \subseteq \mathbb{Z} and a semi-character \alpha: \Lambda \to S^1, and c_1(L_\chi) is determined by the alternating form E(\lambda, \mu) = \mathrm{Im}\, H(\lambda, \mu) for \lambda, \mu \in \Lambda.^[24]^[22] For the case of a principal polarization, the Hermitian form H is positive definite with \det(\mathrm{Im}\, H) = 1 in a suitable basis of \Lambda, and the first Chern class c_1(L_\chi) corresponds precisely to the imaginary part \mathrm{Im}\, H, which defines the principal polarization on X. In this setting, the explicit representative of c_1(L_\chi) in de Rham cohomology is given by the Kähler form associated to H, up to scaling by i / 2\pi.^[24] Examples illustrate these constructions: if \chi is a coboundary, meaning there exists a holomorphic function f: \mathbb{C}^n \to \mathbb{C}^\times such that \chi(\lambda, z) = f(z + \lambda) / f(z), then L_\chi is the trivial line bundle on X. Conversely, if the associated Hermitian form H is positive definite, then L_\chi is an ample line bundle, ensuring that sufficiently high tensor powers embed X into projective space.^[24]

Theta Functions

Theta functions on a complex torus X = \mathbb{C}^g / \Lambda are defined as holomorphic sections of a line bundle L_\chi over X, where \chi: \Lambda \times \mathbb{C}^g \to \mathbb{C}^\times is a factor of automorphy satisfying the cocycle condition \chi(\lambda + \mu, z) = \chi(\lambda, z + \mu) \chi(\mu, z) for \lambda, \mu \in \Lambda and z \in \mathbb{C}^g.^[25]^[22] Such a section \theta: \mathbb{C}^g \to \mathbb{C} is called a theta function with respect to \chi if it transforms under the lattice action via the functional equation

\theta(z + \lambda) = \chi(\lambda, z) \theta(z)

for all z \in \mathbb{C}^g and \lambda \in \Lambda, and descends to a holomorphic section on the quotient X.^[25] These functions play a central role as automorphic forms on X, providing explicit bases for the spaces of global sections H^0(X, L_\chi) when the bundle is ample.^[26] For the case of dimension g=1, corresponding to elliptic curves, the classical Jacobi theta function arises as the unique (up to scalar) nontrivial section of the line bundle associated to the principal polarization. It is given explicitly by the infinite sum

\vartheta(z \mid \tau) = \sum_{n=-\infty}^\infty \exp\left( \pi i n^2 \tau + 2 \pi i n z \right),

where \tau \in \mathbb{H} (the upper half-plane) parameterizes the lattice \Lambda = \mathbb{Z} + \tau \mathbb{Z}.^[27] This function satisfies the automorphy relations \vartheta(z+1 \mid \tau) = \vartheta(z \mid \tau) and \vartheta(z + \tau \mid \tau) = \exp(- \pi i \tau - 2 \pi i z) \vartheta(z \mid \tau), confirming its role as a section of the bundle with the corresponding factor.^[27] In higher dimensions g > 1, the Riemann theta function generalizes this construction for principally polarized abelian varieties, serving as the canonical section of the associated ample line bundle. It is defined by the multivariable series

\theta(z \mid \Omega) = \sum_{n \in \mathbb{Z}^g} \exp\left( 2\pi i \left( \frac{1}{2} n^t \Omega n + n^t z \right) \right),

where z \in \mathbb{C}^g, \Omega is a g \times g symmetric complex matrix with positive definite imaginary part (the period matrix defining \Lambda = \mathbb{Z}^g + \Omega \mathbb{Z}^g), and the sum converges absolutely due to the positivity condition.^[28] This function exhibits the required automorphy with respect to the principal factor \chi(\lambda, z) = \exp(\pi i H(\lambda, z) + \pi i E(\lambda, \lambda)/2), where H is the Hermitian form inducing the polarization and E = \operatorname{Im} H.^[25] The zero locus of a theta function provides the theta divisor on X, which for a principal polarization is an ample hypersurface embedding X into projective space. Specifically, if \dim H^0(X, L) = 1 for the ample line bundle L of the polarization, the zero set \Theta = \{ x \in X \mid \theta(x) = 0 \} of the unique nontrivial section \theta (up to scalar) is the theta divisor, well-defined up to translation and ample by the positivity of the polarization.^[29]^[26]

Hermitian Forms

Appell-Humbert Theorem

The Appell-Humbert theorem establishes a bijective correspondence between the isomorphism classes of holomorphic line bundles on a complex torus X = \mathbb{C}^n / \Lambda, where \Lambda is a full-rank lattice in \mathbb{C}^n, and the set of pairs (H, \chi) consisting of a Hermitian form H: \mathbb{C}^n \times \mathbb{C}^n \to \mathbb{C} such that \operatorname{Im} H(\lambda, \mu) \in \mathbb{Z} for all \lambda, \mu \in \Lambda, and a semi-character \chi: \Lambda \to S^1 satisfying \chi(\lambda_1 + \lambda_2) = \chi(\lambda_1) \chi(\lambda_2) \exp(\pi i \operatorname{Im} H(\lambda_1, \lambda_2)) for all \lambda_1, \lambda_2 \in \Lambda.^[22]^[13] This classification captures all line bundles via algebraic data tied to the torus's underlying vector space and lattice structure.^[30] Given such a pair (H, \chi), the corresponding line bundle L(H, \chi) on X is constructed explicitly as the quotient ( \mathbb{C}^n \times \mathbb{C} ) / \sim, where the equivalence relation is induced by the \Lambda-action \lambda \cdot (z, \zeta) = (z + \lambda, \chi(\lambda) \exp( \pi H(z, \lambda) + \frac{\pi}{2} H(\lambda, \lambda) ) \cdot \zeta ).^[13]^[22] This defines a factor of automorphy for the bundle, ensuring holomorphy descends from the trivial bundle on \mathbb{C}^n. The construction is functorial, preserving tensor products of line bundles via addition of Hermitian forms and multiplication of semi-characters adjusted by the exponential factor.^[22] The pair (H, \chi) is uniquely determined up to isomorphism by the line bundle: if two pairs (H, \chi) and (H', \chi') yield isomorphic bundles, then H = H' and \chi = \chi' after adjusting by a constant phase in S^1.^[13]^[22] Equivalence of data follows from the theorem's proof, which involves lifting bundles to the universal cover and normalizing automorphy factors via cohomology classes in H^1(\Lambda, \mathcal{O}^\times(\mathbb{C}^n)).^[22] The theorem originated with Paul Appell's work around 1900 on two-dimensional tori and was generalized to arbitrary dimensions by Maurice Humbert in his subsequent contributions on analytic transformations and automorphic functions.^[22]

Nerón-Severi Group

The Néron-Severi group of a complex torus X = V / \Lambda, denoted \mathrm{NS}(X), is defined as the subgroup of the Picard group \mathrm{Pic}(X) consisting of isomorphism classes of holomorphic line bundles L such that the first Chern class c_1(L) lies in H^2(X, \mathbb{Z}). Equivalently, it is the quotient \mathrm{Pic}(X) / \mathrm{Pic}^0(X), where \mathrm{Pic}^0(X) is the connected component of the identity in \mathrm{Pic}(X), comprising line bundles with vanishing Chern class. This group captures the algebraic structure of divisors up to algebraic equivalence on X.^[17]^[12] The structure of \mathrm{NS}(X) can be computed using Hermitian forms on the vector space V. Specifically, \mathrm{NS}(X) is isomorphic to the group of Hermitian forms H: V \times V \to \mathbb{C} satisfying \mathrm{Im}(H)(\Lambda \times \Lambda) \subset \mathbb{Z}, where the imaginary part \mathrm{Im}(H) is an alternating bilinear form on \Lambda taking integer values. The Chern class c_1(L) for a corresponding line bundle L is given by this alternating form \mathrm{Im}(H).^[31]^[32] In the algebraic case, where the complex torus X is a projective abelian variety admitting a principal polarization \theta, the Néron-Severi group \mathrm{NS}(X) is generated by the class [\mathcal{O}_X(\theta)] of the associated ample line bundle. For a principally polarized abelian variety of dimension g, the rank of \mathrm{NS}(X) is 1 in the generic case, reflecting the minimal algebraic structure beyond the polarization itself.^[12]^[33] The full Picard group admits a decomposition \mathrm{Pic}(X) \cong \mathrm{NS}(X) \oplus \mathrm{Hom}(\Lambda, \mathbb{C}^*), where \mathrm{Hom}(\Lambda, \mathbb{C}^*) is the group of unitary characters of the lattice \Lambda, corresponding to the topologically trivial holomorphic line bundles in \mathrm{Pic}^0(X). This split extension highlights how \mathrm{NS}(X) isolates the non-trivial Chern class components from the character group.^[32]^[31]

Semi-Characters and Examples

In the Appell-Humbert framework for describing line bundles on a complex torus X = \mathbb{C}^g / \Lambda, a semi-character \chi: \Lambda \to \mathbb{C}^* associated to a Riemann form H (a Hermitian form on \mathbb{C}^g with \operatorname{Im} H(\Lambda, \Lambda) \subseteq \mathbb{Z}) satisfies |\chi(\lambda)| = 1 for all \lambda \in \Lambda and is compatible with H via the relation \chi(\lambda + \mu) = \chi(\lambda) \chi(\mu) \exp(\pi i \operatorname{Im} H(\lambda, \mu)) for \lambda, \mu \in \Lambda, reflecting partial additivity adjusted by the bilinear form \operatorname{Im} H. This compatibility ensures that the pair (H, \chi) defines a unique holomorphic line bundle L(H, \chi) up to isomorphism on X, constructed via the factor of automorphy j(\lambda, z) = \chi(\lambda) \exp(\pi H(z, \lambda) + \frac{\pi}{2} H(\lambda, \lambda)) for z \in \mathbb{C}^g and \lambda \in \Lambda. On an elliptic curve E = \mathbb{C} / (\mathbb{Z} + \tau \mathbb{Z}) with \operatorname{Im} \tau > 0, the principal polarization admits the Riemann form H(z, w) = \frac{\bar{z} w}{\operatorname{Im} \tau}, under which semi-characters yield line bundles of various degrees. For example, bundles of degree k correspond to appropriate choices of \chi compatible with this H.^[34] In higher dimensions (g > 1), principal polarizations derive from Riemann bilinear relations on the period matrix \Omega \in \mathbb{H}_g, where H(z, w) = z^\dagger (\operatorname{Im} \Omega)^{-1} \bar{w} satisfying \operatorname{Im} H(\Lambda, \Lambda) \subseteq \mathbb{Z} and positive definiteness; semi-characters \chi are then maps on \Lambda = \mathbb{Z}^{2g} obeying the same compatibility, parameterizing line bundles over the Néron-Severi group of polarization classes.^[35]

Dual Torus

Definition

In the context of complex tori, the dual torus \hat{X} of a complex torus X = \mathbb{C}^g / \Lambda, where \Lambda is a lattice of rank $2g in \mathbb{C}^g, is defined as the Pontryagin dual \hat{X} = \Hom(X, S^1), the group of continuous homomorphisms from X to the circle group S^1 = \{ z \in \mathbb{C}^\times \mid |z| = 1 \}.^[36] This dual arises from Pontryagin duality for compact abelian Lie groups, identifying characters of X that are trivial on \Lambda.^[37] Algebraically, \hat{X} \cong \Hom(\Lambda, \mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{R} / \mathbb{Z}, but as a complex torus, it is isomorphic to \mathbb{C}^g / \Lambda^*, where \Lambda^* = \{ z \in \mathbb{C}^g \mid \exp(2\pi i \langle z, \lambda \rangle) = 1 \ \forall \lambda \in [\Lambda](/page/Lambda) \} is the annihilator lattice with respect to a suitable \mathbb{R}-bilinear pairing \langle \cdot, \cdot \rangle: \mathbb{C}^g \times \mathbb{C}^g \to \mathbb{R} (often taken as \langle z, \lambda \rangle = \Im(z \cdot \bar{\lambda})).^[37] Equivalently, \Lambda^* \cong \Hom([\Lambda](/page/Lambda), \mathbb{Z}(1)), where \mathbb{Z}(1) = \ker(\exp: \mathbb{C} \to S^1) = 2\pi i \mathbb{Z} is the lattice of periods for the exponential map, ensuring \Lambda^* inherits a complex structure compatible with that of X.^[36] For abelian varieties, there is a natural isomorphism between the dual complex torus and the dual abelian variety \hat{A}, realized via the Hom scheme \Pic^0(A) \cong \underline{\Hom}(A, \mathbb{G}_m), which parameterizes translation-invariant line bundles of degree zero on A and extends the analytic duality to the algebraic category.^[36] This identifies \hat{A} with the moduli space of principal homogeneous spaces under A, preserving the group structure. A representative example is the elliptic curve E = \mathbb{C} / \Lambda with \Lambda = \mathbb{Z} \oplus \tau \mathbb{Z} (\tau \in \mathbb{H}), whose dual \hat{E} \cong E up to isomorphism via the principal polarization induced by the Riemann form on H_1(E, \mathbb{Z}).^[36]

Poincaré Bundle

The Poincaré bundle on a complex torus X = \mathbb{C}^g / \Lambda is constructed as a universal line bundle P over the product X \times \hat{X}, where \hat{X} denotes the dual torus. For each point \phi \in \hat{X}, the restriction P_\phi = P|_{X \times \{\phi\}} is the line bundle on X associated to the character \phi, or more directly through the representation of line bundles via characters on the lattice \Lambda.^[38] An explicit construction of P proceeds on the covering space \mathbb{C}^g \times \mathbb{C}^g quotiented by the lattice \Lambda \times \Lambda^*, where \Lambda^* = \{\phi \in \mathbb{C}^g \mid \langle \phi, \lambda \rangle \in \mathbb{Z} \ \forall \lambda \in \Lambda\} is the dual lattice. The bundle descends from the trivial line bundle on \mathbb{C}^g \times \mathbb{C}^g equipped with automorphy factors given by

\chi(\lambda, z, \phi) = \exp\left(2\pi i \langle \phi, \lambda \rangle \right)

for \lambda \in \Lambda, z \in \mathbb{C}^g, and \phi \in \mathbb{C}^g, ensuring compatibility with the lattice action and yielding a well-defined holomorphic line bundle on the product torus.^[24] This factorization incorporates the bilinear pairing \langle \cdot, \cdot \rangle between the dual spaces, with the exponential term accounting for the character twist under lattice translations. Key properties of P include its restriction to a degree-0 line bundle along each fiber \{ \mathrm{pt} \} \times \hat{X} and along each X \times \{ \mathrm{pt} \}, reflecting its universal nature in parametrizing degree-0 line bundles on X. These restrictions ensure that P is normalized such that P|_{X \times \{0\}} and P|_{\{0\} \times \hat{X}\}} are trivial, while maintaining the universal property for the moduli of line bundles on X.^[38] The Poincaré bundle serves as the essential kernel in the Fourier-Mukai transform for complex tori, establishing an equivalence between the derived categories of coherent sheaves D^b(X) and D^b(\hat{X}). Introduced by Mukai, this transform uses P to duality-pair cohomology on X with sheaf theory on \hat{X}, interchanging skyscraper sheaves at points with line bundles in \mathrm{Pic}^0(X) and enabling applications such as the study of Picard sheaves and semihomogeneous bundles.

References

[1]
[PDF] Abelian varieties - Purdue Math
A complex torus is quotient V/L of a finite dimensional complex vector space. V ⇠= Cn by a lattice L ⇠= Z2n. This is a complex Lie group, that it is a ...
[2]
[PDF] abelian varieties - brian conrad lecture notes by tony feng
Complex tori. Definition 1.2.1. A complex torus is a connected compact Lie group over C. These are the analytic analogues of abelian varieties over C ...
[3]
Line Bundles on Complex Tori (Part 1) | The n-Category Café
Mar 13, 2022 · A complex torus is the same as the quotient of a finite-dimensional complex vector space by a lattice. Every abelian variety is a complex torus, ...
[4]
[PDF] COMPLEX TORI This writeup gives a quick sketch of results about ...
D. A complex torus is a quotient of the complex plane by a lattice,. C/Λ = {z +Λ: z ∈ C}. Algebraically a complex torus is an Abelian group under the ...
[5]
[PDF] Elliptic Curves, Complex Tori, Modular Forms, and
Jul 13, 2020 · We say a complex torus C/Λ has complex multiplication or CM if there exists an endomorphism φ : C/Λ → C/Λ defined by some noninteger complex ...
[6]
[PDF] 17 Complex multiplication
Apr 12, 2017 · A complex torus C/L is both a complex manifold and a group in which the group oper- ations are defined by holomorphic maps (this makes it a ...Missing: mathematics | Show results with:mathematics
[7]
[PDF] Abelian Varieties and Moduli - Purdue Math
Apr 19, 2012 · A complex torus is an abelian variety if and only if it is a projective algebraic variety. In the purely algebraic theory of abelian ...
[8]
[PDF] a beginner's guide to modular curves - UChicago Math
Complex tori are compact Riemann surfaces of genus 1 that are constructed as quotient spaces of C. Through this construction, we will see that every complex.
[9]
[PDF] abelian varieties: geometry, parameter spaces, and arithmetic
Jul 8, 2005 · This map is an isogeny if and only if there exists a map of complex tori f0 : X0 → X such that f0 ◦ f = [n]X for some nonzero n ∈ Z, in which ...
[10]
[PDF] Contents - University of Washington Math Department
Let X = V/Λ with V having a C-basis {e1,...,eg} and Λ having a Z-basis {λ1,...,λ2g}, which gives its period matrix. Theorem (Riemann relations): X is an abelian ...
[11]
[PDF] Abelian Varieties - James Milne
Mar 16, 2008 · A complex torus X is of the form A.C/ if and only if it is ... elliptic curve is not an abelian variety, but a polarized abelian variety.
[12]
[PDF] Solutions to Midterm 1 Problem 1. Prove that R 2/Z2 is ...
Page 1. Math 310 Topology, Spring 2000. Solutions to Midterm 1. Problem 1. Prove that R2/Z2 is homeomorphic to the 2-torus T2. (Here R2 is regarded as a group ...<|control11|><|separator|>
[13]
[PDF] Beginning compact Riemann surface theory
The Jacobian. The period matrix Ω has an associated lattice. L = LΩ = Zg + ΩZg ⊂ Cg. The complex torus quotient Cg/L is the Jacobian of X, Jac(X). The Jacobian.
[14]
[PDF] Basic Theory of Abelian Varieties 1. Definitions - James Milne
Lange, H., and Birkenhake, Ch., Complex Abelian Varieties, Springer, 1992. ... The book by Lange and Birkenhake is a very thorough and complete treatment ...
[15]
[PDF] Lectures on Abelian Varieties, Milan, February 2014
(see Proposition (5.1,2) in Lange-Birkenhake,. Complex Abelian Varieties, cited [CAV] in the future). In particular,. Pr(t) = Pa(f)Pa( ¯f). An endomorphism f ...
[16]
[PDF] 1 Complex Theory of Abelian Varieties - Mathematics
Our next goal is to give a classification of line bundles over a complex torus. We first recall the definition of a line bundle, also known as an invertible.
[17]
[PDF] 1.3 Abelian Varieties as Complex Tori (10/08/03 notes for Math 252 ...
In this section we introduce extra structure on a complex torus T = V/L that will enable us to understand whether or not T is isomorphic to A(C), ...
[18]
[PDF] Complex Multiplication
An isogeny of complex tori is a surjective homomorphism with finite kernel. By an. “isogeny” we mean an invertible element of. Hom. 0 .M;N/ def. D Hom.M;N/˝Q ...
[19]
[PDF] Chapter V. Isogenies. In this chapter we define the notion of an ...
Sep 15, 2011 · In this chapter we define the notion of an isogeny, and we discuss some basic examples, in- cluding the multiplication by an integer n !=
[20]
[PDF] Abelian Varieties - Lecture Notes - Berkeley Math
E.g. if g = 1, then any complex torus of dimension 1 (i.e. any elliptic curve) is isomorphic ... The simplest nontrivial example of an abelian variety is an ...<|control11|><|separator|>
[21]
[PDF] arXiv:2401.13277v1 [math.AG] 24 Jan 2024
Jan 24, 2024 · Observe that if (Ig Zi) is the period matrix for Ti then an isomor- phism T1 → T2 is given by invertible matrices. M ∈ GL(g, C) and R ∈ GL(2g, Z) ...
[22]
[PDF] Siegel Modular Forms Lecture #2 - Gerard van der Geer Home Page
Oct 15, 2020 · This orbifold is a moduli space, namely of complex principally polarized abelian varieties, that is, of complex tori V/Λ with V a. C-vector ...
[23]
[PDF] 16 Elliptic curves over C (part 2)
Apr 10, 2017 · Thus homethety classes of lattices correspond to isomorphism classes of elliptic curves over C, and both are classified by the j-invariant.
[24]
[PDF] Morphisms of complex tori and abelian varieties
Morphisms of complex tori and abelian varieties. Exercise 1. Homomorphisms of complex tori. Let X1 = V1/Λ1 and X2 = V2/Λ2 be two complex.
[25]
[PDF] THE APPEL-HUMBERT THEOREM 1. Proof 1.1. Introduction. The ...
A complex torus is a complex Lie group isomorphic to X = V/Λ. Definition 1.2 ... Step 1 - factors of automorphy. Proposition 1.4.1. There exist a ...<|control11|><|separator|>
[26]
[PDF] Vector Bundles on Elliptic Curves and Factors of Automorphy
VECTOR BUNDLES AND FACTORS OF AUTOMORPHY. 81. 5. Classification of Vector Bundles over a Complex Torus. Here we work with factors of automorphy depending only ...
[27]
[PDF] Line bundles and cohomology of complex tori - Emre Sertöz
Oct 29, 2014 · Therefore we may say that the Chern classes of line bundles on X are in one to one correspondance with Hermitian forms on V whose imaginary part.
[28]
[PDF] Riemann's theta function - Penn Math
(b) Geometrically these theta functions can be identified with sections of ample line bundles on abelian varieties and can be thought of as projective ...
[29]
[PDF] Two or three things I know about abelian varieties
Apr 19, 2017 · [D]. Debarre, O., Complex Tori and Abelian Varieties, SMF/AMS Texts and Monographs 11, American. Mathematical Society, 2005. [M]. Milne,. J.S. ...
[30]
[PDF] lecture 9: theta functions
A theta function is an analytic function of z in C, defined as ϑ(z,τ) = X eπin2τ+2πinz, where parameter τ is in the upper-half plane.
[31]
[PDF] Computing Riemann Theta Functions - FSU Math
Jun 7, 2002 · All terms remaining in the sum are either oscillating (the first factor of every term), or a damped exponential (the second factor of every term) ...Missing: multivariable | Show results with:multivariable
[32]
[PDF] LECtURE 7 (FEBRUARY 18) Principally polarized abelian varieties ...
Feb 18, 2025 · The principal polarization is E(т, 1) = 1, and then H(1, 1) = 1 ... Let X and Y be simple compact complex tori. Then any holomorphic.
[33]
[PDF] Lecture 4 (February 6) The Appel-Humbert theorem. Last time, we ...
Feb 6, 2025 · Along the way, we'll prove the following interesting fact: If L is a holomorphic line bundle on a compact complex torus X, and if H0(X, L) 6= 0, ...
[34]
Line Bundles on Complex Tori
### Summary of Line Bundles on Complex Tori (Chapter 4)
[35]
[PDF] A Quick Introduction to Complex Tori - Parth Shimpi
It can be shown that there is an isomorphism ^T ∼= Hom(Λ, U(1)), i.e. the dual torus is the character-group of the period of T. One can also define the group P( ...
[36]
Picard number of principally polarized abelian varieties
Dec 16, 2013 · Over C, at least, it is known that the Picard number (that is, the rank of the Néron-Severi group of A) is less than or equal to n2, with ...Conceptual understanding of the Néron–Severi group - MathOverflowNéron–Severi group of Abelian surfaces - MathOverflowMore results from mathoverflow.net
[37]
[PDF] Abelian Varieties, Theta Functions and the Fourier Transform
Definition. The complex manifold T = V/ is called a complex torus. As a topological space T is just a product of circles, so the cohomology.