04.10.16 · algebraic-geometry / moduli

Abelian varieties: group law, polarizations, and the dual

shipped3 tiersLean: none

Anchor (Master): Weil 1948 *Variétés abéliennes et courbes algébriques* (originator of the abstract algebraic theory); Mumford *Abelian Varieties* (TIFR 1970) — the rigidity theorem, the theorem of the cube, the dual variety, the theta group; Birkenhake-Lange *Complex Abelian Varieties* (Grundlehren 302); Milne *Abelian Varieties*

Intuition Beginner

An abelian variety is a higher-dimensional doughnut that is also a projective variety and an abelian group — the dimension- $g$ generalisation of an elliptic curve. An elliptic curve 04.04.03 is a one-dimensional example: a torus shaped like a doughnut surface, on which you can add points by a geometric rule, and which sits inside projective space as a smooth cubic.

Now raise the dimension. Over the complex numbers a $g$ -dimensional torus looks like $g$ -fold complex space wrapped up by a lattice of translations. Some of these tori can be drawn inside projective space as smooth varieties, and exactly those carry a compatible group law. We call them abelian varieties. The name "abelian" is a promise the geometry keeps for free: the group law is always commutative.

Visual Beginner

Picture a flat square with opposite edges glued: that is a two-dimensional torus, the surface of a doughnut. An abelian variety of dimension $g$ is the same idea with $g$ complex directions glued by a lattice, and then realised as a smooth shape inside projective space. Addition is geometric: slide the whole picture by a fixed amount and every point moves the same way, so the group operation is a smooth self-map of the variety.

The key extra ingredient over a plain torus is a polarisation: a positive measurement on the lattice that lets the torus fit inside projective space. A doughnut with no such measurement stays an abstract complex torus and never becomes a variety once the dimension passes one.

Worked example Beginner

Take two elliptic curves $E_{1}$ and $E_{2}$ . Each one is a one-dimensional abelian variety: a complex torus that sits in the plane as a cubic curve with a point-addition rule. Form the pair $A = E_{1} \times E_{2}$ , the set of all choices of a point on the first curve together with a point on the second.

This product is a two-dimensional abelian variety. You add two pairs by adding the first coordinates on $E_{1}$ and the second coordinates on $E_{2}$ separately, and the result is again a pair. The group law is commutative because each factor's law is. It is projective because each elliptic curve is, and a product of projective varieties is projective.

The lattice of $A$ is the product of the two lattices, one in each complex line, so $A$ looks like a four-real-dimensional doughnut. Yet most two-dimensional abelian varieties are not products of curves: a generic one is a simple abelian surface with no curve factors at all. The product example is the easy gateway; the rich theory lives in the ones that do not split.

Check your understanding Beginner

Formal definition Intermediate+

Let $k$ be a field. An abelian variety over $k$ is a complete connected group variety $A$ over $k$ : a variety equipped with morphisms $m : A \times A \to A$ (multiplication), $i : A \to A$ (inverse), and a $k$ -point $e \in A (k)$ (identity) satisfying the group axioms, such that the underlying variety is complete (proper) and geometrically connected. Over a field, completeness is equivalent to projectivity for such a variety.

Commutativity (consequence of rigidity). The group law of an abelian variety is automatically commutative; we write it additively, $m (x, y) = x + y$ . This is forced by the rigidity lemma below and is not an extra hypothesis.

Complex-analytic picture. Over $k = C$ , every abelian variety $A$ of dimension $g$ is, as a complex Lie group, a quotient $$ A = \mathbb{C}^g / \Lambda, \qquad \Lambda \cong \mathbb{Z}^{2g} \text{ a full-rank lattice}, $$ a complex torus. The converse fails for $g \geq 2$ : a complex torus $C^{g} /Λ$ is an abelian variety if and only if it admits a polarisation — a positive-definite Hermitian form $H$ on $C^{g}$ whose imaginary part $E = Im H$ is integer-valued on $Λ$ . These are the Riemann bilinear relations: $E : Λ \times Λ \to Z$ is alternating and $H (v, w) = E (i v, w) + i E (v, w)$ is positive-definite Hermitian. Equivalently, the torus carries an ample line bundle 04.05.05.

Isogeny. An isogeny $f : A \to B$ of abelian varieties is a surjective homomorphism with finite kernel (equivalently, a finite surjective homomorphism); its degree is $de g f = # ker f$ (counted with multiplicity in characteristic $p$ ). Isogeny is an equivalence relation: for every isogeny $f$ of degree $n$ there is a dual isogeny $g$ with $g \circ f = [n]$ , multiplication by $n$ .

Dual abelian variety. The dual of $A$ is $$ \hat A = \mathrm{Pic}^0(A), $$ the connected component of the identity of the Picard group 04.05.02, parametrising line bundles algebraically equivalent to zero. It is again an abelian variety of the same dimension $g$ , and there is a canonical isomorphism $A \sim \hat{\hat{A}}$ .

Polarisation as an isogeny. An ample line bundle $L$ on $A$ defines a homomorphism $$ \varphi_L : A \to \hat A, \qquad x \mapsto t_x^* L \otimes L^{-1}, $$ where $t_{x}$ is translation by $x$ . When $L$ is ample, $φ_{L}$ is an isogeny; such an isogeny arising from an ample class is a polarisation. It is a principal polarisation when $φ_{L}$ is an isomorphism.

Theta divisor and $n$ -torsion. When $A$ is principally polarised, the polarisation is represented by an effective divisor $Θ$ , the theta divisor, with $dim H^{0} (A, O_{A} (Θ)) = 1$ . The $n$ -torsion subgroup, for $n$ prime to the characteristic, is $$ A[n] = \ker([n] : A \to A) \cong (\mathbb{Z}/n)^{2g}. $$ The inverse limit over powers of a prime $ℓ$ assembles these into the Tate module $T_{ℓ} A = lim A [ℓ^{m}]$ , on which a polarisation induces the Weil pairing, an alternating non-degenerate pairing $T_{ℓ} A \times T_{ℓ} \hat{A} \to Z_{ℓ} (1)$ .

Key theorem with proof Intermediate+

Theorem (Rigidity lemma; Mumford, Abelian Varieties §II.4). Let $X$ be a complete variety, $Y$ and $Z$ any varieties, and $f : X \times Y \to Z$ a morphism such that, for some point $y_{0} \in Y$ , the restriction $f ∣_{X \times {y_{0}}}$ is constant, equal to $z_{0} \in Z$ . Then there is a morphism $g : Y \to Z$ with $f (x, y) = g (y)$ for all $x, y$ ; that is, $f$ factors through the projection to $Y$ .

Corollary (commutativity). The group law of an abelian variety is commutative, and every morphism of abelian varieties fixing the identity is a homomorphism.

Proof of the lemma. Pick a point $x_{0} \in X$ and an affine open neighbourhood $U ∋ z_{0}$ in $Z$ . Because $X$ is complete, the projection $p : X \times Y \to Y$ is a closed map. The set $W = f^{- 1} (Z ∖ U)$ is closed in $X \times Y$ , so $p (W)$ is closed in $Y$ . The fibre over $y_{0}$ maps into ${z_{0}} \subset U$ , so ${y_{0}}$ avoids $p (W)$ ; let $V = Y ∖ p (W)$ , an open neighbourhood of $y_{0}$ . For $y \in V$ the whole image $f (X \times {y})$ lies in the affine $U$ . A morphism from the complete variety $X \times {y}$ to an affine has image a single point, since global functions on a complete connected variety are constants. So $f (x, y) = f (x_{0}, y)$ for all $x \in X$ and all $y \in V$ . The two morphisms $f$ and $(x, y) \mapsto f (x_{0}, y)$ agree on the dense open $X \times V$ , hence everywhere. Setting $g (y) = f (x_{0}, y)$ gives the factorisation. $□$

Proof of the corollary. Let $A$ be an abelian variety with multiplication $m$ . Consider $f : A \times A \to A$ defined by $f (x, y) = x y x^{- 1} y^{- 1}$ , the commutator. Restricting to $A \times {e}$ gives $f (x, e) = e$ , constant. By rigidity, $f (x, y) = g (y)$ depends on $y$ alone; but $f (e, y) = e$ , so $g \equiv e$ , hence $x y x^{- 1} y^{- 1} = e$ for all $x, y$ : the group is commutative. A similar commutator argument shows any pointed morphism is a homomorphism. $□$

Bridge. The rigidity lemma builds toward the entire structure theory of abelian varieties: it is the foundational reason the group law is commutative, and the same completeness-forces-constancy argument appears again in the proof that $Pic^{0}$ is itself an abelian variety and in the theorem of the cube. This is exactly the rigidity that the one-dimensional case 04.04.03 enjoyed silently — an elliptic curve's chord-and-tangent law was commutative because it equals $Pic^{0}$ — and the higher-dimensional statement generalises that coincidence into a theorem about all complete group varieties. The polarisation $φ_{L} : A \to \hat{A}$ is dual to the line-bundle class $L$ in a precise sense the next section makes explicit, and putting these together, the central insight is that completeness alone, with no positivity assumed, already pins down the algebraic shape of the group, while positivity (the polarisation) is the separate ingredient that makes the torus projective.

Exercises Intermediate+

Exercise 3 (medium, proof).

State the Riemann bilinear relations and use them to explain why a generic complex torus of dimension $2$ is not an abelian variety.

Hint

A polarisation is an alternating integral form $E$ on $Λ$ whose associated Hermitian form is positive-definite. Count conditions against the moduli of the lattice.

Answer

A complex torus $C^{g} /Λ$ is projective if and only if there is an alternating form $E : Λ \times Λ \to Z$ such that $E (i v, i w) = E (v, w)$ and the Hermitian form $H (v, w) = E (i v, w) + i E (v, w)$ is positive-definite. These are the Riemann relations.

For $g = 1$ such a form always exists, so every one-dimensional complex torus is an elliptic curve. For $g = 2$ the existence of even one nonzero integral alternating $E$ compatible with the complex structure imposes algebraic conditions on the period matrix; the space of period matrices admitting such an $E$ is a proper countable union of subvarieties of the full moduli space of complex tori. A torus with very general periods satisfies none of these conditions, so it carries no polarisation and is not an abelian variety. $□$

Exercise 4 (hard, conceptual).

Explain the difference between the dual abelian variety $\hat{A}$ and a polarisation $φ_{L} : A \to \hat{A}$ , and what "principal" adds.

Hint

$\hat{A}$ is a variety; $φ_{L}$ is a map. One is data of $A$ alone, the other depends on a line bundle.

Answer

The dual $\hat{A} = Pic^{0} (A)$ is an abelian variety canonically attached to $A$ , with no choices: it parametrises degree-zero line bundles, and $dim \hat{A} = dim A$ . A polarisation is additional data: a choice of ample line bundle $L$ produces an isogeny $φ_{L} : A \to \hat{A}$ , $x \mapsto t_{x}^{*} L \otimes L^{- 1}$ . Different ample bundles give different polarisations. The polarisation is principal when $φ_{L}$ is an isomorphism, not merely an isogeny; equivalently $ker φ_{L} = 0$ , equivalently (in the analytic picture) the alternating form $E$ has elementary divisors all equal to $1$ . Jacobians of curves carry a canonical principal polarisation from the theta divisor; a general polarised abelian variety has type $(d_{1}, \dots, d_{g})$ with $de g φ_{L} = (d_{1} \dots d_{g})^{2}$ , principal exactly when all $d_{i} = 1$ .

Advanced results Master

Theorem of the cube and the seesaw principle. On $A \times A \times A$ the line bundle obtained by alternating pullbacks of a bundle $L$ over the seven coordinate sums is isomorphic to the structure sheaf; this theorem of the cube (Mumford §II.6) is the source of the bilinearity of $φ_{L}$ and of the formula $[n]^{*} L ≅ L^{\otimes n^{2}}$ (for symmetric $L$ ). The companion seesaw principle — a family of line bundles restricting to the identity class on each fibre and on one section is globally the identity class — is the bookkeeping that turns rigidity into these identities.

Lefschetz embedding theorem. For an ample line bundle $L$ on an abelian variety, $L^{\otimes 2}$ is globally generated and $L^{\otimes 3}$ is very ample: it embeds $A$ in projective space (Lefschetz ^{[Lefschetz 1921]}). This is the precise mechanism realising a polarised complex torus as a projective variety, and it is why the third power of the theta bundle appears throughout the explicit theory of theta functions.

Theta functions. Sections of $L^{\otimes n}$ on $A = C^{g} /Λ$ are realised analytically by theta functions — quasi-periodic holomorphic functions on $C^{g}$ with a prescribed automorphy factor. The dimension of $H^{0} (A, L^{\otimes n})$ equals $n^{g} det E$ by the Riemann-Roch theorem for abelian varieties, recovering the classical theta with characteristics when the polarisation is principal.

Tate module and the Weil pairing. For each prime $ℓ$ not equal to the characteristic, $T_{ℓ} A ≅ Z_{ℓ}^{2 g}$ carries a continuous Galois action, and a polarisation induces the alternating non-degenerate Weil pairing $e_{ℓ} : T_{ℓ} A \times T_{ℓ} \hat{A} \to Z_{ℓ} (1)$ . Over finite fields the characteristic polynomial of Frobenius on $T_{ℓ} A$ controls point counts; this is the engine of the Weil conjectures for abelian varieties (a pointer, treated in depth elsewhere).

Moduli. Principally polarised abelian varieties of dimension $g$ are parametrised by a moduli space $A_{g}$ , constructed analytically as the quotient of the Siegel upper half-space $H_{g}$ by $Sp (2 g, Z)$ and algebraically by Mumford's GIT 04.10.02 via theta-level structure. For $g = 1$ this is the modular curve $A_{1} = M_{1, 1}$ .

Synthesis. The abelian variety packages four ingredients into one object, and putting these together is the central insight of the theory. The first is rigidity: completeness forces commutativity, so the group law is determined by the geometry — this is exactly the structural feature that makes morphisms automatically homomorphisms and is the foundational reason the theory is so constrained. The second is projectivity through positivity: not every complex torus is a variety, and the bridge is the polarisation, an ample class whose Riemann relations cut out the projective ones, with $L^{\otimes 3}$ very ample realising the embedding. The third is duality: $\hat{A} = Pic^{0}$ is an abelian variety dual to $A$ , and a polarisation is precisely an isogeny $A \to \hat{A}$ , so positivity and duality are two faces of one datum. The fourth is arithmetic linearisation: the $n$ -torsion $A [n] ≅ (Z / n)^{2 g}$ and the Tate module with its Weil pairing generalise the elliptic-curve case 04.04.03 to dimension $g$ , and this is the structure the moduli space $A_{g}$ reads off.

Full proof set Master

Proposition. Let $A$ be an abelian variety over an algebraically closed field $k$ and $n$ an integer prime to $char k$ . Then $[n] : A \to A$ is a separable isogeny of degree $n^{2 g}$ , and $A [n] ≅ (Z / n)^{2 g}$ .

Proof. First, $[n]$ is a homomorphism by the rigidity corollary, and it is surjective: its image is a closed subgroup of the connected variety $A$ of the same dimension (since $[n]$ has finite kernel, shown below), hence all of $A$ . To see the kernel is finite and compute the degree, work analytically over $k = C$ first, where $A = C^{g} /Λ$ and $[n]$ is induced by multiplication by $n$ on $C^{g}$ . Then $$ A[n] = \tfrac{1}{n}\Lambda \big/ \Lambda \cong \Lambda / n\Lambda \cong (\mathbb{Z}/n)^{2g}, $$ since $Λ ≅ Z^{2 g}$ . So $# A [n] = n^{2 g}$ and the degree is $n^{2 g}$ .

For general $k$ of characteristic prime to $n$ , the degree is computed by the action of $[n]$ on the tangent space at the identity: the differential $d [n]_{0} = n \cdot id$ on $T_{0} A ≅ k^{g}$ , which is invertible because $n$ is a unit in $k$ . Hence $[n]$ is étale, so separable, and its degree as a covering equals the generic fibre cardinality. The degree of $[n]$ equals the top self-intersection computation $de g [n] = n^{2 g}$ by the projection formula applied to an ample $L$ with $[n]^{*} L ≅ L^{\otimes n^{2}}$ , giving $de g [n] \cdot (L^{g}) = ([n]^{*} L)^{g} = (n^{2})^{g} (L^{g})$ , so $de g [n] = n^{2 g}$ . Separability plus this degree forces $# A [n] = n^{2 g}$ , and the group-scheme structure makes $A [n]$ a finite abelian group annihilated by $n$ of order $n^{2 g}$ , with $2 g$ cyclic factors by the analytic comparison (or by the étale-fundamental-group structure), giving $A [n] ≅ (Z / n)^{2 g}$ . $□$

Remark. When $n = p = char k$ the statement fails: $[p]$ is inseparable, $A [p]$ has order $p^{2 g}$ as a group scheme but at most $p^{g}$ points, and the $p$ -divisible group encodes the supersingular/ordinary dichotomy. This is the point where the analytic and algebraic theories genuinely diverge.

Connections Master

Elliptic curves 04.04.03. An elliptic curve is precisely a one-dimensional abelian variety, and every structural feature here specialises to a familiar fact: the chord-and-tangent law is the group law forced by rigidity, the dual $\hat{E} ≅ E$ is the self-duality of the principal polarisation given by the origin as theta divisor, and $E [n] ≅ (Z / n)^{2}$ is the $g = 1$ case of $A [n] ≅ (Z / n)^{2 g}$ . The whole of this unit is the program of lifting that one-dimensional theory to arbitrary dimension $g$ .
Picard group 04.05.02. The dual abelian variety is the identity component $\hat{A} = Pic^{0} (A)$ of the Picard group, and a polarisation is the isogeny $φ_{L} : A \to \hat{A}$ built from an ample class in $Pic (A)$ . The Néron-Severi group $NS (A) = Pic (A) / Pic^{0} (A)$ is exactly the home of polarisation types, so the divisor-theoretic machinery of that unit is what classifies the positivity data here.
Ample line bundle 04.05.05. Projectivity of a complex torus is equivalent to the existence of an ample line bundle satisfying the Riemann relations, and the Lefschetz theorem that $L^{\otimes 3}$ is very ample is the explicit embedding. Ampleness is therefore not a side condition but the defining boundary between abstract complex tori and genuine abelian varieties once $g \geq 2$ .
Geometric invariant theory 04.10.02. The moduli space $A_{g}$ of principally polarised abelian varieties is constructed as a GIT quotient using Mumford's theta-level structure and the theta group, so the quotient machinery of that unit consumes the abelian variety of this one as its input geometric object — the moduli of abelian varieties is one of the headline applications of the GIT program.

Historical & philosophical context Master

The analytic theory came first: nineteenth-century work on theta functions and abelian integrals (Abel, Jacobi, Riemann) studied $C^{g} /Λ$ and the conditions on periods making theta functions exist — what we now call the Riemann bilinear relations. Lefschetz ^{[Lefschetz 1921]} gave the projective-embedding theorem showing the third power of a positive bundle embeds the torus, anchoring the bridge from transcendental tori to projective varieties. The decisive abstraction was due to André Weil ^{[Weil 1948]}, who in Variétés abéliennes et courbes algébriques constructed abelian varieties and Jacobians over arbitrary fields, dispensing with the complex-analytic definition entirely so that the theory could serve number theory and the study of curves over finite fields — the foundation of his proof of the Riemann hypothesis for curves.

David Mumford's Abelian Varieties ^{[Mumford 1970]} then rebuilt the subject on the rigidity lemma and the theorem of the cube, giving clean scheme-theoretic proofs of commutativity, the duality $A ≅ \hat{\hat{A}}$ , and the structure of the theta group that linearises the polarisation for GIT. Philosophically the abelian variety sits at a triple junction: it is a group (so representation-theoretic and arithmetic methods apply), a projective variety (so algebraic geometry applies), and a complex torus (so Hodge theory and theta functions apply), and the depth of the subject comes from these three viewpoints agreeing on one object. The conservative reading is that abelian varieties are the linear algebra of algebraic geometry — the objects on which everything is as computable as cohomology lattices and bilinear forms allow.

Bibliography Master

Weil, A., Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.
Mumford, D., Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford University Press 1970 (2nd ed. 1974).
Birkenhake, C. & Lange, H., Complex Abelian Varieties, Grundlehren der mathematischen Wissenschaften 302, 2nd ed., Springer 2004.
Milne, J. S., Abelian Varieties, in Cornell, G. & Silverman, J. (eds.), Arithmetic Geometry, Springer 1986, 103–150.
Lefschetz, S., On certain numerical invariants of algebraic varieties with application to abelian varieties, Trans. Amer. Math. Soc. 22 (1921), 327–406.

Prerequisites

04.05.05
04.05.02
04.04.03

Tier anchors

beginner: Mumford *Abelian Varieties* §I informal — a projective complex torus; Silverman §VI (complex elliptic curves) as the 1-dimensional model
intermediate: Mumford *Abelian Varieties* (TIFR 1970, 2nd ed. 1974) §I–§II; Birkenhake-Lange *Complex Abelian Varieties* §1–§4; Milne *Abelian Varieties* (lecture notes) §I
master: Weil 1948 *Variétés abéliennes et courbes algébriques* (originator of the abstract algebraic theory); Mumford *Abelian Varieties* (TIFR 1970) — the rigidity theorem, the theorem of the cube, the dual variety, the theta group; Birkenhake-Lange *Complex Abelian Varieties* (Grundlehren 302); Milne *Abelian Varieties*

References

Weil 1948 — Variétés abéliennes et courbes algébriques · Hermann, Paris; first abstract (field-independent) construction of the Jacobian and of abelian varieties, replacing the analytic definition
Mumford — Abelian Varieties · Tata Institute of Fundamental Research Studies in Mathematics 5, Oxford 1970 (2nd ed. 1974); rigidity, theorem of the cube, dual variety, theta group, $L^{\otimes 3}$ very ample
Birkenhake-Lange — Complex Abelian Varieties · Grundlehren der mathematischen Wissenschaften 302, 2nd ed. Springer 2004; Riemann relations, polarisations, theta functions
Milne — Abelian Varieties · in Cornell-Silverman-Stevens (eds.) Arithmetic Geometry, Springer 1986, 103-150; and standalone lecture notes; Tate module, isogenies, Weil pairing
Lefschetz 1921 — On certain numerical invariants of algebraic varieties · Trans. Amer. Math. Soc. 22, 327-406; very-ampleness of $L^{\otimes 3}$ on a polarised complex torus

Estimated time

beginner: 16m
intermediate: 45m
master: 80m