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Home , Abelian variety

NOTES ON ABELIAN VARIETIES [PART I]

TIM DOKCHITSER

Contents 1. Affine varieties 2 2. Affine algebraic groups 5 3. General varieties 9 4. Complete varieties 10 5. Algebraic groups 13 6. Abelian varieties 16 7. Abelian varieties over C 17 8. and endomorphisms of complex tori 20 9. 22 10. Differentials 24 11. Divisors 26 12. Jacobians over C 27

1 2 TIM DOKCHITSER

1. Affine varieties We begin with varieties defined over an algebraically closed field k = k¯. n n n By Affine space A = Ak we understand the set k with Zariski topology: n V ⊂ A is closed if there are polynomials fi ∈ k[x1, ..., xn] such that n V = {x ∈ k | all fi(x) = 0}.

Every ideal of k[x1, ..., xn] is finitely generated (it is Noetherian), so it does not matter whether we allow infinitely many fi or not. Clearly, arbitrary intersections of closed sets are closed; the same is true for finite unions: {fi = 0} ∪ {gj = 0} = {figj = 0}. So this is indeed a topology. n A closed nonempty set V ⊂ A is an affine variety if it is irreducible, that is one cannot write V = V1 ∪ V2 with closed Vi ( V . Equivalently, n in the topology on V induced from A , every non-empty open set is dense (Exc 1.1). Any closed set is a finite union of irreducible ones. n Example 1.1. A hypersurface V : f(x1, ..., xn) = 0 in A is irreducible precisely when f is an irreducible polynomial. 1 1 Example 1.2. The only proper closed subsets of A are finite, so A and points are its affine subvarieties. 2 2 Example 1.3. The closed subsets in A are ∅, A and finite unions of points and of irreducible curves f(x, y) = 0. n With topology induced from A , a closed set V becomes a topological space on its own right. In particular, we can talk of its subvarieties (irre- ducible closed subsets). The Zariski topology is very coarse; for example, 2 every two irreducible curves in A have cofinite topology, so they are homeo- morphic. So to characterise varieties properly, we put them into a category. A map of closed sets n m φ : A ⊃ V −→ W ⊂ A is a (also called a regular map) if it can be given by x 7→ (fi(x)) with f1, ..., fm ∈ k[x1, ..., xn]. are continuous, by definition of Zariski topology. We say that φ is an if it has an inverse that is also a morphism, and we write V =∼ W in this case. 1 A morphism f : V → A is a regular function on V , so it is simply a function V → k that can be given by a polynomial in n variables. The n regular functions on V ⊂ A form a ring, denoted k[V ], and clearly ∼ k[V ] = k[x1, ..., xn]/I, I = {f s.t. f|V = 0}. Composing a morphism φ : V → W with a regular function on W gives a regular function on V , so f determines a ring homomorphism φ∗ : k[W ] → k[V ], the pullback of functions. Conversely, it is clear that every k-algebra homomorphism k[W ] → k[V ] arises from a unique f : V → W . In other words, V → k[V ] defines an anti-equivalence of categories Zariski closed sets −→ finitely generated k-algebras with no nilpotents. NOTES ON ABELIAN VARIETIES [PART I] 3

In particular, the ring of regular functions determines V uniquely. Now suppose V is a variety. Then k[V ] is an integral domain (Exc 1.3), and the (anti-)equivalence becomes affine varieties over k −→ integral finitely generated k-algebras. The field of fractions of k[V ] is called the field of rational functions k(V ). Generally, n m φ : A ⊃ V W ⊂ A is a rational map if it can be given by a tuple (f1, ...fm) of rational functions fi ∈ k(x1, ..., xn) whose denominators do not vanish identically on V . In other words, the set of points where φ is not defined is a proper closed subset of V , equivalently φ is defined on a non-empty (hence dense) open. 1 So rational functions f ∈ k(V ) are the same as rational maps V P . n n Example 1.4. The ring of regular functions on A is k[A ] = k[x1, ..., xn], n and k(A ) = k(x1, ..., xn) It is important to note that the of a variety under a morphism is not in general a variety: 1: 2 1 1 Example 1.5. The first projection p : A → A takes xy = 1 to A \{0}, 1 which is not closed in A . 2 (xy,y) 2 2 Example 1.6. The map Ax,y −→ At has image A \{x-axis} ∪ {(0, 0)}. The first example can be given a positive twist, in a sense that it actually 1 gives U = A \{0} a structure of an affine variety. Generally, for a rational 0 map φ : V V and U ⊂ V open, say that φ is regular on U if it is defined at every point of U. (For U = V it coincides with the notion of a regular map as before.) If φ has a regular inverse ψ : V 0 → V with ψ(V 0) ⊂ U, we can think of U as an affine variety isomorphic to V 0. In the example above take V 0 : xy = 1 with φ(t) = (t, t−1) and ψ(x, y) = t. n Example 1.7. If V ⊂ A is a hypersurface f(x1, ...xn) = 0, then the com- n plement U = A \ V has a structure of an affine variety with the ring of regular functions k[x1, ..., xn, 1/f]. Many properties of V have a ring-theoretic interpretation. Two very important ones are: The dimension d = dim V is the length of a longest chain of subvarieties

∅ ( V0 ( ··· ( Vd ⊂ V. (For k = C this agrees with the usual dimension of a complex .) With k[Vi] = k[x1, ..., xn]/Pi, this becomes the length of a longest chain of prime ideals k[V ] ) P0 ) ··· ) Pd = {0},

1What is true is that the image f(X) ⊂ Y always contains a dense open subset of the closure f(X) ([?] Exc II.3.19b). 4 TIM DOKCHITSER which is by definition the ring-theoretic dimension of the ring k[V ]. For a variety V it is, equvalently, the transcendence degree of the field k(V ) over k. n Example 1.8. dim A = dim k[x1, ..., xn] = n. n Example 1.9. A hypersurface H ⊂ A has dimension n − 1. Let V be a variety and x ∈ V a point. A regular function on V may be evaluated at x, and the of this evaluation map k[V ] → k is a maximal ideal. (Conversely, every maximal ideal of k[V ] is of this form.) The local ring Ox = OV,x is the localisation of k[V ] at this ideal. In other words,  f Ox = g ∈ k(V ) f, g ∈ k[V ], g(x) 6= 0

This is a local ring, and its maximal ideal mx is the set of rational functions that vanish at x. Write d = dim V . We say that x ∈ V is non-singular if, equivalently, mx (1) dimk 2 = n − d. (‘≥’ always holds.) mx (2) the completion Oˆ = limO /mj is isomorphic to k[[t , ..., t ]] over k. x ←− x x 1 d (3) If V is given by f = ... = f = 0, the matrix ( ∂fi (x)) has rank d. 1 n ∂xj i,j We say that V is regular (or non-singular) if every point of it is non-singular. Generally, a morphism f : V → W is smooth of relative dimension n if for ∗ all points f(x) = y the pullback of functions f : Ox ← Oy identifies f ∗ ∼ Oˆx ←− Oˆy = Ox[[t1, ..., tn]]. So V is regular if and only if V → {pt} is smooth and, in general, a smooth morphism has regular fibres (preimages of points). 2 2 2 2 Example 1.10. The curves C1 : y = x and C2 : y + x = 1 in A are 2 3 2 3 2 non-singular, and C3 : y = x and C4 : y = x + x are singular at (0, 0).

It follows from (3) that the set of non-singular points Vns ⊂ V is open (Exc 1.5), and it turns out it is always non-empty ([?] Thm. I.5.3); in particular, it is dense in V . Finally, there are products in the category of varieties, and they corre- m spond to tensor products of k-algebras. In other words, if V ⊂ A and W ⊂ n m n m+n A are closed sets (resp. varieties) then so is V × W ⊂ A × A = A , ∼ and k[V × W ] = k[V ] ⊗k k[W ].

Exc 1.1. A topological space is irreducible if and only if every non-empty open set is dense. 2 1 1 Exc 1.2. Zariski topology on A = A × A is not the product topology. Exc 1.3. Prove that the ring of regular functions k[V ] of an affine variety V is an integral domain. 2 3 2 3 2 2 3 2 Exc 1.4. Take the curves C : y =x ,D : y =x +x and E : y = x + x in A , and the 2 3 point p = (0, 0) on them. Prove that OˆC,p =∼ k[[t , t ]], OˆD,p =∼ k[[s, t]]/st, OˆE,p =∼ k[[t]] and that they are pairwise non-isomorphic. NOTES ON ABELIAN VARIETIES [PART I] 5

Exc 1.5. Suppose V is given by f1 = ... = fn = 0. Looking at the determinants of the minors of the matrix ( ∂fi ), prove that the set of singular points of V is closed in V . ∂xj

2. Affine algebraic groups In the same way as topological groups (Lie groups, ...) are topological spaces (, ...) that happen to have a structure, affine algebraic groups are simply closed affine sets with a group structure. Definition 2.1. A group G is an affine over k if it has a n structure of a Zariski closed set in some Ak , and multiplication G × G → G and inverse G → G are morphisms. n Equivalently, starting with a closed set V ⊂ A instead, we require (1) A point e ∈ V (unit element), (2) A morphism m : V × V → V (multiplication), (3) A morphism i : V → V (inverse), which satisfy the usual group axioms2. Example 2.2. 1 (1) The Ga = A , group operation (x, y) 7→ x + y. 1 (2) The Gm = A \{0}, group operation (x, y) 7→ xy. 1 2 Recall that A \{0} is a variety via its identification with {xy = 1} ⊂ A . In this notation, multiplication becomes (x1, y1), (x2, y2) 7→ (x1x2, y1y2). This example naturally generalises to GLn (GL1 being Gm):

Example 2.3. Write Mn = An2 for the set of n × n-matrices over k, and In ∈ Mn for the identity matrix. The classical groups  GLn = A ∈ Mn det A 6= 0 ,  SLn = A ∈ Mn det A = 1 ,  t On = A ∈ Mn A A = In , Sp = A ∈ M AtΩA = Ω Ω = 0 In . 2n 2n −In 0 are affine algebraic groups. This is clear for G = SLn, On, Sp2n, because the defining conditions are polynomial in the variables, so G ⊂ An2 is closed. As for GLn, it is the complement to the hypersurface det(aij) = 0, hence affine by Example 1.7 (cf. also Exc 2.1). A homomorphism of affine algebraic groups is a morphism that is also a , an isomorphism is a homomorphism that has an inverse, and and normal subgroups will always refer the ones that are closed in Zariski topology. If H ⊂ G is any ‘abstract’ , its Zariski closure is a subgroup in this sense (Exc 2.2).

(e,id) 2So V −→ V ×V −→m V is identity and same for (id, e) (unit), m ◦ (m×id) = m ◦ (id×m) diag (id,i) as maps V ×V ×V → V (associativity), and V −→ V ×V −→ V ×V −→m V is constant e (inverse). 6 TIM DOKCHITSER

It is clear that the product of two algebraic groups is an algebraic group, and that the kernel of a homomorphism φ : G1 → is an algebraic group. It is also true that the image φ(G1) is an algebraic group (Exc 2.3).

Example 2.4. The classical groups G = SLn, On, Spn are subgroups of GLn, and the embeddings G,→ GLn and the determinant map G → Gm are homomorphisms.

For g ∈ G, the left translation-by-g map lg : G → G is an isomorphism. Because these maps act transitively on G, every point of G ‘looks the same’. For instance, the irreducible components Gi of G cannot meet. Also, because the set of non-singular points Gi,ns ⊂ Gi is non-empty, Gi is non-singular (use left translations again). In particular, the connected component of identity G0 ⊂ G is a non- singular variety. It is a of G, and its left cosets are the 0 0 connected components of G (Exc 2.7). So G/G is finite and G = G o ∆ for some finite group ∆. Thus it suffices to understand connected groups; the classical matrix groups are connected (Exc 2.8). Example 2.5. Every finite group G is an affine algebraic group, via the regular representation G ⊂ Aut k[G] = GLn. In particular, every finite affine algebraic group is a closed subgroup of some GLn. (In fact, any faithful k-representation of G, not just the regular representation, defines such an embedding.) Somewhat surprisingly, it turns out that this is true for all affine algebraic groups:

Theorem 2.6. Every affine algebraic group G is a closed subgroup of GLn for some n. We will prove this, if only to illustrate that questions about affine va- rieties are really questions about k-algebras. First, some preliminaries are necessary. An action of G on a variety V is a group action α : G×V → V which is a n morphism. If V = A and the action is linear (i.e αg : V → V is in GLn(k) for all g ∈ G), we say that V is a representation of G. Equivalently, it is a homomorphism G → GLn of algebraic groups. To prove the theorem, we need to find Σ : G → GL(V ) whose correspond- ing ring map Σ∗ : k[GL(V )] → A is surjective. (Then Σ is an isomorphism of G onto Σ(G).) Putting aside the question of surjectivity, where can we possibly find a non-trivial representation in the first place? Let us reformulate this on the level of the ring A = k[G]. Write3 m∗ : A −→ A ⊗ A, i∗ : A −→ A, e∗ : A → k

3A k-algebra A with such maps m∗, i∗ and e∗ that satisfy the axioms (e∗ ⊗id)◦m∗ = id, (id ⊗m∗) ◦ m∗ =(m∗ ⊗ id) ◦ m∗ and (i∗ ⊗ id) ◦ m∗ = e∗ (dual to the previous footnote) is called a Hopf algebra, and the maps comultiplication, counit and coinverse respectively. NOTES ON ABELIAN VARIETIES [PART I] 7 for the ring maps corresponding to the multiplication m : G × G → G, the inverse i : G → G and the identity e : {pt} → G. To give a representation Σ: G → GL(V ) is equivalent to specifying a k-linear map σ : V −→ V ⊗ A such that (id ⊗e∗) ◦ σ = id and (id ⊗m∗) ◦ σ = (σ ⊗ id) ◦ σ. We say that σ makes V into an A-comodule. There is only one obvious A-comodule, and that is V = A = k[G] itself with σ = m∗, the co-multiplication map. The only problem is that k[G] is an infinite-dimensional k-vector space4, so what we need is the following Lemma 2.7. Every finite-dimensional k-subspace W ⊂ A is contained in a finite-dimensional subcomodule V ⊂ A (so m∗(V ) ⊂ V ⊗ A). Proof. A sum of subcomodules is again one, so we may assume W = hwi ∗ Pn is one-dimensional. Write m (w) = i=1 vi ⊗ ai with a1, ..., an linearly independent over k and complete them to a k-basis {ai}i∈I of A. We claim that V = hw, v1, ..., vni is a comodule. ∗ P Indeed, suppose m (ai) = cijk aj ⊗ ak. Then X ∗ ∗ ∗ ∗ ∗ X m (vi) ⊗ ai = (m ⊗ id)m (w) = (id ⊗m )m (w) = vi ⊗ cijkaj ⊗ ak ∗ P in V ⊗A⊗A. Comparing the coefficients of ak, we get m (vk) = vi⊗cijkaj, ∗ so m (V ) ⊂ V ⊗ A.  Proof of Theorem 2.6. Pick some k-algebra generators of A = k[G], and let W ⊂ A be their k-span. Take an A-comodule W ⊂ V ⊂ A as in the lemma, and let v1, ..., vn be a k-basis of V . The image of ∗ Σ : k[GL(V )] = k[x11, ..., xnn, 1/ det] −→ A P ∗ ∗ ∗ ∗ ∗ contains i e (vi)Σ (xij) = (e ⊗ id)m (vj) = vj, so Σ is surjective.  Note that the proof is completely explicit.

Example 2.8. Take G = Ga. Then A = k[G] = k[t] is generated by t, so take W = hti. The comultiplication ∗ ∼ m : k[t] −→ k[t] ⊗ k[t](= k[t1, t2]) ∗ maps t 7→ t ⊗ 1 + 1 ⊗ t (=t1 +t2), so m (W ) 6⊂ W ⊗ A. In other words, W is not a comodule. But V = h1, ti is one (cf. proof of Lemma 2.7), and the corresponding embedding Ga → GL2 is 1 t t 7−→ 0 1 . In view of Theorem 2.6, affine algebraic groups are also called linear algebraic groups. The statement of the theorem can even be refined so that a given H < G can be picked out as a stabiliser of some linear subspace:

4unless G is finite, in which case the construction recovers Example 2.5 8 TIM DOKCHITSER

Theorem 2.9 (Chevalley). Let H < G be a (closed) subgroup of an affine algebraic group G. There is a linear representation G → GL(V ) and a linear subspace W ⊂ V whose stabiliser is H.

Proof. Exc 2.10.  Extending this further, one proves that if H < G is normal, it is possible to find V such that H is precisely the kernel of φ : G → Aut V (see [?] §16.3). The image φ(G) is then an algebraic group, so factor groups exist for algebraic groups. (In positive this factor group is not unique, because there are injective homomorphisms of algebraic groups that are not , see Exc 2.12; the question whether the quotient G/H exists in the sense of category theory is subtle, see [?] §15–16.)

5 Exc 2.1. Write down explicitly the multiplication map and the inverse for GL2 ⊂ A . Exc 2.2. Suppose G is an algebraic group and H ⊂ G is a subgroup in the ‘abstact group sense’. Then its (Zariski) closure H¯ ⊂ G is a subgroup in the algebraic group sense. Exc 2.3. Show that the image of an algebraic group homomorphism is an algebraic group. (The image of a variety under a morphism is not always a variety, see Exc 3.5.) Exc 2.4. (Closed orbit lemma) Suppose G × V → V is an action of G on a variety V , and let U = Gv be an orbit. Then the closure U¯ ⊂ V is a variety, U ⊂ U¯ is open and non-singular, and U¯ \ U is a union of orbits (of strictly smaller dimension). In particular, the orbits of minimal dimension are closed. Exc 2.5. Let G be an algebraic group and write Aut G for the set of isomorphisms G → G (as algebraic groups). Determine Aut G for G = Ga and G = Gm. Does Aut G have a structure of an algebraic group in these cases, and is it true that its action on G is an algebraic group action? Exc 2.6. If G is connected and H/G is finite, then H ⊂ Z(G), in particular H is abelian. Exc 2.7. Prove that the connected component of identity G0 ⊂ G is a normal subgroup and its left cosets are the connected components of G.

Exc 2.8. Show that the classical groups GLn, SLn, On and Sp2n are connected. Exc 2.9. Do Example 2.8 for G = Gm. Exc 2.10. Prove Theorem 2.9 (Modify the proof of Theorem 2.6.) Exc 2.11. A character of G is a 1-dimensional representation of G, equivalently a ho- momorphism G → Gm. Prove that characters are in 1-1 correspondence with invertible elements x ∈ k[G]× such that m∗(x) = x ⊗ x. What does the product of characters correspond to? Compute the character group for G = Gm and G = Ga. n ¯ Exc 2.12. Let G ⊂ A be an affine algebraic group of dimension > 0 over k = Fp. (p) Suppose G is given by the equations fi(x1, ..., xn) = 0. For a polynomial f write f for the polynomial whose coefficients are those of f raised to the pth power. Prove that (p) (p) p G = {x|fi (x) = 0} is also an algebraic group, and that the Frobenius map F (x) = x is a homomorphism from G to G(p). Prove that F is bijective but not an isomorphism of algebraic groups. NOTES ON ABELIAN VARIETIES [PART I] 9

3. General varieties Recall that to define a topological (C∞, analytic, ...) manifold one takes S a topological space covered by open sets V = i Vi, such that n n (1) Each Vi is identified with a standard open ball in R (or C ). (2) The transition functions between charts Vi ⊃ Vi ∩ Vj → Vj ∩ Vi ⊂ Vj are continuous (C∞, analytic,...). (3) V is Hausdorff and second countable. The Hausdorff condition is necessary to avoid unpleasanties like glueing R with R along R − {0}: ∪ = £ The two origins cannot be separated¢ by¡ open sets, so thec resulting space is not Hausdorff although both charts are. We do not want this. We now copy this definition to glue affine varieties together, and we only allow finitely many charts. An algebraic set V is a topological space covered by finitely many open sets V = V1 ∪ ... ∪ Vn (affine charts), such that

(1) Each Vi has a structure of an affine variety. 5 (2) The transition maps Vi ⊃ Vi ∩ Vj → Vj ∩ Vi ⊂ Vj are isomorphisms . (3) V is closed in V × V (V is separated).6 An (algebraic) variety is an irreducible algebraic set; in particular, varieties are connected. The separatedness condition (3) is equivalent to Hausdorffness for topo- logical spaces, but makes sense for varieties. (Because open sets are usually dense in Zariski topology, varieties are never Hausdorff). See Excs 3.1–3.4. Here is the main example:

n n Example 3.1. P = Pk is a set of tuples [x0 : ··· : xn] with xi ∈ k and not all 0, modulo the relation that [ax0 : ··· : axn] defines the same point for all a ∈ k×. n A subset V ⊂ P is closed if there are homogeneous polynomials fi in x0, ..., xn such that n V = {x ∈ P | all fi(x) = 0}.

As fi are homogeneous, the condition fi(x) = 0 is independent of the choice of a tuple representing x. n n n n To give P a structure of a variety, cover it P = A(0) ∪ · · · ∪ A(n) with n n A(j) = {[x0 : . . . xj−1 : 1 : xj+1 : . . . xn]} ⊂ P

5 That is, rational functions φij : Vi → Vj , defined everywhere on Vi ∩ Vj ⊂ Vi and mapping it to Vj ∩ Vi ⊂ Vj . 6 The topology on V × V comes from Zariski topology on affine varieties Vi × Vj that cover it. 10 TIM DOKCHITSER

The transition maps between charts are indeed morphisms

n n xj −{xk =0} −→ −{xj =0}, (xi) 7→ (xi ), A(j) A(k) xk n n so P becomes an . Closed irreducible subsets of P are called projective varieties; see Exc 3.7. Example 3.2. If X is an affine variety and V ⊂ X an open set, then V is general not an affine variety. However, it can always be covered by finitely many affine subvarieties of X, so it is a variety. Generally, both open and irreducible closed subsets of a variety are varieties (Exc 3.6). A morphism X → Y of algebraic sets is a continuous map which is a morphism when restricted to affine charts. A rational map X Y is a morphism from a dense open set. As before, regular and rational functions 1 1 on X as morphisms X → A and rational maps X P , respectively. The former form a ring k[X], and the latter a field k(X) if X is a variety. However, unless X is affine, k(X) is usually much larger than the field of fractions of k[X] (Exc 3.8). We refer to 1-dimensional varieties as curves and 2-dimensional varieties as surfaces. A 0-dimensional variety is a point.

Exc 3.1. Every variety X is compact, and X is not Hausdorff unless X is a point. Exc 3.2. A topological space X is Hausdorff if and only if the diagonal X ⊂ X × X is closed in the product topology. Exc 3.3. (‘Valuative criterion of separatedness’) Suppose V satisfies (1) and (2) in the definition of an algebraic set. Then it satisfies (3) if and only if for every curve C and a non-singular point P ∈ C, any morphism C \{P } → X has at most one extension to a morphism C → X. Exc 3.4. If f, g : X → V are morphisms of varieties, the set of points x ∈ X where f(x) = g(x) is closed in X. In fact, for a fixed V this holds for all X, f, g if and only if V is separated. 2 2 2 Exc 3.5. Show that the image of A under (xy, x): A → A is not a variety. Exc 3.6. 2 (a) Prove that A − {0} is not isomorphic to an affine variety. (b) Suppose X is an affine variety and ∅= 6 V ⊂ X open. Then V can be covered by finitely many affine subvarieties of X, so it is a variety. (c) Prove that both open and irreducible closed subsets of a variety are varieties. n n Exc 3.7. Prove that A and P are varieties, that is satisfy the separatedness condition. In particular, affine varieties and projective varieties are actually varieties. 1 1 Exc 3.8. Show that k[P ] = k and k(P ) =∼ k(t). Exc 3.9. Prove that every curve has cofinite topology.

4. Complete varieties The nicest manifolds are the compact ones, but Zariski compactness is not the right notion for varieties — from finite-dimensionality it follows easily that every affine variety is compact (Exc 3.1). NOTES ON ABELIAN VARIETIES [PART I] 11

p A variety X is complete if for every variety Y , the projection X ×Y −→2 Y maps closed sets to closed sets7. For k = C this is the same as requiring X to be compact in the usual topology. Also, in the same way as separatedness is a reformulation of being Hausdorff, completeness is equivalent to compactness for topological spaces (Exc 4.1). Any topological space can be compactified, and the same is true in our setting: Theorem 4.1 (Nagata). Every variety can be embedded in a complete va- riety as a dense open subset. Here are two alternative definitions of completeness that are a bit more natural. Both say that X is as large as possible in some sense, and has no missing points. For the proof of equivalence, see Exc 4.2. Lemma 4.2. For a variety X the following conditions are equivalent: (1) (‘Universally closed’) X is complete. (2) (’Maximality’) If X ⊂ Y is open with Y a variety, then X = Y . (3) (’Valuative criterion’) For every curve C and a non-singular point P ∈ C, any morphism C\{P } → X extends to a morphism8 C → X.

1 Example 4.3. A is not complete: 1 1 p2 1 1 (1) fails because under A × A −→ A , the set xy = 1 projects to A \{0}. 1 1 (2) fails because A may be embedded in P as a dense open subset. −1 1 x7→x 1 1 1 (3) fails because A \{0} −→ A does not extend to A → A . Here are some important consequences of completeness: Lemma 4.4. Suppose X is a . (a) Every closed subvariety Z ⊂ X is complete. (b) For any morphism f : X → Y the image f(X) is complete and closed in Y . (c) k[X] = k, in other words X has no non-constant regular functions. (d) If X is affine, then X is a point. Proof. (a) Clear from the definition. (b) f(X) is the same as p2 of the “graph of f”(x, f(x)) ⊂ X × Y . f 1 1 (c) The image of X under the composition X → A ,→ P is connected, closed and misses ∞, so it must be a point. (d) Affine varieties are characterised by k[X].  n The main example of a complete variety is P (Exc 4.4). By (a), all projective varieties are therefore complete, and these are the only complete

7 One says that π : X → {pt} is ‘universally closed’ (π × idY is a closed map for all Y ). 8necessarily unique as X is a variety and is therefore separated 12 TIM DOKCHITSER

varieties we will ever encounter.9 Incidentally, note that this proves Nagata’s n n theorem for affine varieties: embed V ⊂ A ⊂ P and take the closure V¯ n n of V in P (the intersection of all closed subsets of P containing V ). Then V ⊂ V¯ is dense open and V¯ is a , hence complete.10 Here is one standard illustration of the power of completeness:

Lemma 4.5. Suppose C1 and C2 are complete non-singular curves. (i) Any morphism C1 \{P1, ..., Pn} → C2 extends uniquely to C1 → C2. (ii) Every non-constant map f : C1 → C2 is surjective.

Proof. (i) use 4.2 (3); uniqueness follows from separatedness of C2. (ii) Im f is closed, so it is either a point or the whole of C. 

What (i) says is that every rational map C1 → C2 extends to a unique morphism, so for non-singular complete curves there is no real distinction between rational maps and morphisms. From here it is not hard to get that C 7→ k(C) defines an equivalence of categories complete non-singular Finitely generated field extensions −→ curves over k K/k of transcendence degree 1. 2 1 1 In higher dimensions this is not true — for instance P and P × P have the same field of rational functions, but are not isomorphic (Exc 4.8). When k = C, complete non-singular curves are the same as (compact) Riemann surfaces. In particular, every is algebraizable (arises from such a curve), but this is not true for higher-dimensional com- pact complex manifolds. Nevertheless, for complete varieties there is a very close connection between the usual complex topology and Zariski topology:

Theorem 4.6 (Chow). Let X,Y be complete varieties over C. (1) Every analytic subset11 of X is closed in Zariski topology. (2) Every holomorphic map f : X → Y is induced by a morphism of varieties. n Chow proved (1) for X = P and the rest of the assertions follow relatively easily (using Chow’s lemma for (1) and applying (1) to the graph of f in (2); see [?] §1.3). In particuar, the only meromorphic functions on a complete variety over 1 C are rational functions (take Y = P ).

p ∗ Exc 4.1. A topological space X is compact if and only if X × Y −→2 Y takes closed sets to closed sets for every topological space Y . Exc 4.2. Prove Lemma 4.2.

9In fact, complete curves and surfaces are always projective, and it is quite non-trivial to construct a non-projective complete 3-dimensional variety [?]. 10Thus Nagata’s theorem for arbitrary varieties becomes a question of arranging the completions of affine charts so that they can be glued together. This may be done using ‘blowing-ups’ and ‘blowing-downs’, see [?]. 11Locally (in the usual complex topology) a zero set of holomorphic functions NOTES ON ABELIAN VARIETIES [PART I] 13

Exc 4.3. Let C be a curve and P ∈ C a non-singular point. Show that the local ring

OC,P ⊂ k(C) of functions defined at P is a discrete valuation ring. n Exc 4.4. Use Exc 4.3 and the ‘valuative criterion’ to show that P is complete. n n Exc 4.5. Determine k[P ] and k(P ). Exc 4.6. Suppose C is a curve. (a) Show that there is a non-singular complete curve C˜ birationally isomorphic to C. ˜ ˜ In other words, there are rational maps φ : C C and ψ : C C with φψ = id and ψφ = id. Show that any two such C˜ are isomorphic. (b) If C is complete, then ψ is a surjective morphism. (c) If C is non-singular, then φ is an injective morphism identifying C with an open set of the form C˜ \{P1, ..., Pn}. Exc 4.7. Explain why Lemma 4.5 fails if one of the words ‘complete’, ‘non-singular’ or ‘curves’ is omitted. 2 1 1 Exc 4.8. Show that P and P × P have the same field of rational functions but are 2 not isomorphic. (You may want to use that a non-singular curve C ⊂ P defined by a homogenous equation f(x0, x1, x2) = 0 of degree d has g = (d − 1)(d − 2)/2.) Exc 4.9. Prove that the complete variety in Nagata’s theorem is not necessarily unique. Exc 4.10. Show that for complex varieties that are not complete Chow’s theorem may fail. Exc 4.11. Prove that a compact complex manifold has at most one .

5. Algebraic groups As before, k = k¯ is an algebraically closed base field. We define algebraic groups from varieties exactly as in the affine case (Def. 2.1). Definition 5.1. A group G is an algebraic group if it has a structure of an algebraic set, and multiplication G × G → G and inverse G → G are morphisms. As before, homomorphisms refer to morphisms that are group homomor- phisms, isomorphisms are isomorphisms of both groups and varieties, and subgroups and normal subgroups always refer to closed ones.

Example 5.2. Affine algebraic groups Gm, Ga, GLn, ... are algebraic groups. Example 5.3. Elliptic curves and their products are algebraic groups. Example 5.4. The multiplication-by-m map [m]: G → G is a homomor- phism for any commutative algebraic group G and m ∈ Z (Exc 5.1). Basic properties of algebraic groups carry over immediately from the affine 0 case: an algebraic group G is a G = G o ∆ of its con- nected component of identity G0 and a finite ∆. Again, G0 is a non-singular variety. Kernels and images of algebraic group homomor- phisms exist, and so do factor groups in the same ‘na¨ıve’ sense as before. Example 5.5. Algebraic groups often occur naturally as automorphism groups of varieties (see Exc 5.2 though). For example, suppose C is a com- plete non-singular curve of genus g. ∼ 1 ∼ (g = 0) C = P , and Aut C = PGL2 = GL2 /Gm is a M¨obiusgroup (Exc 5.3). 14 TIM DOKCHITSER

(g = 1) Choosing a point O ∈ C makes C into an , and from ∼ the theory of elliptic curves it follows that Aut C = C n Aut(C,O) with Aut(C,O) finite of ≤ 24 (usually just {id, [−1]}.) (g ≥ 2) Aut C is finite.12 Proposition 5.6. The only one-dimensional connected algebraic groups are Ga, Gm and elliptic curves.

Proof. Write G = C \{P1, ..., Pn} with C a non-singular complete curve (Exc 4.6), and take x ∈ G. The left translation map lx : y 7→ xy on G extends to an automorphism

lx : C −→ C, because C is non-singular and complete. So C has infinitely many auto- morphisms that are (a) fixed point free on G, and (b) preserve the set of ‘missing points’ {P1, ..., Pn}. Write g for the genus of C, and e ∈ G for the identity element. g ≥ 2: Aut C is finite, so this is impossible. g = 1: If n ≥ 1 then | Aut(C,P1)| ≤ 24, so n = 0 and G is complete. For x ∈ G there is a unique fixed point free map taking the identity element e to x, which must be lx in every group law on G which has e as the identity element. So the group law must be the same one as the standard one on an elliptic one. ∼ ∼ g = 0: Now C = P1 and Aut C = PGL2(k) is the group of M¨obius transformations. These are uniquely determined by what they do to 3 given points, in particular Aut G 6= {1} implies n ≤ 2. 1 1 n = 0: Then lx : P → P has no fixed points, which is impossible. 1 n = 1: Change the coordinate on P to move P1 to ∞ and e to 0. The 1 fixed point free automorphisms of P −{∞} are translations z 7→ z + a, again there is a unique one taking 0 to a given a ∈ G, and G = Ga. n = 2: Similarly, move P1 7→ 0,P2 7→ ∞ and e 7→ 1. The automorphisms 1 of P \{0, ∞} are z 7→ az and z 7→ a/z. Only the former ones are fixed point free, and there is a unique one taking e → x for a given x. So the group law is unique, G = Gm.  Suppose G is an algebraic group over k. Recall that if G is an affine variety, G is also called a . Definition 5.7. An abelian variety is a complete connected algebraic group.

12 Hurwitz (1893) showed that | Aut(C)| ≤ 84(g − 1) over C, and the bound is sharp for infinitely many g (Macbeath 1961). Schmid (1938) proved finiteness when char k = p > 0 and noted that Hurwitz’ bound fails for small p. It still holds when p > g + 1, except for p 2 p−1 y − y = x which has g = 2 and | Aut(C)| = 8g(g + 1)(2g + 1) (Roquette 1970). NOTES ON ABELIAN VARIETIES [PART I] 15

We have seen that 1-dimensional algebraic groups are either linear (Ga, Gm) or abelian varieties (elliptic curves). Much more generally, these two extremes build all algebraic groups: Theorem 5.8 (Barsotti-Chevalley). Every connected algebraic group G fits into an exact sequence 1 −→ H −→ G −→ A −→ 1 with H/G the unique largest linear connected subgroup of G, and A an abelian variety. Remark 5.9. With the theory of linear groups thrown in, the classification can be extended. There is a unique filtration of G of the form

finite AV semisimple G G0 G1 G2 G3 1 connected linear solvable with Gi connected and normal in Gi−1. Here: A torus is an algebraic group isomorphic to Gm × · · · × Gm; A unipotent group is a subgroup of upper-triangular matrices with ones on the diagonal; A is one admitting a filtration 1 = H0 /H1 / ··· /Hk = G with Hi/Hi−1 commutative; A semisimple group is one whose radical G2 (the unique maximal connected linear solvable normal subgroup) is trivial; a semisimple group admits a finite covering G1 × · · · × Gn → G with Gi almost simple (finite centre C and G/C simple). Every almost is isomorphic to either SLn+1 (type An), Sp2n (type Cn), ,,,,G2 (exceptional groups) or isogenous to an (types Bn,Dn). See [?] Ch. X for a more extended summary and references.

Example 5.10. G = GLn. Then G = G1, G2 = Z(G) = Gm and G/G2 = PGLn is simple. Example 5.11. Is it not hard to deduce that the only connected commuta- ∼ m n tive linear groups are U = (Gm) ×(Ga) . So every commutative connected algebraic group G has the form G = A o U with A an abelian variety, U as above and acting trivially on A.

Exc 5.1. Prove that the multiplication-by-m map [m]: G → G is a homomorphism for any commutative algebraic group G and m ∈ Z. Exc 5.2. Give an example of a variety V such that Aut V has no natural structure of an algebraic group. 1 at+b Exc 5.3. Show that every automorphism of Pk is of the form t 7→ ct+d . Deduce that 1 ∼ ∗ Aut P = PGL2(k) (= GL2(k)/k ). Exc 5.4. Show that there are no non-constant algebraic group homomorphisms from an abelian variety to a linear algebraic group. 16 TIM DOKCHITSER

6. Abelian varieties Suppose k = k¯ as before. Recall that an abelian variety A/k is an alge- braic group over k, which is a complete variety. Let us validate ‘abelian’ and prove that abelian varieties are always commutative. This must clearly rely on completeness, and we follow Mumford’s approach using rigidity: Lemma 6.1 (Rigidity). Suppose f : V ×W → U is a map of varieties, V is complete, and

f({v0} × W ) = f(V × {w0}) = {u0} for some points v0, w0 and u0. Then f is constant, f(V × W ) = {u0}. 13 −1 Proof. Let U0 be an open affine neighbourhood of u0 and Z = f (U −U0). This is a closed set, and so is its image under the projection p2 : V ×W → W , as V is complete. As w0 ∈/ p2(Z), the complement W0 = W − p2(Z) is open dense in W . But for all w ∈ W0 the image f(V × {w}) ⊂ U0 must be a point, as V × {w} is complete and U0 is affine. In other words, f(V × {w}) = f((v0, w)) = u0. −1 −1 So f (u0) contains a dense open V × W0; as f (u0) is also closed, it must be the whole space, so f is constant.  Corollary 6.2. If U, V, W are varieties, V is complete, U is an algebraic group, and f1, f2 : V × W → U are morphisms that agree on {v0} × W and on V × {w0}, then they agree everywhere. −1 Proof. The map x 7→ f1(x)f2(x) is constant by the rigidity lemma.  Corollary 6.3. Abelian varieties are commutative. Proof. The maps xy and yx from X × X to X agree on X × {e} and e × X, so they must agree everywhere by the previous corollary.  From now on let us write the group operation on abelian varieties as addition, and denote the identity element by 0. Corollary 6.4. If A, B are abelian varieties, then every morphism of vari- eties f : A → B that takes 0 to 0 is a homomorphism of abelian varieties. Proof. The morphisms f(x) + f(y) and f(x + y) from A × A to B agree on {0} × A and on A × {0}, so they are equal.  Corollary 6.5. Every morphism f : A → B between abelian varieties is a composition of a translation on B and a homomorphism A → B.

13 Over k = C, this works as follows: if w is close to w0, then f(V × {w}) is close to u0, by the compactness of V and continuity of f. So f(V × {w}) is contained in some open ball around u0. But there are no non-constant analytic maps from V to an open ball (maximum principle), so f(V × {w}) is a point for such w, namely f((v0, w)) = u0. This proves that the set of such w is open; but it is also closed, so f is constant. NOTES ON ABELIAN VARIETIES [PART I] 17

In fact, in the two preceding corollaries B could be any algebraic group. Rigidity has another curious consequence: in defining an abelian variety we could have dropped the associativity condition, as it also follows auto- matically from rigidity! (Exc 6.1) For instance, for elliptic curves this gives a quick proof of the associativity of the group law, that only relies on E being complete. Remark 6.6. It is possible to extend Corollary 6.5 slightly: any rational map φ : G A from a connected algebraic group to an abelian variety is a composition of a translation with a homomorphism G → A (in particular, φ a morphism).

Exc 6.1. Suppose V is a complete variety, e ∈ V (k) a point, and we have a morphism ∗ : V ×V → V and an isomorphism i : V → V . If x∗e = e∗x = x and x∗i(x) = i(x)∗x = e, then ∗ is associative, so V is an abelian variety. 1 Exc 6.2. Prove that for an abelian variety A, every rational map P A is a constant n morphism (hint: 6.6). Deduce that every rational map P A is also constant.

7. Abelian varieties over C If the base field k = k¯ has characteristic 0, by Lefschetz principle (e.g. [?] §VI.6) understanding varieties over k essentially reduces to understanding those over C. Suppose k = C and X/C is a g-dimensional abelian variety. As X is regular and complete, in classical topology (also referred to as Euclidean or strong topology) it is a compact complex manifold with a group structure. In other words, it is a compact complex-analytic of C-dimension g. Theorem 7.1. Let X/C be a g-dimensional abelian variety. As complex ∼ g g Lie groups, X = C /Λ for some Λ ⊂ C of rank 2g.

Proof. Suppose X is any complex Lie group, and write V = TeX for the tangent space at identity. From the theory of Lie groups, for every v ∈ V there is a unique analytic group homomorphism (“one-parameter subgroup”)

φv : C −→ X with dφv(0) = v, and φ·(·): V × C → X analytic. Define the exponential map exp:V →X by exp v = φv(1). It is analytic, and we have (1) φv(t) = exp(tv). [φv(st) = φtv(s) by uniqueness of φv; now put s = 1.] (2) If X is commutative then exp : V → X is a group homomorphism. [As X is abelian, ψ : t 7→ exp(tx) exp(ty) is a homomorphism C → X. But dψ(1) = x + y, so ψ = φx+y again by uniqueness; now put t = 1.] Now if X is an abelian variety, then exp : V → X is surjective because exp(V ) ⊂ X is an open subgroup (Exc 7.1). Finally, ker exp ⊂ V is a discrete subgroup with compact quotient, so it is a lattice of maximal rank.  18 TIM DOKCHITSER

Corollary 7.2. Let X be a g-dimensional abelian variety over C, and write X[n] for the n- subgroup of X (all x with nx = 0). Then14 ∼ 2g X[n] = (Z/nZ) (n ≥ 1). ∼ 2g ∼ 1 ∼ 1 2g Proof. X = C /Λ, so X[n] = n Λ/Λ = ( n Z/Z) as a group.  The proof of Theorem 7.1 does not use compactness of X until the last line, so all conclusions except Λ = ker exp having maximal rank remain valid for all commutative Lie groups over C:

Example 7.3. For X = Ga, exp : C → C = Ga is identity, and Λ = {1}. ∗ Example 7.4. For X = Gm, exp : C → C = Gm is the usual exponential z function z 7→ e , and Λ = 2πiZ is a lattice of rank one; in other words, ∼ Gm = C/Z as an analytic Lie group. Example 7.5. (Elliptic curves) If Λ ⊂ C is a lattice, the parametrisation 0 (℘,℘ ,1) 2 3 2 C/Λ −−−−−→ E : y = 4x −g2x−g3 ⊂ P is given by the elliptic functions g2 = g2(Λ), g3 = g3(Λ) and ℘(z) = ℘(Λ, z): P 1 P 1 1 P 1 1  g2 = 60 w4 , g3 = 140 w6 , ℘(z) = z2 + (z−w)2 − w2 . w∈Λ w∈Λ w∈Λ w6=0 w6=0 w6=0 Note that ℘(z) is a transcendental function, in other words the morphism 1 = → E is by no means algebraic.15 AC C The three examples combine to an analytic analogue of the classification of 1-dimensional algebraic groups (Prop. 5.6): a connected 1-dimensional commutative complex Lie group is either Ga = C, Gm = C/Z or an elliptic curve E = C/Zτ1 ⊕ Zτ2. g A compact analytic Lie group of the form C /Λ is called a (g-dimensional) . By the example above, every 1-dimensional complex torus is an elliptic curve. Unfortunately, this fails for g > 1: not every complex torus is an abelian variety. The ones that are varieties are called algebraizable, and we give a criterion below. Everything else goes well though: g 0 g0 0 Theorem 7.6. Suppose A = C /Λ and A = C /Λ are abelian varieties. (1) Abelian subvarieties U ⊂ A are in one-to-one correspondence with g the C-subspaces V ⊂ C for which rkZ(V ∩ Λ) = 2 dim V . 0 0 (2) HomAV(A, A ) = {M ∈ Matg0×g(C) | MΛ ⊂ Λ }. ∼ 0 0 (3) A = A if and only if MΛ = Λ for some M ∈ GLg(C). Proof. (3) is a special case of (2). Half of (2) is also easy: a homomorphism

14By Lefschetz principle, the same holds over any k = k¯ of characteristic 0 (Exc 7.2) 15E.g. its kernel Λ is not Zariski closed; in fact, there are no non-constant morphisms 1 A → E by Exc 6.2. NOTES ON ABELIAN VARIETIES [PART I] 19

0 φ : A → A induces a C-linear map on the tangent spaces at e, g g0 dφ : C → C and exp ◦ dφ = φ ◦ exp by the uniqueness of exp, so φ(Λ) ⊂ Λ0. Applying this to the inclusion U ⊂ A gives half of (1) as well. The other half of (1) and (2) is a special case of Chow’s theorem 4.6.  g 0 g0 0 Corollary 7.7. If A = C /Λ and A = C /Λ are abelian varieties, then 0 0 HomAV(A, A ) is a free Z-module of rank ≤ 2gg . g Proof. Let L ⊂ Λ be a rank g sublattice which spans C over C. Then 0 0 HomAV(A, A ) ,→ HomZ(L, Λ ) from 7.6 (2).  This bound is best possible for all g and g0 (Exc 8.3). Example 7.8. If E = C/Λ is an elliptic curve, then End E = Hom(E,E) is either Z or has rank 2. In the latter case, it is an order in an imaginary quadratic field (Exc 7.3) and we say that E has . Another immediate corollary of Chow’s theorem is a description of ratio- g nal functions on an abelian variety A = C /Λ: Rational functions on A = Meromorphic functions on A = g Λ-periodic meromorphic functions on C . It turns out that algebraicity of a complex torus amounts to requiring that it has ‘enough’ meromorphic functions, and has the following explicit description: g Theorem 7.9. (Lefschetz) Let Λ ⊂ C be a Z-lattice of rank 2g. For X = g C /Λ the following conditions are equivalent: (1) X is an abelian variety. (2) X is a projective variety. (3) X has g algebraically independent meromorphic functions. g g (4) There is a positive-definite Hermitian form H : C × C → C such that E = Im H takes values in Z on Λ × Λ. A form H as in (4) is called a Hermitian , and E = Im H an alternating Riemann form. Note that E :Λ × Λ → Z is alternating and E(iu, iv) = E(u, v) (use that H is Hermitian). Conversely, such a E recovers H by H(u, v) = E(iu, v) + iE(u, v), see Exc 7.4. Example 7.10. (Elliptic curves) For 1-dimensional complex tori X = C/Λ the conditions are always satisfied: (1) Every compact Riemann surface in algebraizable. (2) The field of meromorphic functions is generated by ℘(z) and ℘0(z) with one relation (Example 7.5). 1 (3) Rescale the lattice so that Λ = Z ⊕ Zτ. Then H(z, w) = Im τ zw¯ is a Hermitian Riemann form. 20 TIM DOKCHITSER

g For g > 1, a general lattice Λ ⊂ C does not admit any alternating Riemann forms: if {ei} is a Z-basis of Λ, an alternating form E :Λ × Λ → Z is determined by E(ei, ej) ∈ Z. The condition E(iu, iv) = E(u, v) forces linear relations with real coefficients upon them, and these will have no non- zero solutions in general. Thus, ‘most’ tori are not algebraizable.

Exc 7.1. If G is a connected and U ⊂ G an open subgroup, then U = G. 2g Exc 7.2. Use Lefschetz principle to show that A[n] =∼ (Z/nZ) for a g-dimensional abelian variety A over a field k = k¯ of characteristic 0. Exc 7.3. Suppose E = C/(Z + τZ) is an elliptic curve with End E 6= Z. Show that K = Q(τ) is an imaginary quadratic field, and End E,→ K is naturally a subring which has rank 2 over Z (so it is an order of K). Exc 7.4. Show that H 7→ Re H defines a 1-1 correspondence between Hermitian forms H g g on C and real-valued alternating forms E on C satisfying E(iu, iv) = E(u, v). 2g 2g Exc 7.5. Let E : Z × Z → Z be an alternating non-degenerate form. There is a basis 2g 0 M for Z in which E has the shape −M 0 with M a diagonal matrix with positive integer entries on the diagonal. Exc 7.6. Write down an explicit example of a non-algebraizable complex torus.

8. Isogenies and endomorphisms of complex tori g g0 0 Recall that by 7.6(2), homomorphisms φ : C /Λ → C /Λ of abelian g g0 0 varieties are just C-linear maps C → C mapping Λ to Λ . Let us call the latter homomorphisms of complex tori even when they are not algebraizable. (Equivalently, it is a homomorphism as analytic Lie groups.) Definition 8.1. A homomorphism φ : X → X0 of complex tori is an of degree m if it is surjective with kernel of size m. We write deg φ = m, and we say that the two tori are isogenous (denoted X ∼ X0). Equivalently, φ is a homomorphism which is a covering map of degree m g g0 0 0 in the topological sense. If φ : C /Λ → C /Λ is such an isogeny, then g = g g g0 0 and we identify C = C via φ, it gives inclusion of lattices φ :Λ ,→ Λ , identifying Λ with a sublattice of index m in Λ0. Here are some immediate properties of isogenies: φ ψ • For isogenies X−→X0−→X00 of complex tori, deg(ψφ) = deg φ deg ψ. [Consider Λ ,→ Λ0 ,→ Λ00]. g g • The multiplication-by-m map [m]: C /Λ → C /Λ is an isogeny of degree m2g [cf. 7.2]. • There is an isogeny X → X0 if and only if there is one X0 → X. [If Λ ⊂ Λ0 has index m, then mΛ0 ⊂ Λ.] Thus, being isogenous is an . Specifically, for a degree m isogeny φ : X → X0 we constructed a unique isogeny φ˜ : X0 → X (of degree m2g−1) with φφ˜ = [m], called the conjugate isogeny. For elliptic curves φ˜=˜ φ, but this fails in higher dimensions (wrong degree). φ ψ • For isogenies X −→ X0 −→ X00 of complex tori, ψφf = φ˜ ◦ ψ˜. NOTES ON ABELIAN VARIETIES [PART I] 21

• A complex torus isogenous to an abelian variety is an abelian variety. [Restrict the Riemann form to Λ ⊂ Λ0 16]. The End A of a complex abelian variety A is finitely 0 generated and free as a Z-module (7.7), and we call End A = End A ⊗Z Q the endomorphism algebra of A. Its units are isogenies A → A. We call A simple if it has no abelian subvarieties apart from {0} and A. Lemma 8.2. (1) A is simple if and only if End0 A is a division algebra. (2) If A and A0 are simple, then either A ∼ A0 or Hom(A, A0) = 0. (3) If A ∼ A0 then End0 A =∼ End0 A0. Proof. (1, 2) Exc 8.5. (3) If φ : A → A0 is an isogeny of degree m, the maps α : f 7→ φfφ˜ : End A → End A0 and β : f 7→ φfφ˜ : End A0 → End A compose to αβ = [m] ∈ End A and βα = [m] ∈ End A0. Because [m] becomes 0 a unit after ⊗Q, α and β are isomorphisms on the level of End .  Theorem 8.3 (Poincare reducibility). Suppose A0 ⊂ A is an abelian sub- variety of a complex abelian variety. Then there is an abelian subvariety A00 ⊂A such that A=A0+A00 and A0 ∩A00 is finite; A is isogenous to A0 ×A00.

Proof. Exc 8.6.  Thus, ever non-simple abelian variety is isogenous to a product A0 × A00 of abelian varieties of smaller dimension. Continuing this process, we get Corollary 8.4 (Complete reducibility). Every complex abelian variety is isogenous to a product simple abelian varieties,

n1 nk A ∼ A1 × · · · × Ak . For such an A, 0 ∼ 0 End A = Matn1×n1 (D1) × · · · × Matn1×nk (Dk),Di = End (Ai), in particular it is a semisimple Q-algebra (product of matrix algebras over division algebras). A formal way of saying this is that the category of ‘abelian varieties isogeny’ (objects = abelian varieties, morphisms = HomAV ⊗ZQ) is a semisimple Q-linear category (Homs are Q-vector spaces and every object is a direct sum of simple objects). It remains to understand algebraically which division rings may actually occur as endomorphism algebras of simple abelian varieties. For this we first need to reformulate the algebraizability condition (existence of a Riemann form).

16The same argument also gives an alternative proof that subtori of abelian varieties are abelian varieties (half of Thm. 7.6(2)), avoiding the use of Chow’s theorem. 22 TIM DOKCHITSER

Exc 8.1. Show that the elliptic curves En = C/(Z+ni Z) for n ≥ 1 are pairwise isogenous. 0 Determine End En and End En. Exc 8.2. ‘A generic complex torus has endomorphism ring Z’. Explain. Exc 8.3. For any g, g0 ≥ 1 give an example of complex abelian varieties with dim A = g, dim A0 = g0 and dim Hom(A, A0) = 2gg0. Give also an example with Hom(A, A0) = {0}. Exc 8.4. Give an example of an abelian variety which is isogenous to a product of two (positive-dimensional) abelian varieties, but is itself not such a product. Exc 8.5. Prove that A is a simple abelian variety if and only if End0 A is a division algebra. Show that if A and A0 are simple abelian varieties, then either A ∼ A0 or Hom(A, A0) = 0. Exc 8.6. Prove the Poincare reducibility theorem (8.3). Hint: Take the orthogonal com- plement V of Λ0 ⊂ Λ with respect to the Hermitian Riemann form H, and let Λ00 = V ∩Λ.

9. Dual abelian variety ∼ g Let X = V/Λ be a complex torus of dimension g (so V = C ). Define ∗ V = {f ∈ HomR(V, C) | f(αv) =αf ¯ (v), all α ∈ C, v ∈ V }, ∗ ∗ Λ = {f ∈ V | Im f(v) ∈ Z all v ∈ Λ}. ∗ Because (v, f) 7→ Im f(v) is a non-degenerate R-linear pairing V × V → R, Λ∗ ⊂V ∗ is again a lattice of rank 2g. Call X∗ =V ∗/Λ∗ the dual torus of V/Λ. Clearly, • X∗∗ = X canonically; • a homomorphism f : X → Y induces f ∗ : Y ∗ → X∗. Moreover, f is an isogeny if and only if f ∗ is; if they are, then deg f = deg f ∗; φ ψ • (ψ ◦ φ)∗ = φ∗ ◦ ψ∗ for homomorphisms X −→ X0 −→ X00; • [m]∗ = [m] for the multiplication-by-m map [m]: X → X, the properties expected from a decent duality. If X = V/Λ is an abelian variety, then a Hermitian Riemann form on X ∗ defines a C-linear isomorphism λH : v 7→ H(v, ·) from V to V which maps Λ to Λ∗, because Im H is integer-valued on Λ × Λ by definition. So ∗ λH : X → X is an isogeny17. Such as isogeny λ : X → X∗ induced by a positive-definite Hermitian form H is called a polarisation on X, and Lefschetz’ Theorem 7.9 asserts that abelian varieties are precisely those tori that admit a polarisa- tion. Note also that λ∗ = λ (identifying X∗∗ with X). We call λ a principal polarisation if it has degree 1, in other words if it is an isomorphism X−→∼ X∗. By Exc 7.5, a principal polarisation comes from 0 I a Riemann form −I 0 in some basis of Λ (I is the g × g identity matrix). Not every abelian variety admits a principal polarisation, but it is always isogenous to one that does (Exc 9.2).

Example 9.1. On an elliptic curve X = C/(Z + Zτ), any Riemann form is 1 as integer multiple of the standard one H(z, w) = Im τ zw¯. An elliptic curve

17whose degree is p| det E|, where E = Im H (Exc 9.1) NOTES ON ABELIAN VARIETIES [PART I] 23

∼ ∗ is principally polarised via λH : X−→X , and every other polarisation has the form [n]λH for some non-zero n. Now we return to the question of understanding the endomorphism al- gebra End0 A of an abelian variety A. Fix a polarisation λ : A → A∗. As it is an isogeny, it induces a ring isomorphism End0 A → End0 A∗ via φ 7→ λφλ−1. But also the duality map f 7→ f ∗ : End A −→∼ End(A∗), induces an anti18-isomorphism End0 A → End0 A∗. Combining the two gives an anti-involution on End0 A: Definition 9.2. The anti-involution ι : φ 7→ φι = λφ∗λ−1 on End0 A is called the Rosati involution induced by the polarisation λ. Note that ι([m]) = λ[m]∗λ−1 = λ[m]λ−1 = [m]λλ−1 = [m] (isogenies 0 commute with addition), so ι acts trivially on Q ⊂ End A. 0 Example 9.3. If E is a non-CM elliptic curve, then End E = Q and the Rosati involution induced by any λ is the identity map ι : Q → Q. Example 9.4. If E is a CM elliptic curve, then the Rosati√ involution in- 0 ∼ duced by any λ is the complex conjugation on End E = Q( −d) (check). Now suppose A is simple, so D = End0 A is a finite-dimensional division 19 algebra over Q. Denote by Tr = TrD/Q the trace from D to Q. It is easy to see what the positive-definiteness of the Riemann form corresponds to: Lemma 9.5. The quadratic form φ 7→ Tr(φφι) is positive definite on End0 A. Division algebras with these properties can be classified completely: Theorem 9.6 (Albert). Let D be a finite-dimensional division algebra over Q with a positive-definite anti-involution ι. Write K for the centre of D, and K0 for the subfield of K of ι-invariants. Then K is a field, K0 is a totally real field, and the pair (D, ι) belongs to one of the following types: Type I: D = K = K0 and ι is identity. Type II: K = K0, D is a quaternion algebra over K, and there is an isomorphism ∼ D ⊗Q R = Mat2×2(R) × · · · × Mat2×2(R), t t taking ι to the involution (x1, ..., xe) 7→ (x1, ..., xe). Type III: K = K0, D is a quaternion algebra over K, and there is an isomorphism ∼ D ⊗Q R = H × · · · × H taking ι to (ιH, ..., ιH) with ιH(x) = TrH/R(x) − x, the usual involution on the standard (Hamiltonian) quaternions H.

18i.e. it preserves addition and changes the order in multiplication 19 n =trace of the left multiplication by α ∈ D as a Q-linear endomorphism of D(=∼ Q ) 24 TIM DOKCHITSER

Type IV: K/K0 is a totally imaginary quadratic extension, D is a division algebra over K (of some rank d2) whose Brauer invariants above any fixed finite place of K0 sum to 0, and there is an isomorphism ∼ D ⊗Q R = Matd×d(C) × · · · × Matd×d(C), t t taking ι to the involution (x1, ..., xe0 ) 7→ (¯x1, ..., x¯e0 ).

Let D be a division algebra as in the theorem, and denote e0 = [K0 : Q], 2 e = [K : Q], d = dimK D. Can D actually occur as the endomorphism g 2 algebra of some abelian variety A = C /Λ? One restriction is that d e = ∼ 2g dimQ D|2g, since D should act Q-linearly on Λ ⊗ Q = Q . Conversely, suppose g ≥ 1 an integer with dimQ D|2g. It turns out that 2 except when D has Type III with g/2e ∈ {1, 2} or Type IV with g/e0d ∈ {1, 2} there always is a simple complex abelian variety of dimension g with endomorphism algebra D (see [?]). In the exceptional cases there are explicit additional conditions guaranteeing the existence (see [?]).

Exc 9.1. Show that the degree of the polarisation induced by a Hermitian Riemann form 2 H satisfies (deg λH ) = | det E| (the 2g × 2g-determinant of E = Im H on Λ × Λ). Exc 9.2. Give an example of an abelian variety that has no principal polarisation. Show, however, that every abelian variety over C is isogenous to a principally polarized one.

10. Differentials Let X be a variety over a field k = k¯. The rational k-differentials on X P are formal finite sums ω = i fidgi with fi, gi ∈ k(X), modulo the relations d(f + g) = df + dg, d(fg) = fdg + gdf, da = 0 (a ∈ k).

If char k = 0, dim X = n and t1, ..., tn ∈ k(X) are algebraically independent, every differential can be written uniquely as g1dt1+...+gndtn with gi ∈ k(X) (Exc 10.1), so their space is =∼ k(X)n as a k-. 1 Example 10.1. X = P has differentials f(x)dx for f ∈ k(X) = k(x). Example 10.2. On the curve C : y2 = x3 + 1 every differential can be written uniquely as f(x, y)dx, and also as h(x, y)dy with f, h ∈ k(C). (use that 0 = d(y2 −x3 −1) = 2ydy − 3x2dx to transform between the two). P A differential ω is regular at P ∈ X if it has a representation ω = i fidgi 20 with fi, gi regular at P . If ω is regular everywhere, we call it a regular differential, and we write ΩX for the k-vector space of those. A morphism φ : X → Y naturally induces a k-linear map ∗ P P ∗ ∗ φ :ΩY → ΩX , i fidgi 7→ i(φ fi)d(φ gi),

20If P ∈ X is a non-singular point, there is an easy test: pick the algebraically in- 2 dependent functions g1, ..., gd ∈ k(X) so that gi − gi(P ) generate mP /mP , and write P ω = i fidgi. Then ω is regular at P if and only if all the fi are (Exc 10.2) NOTES ON ABELIAN VARIETIES [PART I] 25 the pullback of differentials. For complete varieties dimk ΩX is finite; if, moreover, X is projective then regular differentials are the same as holo- morphic differentials21. 1 n Example 10.3. P (and, generally, P ) has no non-zero regular differentials. 2 3 dx Example 10.4. The curve C : y = x +1 has ΩC = k y , of k-dimension 1. g Example 10.5. An abelian variety A = C /Λ has ΩA = hdz1, ..., dzgi, where zi are the standard coordinates. In fact, dimk ΩA = dim A for an abelian variety over any k, and all ω ∈ ΩA are invariant under translations on A.22

Definition 10.6. The genus g of a complete non-singular curve X is dimk ΩX . 1 2 3 2 Thus, P has genus 0 and y = x + 1 ⊂ P genus 1. Generally: Example 10.7. (Plane curves) A curve given by a non-singular homoge- 2 (d−1)(d−2) neous equation f(x, y, z) ⊂ P has genus g = 2 , d = deg f. Example 10.8. (Hyperelliptic curves, char k 6= 2) Let f(x) be a polynomial of degree 2g + 1 or 2g + 2 with no multiple roots, for some g ≥ 0. The two affine charts 2 2 2g+2 1 y = f(x) and Y = X f( X ) y 1 glue via Y = xg+1 , X = x to a curve C. The curve is complete, non-singular 1 and admits a degree 2 map C → P (via (x, y) 7→ x). Such a curve is called hyperelliptic, and every has a model as above (use that [k(C): k(x)] = 2). The reflection automorphism (x, y) 7→ (x, −y): C → C is called the hyperelliptic involution. Here Ddx xdx xg−1dxE Ω = , ,..., , C y y y so C has genus g. (See Exc 10.4). It is consequence of Riemann-Roch theorem that for g ≤ 2 every complete non-singular curve is hyperelliptic, and genus 3 curves are either hyperellptic or non-singular plane quartics.

Exc 10.1. Suppose X is an n-dimensional variety, t1, ..., tn ∈ k(X) are algebraically independent and the (finite) extension k(X)/k(t1, ..., tn) is separable (e.g. char k = 0).

Then every rational differential on X can be written uniquely as g1dt1 + ... + gndtn with gi ∈ k(X).

21These are special cases of finite-dimensionality of the space of global sections of a on a complete variety, and of Serre’s ‘G´eom´etrieAlg´ebriqueG´eom´etrie Analytique’ (GAGA) principle for complex projective varieties. 22 Otherwise the action by translations would give a non-constant morphism A→GL(ΩA). 26 TIM DOKCHITSER

Exc 10.2. Suppose X is a variety, P ∈ X a non-singular point, and g1, ..., gdim X ∈ k(X) are algebraically independent functions so that gi − gi(P ) generate the k-vector space 2 P mP /mP .(mP ⊂ OX,P is the maximal ideal.) Then ω = i fidgi is regular at P if and only if all the fi are. n Exc 10.3. Prove that P has no non-zero regular differentials. Exc 10.4. Suppose char k 6= 2. Let C : y2 = f(x) with f(x) ∈ k[x] of degree 2g + 1 or dx xdx xg−1dx 2g + 2 with no multiple roots. Prove that ΩC = h y , y ,..., y i.

11. Divisors Let C be a complete non-singular curve over k = k¯.A divisor on C is a finite formal linear combination of points, r X D = ni(Pi), ni ∈ Z,Pi ∈ C. i=1 P The degree of D is ni. If f ∈ k(C)× is a non-zero rational function, define the divisor of f X (f) = ordP (f)(P ). P ∈C

(Recall that the local rings OC,P are discrete valuation rings, and ordP denotes the corresponding valuation on k(X).) Divisors of the form (f) are called principal, and it is not hard to see that they have degree 0. Clearly they form a subgroup, and the quotient groups divisors on C divisors of degree 0 on C Pic C = , Pic0 C = principal divisors principal divisors are called the of C and the degree 0 Picard group of C. Two divisors are called linearly equivalent if they have the same class in Pic C. There is an obvious (split) exact sequence

0 deg 0 −→ Pic C −→ Pic C −→ Z −→ 0. 1 P Example 11.1. On C =P any divisor D = na(a) of degree 0 is principal, a∈P1 Y D = (f), f = (x − a)na . a6=∞ 0 1 1 So Pic P = {0} and Pic P = Z. Example 11.2. On an elliptic curve E with origin O every divisor of degree 0 is linearly equivalent to a divisor of the form (P )−(O) for a unique P ∈ E. 0 ∼ ∼ Thus Pic E = E and Pic E = E × Z as abelian groups. A morphism of non-singular complete curves φ : C → C0 induces a map φ∗ : {divisors on C0} −→ {divisors on C}, NOTES ON ABELIAN VARIETIES [PART I] 27 the pullback of divisors. For a point D = (P ), ∗ X φ (P ) = eQ(Q), f(Q)=P the sum taken over the pre-images Q of P , and the coefficient eQ the rami- fication index23 of f at Q. It is easy to check that deg φ∗D = deg φ deg D and φ∗(f) = (f ◦ φ) for f ∈ k(D)×, in particular φ∗ takes divisors of degree 0 to divisors of degree 0 and principal divisors to principal divisors. So it induces maps φ∗ : Pic C0 → Pic C and Pic0 C0 → Pic0 C. 2 2 1 Example 11.3.√ Let C : y√= x ⊂ P and D = P , with φ :(x, y) 7→ y. Then φ∗(a) = ( a, a) + (− a, a) for non-zero a, and φ∗(0) = 2(0, 0).

12. Jacobians over C Generalising Examples 11.1 and 11.2, it turns out that for any complete non-singular curve C of genus g the group Pic0 C has a natural structure of a principally polarized abelian variety of dimension g, the Jacobian of C. Over the complex numbers this is classical theory of Abel and Jacobi. Thus, suppose k = C (so C is a compact Riemann surface), and choose (1) a basis ω1, . . . , ωg of the space of regular differentials ΩC and (2) a basis γ1, . . . , γ2g of the first homology group H1(C, Z). Fix a point P0 ∈ C and look at the map Z P Z P g   C → C ,P 7→ ω1,..., ωg . P0 P0 This is not well-defined because the integrals are path-dependent; changing P a path by a loop with a class in niγi ∈ H1(C, Z) modifies the integral by P niλi, where Z Z  λi = ω1, ..., ωg , (i = 1,..., 2g). γi γi R (The integrals ωj are called the periods of C). Letting γi g Λ = Zλ1 + ... + Zλ2g ⊂ C (“period lattice”), g the integration map C → C /Λ is now well-defined. Extending it by linearity defines the Abel-Jacobi map g (12.1) {divisors on C} −→ C /Λ. g It turns out that (H1(C, Z) =) Λ ,→ C (= ΩC ) is a lattice, and we have

∗ 23 ordQ φ f 0 Defined by eQ = for any f ∈ k(C ) with ordP f 6= 0. ordP f 28 TIM DOKCHITSER

0 g 24 Theorem 12.2 (Abel–Jacobi). The map (12.1) a bijection Pic C → C /Λ.

The lattice Λ = H1(C, Z) carries a natural non-degenerate alternating form E :Λ×Λ → Z of determinant 1, the intersection pairing for homol- 25 g ogy classes ; one checks that after embedding Λ ,→ C , the form satisfies g g E(iu, iv)=E(u, v) for u, v ∈ C . So E is a Riemann form and gives C /Λ the structure of a principally polarized abelian variety, the Jacobian Jac C of C. 2 6 Example 12.3. Let C be the genus 2 hyperelliptic curve y = x −1 over C. dx xdx Here ΩC = h y , y i, and we let α1, α2, β1, β2 be the standard ‘symplectic’ basis of H1(C, Z) as in Figure 1; thus E(αj, βj) = 1 = −E(βj, αj), and the rest are 0.

Figure 1

To represent C as a Riemann surface, make 3 cuts in C between the roots 6 of x√−1 (6th roots of unity), see Figure 2. There are two continuous branches of x6 − 1 on the complement, in other words every point (x, y) ∈ C with x∈ / {cuts} lies on one of the two sheets. Glueing them together gives C, which is topologically a surface as in Figure 3 (the cuts become joining circles), visibly of genus 2.

Figure 2 Figure 3 Figure 4

2 The period lattice Λ ⊂ C is spanned by 4 vectors R dx , R xdx  = R √ dz , R √zdz , γ ∈ {α , α , β , β }, γ y γ y γ z6−1 γ z6−1 1 2 1 2

24Abel showed that the degree 0 divisors in the kernel of (12.1) are precisely the divisors of meromorphic functions, and Jacobi proved surjectivity. 25To define E(γ, γ0) pick closed loops representing γ and γ0 that intersect each other transversally in a finite set of points. Then follow γ, and count the number of times γ0 meets it from the left minus the ones from the right (C is oriented). This is bilinear and alternating by construction, and also non-degenerate of determinant 1 (look at the standard generators of H1(C, Z), loops going ‘once around the holes’ as in Figure 1). NOTES ON ABELIAN VARIETIES [PART I] 29 the plane integrals taken over the√ four loops in Figure 4, and the dotted lines indicate ‘the other sheet’, i.e. − z6 − 1 in the denominator instead. This is the same as changing the sign and inegrating in the opposite direction. By continuity we can deform all 4 integrals to twice the integrals along the line segments as in Figure 5 (the third one with minus).

Figure 5

The 2 components for the 4 basis vectors of Λ are therefore Z (j+1)/6 e2πitk 2πi√ , j ∈ {0, 4, 1, 3}, k ∈ {1, 2}, 12πit j/6 e − 1 again with minus for the third integral. Evaluated numerically, they are  ζ 1 1 −ζ constant × , ζ = e2πi/3. ζ2i i −i ζ2i 1 0 Move the first 2 vectors to 0 and 1 with an element of GL2(C), we get Jac C =∼ 2/Λ, Λ = 1, 0, √i 2 , √i −1 . C 0 1 3 −1 3 2 Note that Λ contains an index 12 sublattice 1 0 ζ 0 2 0 , 1 , 0 , ζ , so Jac C ∼ E × E (via the corresponding 12-isogeny), where E = C/Z + ζZ is the unique elliptic curve over C with an automorphism of order 6, E : y2 = x3 + 1. 0 ∼ In particular, End (Jac C) = GL2(Q(ζ)). Now we go back to an arbitrary complete non-singular curve C over C with a fixed base point P0 ∈ C. We have defined a map Z P Z P  g P 7→ ω1,..., ωg): C −→ C /Λ = Jac C, P0 P0 which is clearly holomorphic, and therefore a morphism by Chow’s theorem. (Recall that both compact Riemann surfaces and abelian varieties over C are projective varieties.) Seeing the Jacobian as Pic0, the map is 0 C −→ Pic C,P 7→ (P ) − (P0), ∼ 1 and it is an embedding unless C = P . 30 TIM DOKCHITSER

The Jacobian of C is functorial in two ways. A non-constant map of curves φ : C 7→ C0 0 ∗ 0 induces maps φ∗ : Jac C −→ Jac C and φ : Jac C −→ Jac C, given on the level of divisors by the usual pushforward and pullback of divisors ∗ X φ∗(Q) = (φQ), φ (P ) = eQ(Q), f(Q)=P 0 ∗ restricted to Pic = Jac. Clearly φ∗ ◦ φ = [deg φ], in particular φ∗ is onto, and ker φ∗ is finite. On the level of ambient spaces, ∗ 0 g0 0 g φ :ΩC0 /H1(C , Z) = C /Λ −→ C /Λ = ΩC /H1(C, Z) ∗ is induced by the usual pullback of differentials φ :ΩC0 → ΩC . Example 12.4. Let us use these purely functorial properties to compute (up to isogeny) the Jacobian of C : y2 = ax6 + bx4 + cx2 + d. Supposing the right-hand side has no multiple roots, C is a genus 2 hyper- elliptic curve. It admits two non-constant maps to elliptic curves, 2 3 2 φ : C  E1 : y = ax + bx + cx + d 2 3 2 ψ : C  E2 : y = dx + cx + bx + a 2 1 y given by (x, y) 7→ (x , y) and (x, y) 7→ ( x2 , x3 ) respectively. The pullback gives morphisms of abelian varieties with finite kernels ∗ ∗ φ : E1 −→ Jac C, ψ : E2 −→ Jac C.

By Poincare reducibility theorem, Jac C ∼ E1 × E2 if E1 6∼ E2. If E1 ∼ E2, the claim is still true, because ∗ dx d(x2) 2xdx φ y = y = y ∗ dx d(1/x2) 2dx ψ y = y/x3 = − y ∗ ∗ are linearly independent, so φ and ψ map the lattices of E1 and E2 into 2 different complex subspaces of C , and the two image lattices generate that of Jac C up to finite index. Conversely, any genus 2 curve whose Jacobian is not simple is isomorphic to one of the form y2 = ax6 + bx4 + cx2 + d (Exc 12.2).

2 Exc 12.1. Suppose C : y = f(x) with f ∈ C[x] of even degree 2g + 2 with distinct roots αi. (Any genus g hyperelliptic curve over C has such an equation.) For j = 2,..., 2g + 2 fix any path in C from α1 to αj that does not pass through the αk and any continuous branch of pf(z) along the path. Generalising Example 12.3, show that the vectors

g−1 R αj √dz , R αj √zdz ,... R αj z√ dz , j = 2,..., 2g + 2 α1 f(z) α1 f(z) α1 f(z) generate the period lattice of C. Exc 12.2. Suppose C is a hyperelliptic genus 2 curve whose Jacobian is not simple. Then C =∼ y2 = ax6 + bx4 + cx2 + d. (You may use Torelli’s theorem — see below.)