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. Isogenies and endomorphisms of complex tori 20 9. Dual abelian variety 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 2 k[x1; :::; xn] such that n V = fx 2 k j all fi(x) = 0g: 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: ffi = 0g [ fgj = 0g = ffigj = 0g. 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 morphism (also called a regular map) if it can be given by x 7! (fi(x)) with f1; :::; fm 2 k[x1; :::; xn]. Morphisms are continuous, by definition of Zariski topology. We say that φ is an isomorphism 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 = ff s.t. fjV = 0g: 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 2 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 2 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 image 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 n f0g, 1 which is not closed in A . 2 (xy;y) 2 2 Example 1.6. The map Ax;y −! At has image A n fx-axisg [ f(0; 0)g. The first example can be given a positive twist, in a sense that it actually 1 gives U = A n f0g 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 : V 0 ! V with (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 (x; y) = t. n Example 1.7. If V ⊂ A is a hypersurface f(x1; :::xn) = 0, then the com- n plement U = A n 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 manifold.) With k[Vi] = k[x1; :::; xn]=Pi, this becomes the length of a longest chain of prime ideals k[V ] ) P0 ) ··· ) Pd = f0g; 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 2 V a point. A regular function on V may be evaluated at x, and the kernel 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 2 k(V ) f; g 2 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 2 V is non-singular if, equivalently, mx (1) dimk 2 = n − d. (`≥' always holds.) mx (2) the completion O^ = 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 ∗ ∼ O^x − O^y = Ox[[t1; :::; tn]]: So V is regular if and only if V ! fptg 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 O^C;p =∼ k[[t ; t ]], O^D;p =∼ k[[s; t]]=st, O^E;p =∼ k[[t]] and that they are pairwise non-isomorphic.