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Two Or Three Things I Know About Abelian Varieties

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Two Or Three Things I Know About Abelian Varieties TWO OR THREE THINGS I KNOW ABOUT ABELIAN VARIETIES OLIVIER DEBARRE Abstract. We discuss, mostly without proofs, classical facts about abelian varieties (proper algebraic groups defined over a field). The general theory is explained over arbitrary fields. We define and describe the properties of various examples of principally polarized abelian varieties: Jacobians of curves, Prym varieties, intermediate Jacobians. Contents 1. Why abelian varieties? 2 1.1. Chevalley's structure theorem 2 1.2. Basic properties of abelian varieties 2 1.3. Torsion points 4 2. Where does one find abelian varieties? 4 2.1. Complex tori 4 2.2. Jacobian of smooth projective curves 5 2.3. Picard varieties 6 2.4. Albanese varieties 7 3. Line bundles on an abelian variety 7 3.1. The dual abelian variety 7 3.2. The Riemann{Roch theorem and the index 8 4. Jacobians of curves 9 5. Prym varieties 10 6. Intermediate Jacobians 12 7. Minimal cohomology classes 13 References 15 These notes were written for an informal mini-course given at the University of North Carolina, Chapel Hill, April 17 & 19, 2017. 1 2 O. DEBARRE 1. Why abelian varieties? 1.1. Chevalley's structure theorem. If one wants to study arbitrary algebraic groups (of finite type over a field k), the first step is the following result of Chevalley. It indicates that, over a perfect field k, the theory splits into two very different cases: • affine algebraic groups, which are subgroups of some general linear group GLn(k); • proper algebraic groups, which are called abelian varieties and are the object of study of these notes. Theorem 1 (Chevalley). Let G be an algebraic group defined over a perfect field k. There exist a canonical exact sequence of morphisms of algebraic groups 1 −! Gaff −! G −! A −! 0; where Gaff is an affine algebraic group and A is a proper algebraic group, both defined over k. This sequence is not split in general. 1.2. Basic properties of abelian varieties. We prove two fundamental properties of abelian varieties (one of which justifies their name!). Theorem 2. Any abelian variety is a commutative group and a projective variety. Proof. To prove commutativity, we rely on the following rigidity lemma. Lemma 3. Let X be an irreducible proper variety defined over an algebraically closed field, let Y be an irreducible variety, let y0 be a point of Y and let u: X × Y ! Z be a regular map such that u(X × fy0g) is a point. Then u(X × fyg) is a point for all y 2 Y . Proof. Let Γ := f(x; y; z) 2 X × Y × Z j z = u(x; y)g be the graph of u and let q : X × Y × Z ! Y × Z be the projection. Since X is proper, q(Γ) = f(y; z) 2 Y × Z j 9x 2 X z = u(x; y)g is a closed subvariety, which is irreducible by hypothesis. The projection q(Γ) ! Y is surjec- tive and the fibre of y0 is a point. This implies that the variety q(Γ) as same dimension as Y . Let x0 be a point of X; the graph f(y; u(x0; y)) j y 2 Y g of the regular map y 7! u(x0; y) is closed in q(Γ) and has same dimension: they are therefore equal and, for all x in X and all y in Y , we have u(x; y) = u(x0; y). The conclusion of the lemma may not hold if X is not proper, as shown by the regular map u: A1×A1 ! A1 defined by u(x; y) = xy: we have u(A1×f0g) = f0g but u(A1×ftg) = A1 for t 6= 0. In the proof above, q(Γ) = f(y; xy)g is not a variety (only a constructible set). Going back to the proof of the theorem, given an abelian variety A, we work over the algebraic closure of k and apply the lemma to the map u: A × A ! A defined by u(x; x0) = x−1x0x. It contracts A × feg to the point e. The lemma implies that for all x; x0 in A, we have u(x; x0) = u(e; x0) = x0, so that A is an abelian group. Another useful consequence of the lemma is the following. TWO OR THREE THINGS I KNOW ABOUT ABELIAN VARIETIES 3 Proposition 4. Let A be an abelian variety and let G be an algebraic group. A regular map u: A ! G that satisfies u(eA) = eG is a group morphism. Proof. Define a regular map u0 : A × A ! G by u0(x; x0) = u(x−1)u(x0)u(x−1x0)−1: 0 0 We have u (A × feAg) = feGg and the rigidity lemma implies that for all x; x in A, we have 0 0 0 0 u (x; x ) = u (eA; x ) = eG, which implies the proposition. To prove that A is projective, we need to show that it contains an ample divisor. For that, we may assume that the base field k is algebraically closed. Following Weil, we first Tr construct hypersurfaces D1;:::;Dr in A such that i=1 Di = f0g as follows. Let x1 be a (closed) point in A distinct from the origin 0A. The exists an affine open neighborhood V of 0A in A. If z 2 V \ (V + x1), both 0A and x1 are in the affine open subset N N U := V + x1 − z. Embed U in some affine space Ak and choose a hyperplane H ⊂ Ak such that 0A 2 H but x1 2= H. Set D1 be the closure in X of U \ H; this is a hypersurface in A which contains 0A but not x1. If there is a point x2 in D1 r f0Ag, construct, using the same process, a hypersurface D2 ⊂ X which contains 0A but not x2. Repeat the process and construct hypersurfaces D1;D2;::: . We obtain a strictly decreasing chain D1 ) D1 \ D2 ) ··· of closed subsets of A which, since A is noetherian, must stop at some point where D1 \···\ Dr = f0g. We will now prove that the divisor D := D1 \···\ Dr is ample on A. Let a and b be distinct points of A. Since b − a 6= 0A, there exists i 2 f1; : : : ; rg, say i = 1, such that b − a2 = Di. Set a1 = a; then D1 + a1 contains a but not b. Take b1 outside of (b − D1) [ (D1 − a1 − b), so that b2 = (D1 + a1) [ (D1 + b1) [ (D1 − a1 − b1): Similarly, for each j > 1, choose aj outside of b − Dj and bj outside of (b − Dj) [ (Dj − aj − b) [ (b − a1 − D1), so that b2 = (Dj + aj) [ (Dj + bj) [ (Dj − aj − bj): The effective divisor r X (Dj + aj) + (Dj + bj) + (Dj − aj − bj) i=1 contains a but not b. By the theorem of the square (see Theorem 5 below), it is linearly equivalent to 3D. This implies that the map 0 _ 3D : A −! P(H (A; OA(3D)) ) associated with the global sections of OA(3D) is an injective morphism. It is in particular finite and 3D (hence also D) is ample. This finishes the proof of the theorem. The main ingredient of the second part of the proof above is the following crucial (and difficult) theorem (Weil). 4 O. DEBARRE Theorem 5 (Theorem of the square). Let D be a divisor on an abelian variety A defined over a field k. For any x; y 2 A(k), we have (D + x) + (D + y) ≡ D + (D + x + y): lin Given an integer n 2 Z, we will denote the morphism multiplication by n on A by nA : A −! A and by A[n] its kernel (so that A[n](k) is the set of n-torsion k-points). 1.3. Torsion points. Let A be an abelian variety of dimension g defined over a field k and let n be a non-zero integer. When k = C, we will see in Section 2.1 that the morphism nA defined above has degree n2g and that the subgroup A[n](C) of n-torsion points in the group A(C) is isomorphic to (Z=nZ)2g. In positive characteristics, the situation is different. Theorem 6. Let A be an abelian variety of dimension g defined over a field k of char- 2g acteristic p > 0 and let n be a non-zero integer. The map nA is surjective of degree n . Moreover, • if n is prime to p, the map nA is ´etaleand, when k is separably closed, the group A[n](k) is isomorphic to (Z=nZ)2g; • the map pA is not ´etaleand, when k is separably closed, there is an integer r 2 f0; : : : ; gg (called the p-rank of A) such that, for any positive integer m, the group A[pm](k) is isomorphic to (Z=pmZ)r. The p-rank r can take any value between 0 and g but the \typical" value is r = g (the smaller the p-rank, the more special A is). 2. Where does one find abelian varieties? 2.1. Complex tori. Let V be a complex vector space of dimension g and let Γ ⊂ V be a lattice (of rank 2g). Then X := V=Γ is a compact complex manifold (diffeomorphic to (S1)2g) which is a group. It is called a complex torus. Note that for any positive integer n, 2g the morphism nX is surjective of degree n and that its kernel, the group of n-torsion points 1 2g of X, is isomorphic to n Γ=Γ ' (Z=nZ) .
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