Abelian Varieties and Complex Tori: A Tale of Correspondence

Transcription

1 Abelian Varieties and Complex Tori: A Tale of Correspondence Nate Bushek March 12, 2012

2 Introduction: This is an expository presentation on an area of personal interest, not expertise. I will use results found in many texts without citation. I will happily provide any additional references upon request. Basic algebraic geometry will be used, but I will try to provide an in context review for those unfamiliar. We will look at major results with minimal proof. The goal is to tell the story of a two pictures of the same thing.

3 Starting Definitions: (For the remainder of the talk, everything will be considered over C.) Let P n be the set of lines in C n+1 through the origin. We put the Zariski Topology on P n by taking its closed sets to be vanishing sets of any collection of homogeneous polynomials in n + 1 coordinates. A Zariski closed set in P n is called a projective variety. Any Zariski open subset of a projective variety is called a variety. A projective variety is called a irreducible if it cannot be written as a nontrivial union of non-empty Zariski closed sets.

4 Starting Definitions (continued): Briefly, let s say a regular map f : X Y of varieties is a continuous map that locally (i.e., f f 1 (U): f 1 (U) U) has rational functions (with non-vanishing denominator) in its coordinates. (!!Not precise defintion!!) Briefly, let s say a variety is non-singular if the dimension of the tangent space at every point matches the dimension of the variety. Intuitively, these are varieties that look like smooth manifolds. An algebraic group is a variety X with regular maps µ : X X X and i : X X giving X a group structure, i.e., x y = µ(x, y) and x 1 = i(x).

5 The question arises, what happens if when we consider only algebraic groups that are irreducible projective varieties, i.e., irreducible, closed sets in the Zariski topology of P n? Definition: An abelian variety is an irreducible projective algebraic group. Note that we do not require abelian varieties to have commutative group operation. This is a consequence.

6 Starting Definitions (continued): A lattice, Λ, in C n is the Z span of a set of 2n, R-linearly independent vectors in C n. A quotient C n /Λ is called a complex torus. For now, let s say C n /Λ is polarizable, if there exists a special type of bi-linear form on C n that is integer valued on Λ more precise definition later.

7 Examples: Complex Tori Let n = 1, take Λ = Z Zi. Then C/Λ is the standard torus we all know and love obtained by identifying the edges of the unit square. In fact, we can take any two linearly independent vectors ω 1, ω 2 in C to define a lattice Λ = Zω 1 Zω 2. Let M be a genus g, compact, connected Riemann surface. It is known that the dimension of the space of global holomorphic 1-forms is g, let φ 1,..., φ g be a basis. Let α 1,..., α 2g be a set of generators of H 1 (M, Z). Let A i = ( α i φ 1,..., α i φ g ) C g. The Jacobian of M is the complex torus of C g modulo the lattice ZA 1 ZA 2g. All of these examples are polarizable.

8 Examples: Abelian Varieties Elliptic curves are projective varieties in P 2 defined by an equation of the form y 2 z = x 3 + axz 2 + bz 3 4a b 2 0. Elliptic curves have a naturally defined commutative group structure. Every elliptic curve is the embedded image of a one dimensional complex torus. Each elliptic curve is isomorphic to a plane curve in C 2 given by y 2 = x 3 + ax + b, along with a point at infinity. A group law is defined on the affine curves using points of intersection between lines and the curve, the point at infinity is the identity.

9 Figure: P + Q = R on y 2 = x 3 x

10 GOAL: { } abelian varieties, regular maps { } polarizable complex tori, holomorphic maps Note the significant difference in topologies on each side.

11 A GAGA Association: Given a non-singular projective variety X, we can give the set X a canonical structure of a C-analytic manifold such that the Zariski topology is weaker than the associated manifold topology. Let X h denote the C-analyic manifold associated to a non-singular projective variety X. Let X and Y be non-singular projective varieties and X h, Y h the associated manifolds. If f : X Y is a regular map, then f : X h Y h is holomorphic. Conversely, if f : X h Y h is holomorphic, then f : X Y is regular. (The first is obvious, the second is not.) If X h is the associated C-manifold of some non-singular projective variety X, it is unique.

12 Definition: A complex Lie group is a C-analytic manifold with holomorphic multiplication and inversion maps. Using GAGA we see that for an abelian variety X, X h is a complex Lie group. Since X h is a closed subset of P n, it is compact. Since X is irreducible, X h is connected. Then...

13 Proposition: Any compact, connected complex Lie group is commutative. HOORAY!! ABELIAN varieties are commutative! Proposition: Any compact, connected complex Lie group G is isomorphic (as groups and C-analytic manifolds) to a complex torus, i.e. C n /Λ for some n and lattice Λ in C n. CONCLUSION: For each abelian variety X, the associated C-manifold X h is isomorphic to complex torus.

14 Polarizable Complex Tori are Abelian Varieties: It is fairly easy to see that C n /Λ is a compact, connected C-manifold, with a holomorphic group structure induced by the vector space structure of C n. GOAL: When C n /Λ is polarizable, we want to say that C n /Λ is X h for some abelian variety X. SUFFICIENT STEP: To acheive our goal, it suffices to show that C n /Λ is X h for some projective variety X. For, GAGA guarantees that the holomorphic group operations on the torus will be regular group operations on the associated variety.

15 In fact, Theorem: (Chow s Theorem) A closed analytic subset of P N is Zariski closed in P N. Chow s theorem means that any closed submanifold of P N (in the topology as a C-manifold) is X h for a unique projective variety X. Thus, REDUCED SUFFICIENT STEP: Show that C n /Λ is a closed submanifold of some P N, or equivalently, find a holomorphic immersion of C n /Λ into P N.

16 We seek an immersion F (z) = [f 0 (v); ; f N+1 (v)], F : C n P N. A few desired properties of F are: f i are all holomorphic. F (v + l) = λ(v + l)f (v) for v C n and l Λ, 0. Since λ[z 0 ; ; z N+1 ] = [z 0 ; ; z N+1 ] in the homogeneous coordinates of P N, this will allow F to descend to a well-defined map on C n /Λ. Putting these together, we want N + 1 holomorphic functions f i on C n such that f i (v + l) = λ(v + l)f i (v).

17 Definition: Let Λ be a lattice in C n. A theta function of type (L, J) is a (not identically zero) meromorphic function θ : C n C such that for all z C n and l Λ, θ(z + l) = θ(z) exp(2πi(l(z, l) + J(l)), where L, J are complex functions on Λ Λ and Λ respectively, and is L is C-linear in the first entry (thus, L is actually a function on C n Λ). Holomorphic theta functions of the same type will be the candidates for the holomorphic embedding C n /Λ P N. Are there any?

18 Definition: Let Λ be a lattice in C n. A non-degenerate Riemann form E on C n with respect to Λ is a alternating R-bilinear form E : C n C n R that is integer valued on Λ Λ and satisfies the property that the form S : C n C n R given by S(u, v) = E(iu, v) is symmetric and positive definite. If there exists a non-degenerate Riemann form on C n with respect to Λ, we say C n /Λ is polarizable.

19 To each type of theta function on C n with respect to Λ, we can associate a Riemann form E. A theorem of Frobenius states that for each type of theta function (L, J) associated to a non-degenerate Riemann form E, the C vector space of holomorphic theta fuctions of that type has dimension det E. In particular, there is at least one holomorphic theta function with respect to a polarizable torus.

20 A theorem of Lefschetz states that if θ is a holomorphic theta function on C n with respect to Λ, then then the map F (v) = [θ 0 (v),, θ N+1 (v)] gives a holomorphic immersion of C n /Λ P N, where θ i are N + 1 linearly independent holomorphic theta functions of the same type as θ 3. This acheives REDUCED SUFFICIENT STEP. Corollary: If C n /Λ is polarizable, then C n /Λ = X h for an abelian variety X.

21 n=1, All Complex Tori are Abelian Varieties When n = 1, it turns out that all complex tori C/Λ, Λ = Zω 1 Zω 2, are polarizable, and hence Abelian varieties. These are elliptic curves. A polarization can always be given by E(u, v) = I(uv) ω 1 ω 2. For n = 1, can always embedd into C 2 with the Weierstrass -function. (z) = 1 z 2 + l Λ l 0 ( 1 (z l) ) l 2, via F (z) = [ (z); (z); 1], add point at infinity. Group structures agree, zero goes to point at infinity.

22 Thank You!