Curves, abelian varieties and the Schottky problem

Samuel Grushevsky

1 Abelian varieties and their moduli space Ag

1.1 Definitions

Definition 1. An abelian variety is a projective variety which is a group object with holomorphic structure maps. Exercise 1. Any abelian variety is an abelian group.

Example 1. The torus C/(Z + Zτ) for any τ ∈ C\R.

We’d like to study the moduli of abelian varieties; we consider A =∼ A0 if there is a biholomorphism preserving the group structure.

1.2 Dimension 1

Question 1. Is C/(Z + Zi) isomorphic to C/(Z + Z(2i))? Answer 1. No; the first has an automorphism (rotation), whereas the second does not.

Theorem 1. Let A1 be the moduli of 1-dimensional abelian varieties (i.e. complex tori). Then A1 is the usual “critical strip” in the upper half-plane given by {z ∈ C : |Re(z) ≤ 1/2, |z| ≥ 1}.

Proof. Any such object is /( + τ) for some τ ∈ \ . First, /( + τ) = /( + (τ + 1). Moreover, we can C∼ Z Z C R C Z Z C Z Z define f : C/(Z + Z(−1/τ) −→ C/(Z + Zτ) via z 7→ τz; then 1 7→ τ and −1/τ 7→ −1. This all tells us that we can reduce to a single vertical strip of widge 1, and moreover that we can require that |τ| ≥ 1.

In fact, we can write A1 = UHP/SL(2, Z).

1.3 Dimension g

Here is an important definition. Definition 2. A polarization on a complex manifold A is the first Chern class of an ample line bundle L on A.

Here are two explanatory definitions.

Definition 3. A line bundle L over A is called ample if there is some N > 0 such that |NL| : A,→ Pn+1, or equivalently if for all Y ⊂ A of dimension d, Ld · Y > 0.

2 1,1 Definition 4. The first Chern class of a line bundle L over A is an element c1(L) ∈ H (A, Z) ∩ H>0 (A, C).

Definition 5. A polarization c1(L) of A is called principal if L only has one section (up to constants), i.e. dim H0(A, L) = 1.

This is really nice, because it essentially fixes a line bundle with a section. Here is our real target. Definition 6. A principally polarized (complex) g-dimensional abelian variety is exactly what it sounds like. We generally denote this (A, c1(Θ)) or (A, Θ). We will call these ppav’s for short, or even just “abelian varieties”.

1 Let’s relate this back to our definition of A1. We write Ag for the moduli space of ppav’s, where we consider 0 0 0 ∗ 0 0 (A, Θ) ∼ (A , Θ ) if there is some biholomorphism f : A → A such that f Θ = Θ (or equivalently f∗Θ = Θ ). Remark 1. This choice of equivalence disregards the group structure; we can recover it by translation, but this definition yields better stack-theoretic properties. More precisely, let a ∈ A be a point and let ta : A → A be the translation by a, i.e. z 7→ z + a. But since ta can be continuously deformed to id, there’s no way it can change 2 c1(L) ∈ H (A, Z), since this is a discrete group.

1.4 Analytic approach

Exercise 2. The universal cover of a dimension-g abelian variety is Cg.

Thus, any abelian variety is a quotient of Cg by a lattice Λ. This immediately implies that Λ is discrete. Also, Λ must be generated by a finite number of vectors, Λ = Ze1 + ... + Zen. But A is compact (since it is projective), g g so in fact n = 2g. This means that Λ ⊗Z R = C . Thus, Λ cannot lie in any proper real subspace of C . g ∼ g So, we can write Λ = Ze1 + ... + Ze2g. Suppose we have some M ∈ GL(g, C). Then C /Λ −→ C /M(Λ) via z 7→ Mz. Thus, we can choose e1, . . . , eg to be the unit vectors in the coordinate directions, and then write g g g τ = (eg+1, . . . , e2g), a g × g complex matrix. So finally, we have A = C /(Z + Z τ). Let’s denote this by Aτ .

Question 2. For which τ is Aτ a projective variety? (So far, it’s only a complex manifold.)

Answer 2 (Riemann’s bilinear relation). Aτ is a projective variety iff τ is symmetric and Im(τ) is positive-definite.

We will take this as a black box.

Definition 7. The Siegel upper half-space, denoted Hg, is the set of all complex symmetric g × g matrices τ with positive-definite imaginary part.

We haven’t said the word “polarization” yet. We’d like to directly construct a principal polarization on Aτ . g Definition 8. The theta function is the map θ : Hg × C → C given by X θ(τ, z) = exp(πinT · (τn + 2z)). g n∈Z (As n is a vector, nT denotes its transpose.)

This is useful for the following reason. Exercise 3. For any integral m, θ(τ, z + m) = θ(τ, z); that is, θ is periodic with integral period in the second variable. Moreover, θ(τ, z + mτ) = ef (m, z)θ(τ, z) for some suitable function f.

This tells us that actually, θ should be thought of as a section of a certain bundle.

g f(m,z) Definition 9. Let Θτ = {z ∈ Aτ : θ(τ, z) = 0}. (This makes sense; we can pass from C down to Aτ since e is always nonzero.)

Proposition 1. Θτ ⊂ Aτ defines a principal polarization.

Proof sketch. It is relatively easy to see that this defines an ample line bundle. To see that it is principal, we can g just check that its self-intersection number is Θτ = g!: the intersection number is constant in families, so we can g Pg ∗ g compute it on E (for E an elliptic curve), and then ΘEg = ( i=1 pri Θ) , and there are g! ways of choosing. 0 Proposition 2. (Aτ , Θτ ) ∼ (Aτ 0 , Θτ 0 ) iff there is some γ ∈ Sp(2g, Z) such that γτ = τ . Recall that the element γ of the symplectic group can be thought of as a block  AB  γ = CD such that  0 1   0 1  γT γ = . −1 0 −1 0

2 Here, A, B, C, D are g × g matrices over Z, and we have the action Aτ + B γτ = . Cτ + D

Thus, Ag = Hg/Sp(2, Z). (Properly speaking, this is an orbifold with finite stabilizers.)

Theorem 2 (Borel). PicQ(Ag) = Q (where we’re over Q in order to throw out the fact that some points have finite stabilizers).

∗ ∗ 2 The idea here is that Hg is contractible, so H (Ag) = H (Sp(2g, Z)), and one checks that H (Sp(2g, Z), Q) = Q.

2 Kodaira dimension of Ag

Recall that Ag = Hg/Sp(2g, Z). Borel’s theorem (which is really a statement about group cohomology) tells us that there is an essentially unique line bundle over Ag; we aim to find that line bundle.

Definition 10. A Siegel modular form of weight k is a holomorphic function F : Hg → C such that for any τ ∈ Hg and any γ = (A, B; C,D) ∈ Sp(2g, Z), we have the relation F (γ ◦ tau) = det(Cτ + D)k · F (τ).

Of course, these should be thought of as sections of some line bundle on Ag. Let L be the line bundle of modular forms of weight 1. Then, Borel’s theorem can be written as

PicQ(Ag) = Q · L (in a “morally correct” way; of course this would still be true if we replaced L by a nonzero rational multiple).

Definition 11. The Hodge vector bundle E → Ag is the rank g vector bundle such that its fiber over [A] is H1,0(A, C). Lemma 1. det E = L.

(Of course, if we’re purely algebraic (and not analytic) geometers, we might take this as a definition of L.)

Proof. Note that dz1 ∧ ... ∧ dzg is a section of det E. So, we need to compare this at Aτ and Aγ◦τ . Recall that

g g g g g g Aτ = C /(Z + Z τ),Aγτ = C /(Z + Z (γτ)) given by z 7→ (Cτ + D) · z. Thus,

dz1 ∧ ... ∧ dzg 7→ d((Cτ + D)z)1 ∧ ... = det(Cτ + D) · (dz1 ∧ ... ∧ dzg).

We may also call L the Hodge line bundle.

Proposition 3. The canonical bundle is given by KAg = (g + 1)L. (Note that we’re using additive notation, so coefficients really mean tensor powers.) V Proof. Observe that ω(τ) = dτ11 ∧ dτ12 ∧ dτ13 ∧ ... ∧ dτgg = 1≤i≤j≤g dτij is a section of KHg . We must compare ω(τ) and ω(γ ◦ τ). Of course we can do this as before (and indeed, this is an exercise!), but here we can also cheat by noting that Sp(2g, Z) is generated by certain elements of the form  1 B   A 0   0 1  , , . 0 1 0 A−T −1 0

We can easily check that the first matrix transforms τ to τ + B, so it preserves dτij. Similarly, the second matrix T V T g+1 V transforms τ to AτA , and we compute that d(AτA )ij = (det A) · dτij. The last one is also an easy calculation.

3 Question 3. Is Ag quasiprojective?

To determine this, we need to see whether it has an ample line bundle. Of course, there’s essentially just one, namely L. Let’s try to construct sections of NL (for some arbitrary number N). g Recall that we had the theta function θ : Hg × C → C given by X θ(τ, z) = exp(πi((nt(τn + 2z)). g n∈Z We’d like to somehow use this to construct sections, which motivates the following definition.

Definition 12. For any ε, δ ∈ (Z/2Z)g, define the theta constant with characteristic ε, δ to be

 ε  X   ε t   ε  δ  θ (τ) = exp πi n + τ n + + 2 . δ g 2 2 2 n∈Z

 ε  Note that θ (τ) ' θ(τ, ετ+δ ). Thus, it is reasonable to ask whether these are modular forms. δ 2

Theorem 3 (Igusa, probably others too). For all γ ∈ Sp(2g, Z),    √   ε 8 ε θ γ (γ ◦ τ) = 1 det(cτ + d)1/2θ (τ), δ δ where the action of Sp(2g, Z) on ((Z/2Z)g)2 is given by  ε   AB   ε   AT B  γ = + diag . δ CD δ CT D

In particular, if we define the level subgroup

Γg(2) = ker(Sp(2g, Z) → Sp(2g, Z/2)), then for any γ ∈ Γg(2),   √   ε 8 ε θ (γτ) = 1 det(Cτ + D)1/2θ (τ). δ δ

Thus, for example, Y  ε  θ8 (τ) δ ε,δ even is a modular form (for Sp(2g, Z)). Definition 13. The pair ε, δ is called even or odd depending on the parity of εt · δ.

 ε  Thus, θ (τ) ≡ 0 iff ε, δ is odd. δ  ε  So we can use θ8 (τ) to construct many modular forms. This gives us the following result. δ

Theorem 4 (Igusa,Salvati Manni,Mumford). L is ample on Ag. (More precisely, NL is very ample on Ag when we have N ≈ #Sp(2g, Z/3).)

Of course, now that we know that L is ample, then for n >> 0 we have that |nL| gives an embedding of Ag into some projective space. The natural question then becomes: What is its image, i.e. what is the compactification of Ag induced by this embedding?

Definition 14. The Satake-Baily-Borel minimal compactification of Ag is given by, when certain entries blow up, deleting their rows and columns.

4 SBB ` ` ` Thus, Ag = Ag Ag−1 ... A0. Note that this is highly singular, since the boundary has codimension g.

SBB Theorem 5. L extends to an ample line bundle over Ag .

Now let’s move into the world of birational geometry. Definition 15. For a line bundle L on a projective variety X, the Iitaka dimension is given by

ln dim H0(X, nL) κ(X,L) = lim . n→∞ ln n (The number dim H0(X, nL) will grow like some power of n, and this picks out what that power is.) If κ(X,L) = dim X, then L is called big; this is equivalent to saying that for all n >> 0, |nL| gives a birational map from X to a projective space (i.e. dim |nL|(X) = dim X).

Definition 16. The Kodaira dimension of a variety X is by definition Kod(X) = κ(X,KX ). X is called of general type if Kod(X) = dim X, iff KX is big.

Question 4. What is Kod(Ag)?

There is a trick, called Zariski decomposition, which says “big + effective = big”. Thus, if we can write KAg as the sum of something big and something effective, then Ag is of general type. Now, our line bundle L can serve as the big line bundle. From here, the idea is to use toroidal compactifications of Ag. We see that

 t  it b 0 lim 0 = (τ , b) ∈ Ag−1 × Aτ 0 . t→∞ b τ

So, let us write Xg → Ag be the universal family of ppav’s. Then the partial toroidal compactification is by definition

0 a Ag = A Xg−1.

Theorem 6. There exist many toroidal compactifications Ag of Ag such that:

0 1. If Ag ⊃ Ag, then codim ∂Ag = 1. 2. extends to a vector bundle on A , where = π Ω1 . E g E ∗ Xg /Ag

perf perf 0 3. There exists Ag (the perfect cone compactification) such that codim (Ag − Ag) = 2.

perf 4. (Shepherd-Barron 2004) For g ≥ 12, Ag is the canonical model of Ag.

V or 5. (Alexeev 2002) There exists Ag (the second Voronoi compactification) over which there is a universal family.

0 Thus, for studying Kod(Ag), looking at Ag is good enough.

Our goal for next time will be to show that Ag is of general type for all g ≥ 7.

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