11 Holomorphic Function Theory on C and Complex Tori

11 Holomorphic Function Theory on Cn and Complex Tori

Difference for holomorphic functions on domains in C and in Cn Let us mention some difference for holomorphic functions on domains in C and in Cn. • A one variable holomorphic function has isolated zeros. For example, f(z)= z2 has zero z = 0 with oder 2. When n ≥ 2, zeros of a holomorphic function f(z) defined on a domain n in C are no longer isolated points. For example, f(z1, z2)= z1, its zero set is a complex line {(z1, z2) | z1 =0}. In general, the zeros of a holomorphic function f(z) defines a hypersurface (i.e. complex dimension n − 1) submanifold except having some singularities. n • For several holomorphic functions f1, ..., fm defined on a domain of C , the intersection of their common zeros

V = {z ∈ D | f1(z)= f2(z)= ... = fm(z)=0} has complicated geometry. When n = 1, this does not happen. We have to use other approach to study it, e.g., differential geometry and algebraic geometry. f • When n = 1, two holomorphic functions f and g define a meromorphic function g . At each point z0, we have k f(z) (z − z0) u(z) = m g(z) (z − z0) v(z) where k ≥ 0, m ≥ 0 are integers, u(z) and v(z) are holomorphic funcions defined near z0 f with u(z0) =6 0 and v(z0) =6 0. Consequently, g is either holomorphic at z0, or has a pole at z0. When n> 1, consider the following example

f(z , z ) z 1 2 = 1 . g(z1, z2) z2

It is clear that as (z1, z2) goes to (0, 0), there is no limit for f(z1, z2) (For some sequence (n) (n) (n) (n) C (z1 , z2 ), the function values f(z1 , z2 ) converges to a finite point in , while for some (n) (n) (n) (n) other sequence (z1 , z2 ), the function values f(z1 , z2 ) goes to infinity), i.e., it is unde- f termined for the function value of g at (0, 0). In general, for holomorphic functions f, g define on a domain of Cn with n > 1, the subset Zero(f) ∩ Zero(g)

64 is the determinant set, i.e., when z goes to any point of Zero(f) ∩ Zero(g), the function f(z) value g(z) has no limit. • There are many phenomena showing big diﬀerences between n = 1 and n > 1. For example,

Theorem 11.1 (H. Alexander, 1977) When n > 1, any proper 23 holomorphic function f : Bn → Bn must be biholomorphic.

n n 2 2 n Here B = {z ∈ C | |z1| + ... + |zn| < 1} is the unit ball in C . However, when n = 1, there are many proper holomorphic map

m z − aj f(z)= Y , m> 1 1 − ajz j=1 from ∆(1) to itself which are not biholomorphic. Automorphism groups for simply connected Riemann surfaces An one dimensional complex manifold is called a Reimann surface. There are only three simply connected Riemann surfaces: C, B1 and CP1. iθ z−a By Schwarz Lemma, we have understood Aut(∆(1)) = {e 1−a z} where |a| < 1 and θ ∈ R. Recall a linear transformation az + b f(z)= , ad − bc =06 cz + d can be regarded as a biholomorphic map f : Cˆ → Cˆ. Conversely, we prove

Theorem 11.2 (i) Any biholomorphic map f : Cˆ → Cˆ is a linear transformation.

(ii) Any biholomorphic map g : C → C is a linear transformation of the form

g(z)= az + b with a =06 .

Proof: (i) Assume f : (∞) = ∞. Otherwise suppose f(∞) = c =6 ∞. We can take 1 ψ(w)= w−c such that ψ ◦ f(∞)= ∞.

23i.e., for any compact set A, f −1(A) is compact.

65 Since f(∞)= ∞, it implies

f|C : C → C is biholomorphic.

Then it is reduced into (ii). In other words, we only need to prove (ii).

∞ n (ii) Write g(z)= Pn=0 anz as a power series. Claim: g is a polynomial.

Suppose that g is not a polynomial. Then an =6 0 for inﬁnitely many n. We can regard g is a function deﬁned on Cˆ with isolated singularity ∞. If we use the coordinate (V∞, z), −1 it means that g ◦ zV∞ has isolated singularity at z = 0, i.e., e e ∞ 1 a (g ◦ z−1 )(z)= g( )= n V∞ z X zn e n=0 has an essential singularity at 0. −1 Cˆ Then g ◦ zV∞ (∆(0,r)) is dense in for any r> 0 by Casorati-Weierstrass theorem. But this is impossible because g is 1-to-1 and onto, which is a contradiction.

We have proved that g must be a polynomial:

2 n g(z)= b0 + b1z + b2z + ... + bnz .

We need to prove that g(z) is linear: deg(g)=1. In fact, if deg(g) > 1, by the fundamental theorem of algebra, ∀w ∈ C, the equation g(z)= w has deg(g) roots, but it is a contraction because g is one-to-one.

Compact Riemann surfaces For a compact Riemann surface, we can deﬁne by genus the number of “holes” of this manifold. If the genus is 0, it is biholomorphic to Riemann sphere S = CP1 = C ∪ {∞}. The higher dimensional generalization is the n-dimensional complex projective space CPn.

If the genus is 1, it is biholomorphic to a complex torus T := C/Λ. A higher dimensional generalization leads to Tn := Cn/Λ.

If the genus ≥ 2, it is a hyperbolic surface. For every hyperbolic Riemann surface, the universal covering space is B1, the unit disk, or H1, the upper-half plane, and the fundamental group is isomorphic to a Fuchsian group, and thus the surface can be modeled

66 by a Fuchsian model H/Γ where H is the upper half-plane and Γ is the Fuchsian group. The set of representatives of the cosets of H/Γ are free regular sets and can be fashioned into metric fundamental polygons. Quotient structures as H/Γ are generalized to Shimura varieties.

Dimension two complex manifolds An two dimensional complex manifold is called a complex surface.

For dimension 2 compact complex manifolds, we have the Enriques-Kodaira classiﬁcation which states that every non singular minimal compact complex surface is of exactly one of the 10 types of rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.

For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on the global spherical shell conjecture, still unproved in 2009). For surfaces of general type not much is known about their explicit classiﬁcation, though many examples have been found.

Lattice and quotient spaces We have discussed complex projective space CPn. Let us discuss complex tori T = Cn/Λ. A torus can be descripted by notion of quotient space. On the complex plan C, if we identity a point z with all numbers z + m + in where m, n ∈ Z are integers. Namely, we deﬁne z ∼ w if and only if z = w + m + in for some integers m and n. We obtain the quotient space C/ ∼ which is the torus. From topology, a quotient space C/ ∼ has the induced topology from C. In a natrual way, it is a Riemann surface. We may call L = {m + in | m, n ∈ Z} a lattice, which is a group with the standard addition. A torus can be written as the quotient space

C/L.

More general, let Ω = C and G be the subgroup generated by z 7→ z + ω1 and z 7→ z + ω2 where ω1,ω2 ∈ C are R-linearly independent. We can identify z ∼ w if and only if z = w + mω1 + nω2 Then we can deﬁne general torus

C/hz 7→ z + ω1, z 7→ z + ω2i

67 is a Riemann surface. It is the same as C/L with the lattice L = {mω1 + nω2 | m, n ∈ Z}. Similarly, we can deﬁne n dimensional complex torus Tn = Cn/Λ. Quotient space and covering space For any topological space X, if there is equivalence relation ∼ on X in which we denote [x] as the equivalence class of x, then the quotient space X/ ∼:= {[x] | x ∈ X}. There is a natural projection map q : X → X/ ∼, x 7→ [x]. The quotient space X/ ∼ is naturally equipped with the topology, called the quotient topology, where the open sets on X/ ∼ are deﬁned to be those sets of equivalence classes whose unions are open sets in X, namely, a subset V ⊂ X/ ∼ is open if and only if q−1(V ) is open in X.

A complex torus C/L where L = Zω1 + Zω2 is a topological space. Furthermore, a complex torus has a “covering space” C. Let us introduce the definition. Let X,Y be topological spaces and f : Y → X is called a covering map if and only if every −1 x ∈ X has a neighborhood U in X so that f (U) = ∪jVj with open and disjoint Vj ⊂ Y so that f : Vj → U is a homeomorphism for each j. We call Y a covering space of X. Clearly, the quotient map q : C → C/L is a covering map. Torus is a Riemann surface We also can see that C/L is a Riemann surface. In fact, consider the quotient map q : C → C/L. For any x ∈ C/L, we can take a small neighborhood −1 Ux in C/L such that the inverse q (Ux) is the union of infinitely many disjoint open subsets C Z+ Ux,j ⊂ , j ∈ . We then use any one (Ux,j, zUx,j ) to define the coordinate map for (Vx,dVx ) which is independent of choice of j. Biholomorphic maps between complex tori A compact Riemann surface with genus one is biholomorphic to a tours C/Γ where Γ = {g(z) = z + mω1 + nω2 | m, n ∈ Z} is a subgroup of Aut(C), ω1,ω2 are R-linearly independent. After considering maps between Riemann Spheres, let us consider maps between complex tori.

f Lemma 11.3 C/Γ1 −−−−→ C/Γ2 is a biholomorphic map f iﬀ ∃ F (z) = az with a =6 0 such that F maps the equivalent classes w.r.t Γ1 to equivalent classes w.r.t Γ2.

f Proof: Suppose C/Γ1 −→ C/Γ2 is a biholomorphic map. The quotient projection π : C → C/Γj is a universal covering map, j =1, 2. We consider the diagram

F C −→ C ↓ π1 ↓ π2 f C/Γ1 −→ C/Γ2

68 Fixing a point z0 ∈ C, we have π1(z0) ∈ C/Γ1 and f(π1(z0)) ∈ C/Γ2. Then we ﬁx a point −1 w0 ∈ π2 f(π(z0)). By local homeomorphic property (local biholomorphic), we obtain a −1 local map F = π2 ◦ f ◦ π1 such that F (z0)= w0. We claim that

F can extend holomorphically on C.

In fact, for any point z ∈ C and take a curve C in C with its initial point z0 and the terminal e point z. Then C := π1(C) is a curve in C/Γ1 with the initial point π1(z0) and the terminal −1 point π1(z). Since f is well-deﬁned along the curve C, F = π2 ◦ f ◦ π1 can extend along the curve C. Since C is simply connected, such extension is independent of choice of the curve C. Thereforee we claim is proved. e −1 Let g = f . Then we can similarly show that ∃ a unique G with G(w0)= z0 and G is holomorphic. Moreover, F ◦ G = G ◦ F = Id. This proves F ∈ Aut(C) and hence F = az + b for some a =6 0. By changing coordinate (more precisely, by a translation), we can assue F = az.

Conversely, let F = az with a =6 0. If F maps equivalent classes of Γ1 to equivalent classes w.r.t Γ2, then f([z]) = [F ([z])] deﬁnes a holomorphic equivalent map of C/Γ1 to C/Γ2.