Dependence and Unicity of Meromorphic Mappings

Yoshihiro AIHARA

Introduction.

Let π : X → ℂ^{m} be a finite analytic covering space and M a projective algebraic manifold. Let f_{1}, ⋯, f_{l} be dominant meromorphic mappings from X into M. Suppose that they have the same inverse images of given divisors on M. In this talk, we give conditions under which f_{1}, ⋯, f_{l} are algebraically related. Roughly our result says that if these mappings satisfy the same algebraic relation at all points of the set of the inverse images of divisors and if the given divisors are sufficiently ample, then they must satisfy this relationship identically. We first give criteria for algebraic dependence under a condition on the existence of meromorphic mappings separating the fibers of π : X → ℂ^{m}. Next, we give applications of these criteria. In particular, we give some conditions under which two holomorphic mappings are related by endomorphism of elliptic curves.

§1 Criteria for propagation of algebraic dependence.

We first give a definition of algebraic dependence of meromorphic mappings. We set M^{l} = M × ⋯ × M (l-times). For meromorphic mappings f_{1}, ⋯, f_{l} : X → M, we define a meromorphic mapping f_{1} × ⋯ × f_{l} : X → M^{l} by

(f_{1} × ⋯ × f_{l})(z) = (f_{1}(z), ⋯, f_{l}(z)), z ∈ X − (I(f_{1}) ∪ ⋯ ∪ I(f_{l})),

where I(f_{j}) are the indeterminacy loci of f_{j}. A proper algebraic subset Σ of M^{l} is said to be decomposable if, for some positive integer s not grater than l, there exist positive integers l_{1}, ⋯, l_{s} with l = l_{1} + ⋯ + l_{s} and algebraic subsets Σ_{j} ⊆ M^{l_{j}} such that Σ = Σ_{1} × ⋯ × Σ_{s}. We denote by B the ramofication divisor of π : X → ℂ.

Definition 1. Let S be an analytic subset of X. Nonconstant meromorphic mappings f_{1}, ⋯, f_{l} : X → M are said to be algebraically dependent on S if there exists a proper algebraic subset Σ of M^{l} such that (f_{1} ×⋯×f_{l})(S) ⊆ Σ and Σ is not decomposable. In this case, we also say that f_{1}, ⋯, f_{l} are Σ-related on S.

Let L → M be an ample line bundle over M Let D_{1}, ⋯, D_{q} be divisors in |L| such that D_{1} + ⋯ + D_{q} has only simple normal crossings, where |L| is the complete linear system defined by L. Let S_{1}, ⋯, S_{q} be hypersurfaces in X such that dim S_{i}∩S_{j} ≤ m−2 for any i ≠ j. We define a hypersurface S in X by S = S_{1}∪⋯∪S_{q}. Let E be an effective divisor on X, and let k be a positive integer. If E = ∑_{j} ν_{j}E′_{j} for distinct irreducible hypersurfaces E′_{j} in X and for nonnegative integers ν_{j}, then we define the support of E with order at most k by

Supp_{k} E = ⋃_{0<ν_{j}≤k} E′_{j}.

Assume that Supp_{k_{j}}f_{0}^{∗}D_{j} coincides with S_{j} for all j with 1 ≤ j ≤ q, where k_{j} is a fixed positive integer. Let ℱ be the set of all dominant meromorphic mappings f : X → M such that Supp_{k_{j}}f^{∗}D_{j} is equal to S_{j} for each j with 1 ≤ j ≤ q. Let F_{1}, ⋯, F_{l} be big line bundles over M. We define a line bundles F̃ over M^{l} by

F̃ = π_{1}^{∗}F_{1} ⊗ ⋯ ⊗ π_{l}^{∗}F_{l},

where π_{j} : M^{l} → M are the natural projections on j-th factor. Let L̃ be a big line bundle over M^{l}. In the case of L̃ ≠ F̃, we assume that there exists a positive rational number γ̃ such that γ̃F̃ ⊗ L̃^{−1} is big. If L̃ = F̃, then we take γ̃ = 1. Let ℛ be the set of all hypersurfaces Σ in X such that Σ = Supp D̃ for some D̃ ∈ |L̃| and Σ is not decomposable. We denote by B the ramification divisor of π : X → ℂ^{m}.

Definition 2. Let Y be a compact complex manifold. We say that a meromorphic mapping f : X → Y separates the fibers of π : X → ℂ^{m}, if there exists a point z in ℂ^{m} −(Supp π_{∗}B ∪π(I(f))) such that f(x) ≠ f(y) for any distinct points x, y ∈ π^{−1}(z).

Let s_{0} be the sheet number of π : X → ℂ^{m}. Assume that f : X → M separates the fibers of π : X → M. Since L is ample, there exist a positive integer μ and a pair of sections σ_{0}, σ_{1} ∈ H^{0}(M, μL) such that a meromorphic function f^{∗}(σ_{0}/σ_{1}) separates the fibers of π : X → ℂ^{m} for all such mappings f. We denote by μ_{0} the least positive integer among those μ′s. We assume that there exists a line bundle, say F_{0}, in {F_{1}, ⋯, F_{l}} such that F_{0} ⊗ F_{j}^{−1} is either big or trivial for all j. Set k_{0} = max_{1≤j≤q} k_{j}. We define L_{0} ∈ Pic(M) ⊗ ℚ by

L_{0} = (∑_{j=1}^{q} \frac{k_{j}}{k_{j} + 1} − 2μ_{0}(s_{0} − 1)) L ⊗ (−\frac{γ̃lk_{0}}{k_{0} + 1}F_{0}) .

Then we have our basic result, from which we see that, if L_{0} is sufficiently big, then the algebraic dependence on S propagates to the whole space X.

Lemma 1. Let f_{1}, ⋯, f_{l} be arbitrary mappings in ℱ and Σ ∈ ℛ. Suppose that f_{1}, ⋯, f_{l} are Σ-related on S. If L_{0} ⊗ K_{M} is big, then f_{1}, ⋯, f_{l} are Σ-related on X.

Now, let us consider a more general case. Let L_{1}, ⋯, L_{l} be ample line bundles over M. Let q_{1}, ⋯, q_{l} be positive integers and assume that D_{j} = D_{j1} +⋯+D_{jq_{j}} ∈ |q_{j}L| has only normal crossings, where D_{jk} ∈ |L|. Let Z be a hypersurface in X. Let 𝒢 be a family of dominant meromorphic mappings f : X → M such that

Supp_{k_{j}}f^{∗}D_{j} = Z

for some 1 ≤ j ≤ l. In the case where L_{j} = L for all j, we define G_{0} ∈ Pic(M) ⊗ ℚ by

G_{0} = ( min_{1≤j≤l} \{\frac{q_{j}k_{j}}{k_{j} + 1}\} − 2μ_{0}(s_{0} − 1)) L ⊗ (−\frac{γ̃lk_{0}}{k_{0} + 1}F_{0}) .

Then we have the following:

Lemma 2. Let f_{1}, ⋯, f_{l} be arbitrary mappings in 𝒢 and Σ ∈ ℛ. Suppose that f_{1}, ⋯, f_{l} are Σ-related on Z. If G_{0} ⊗ K_{M} is big, then f_{1}, ⋯, f_{l} are Σ-related on X.

For F ∈ Pic(M) ⊗ ℚ, we define [F/L] by

[F/L] = inf {γ ∈ ℚ; γL ⊗ F^{−1} is big} .

Set

n_{1} = q_{1} − [K_{M}^{−1}/L] − 2μ_{0}(s_{0} − 1) and n_{j} = q_{j} − [K_{M}^{−1}/L] (2 ≤ j ≤ l).

We also set

p_{j} = \frac{q_{j}k_{j}}{1 + k_{j}} − [K_{M}^{−1}/L] − 2μ_{0}(s_{0} − 1)

for all j with 1 ≤ j ≤ l and e_{0} = 2μ_{0}(s_{0} − 1) + 1. Then we have the following criterion, which is a corollary of Lemma 2:

Corollary. Let f_{1}, ⋯, f_{l} be arbitrary mappings in 𝒢 and Σ ∈ ℛ. Suppose that f_{1}, ⋯, f_{l} are Σ-related on Z. If all n_{j} > 0 and if

p_{1} − \frac{γ̃lk_{0}}{k_{0} + 1}[F_{1}/L] + ∑_{j=2}^{l} (n_{1}p_{j} − \frac{γ̃le_{0}k_{0}}{n_{j}(k_{0} + 1)}[F_{j}/L]) > 0,

then f_{1}, ⋯, f_{l} are Σ-related on X.

Remark. We give a remark on the assumptions in the above lemmas. In Lemma 1, we assume that D_{1}, ⋯, D_{q} are linearly equivalent. We consider the case where D_{i} and D_{j} are not linearly equivalent for some pair (i, j) but all the Chern classes c_{1}([D_{j}]) are identical. In this case, the conclusion of the Lemma 1 remains valid provided that the line bundle

(∑_{j=1}^{q} \frac{k_{j}}{k_{j} + 1} − 2μ_{0}(s_{0} − 1)) [D_{1}] ⊗ (−\frac{γ̃lk_{0}}{k_{0} + 1}F_{0})

is ample. We next consider Lemma 2. In the case where L_{i} and L_{j} are not same for some i and j but all the Chern classes c_{1}(L_{j}) are identical, the conclusion of Lemma 2 is still valid if the line bundle

( min_{1≤j≤l} \{\frac{q_{j}k_{j}}{k_{j} + 1}\} − 2μ_{0}(s_{0} − 1)) L_{1} ⊗ (−\frac{γ̃lk_{0}}{k_{0} + 1}F_{0})

is ample.

§2. Unicity theorems for meromorphic mappings.

In this section we give some unicity theorems as an application of criteria for dependence by taking line bundles F_{j} of special type. Let Φ : M → ℙ_{n}(ℂ) be a meromorphic mapping with rank Φ = dim M. We denote by H the hyperplane bundle over ℙ_{n}(ℂ). Now let l = 2 and take F_{1} = F_{2} = Φ^{∗}H. We also take L̃ = F̃. Then we see

L_{0} = (∑_{j=1}^{q} \frac{k_{j}}{k_{j} + 1} − 2μ_{0}(s_{0} − 1)) L ⊗ (−\frac{2k_{0}}{k_{0} + 1} Φ^{∗}H) .

A set {D_{j}}_{j=1}^{q} of divisors is said to be generic with respect to f_{0} and Φ provided that

R_{j} := f_{0}(X − I(f_{0})) ∩ Supp D_{j} ∩ {w ∈ M; rank dΦ(w) = dim M} ≠ ∅

for at least one j with 1 ≤ j ≤ q. We assume that {D_{j}}_{j=1}^{q} is generic with respect to f_{0} and Φ. Let ℱ_{1} be the set of all mappings f ∈ ℱ such that f = f_{0} on S. Then we have the following unicity theorems by Lemma 1:

Theorem 1. Suppose that L_{0} ⊗ K_{M} is big. Then the family ℱ_{1} contains just one mapping f_{0}.

We next consider the case dim M = 1. Assume that M is a compact Riemann surface with genus g_{0}. In the case g_{0} = 0, we have the following unicity theorem for meromorphic functions on X by Theorem 1 as follows:

Theorem 2. Let f_{1}, f_{2} : X → ℙ_{1}(ℂ) be nonconstant holomorphic mappings. Let a_{1}, ⋯, a_{d} be distinct points in ℙ_{1}(ℂ). The following hold.

(1) Suppose that Suppf_{1}^{∗}a_{j} = Suppf_{2}^{∗}a_{j} for all j. If d ≥ 2s_{0} + 3, then f_{1} and f_{2} are identical on X.

(2) Suppose that Supp_{1}f_{1}^{∗}a_{j} = Supp_{1}f_{2}^{∗}a_{j} for all j. If d ≥ 4s_{0} + 3, then f_{1} and f_{2} are identical on X.

§3. Holomorphic mappings into smooth elliptic curves.

Let E be a smooth elliptic curve and let f_{1}, f_{2} : X → E be nonconstant holomorphic mappings. We consider the problem to determine the condition which yields f_{2} = φ(f_{1}) for an endomorphism φ of the abelian group E. We first note the following fact: If f : X → E separates the fibers of π : X → ℂ^{m}, then we can take μ_{0} = 2. We denote by [p] the point bundle determined by p ∈ E. Let F_{1} = F_{2} = [p]. Let φ ∈ End(E) and consider a curve

S̃ = {(x, y) ∈ E × E; y = φ(x)}

in E ×E. Let L̃ be the line bundle [S̃] determined by S̃. In this section, γ̃ denotes the infimum of rational numbers such that γF̃ ⊗[S̃]^{−1} is ample. Then we have γ̃ = deg φ+1. This result proved by T. Kastura. By making use of Theorem 1, we have the following theorem:

Theorem 3. Let f_{1}, f_{2} and φ be as above. Let D_{1} = {a_{1}, ⋯, a_{d}} be a set of d points and φ a endomorphism of E. Set D_{2} = φ(D_{1}). Assume that the number of points in D_{2} is also d. Suppose that Supp_{k} f_{1}^{∗}D_{1} = Supp_{k} f_{2}^{∗}D_{2} for some k. If d > 2(deg φ + 1) + 8(s_{0} − 1)(1 + k^{−1}), then f_{2} = φ(f_{1}).

In the case where ♯D_{2} < d, we have the following theorem by Corollary of Lemma 2:

Theorem 4. Let f_{1}, f_{2} : ℂ^{m} → E be as above. Let D_{1} = {a_{1}, ⋯, a_{d}} be a set of d points and φ ∈ End(E). Set D_{2} = φ(D_{1}). Assume that the number of points in D_{2} is d′. Suppose that Supp_{1} f_{1}^{∗}D_{1} = Supp_{1} f_{2}^{∗}D_{2}. If dd′ > (d + d′)(deg φ + 1), then f_{2} = φ(f_{1}).

The following unicity theorem is a direct conclusion of Theorem 3:

Theorem 5. Let a_{1}, ⋯, a_{d} be distinct points in E. Let f_{1}, f_{2} : X → E be nonconstant holomorphic mappings. Suppose that Supp_{1} f_{1}^{∗}a_{j} = Supp_{1} f_{2}^{∗}a_{j} for all j. If d > 16s_{0} − 12, then f_{1} and f_{2} are identical on X.

Numazu College of Technology Shizuoka 410–8501 E-mail address: aihara@la.numazu-ct.ac.jp