A note on the generalization of the Kodaira embedding theorem
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https://doi.org/10.1007/s11425-020-1755-9
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A note on the generalization of the Kodaira embedding theorem In Memory of Professor Zhengguo Bai (1916–2015)
Chao Li, Xi Zhang∗ & Qizhi Zhao School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China Email: [email protected], [email protected], [email protected] Received February 5, 2020; accepted August 3, 2020 Abstract
In this paper, we generalize the famous Kodaira embedding theorem.
Keywords MSC(2010)
Kodaira embedding theorem, holomorphic fibration, ampleness 53C07, 58E15
Citation: Li C, Zhang X, Zhao Q Z. A note on the generalization of the Kodaira embedding theorem. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1755-9
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Introduction
The famous Kodaira embedding theorem (see, e.g., [5]) asserts that any compact complex manifold admitting a positive line bundle can be holomorphically embedded in some projective space CP N . In this paper, we consider a generalization associated with a holomorphic fibration. Let f : X → Y be a holomorphic fibration between complex manifolds. In this paper, the fibration is always assumed to be smooth. Definition 1.1 (Fiber-wisely positive). A holomorphic line bundle L over X is f -fiber-wisely positive, if the restriction of L on any fiber is positive. Definition 1.2 (Projective fibration). The fibration f is projective, if there exists a holomorphic vector bundle E → Y together with a holomorphic embedding j : X → P(E ∗ ), such that f = πP(E ∗ ) ◦j. Here, πP(E ∗ ) is the natural projection from P(E ∗ ) to Y . We have the following generalization of the famous Kodaira embedding theorem. Theorem 1.3. Let f : X → Y be a holomorphic fibration with X being compact and Y compact and connected. If X admits an f -fiber-wisely positive holomorphic line bundle, then f is projective. We consider the following disjoint union: ⊔ Ek = H 0 (f −1 (y), Lk |f −1 (y) ),
(1.1)
y∈Y
where k > 1. By choosing sufficiently large k, Ek will be naturally a holomorphic vector bundle, and we can construct a holomorphic embedding from X to P(Ek∗ ) fiber by fiber. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Maps associated with holomorphic global sections
Let L be a holomorphic line bundle over a compact complex manifold M . We set EL = H 0 (M, L), and ΣL =
∩
{x ∈ M | u(x) = 0}.
(2.1)
(2.2)
u∈EL
There is a natural map ΦL : L∗ → EL∗ defined by ΦL (e∗ )(u) = e∗ (u)
(2.3)
for any e∗ ∈ L∗ and u ∈ EL . If x ∈ M \ ΣL , then ΦL maps L∗ |x to a 1-dimensional subspace. So ΦL induces a map ϕL : M \ ΣL → P(EL∗ ): x 7→ Φ(L∗ |x ). (2.4) We have the following facts: (1) L is globally generated if and only if ϕL is well defined on whole M , or ΣL = ∅. (2) L is very ample if and only if ϕL is a well-defined embedding on whole M . Assume that dim EL = N + 1 with N > 1. By the canonical equivalence (EL∗ )∗ ∼ = EL , we can also ∗ view any u ∈ EL as a compl
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