Application of Kodaira Embedding Theorem

Question

I am going to give a talk on Kahler manifold. In particular, I will outline a proof of the Kadaira Embedding theorem.

I also wish to give some applications of the theorem. One of the application would be the Riemann bilinear relation on complex torus.

I am searching for other applications. Does anyone has a good suggestion?

Answer 1

Another application is the Ségre map $\mathbb{P}^{n} \times \mathbb{P}^{m} \rightarrow \mathbb{P}^{N}$. You will find this as a comment in 'Principles of algebraic Geometry' by Griffiths and Harris (p. 192).

Answer 2

Let me mention some important theorems about Kodaira embedding theorem

Let $X$ be a compact complex manifold, and $L$ be a holomorphic line bundle over $X$ equipped with a smooth Hermitian metric $h$ whose curvature form (locally given by $−i2π∂\bar ∂\log h$) is a positive definite real $(1,1)$-form, and so defines a Kähler metric $ω$ on $X$. Then the Kodaira embedding theorem states that there is a positive integer $k$ such that $L^k$ is globally generated (i.e. for every $x∈X$ there is a global holomorphic section $s∈H^0(X,L^k)$ with $s(x)≠0$) and the natural map $X\to\mathbb P(H^0(X,L^k)^∗)$, which sends a point $x$ to the hyperplane of sections which vanish at $x$, is an embedding. In particular, $X$ is a projective manifold.

Theorem 1.1 of the this paper extends this theorem of Gang Tian to the case of $X$ not necessarily compact, with compact analytic subvariety $Σ$ and holomorphic-Hermitian line bundle $(L,h)$ such that $h$ is continuous on $X∖Σ$ and has semi-positive curvature current $γ=c_1(L,h)$. In this context the authors consider the spaces of $L^2$-holomorphic sections of the tensor powers $L^p|_{X∖Σ}$, the Bergman density functions Pp associated with orthonormal bases, and the Fubini-Study $(1,1)$-currents $γ_p$ for which the $P_p$ serve as potentials. Under these conditions, it is shown in Theorem 1.1 that each $γ_p$ extends to a closed positive current on $X$, and that $\frac{1}{p}γ_p$ approaches $γ$ weakly if $\frac{1}{p}\log P_p→0$ locally uniformly on $X∖Σ$, as $p→∞$.

We have also the following theorem

If $X$ is a normal compact Kahler variety with isolated singularities that admits a holomorphic line bundle $L$ that is positive when restricted to the regular part of $X$, then $X$ is biholomorphic to a projective-algebraic variety.