Divisors, Picard group and Kodaira embedding theorem

1. Divisors and Picard group
- 1.1. Holomorphic line bundles and first Chern class
- 1.2. Divisors, line bundles and sheaves
  - 1.2.1. From divisors to Picard group
  - 1.2.2. The corresponding line bundle of $ \mathcal{O}(Y) $
2. Example: Projective space
3. Blowing-up
- 3.1. Blowing-up
4. Kodaira vanishing theorem
5. The traditional proof of Kodaira embedding theorem
6. An analytic proof by Donaldson

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1 Divisors and Picard group

1.1 Holomorphic line bundles and first Chern class

A complex line bundle is a 2 dimensional vector bundle with a complex structure on each fiber, i.e. each change of coordinates $ g_{ij}: U_j\cap U_i \times \mathbb{R}^2 \longrightarrow U_i\cap U_j\times\mathbb{R}^2 $ is $ i $-linear, i.e. $ g_{ij} $ can be represented by a function $ U_{i}\cap U_j \longrightarrow \mathbb{C} $.

A holomorphic line bundle is a complex line bundle that is also a complex manifold with the projection being holomorphic. In the same notation, the $ g_{ij} $ are now holomorphic functions.

A hermitian metric on a line bundle $ L $ is a positive sesquilinear form on each fiber. To define the Chern form of $ L $, let $ U $ be an open set of $ X $ over which $ L $ is trivialized and $ s_x $ is a holomorphic section of $ L $ over $ U $ that is non-vanishing, then one defines \[ \omega_{L,h} = \frac{1}{2\pi i}\partial \bar \partial \log |s|_h^2 \] which is independant of $ s $ since the ratio of two different $ s $ is in $ \mathcal{O}^*(U) $.

A Chern form is a real (1,1) form.

The set of isomorphic class of holomorphic line bundle is in one-to-one correspondance to $ H^1(X,\mathcal{O}_X^*) $

The proof of this fact is straightforward, but it is worth to remark that this result convinces us that the natural mapping $ \check{H}^1(\mathcal{U},X) \longrightarrow \check{H}^1(\mathcal{V},X) $ where $ \mathcal{U} $ is a finer open covering than $ \mathcal{V} $ is injective, since a line bundle is completely defined by one set of change of coordinates $ (g_{ij}) $.

Now the Chern class of a holomorphic line bundle is the class of $ \omega_{L,h} $ in $ H^2(X,\mathbb{Z}) $, which turns out to be independent of $ h $ and is in fact lies in the image of $ H^2(X,\mathbb{Z}) \longrightarrow H^2(X,\mathbb{R}) = H^2(X, \mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{R} $. In fact, the class of $ \omega_{L,h} $ can be defined using the following exact sequence: \[ 0 \longrightarrow \mathbb{Z} \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}_X^* \longrightarrow 0 \] where the injective arrow is the multiplication by $ i2\pi$ and the surjective one is exponential. The Chern map is in fact $ H^1(X,\mathcal{O}^*_X) \longrightarrow H^2(X,\mathbb{Z}) $. To prove this, one uses a double complex whose horizontal is the de Rham resolution and vertical is the Čech resolution and diagram chasing.

1.2 Divisors, line bundles and sheaves

A holomorphic line bundle is the same as a locally free $ \mathcal{O}_X $-module of rank 1.
An isomorphic class of line bundles is the same as a locally free isomorphic sheaf of $ \mathcal{O}_X $-module of rank 1.

1.2.1 From divisors to Picard group

A divisor is a formal sum of irreducible hypersurface, which can also be intepreted as an element of $ \check{H}^0(X,K_X^*/\mathcal{O}_X^*) $, which gives a mapping $ Div(X) \longrightarrow \check{H}^0(X,K_X^*/\mathcal{O}_X^*) $ with principal divisors being exactly sent to elements of $ K_X^*/\mathcal{O}_X^* $ coming from $ K_X^* $. multiplicative).

Since the following sequence \[ 0 \longrightarrow \mathcal{O}_X^* \longrightarrow K_X^* \longrightarrow K_X^*/O_X^* \longrightarrow 0 \] is exact, one has an application $\mathcal{O}:\ Div(X) = H^0(X, K_X^*/\mathcal{O}_X^*) \longrightarrow H^1(X, \mathcal{O}_X^*) = Pic(X) $. The kernel of $ \mathcal{O} $ corresponds to the the space of principal divisors. It is however worth having details of the application $\mathcal{O} $.

Let $ D = (U_i,f_i)\in Div(X) $ where $ f_i $ are meromorphic function on $ U_i $ with $ f_i/f_j\in \mathcal{O}_X^* $, then $ \mathcal{O}(D) $ is defined as following: \[ \mathcal{O}(D) (U_i) = f_i^{-1}\mathcal{O}_X(U_i) \]. Note that if $ D $ is effective, i.e. $ f_i\in \mathcal{O}_X(U_i) $ then $ \mathcal{O(D)} $ is the sheaf of holomorphic functions vanishing on $ D $.

To resume, here are some basic consequence of the above discussion: If $ D $ is effective then

If $ D $ is effective then $ H^0(X, \mathcal{O}(D) \ne 0 $.
If $ D $ is effective then $ \mathcal{O}(-D) $ is the sheaf of holomorphic function vanishing on $ D $. Therefore $ \mathcal{O}(-D) $ can be viewed as a ideal subsheaf of $ K_X $ and one has the following exact sequence: \[ 0 \longrightarrow \mathcal{O}(-D) \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O_D} \longrightarrow 0 \] where $ \mathcal{O}_D $ is the sheaf of "regular functions" on $ D $.
If $ L $ is a holomorphic line bundle and $ 0\ne s\in H^0(X,L) $ then $ \mathcal{O}(Z(s)) \equiv L $
If $ D $ is effective then $ \mathcal{O}(D) $ has a non-zero global section, for example section $ 1 = (U_i, f_i) $

1.2.2 The corresponding line bundle of $ \mathcal{O}(Y) $

Let $ Y $ be a hypersurface of $ X $, then the line bundle $ \mathcal{O}(Y) $ is isomorphic to $ \mathcal{N}_{Y,X} $ the normal line bundle of $ Y $ in $ X $.

By consequence, $ K_Y = \restr{(K_X\otimes \mathcal{O}(Y))}{Y} $.

2 Example: Projective space

2.1 $ \mathcal{O}(d) $ and its sections

Let's have some examples for the point of view discussed above, starting with the torsion sheaves $ \mathcal{O}(d) $.

The sheaf $ \mathcal{O}(-1) $, called tautological sheaf, is an invertible sheaf on $ \mathbb{P}^n_{\mathbb{C}} $ such that the fiber over $ l\in \mathbb{P}^n_{\mathbb{C}} $ of the corresponding line bundle is $ l $ itself. Let $ l=[x_0:\dots:x_n] \ in U_i $ then a point in $ l $ is of form $ t_i[\frac{x_0}{x_i}:,\dots:\frac{x_n}{x_i}] $ with coordinates in $ U_i $ being $ t_i $. So the change of coordinates from chart $ U_i $ to $ U_j $ is, since $ \frac{t_j}{x_j} = \frac{t_i}{x_i} $: \[ g_{ji} = \frac{x_i}{x_j} \] One notes by $ \mathcal{O}(1) $ the dual of $ \mathcal{O}(-1) $ and $ \mathcal{O}(d) = \mathcal{O}(1)^{\otimes d} $ and $ \mathcal{O}(-d)=\mathcal{O}(-1)^{\otimes d} $

Now if an invertible sheaf $ \mathcal{L} $ with $ \mathcal{L}(U_i) = \frac{1}{f_i}\mathcal{O}_X(U_i) $, the change of coordinates of the corresponding line bundle from chart $ U_i $ to $ U_j $ is $ g_{ji} = \frac{f_j}{f_i} $. So for $ \mathcal{O}(1) $, one has $ \frac{f_j}{f_i} = \frac{x_j}{x_i} $, i.e. there exists a linear combination $ A $ of $ x_0,\dots, x_n $ such that $ f_i = \frac{A}{x_i} $ is a holomorphic function correponding to the 1-section viewed in chart $ U_i $. As presented in the previous section, $ \mathcal{O}(1) $ is the associated line bundle of a hyperplane defined by the equation $ A = 0 $.

Similarly, $ \mathcal{O}(d) $ is the associated line bundle of a hypersurface $ A_d $ defined by a homogenous equation of degree $ d $, and $ \mathcal{O}(d) $ is the line bundle associated to the sheaf of holomorphic functions vanishing on $ A_d $.

2.2 Line bundles and maps to projective space, Veronese embedding

A linearly independent family $ s_0,\dots, s_N $ of global sections of a holomorphic line bundle $ L $ defines a holomorphic map $X\setminus Bs((s_i)) \longrightarrow \mathbb{P}^N_{\mathbb{C}} $ where $ Bs((s_i)) $ is the set of basepoints of $(s_i)$ where all the sections $ s_i $ vanish.

The global sections of $ \mathcal{O}(d) $ over $ \mathbb{P}^n_{\mathbb{C}} $ are $ H^0(\mathbb{P}^n_{\mathbb{C}}, \mathcal{O}(d) = \mathbb{C}[z_0,\dots,z_n]_d $ the vector space of homogenous polynomial of degree $ d $. The corresponding projective map is in fact a embedding, called Veronese embedding.

2.3 Canonical bundle and Euler sequence

\[ K_{\mathbb{P}^n_{\mathbb{C}} = \mathcal{O}(-n-1)} \]

\[ 0 \longrightarrow \mathcal{O} \longrightarrow \mathcal{O}(1)^{\oplus n} \longrightarrow \mathcal{T}_{\mathbb{P}_{\mathbb{C}}^n} \longrightarrow 0 \]

3 Blowing-up

3.1 Blowing-up

Let $ X $ be a complex manifold, $ x\in X $ and $ \pi:\ \hat X \longrightarrow X $ is the blow-up of $ X $ at $ x $ and $ E $ be the corresponding exceptional divisor, then \[ K_{\hat X} = \pi^* K_X \otimes \mathcal{O}((n-1)E). \] As a consequence, $ \restr{\mathcal{O}(E)}{E} = \mathcal{O}(-1) $ where $ \mathcal{O}(-1) $ is the tautologic sheaf over $ E = \mathbb{P}^{n-1}_{\mathbb{C}} $.

The appearance of the number $ n-1 $ is natural and can be explained as follow. First note that

in $ \hat X\setminus E $, there is no difference between $ K_{\hat X} $ and $ K_X $.
$ \pi_* $ send every tangent vector of $ E $ to the tangent vector $ 0 $ at $ x\in X $,
the pull-back $ \pi^* \omega $ of an $ n $-form on $X $ always vanishes on $ E $ therefore cannot generate $ K_{\hat X} $.

Our correction of this should be dividing $ \pi^* \omega $ by $ f^k $ where $ f $ is the equation defining $ E $ in $ \hat X $, i.e. tensoring $ \pi^* K_X $ by an appropriate multiple of $ \mathcal{O}(E) $ depending on the order of vanishing of $ \pi^* \omega $ at $ E $, which we claim to be $ n-1 $.

Here is the argument I used to convince myself: this vanishing order is that of the ratio of $ \pi^* \omega $ and a non-zero $n $-form (says the standard in the base formed by $ n-1 $ tangent vectors $ e^E_i $ of $ E $ and the normal vector $ v $ of $ E $ in $ X $), i.e. the vanishing order of $ \pi^*\omega(e^E_i, v) $. Each $ e^E_i$ plugged into $ \pi^*\omega $ adds one order of vanishing resulting in $ n-1 $.

Here is the argument I would use to convince others: WLOG, suppose that $ X = \mathbb{C}^n $ and $ x=0 $, then $ \hat X $ can be seen as a subset of $ \mathbb{C}^n\times \mathbb{P}^{n-1} $ with the coordinates in each chart $ U_i = \left\{(x_1, \dots, x_n,[p_1:\dots:p_n]):\ p_i\ne 0\right\}$ being $ (\frac{p_1}{p_i},\dots,\frac{p_n}{p_i},\zeta_i) $ with $ z_k = \frac{p_k}{p_i}\zeta_i $. The map $ \pi $ is given in local coordinates as \[ (\frac{p_1}{p_i},\dots,\frac{p_n}{p_i},\zeta_i) \mapsto (\frac{p_1}{p_i}\zeta_i, \dots \zeta_i, \dots \frac{p_n}{p_i}\zeta_i) \] The pull-back of $ \omega $ is \[ d(\frac{p_1}{p_i}\zeta_i)\wedge\dots\wedge d\zeta_i \wedge \dots\wedge d(\frac{p_n}{p_i}\zeta_i) = \zeta_i^{n-1}d(\frac{p_1}{p_i})\wedge\dots\wedge d\zeta_i\wedge\dots\wedge d(\frac{p_n}{p_i}) \] which vanishes with order $ n-1 $.

For the consequence, note that $ \mathcal{K}_E = \mathcal{O}(-n) = \restr{(K_{\hat X}\otimes \mathcal{O}(E))}{E} = \restr{(\pi^* K_{X}\otimes \mathcal{O}(nE))}{E} $, but $ \pi^* K_X $ is trivial over $ E $, therefore $ \restr{\mathcal{O}(E)}{E} = \mathcal{O}(-1) $.

4 Kodaira vanishing theorem

Let $ X $ be a compact complex manifold of dimension $ n $ and $ L $ is a positive holomorphic line bundle on $ X $, i.e. there exists a hermitian metric $ h $ on $ L $ such that the Chern form $ \omega_{L,h} $ is positive (i.e. a Kahler form). Then \[ H^q(X,\Omega_X^p \otimes \mathcal{L}) = 0 \quad\forall p+q >n. \] In particular, \[ H^i(X, K_X\otimes \mathcal{L}) = 0 \quad \forall i>0. \]

Since $ \mathcal{H}^{0,q}(E) \simeq H^q(X,\mathcal{E}) $ for all hermitian holomorphic vector bundle $ E $ with $ \mathcal{E} $ the corresponding sheaf of holomorphic sections. One needs to prove that all harmonic form in $ \mathcal{A}_X^{p,q}(L) $ vanishes for $ p+q>n $. This comes from the following two identities: Let $ \nabla $ be the Chern connection on $ L $ and $ \nabla = \nabla' + \nabla'' $ be its decomposition to $ (1,0) $ and $ (0,1) $ operators and $ \Delta_L' $ be the Laplacian corresponding to $ \nabla' $ then

$ \Delta_L = \Delta'_L + 2\pi[L,\Lambda] $ where $ L $ and $ \Lambda $ are Lefshetz operators.
$[L,\Lambda] = (k-n)Id_{\mathcal{A}^k_X}$

Therefore $ 0\leq(\alpha,\Delta'a)_{L^2} = 2\pi (n-k)(\alpha,\alpha)_{L^2}\leq 0 $

5 The traditional proof of Kodaira embedding theorem

Let $ X $ be a compact complex manifold with $ L $ a positive holomorphic line bundle on $ X $. Then $ L $ is generated by finitely many of its global sections and $ X $ can be embedded in a projective space $ \mathbb{C}\mathbb{P}^N $ with $ N $ sufficiently large.

The following approache is straight-forward: one shows that at every $ x\in X $, there is a global (holomorphic) section $ s_x $ of $ L^{\otimes m_x} $ such that $ s_x(x)\ne 0 $ then by compactness one can choose finitely many such sections and $ m_x $ which can be guaranteed to generate every germs at $ x $ of $ L^{\otimes m} $ with $ m=\max m_x $. That is one needs to prove that \[ H^0(X,\mathcal{L}^{\otimes m_x}) \twoheadrightarrow H^0({x}, \restr{\mathcal{L}^{\otimes m_x}}{x}) \] is surjective. Let $ \pi: \hat X \longrightarrow X $ be the blow-up of $ X $ at $ x $ and $ E = \pi^{-1}(x) $ be the corresponding exceptional divisor then one has

Figure 1: Insert caption [fig:kodaira-blowup]

where the vertical arrows are isomorphic (easy to see, maybe the right one needs Hartog). So one only needs to see that $H^0(\hat X,\pi^*\mathcal{L}^{\otimes m_x}) \twoheadrightarrow H^0(E,\restr{\pi^*\mathcal{L}^{\otimes m_x}}{E})$. Since $ E $ is a divisor of $ \hat X $, one has $ 0 \longrightarrow \mathcal{O}(E) \longrightarrow \mathcal{O}_{\hat X} \longrightarrow \mathcal{O}_E \longrightarrow 0 $ hence \[ 0 \longrightarrow \mathcal{O}(E)\otimes\pi^* \mathcal{L}^{\otimes m_x} \longrightarrow \pi^* \mathcal{L}^{\otimes m_x} \longrightarrow \restr{\pi^*\mathcal{L}^{\otimes m_x}}{E} \longrightarrow 0. \]

It remains to prove that $ H^1(\hat X, \mathcal{O}(E)\otimes\pi^* \mathcal{L}^{\otimes m_x}) $, or by thm:Kodaira-vanishing, that $ \mathcal{O}(E)\otimes\pi^* \mathcal{L}^{\otimes m_x}\otimes K_{\hat X}^{-1} = \mathcal{O}(-nE)\otimes\pi^* (\mathcal{L}^{\otimes m_x}\otimes K_X^{-1}) $ is positive, where we used the fact that $ K_{\hat X} = \pi^* K_X\otimes \mathcal{O}_{\hat X}((n-1)E) $.

Note that on $ \hat X \setminus E $, one can choose $ m_x $ large enough such that $ \mathcal{L}^{\otimes m_x}\otimes K_X^{-1}\times \mathcal{O}(-nE) $ is positive. It remains to observe $ E\subset \hat X $ which is in fact $ \mathbb{CP}^{n-1} $. But $ \restr{\mathcal{O}(-E)}{E} \equiv \mathcal{O}(1) $ is positive, which concludes the proof.