Chern class of line bundles The first c1(L) of a holomorphic line bundle L ∈ P ic(X) on X is the image of L under the boundary map 1 ∗ 2 Z c1 : P ic(X) ≃ H (X, OX ) → H (X, ). Roughly speaking, P ic(X) can be computed by the discrete part, measured by its image in 2 1 H (X, Z) and the continuous part from the H (X, OX ). The rank of the image c1(P ic(X)) is called the Picard number ρ(X). The Picard number could be very small. For example, it is known that ρ(X) = 0 for a generic with dimension ≥ 2.

To study the continuous part, we consider the kernel of the map P ic(X) → H2(X, Z) which is called Its Jacobian P ic0(X).

Another way to define cohomology group of a defined by flasgue sheaves(cf. [H05], p.290.) A resolution of a sheaf F is a complex 0 →F 0 →F 1 → ... together with a homomorphism F→F 0 such that

0 →F→F 0 →F 1 → ... is an exact complex of sheaves.

A sheaf F is called flasque if for any open subset U ⊂ M the restriction map ru,M : F(M) →F(U) is surjective.

Flasque sheaves are very special because of the following Lemma.

Lemma 4.5 If 0 →F 0 →F 1 →F 2 → 0 is a short exact sequence and F 0 is flasque, then the induced sequence

0 →F 0(U) →F l(U) →F 2(U) → 0 is exact for any open subset U ⊂ M.

Next, one has to ensure that any sheaf can be resolved by flasque sheaves. This will allow to define the cohomology of any sheaf.

Proposition 4.6 Any sheaf F on M admits a resolution

0 →F→F 0 →F 1 →F 2 → ... such that all sheaves F i, i =0, 1, ..., are flasque.

40 The i-th cohomology group Hl(M, F) of a sheaf F is the i-th cohomology of the complex

0 2 φ F φ2 F 0(M) −→F 1(M) −→ (M) −→ ...... induced by a fiasque resolution F→F •. Explicitly,

Ker(φi : F(M) →F i+1(M)) Hi(M, F) := Im(φi−1 : F i−1(M) →F i(M))

Clearly, with this definition, any fiasque sheaf F has vanishing cohomology H(M, T )=0 for i> 0. Moreover, for any sheaf F, one has the space of sections H0(M, F)=Γ(M, F)= F(M). That this definition of cohomology is really independent of the chosen fiasque reso- lution is due to

Proposition 4.7 If F →F • and F → G• are two frasque resolutions of a sheaf F, then both define naturally isomorphic cohomology groups.

Now we are going back to the Checkˇ cohomology.

Proposition 4.8 For any open covering M = ∪iUi, there exists a natural homomorphism

ˇp i H ({Ui}, F) → H (M, F)

These homomorphisms are in general not bijective, as the open covering might be very coarse. However, passing to the limit often results in isomorphisms with the true cohomology groups, i.e. when the topological space is reasonable (e.g. paracompact), the induced maps Hˇ p(M, F) → Hp(M, F) are indeed bijective and hence is a group isomorphism.

41 5 Almost complex manifolds

Let X be a complex manifold. We may regard X as a real manifold so that we have tangent TX (or T M). Since TX is not complex manifold, how to construct a “holomorphic vector bundle” ? Another question, when does a real manifold carry a complex structure so that it is indeed a complex manifold ?

Almost complex structure on a real vector space Let V be a finite dimensional real vector space. An almost complex structure on V is a linear transformation J : V → V with J 2 = −id.

We claim that if J : V → V is an almost complex structure, then dimRV is even and V admits, in a natural way, the structure of a complex vector space. In fact, the complex structure is defined by (a + ib) v = a v + b J(v), ∀a, b ∈ R and ∀u ∈ V . The R-linearity of J and the assumption J 2 = −id yield (a + ib)(c + id) v =(a + ib) (c + id) v and in particular i(i v)= −v. Hence dimR V = 2 dimC V . Our claim is proved.

Let V be a finite dimensional real vector space. We define the complex vector space 17 VC := V ⊗R C, which is called the complexification of V . We have embedding V ⊂ VC, v → v ⊗ 1. V is left invariant under the complex conjugation on VC which is defined by v ⊗ λ := v ⊗ λ, ∀v ∈ V and ∀λ ∈ C.

If V admits an almost complex structure J, J extends to an endomorphism J : VC → VC 1,0 0,1 which is C-linear. Since the eigenvalues of J in VC is ±i, the eigenspace V and V are

1,0 V := {v ∈ VC | J(v)= i v} = {X − iJX | X ∈ V },

0,1 V := {v ∈ VC | J(v)= −i v} = {X + iJX | X ∈ V }. We claim: 1,0 0,1 VC = V ⊕ V , 1,0 0,1 and complex conjugation on VC induces an R-linear isomorphism V ≃ V . In fact, since V 1,0 ∩ V 0,1 = 0, the canonical map

1,0 0,1 1 1 VC → V ⊕ V , v → (v − iJ(v)) ⊕ (v + iJ(v)) 2 2 defines the desired isomorphism.

17 If V has a basis {e1, ..., en} over R, then the complex vector space VC has the basis {e1 ⊗ 1, ..., en ⊗ 1} over C. For more details on of two vector spaces, see www.en.wikipedia.org.

42 Also there is a C-linear isomorphic map

(V,J) → (T 1,0, i) 1 X =2ReZ → 2 (X − iJX)= Z

Similarly V =(V, −J) is C-linearly isomorphic to (T 0,1, −i).

Tangent vector and cotangent bundles over a real manifold Let X be an n- dimensional real manifold which is equipped with coordinate charts {(Uα, xα)} of X such −1 1 n that the transition maps ραβ = xα ◦ xβ = (ραβ, ..., ραβ) are smooth maps from an open subset of Rn onto another open subset of Rn. Then we can define the Jacobain matrix of the transition maps ραβ:

1 1 ∂ραβ ∂ραβ 1 ...  ∂x ∂xn  θαβ := n n  ∂ραβ ∂ραβ   ∂x1 ... ∂xn  The of X is the (real) vector bundle TX on X which is given by the transition matrices θαβ. In fact, by the chain rule, one can check that the transition matrices satisfy the transition conditions (i.e., θαβ = Id,θαβ θβα = Id and θαβ θβγ θγα = Id).

If E is a vector bundle over a manifold X with local transition data {Uα, θαβ), then its ∗ t −1 t −1 dual vector bundle E is a vector bundle defined by {Uα, (θαβ) }. Here (A ) means the inverse of the transpose of a matrix A.

The T ∗X of X is the dual bundle of TX.

43