Complex Surfaces

Andrew Kobin Spring 2019 Contents Contents

Contents

0 Introduction 1

1 Complex Manifolds 2 1.1 Holomorphic Functions ...... 2 1.2 Complex Manifolds ...... 3 1.3 Dolbeault Cohomology ...... 6 1.4 K¨ahlerManifolds ...... 8 1.5 Analytic Varieties ...... 13 1.6 Divisors, Line Bundles and Sections ...... 15 1.7 Cohomology and Chern Classes ...... 21 1.8 Connections on Vector Bundles ...... 30

i 0 Introduction

0 Introduction

These notes are from a course on 4-manifolds taught by Dr. Tom Mark at the University of Virginia in Spring 2019. I stuck around for the beginning of the course to take notes on complex manifolds, complex surfaces and surface classification since these topics are developed in parallel to the algebro-geometric theory of surfaces. The main reference for this course is Griffiths-Harris, Principles of Algebraic Geometry.

1 1 Complex Manifolds

1 Complex Manifolds

1.1 Holomorphic Functions

Recall that a complex function f : C → C is holomorphic if it satisfies the Cauchy-Riemann equations: ∂u ∂v ∂u ∂v = and = − ∂x ∂y ∂y ∂x where f = u + iv. To express this for functions of multiple complex variables, we can reinterpret this definition as follows. Set z = x + iy and define differentials dz = dx + i dy, dz¯ = dx − i dy and ∂f ∂f df = dx + dy. ∂x ∂y 1 i Then dx = 2 (dz + dz¯), dy = − 2 (dz − dz¯) and 1 ∂f ∂f  1 ∂f ∂f  df = − i dz + + i dz¯ 2 ∂x ∂y 2 ∂x ∂y which suggests defining the following complex derivatives: ∂f 1 ∂f ∂f  ∂f 1 ∂f ∂f  = − i and = + i . ∂z 2 ∂x ∂y ∂z¯ 2 ∂x ∂y

∂f ∂f Then df = ∂z dz + ∂z¯ dz¯. Lemma 1.1.1. A complex function in one variable f = u + iv : C → C is holomorphic if ∂f and only if ∂z¯ = 0. n For a multivariable function f : C → C, write zj = xj + iyj, 1 ≤ j ≤ n, for the coordinates on Cn. Then n n n n X ∂f X ∂f X ∂f X ∂f df = dx + dy = dz + dz¯ ∂x j ∂y j ∂z j ∂z¯ j j=1 j j=1 j j=1 j j=1 j where ∂f and ∂f are defined in the same way as their single-variable counterparts above. ∂zj ∂z¯j n Definition. Let f : C → C be a complex function in the variables zj = xj + iyj, 1 ≤ j ≤ n, and define n n X ∂f X ∂f ∂f = dz and ∂f¯ = dz¯ . ∂z j ∂z¯ j j=1 j j=1 j Then f is holomorphic if ∂f¯ = 0.

Lemma 1.1.2. A function f : Cn → C is holomorphic if and only if it satisfies the Cauchy- Riemann equations in each variable zj.

n m Definition. More generally, f = (f1, . . . , fm): C → C is holomorphic if f1, . . . , fm are all holomorphic.

2 1.2 Complex Manifolds 1 Complex Manifolds

1.2 Complex Manifolds

Definition. A complex n-manifold is a Hausdorff topological space that is locally homeo- morphic to (an open subset of) Cn and has holomorphic transition functions. Explicitly, M is a complex manifold if it admits a complex atlas: a choice of open covering {Ui} of X n together with homeomorphisms ϕi : Ui → ϕi(Ui) ⊆ C such that for each pair of overlapping charts Ui,Uj, the transition map

−1 ϕij := ϕj ◦ ϕi : ϕi(Ui ∩ Uj) −→ ϕj(Ui ∩ Uj) and its inverse are holomorphic. A complex structure on M is the choice of such a complex atlas, up to holomorphic equivalence of charts, defined by a similar condition to the above. Let M be a complex n-manifold and fix a point p ∈ M. Then in a chart U ⊆ M around p, the complex structure determines local parameters dx1, . . . , dxn, dy1, . . . , dyn. As in the previous section, there are also complex parameters dzj = dxj + i dyj and dz¯j = dxj − i dyj for 1 ≤ j ≤ n. Each of the dxj, dyj, dzj, dz¯j are R-linear functionals TpM → R. Writing dzj = dxj + i dyj extends these to C-linear functionals TpM → C (and likewise for dz¯j), so ∗ ∂ 1  ∂ ∂  ∂ 1  ∂ ∂  each dzj, dz¯j ∈ T M ⊗ . Similarly, each = − i and = + i p R C ∂zj 2 ∂xj ∂yj ∂z¯j 2 ∂xj ∂yj ∗ are elements of the complexified tangent space TpM ⊗R C. Note that TpM ⊗ C and Tp ⊗ C are both 2n-dimensional complex vector spaces (so 4n-dimensional real vector spaces). The above functionals are cotangent vectors and can be evaluated on the above tangent vectors as follows: for 1 ≤ j, k ≤ n,  ∂  1  ∂ ∂  dzj = (dxj + i dyj) − i ∂zk 2 ∂xk ∂yk 1   ∂   ∂    ∂   ∂  = dxj + dyj + i dyj − dxj 2 ∂xk ∂yk ∂xk ∂yk 1 = [(δ + δ ) + i(0 − 0)] = δ . 2 ij ij ij

 ∂   ∂   ∂  n ∂ ∂ ∂ ∂ o Likewise, dz¯j = δij and dzj = dz¯j = 0. Thus ,..., , ,..., ∂z¯k ∂z¯k ∂zk ∂z1 ∂zn ∂z¯1 ∂z¯n ∗ and {dz1, . . . , dzn, dz¯1, . . . , dz¯n} are dual bases for the vector spaces TpM ⊗ C and Tp M ⊗ C, respectively. As a consequence, there are several different notions of “tangent space” for a complex manifold:

ˆ Real tangent space: the underlying real vector space of TpM, which has real dimen- sion 2n;

ˆ C Complex tangent space: the complex vector space Tp M := TpM ⊗ C, which has complex dimension 2n; n on ˆ 0 ∂ C Holomorphic tangent space: the subspace TpM := Span ∂z of Tp M; C j j=1 n on ˆ 00 ∂ C Antiholomorphic tangent space: the subspace Tp M := Span ∂z¯ of Tp M. C j j=1

3 1.2 Complex Manifolds 1 Complex Manifolds

∗ C,∗ We also define the real and complex cotangent spaces Tp M and Tp M, as well as their 1,0 0 ∗ 0,1 00 ∗ holomorphic and antiholomorphic counterparts Tp M := (TpM) and Tp M := (Tp M) , respectively.

Proposition 1.2.1. Let M be a complex manifold of dimension n and p ∈ M. Then

C 0 00 (1) Tp M = TpM ⊕ Tp M.

C 0 00 (2) Complex conjugation acts C-linearly on Tp M and exchanges TpM and Tp M.

C,∗ 1,0 0,1 (3) Tp M = Tp M ⊕ Tp M, and specifically we have

1,0 C,∗ 00 n Tp M = {α ∈ Tp M | α ≡ 0 on Tp M} = SpanC{dzj}j=1 0,1 C,∗ 0 n Tp M = {α ∈ Tp M | α ≡ 0 on TpM} = SpanC{dz¯j}j=1.

C,∗ 1,0 0,1 (4) Complex conjugation acts C-linearly on Tp M and exchanges Tp M and Tp M. Suppose M m and N n are complex manifolds with open subsets U ⊆ M,V ⊆ N, and f : U → V is an arbitrary function. For 1 ≤ j ≤ m, write the coordinates on U as zj = xj + iyj and similarly, for 1 ≤ k ≤ n, write the components of f on V as wk = uk + ivk. The real Jacobian of f is an m × n matrix   ∂uk ∂uk

 ∂xj ∂yj  J f =   R  ∂vk ∂vk  ∂xj ∂yj

which defines a linear map JRf : TpM → Tf(p)N for any p ∈ U. Extending scalars allows C C us to define the complex Jacobian JCf : Tp M → Tf(p)N given by the same matrix JRf with n o n o respect to the real bases ∂ , ∂ , ∂ , ∂ and ∂ , ∂ , ∂ , ∂ of these tangent spaces. ∂xj ∂x¯j ∂yj ∂y¯j ∂uk ∂u¯k ∂vk ∂v¯k n o n o With respect to the complex bases ∂ , ∂ and ∂ , ∂ , the matrix is ∂zj ∂z¯j ∂wk ∂w¯k   ∂wk ∂wk

 ∂zj ∂z¯j  J f =   . C  ∂w¯k ∂w¯k  ∂zj ∂z¯j

Proposition 1.2.2. Let U ⊆ M,V ⊆ N and f : U → V be as above. Then the following are equivalent:

(1) f is holomorphic.

(2) For all 1 ≤ j ≤ m and 1 ≤ k ≤ n, ∂wk = ∂w¯k = 0. ∂z¯j ∂zj

1,0 1,0 0,1 0,1 (3) For all p ∈ U, JCf takes Tp M to Tf(p)N and Tp M to Tf(p)N.

4 1.2 Complex Manifolds 1 Complex Manifolds

Corollary 1.2.3. Suppose M and N are complex manifolds of the same dimension and f : M → N is holomorphic. Then   2 ∂wk det(JRf) = det(JCf) = det ≥ 0. ∂zj Corollary 1.2.4. A holomorphic map f : M → N between complex manifolds of the same dimension is orientation-preserving. In particular, transition maps are orientation-preserving, so every complex manifold has a naturally defined orientation. Example 1.2.5. Note that the operators ∂ and ∂ act as expected on the coordinates ∂zj ∂z¯j 2 2 3 {zj, z¯j}. For example, consider the map f : C → C, f(z, w) = z + w . Then
JCf = 2z 3w which is surjective whenever (z, w) 6= (0, 0) so for any constant c ∈ C r {0}, the equation z2 + w3 = c defines a smooth submanifold of C2. More generally, there are holomorphic versions of the inverse and implicit function theo- rems from differential geometry:

Theorem 1.2.6 (Inverse Function Theorem). If f : M → N is holomorphic and JCf is an isomorphism at p ∈ M, then there is a neighborhood V ⊆ N of f(p) and an inverse f −1 : V → f −1(V ) to f around p that is holomorphic on V .

Theorem 1.2.7 (Implicit Function Theorem). Let f : M1 × M2 → N be a holomorphic function such that JCf is invertible at (p, q) ∈ M1 × M2 and set w = f(p, q). Then there is a unique holomorphic map g : M → N such that f(p, g(p)) = w. To prove these, one need only prove that f −1 (in the inverse function theorem) and g (in the implicit function theorem) are holomorphic. The rest follows from real differential geometry. Remark. In the above example, the complex submanifold defined by the equation z2+w3 = c is not compact. In fact, a complex submanifold of Cn is never compact. To see this, fix an embedding M,→ Cn of a complex submanifold. Then the composition f : M,→ Cn → C with projection onto the ith component is a holomorphic function, so by the maximum principle, f must be constant. A version of this argument can be used to prove the following important fact. Theorem 1.2.8. If M is a closed complex manifold, then there are no nonconstant holo- morphic functions M → C. Example 1.2.9. Let Pn = CP n be complex projective n-space, defined as the quotient n+1 C r {0}/ ∼ where (z0, . . . , zn) ∼ (w0, . . . , wn) if wj = λzj for some λ ∈ C r {0}. Al- ternatively, Pn is the space of complex lines through the origin in Cn+1. Write the coordi- n n nates on P as [z0, . . . , zn]. There is a natural complex structure on P via the open sets

5 1.3 Dolbeault Cohomology 1 Complex Manifolds

n Uj = {[z0, . . . , zn] ∈ P | zj 6= 0} and charts n ϕj : Uj −→ C   z0 zj−1 zj+1 zn [z0, . . . , zn] 7−→ ,..., , ,..., . zj zj zj zj One can check that the transition maps are holomorphic, so this gives Pn the structure of an n-dimensional complex manifold. For any homogeneous polynomial f ∈ C[z0, . . . , zn], n n the equation f(z0, . . . , zn) = 0 defines a subset V (f) in P , called an algebraic set in P . If   J f = ∂f ··· ∂f is nonzero at every point in V (f) except for the point corresponding C ∂z0 ∂zn to 0 ∈ Cn+1, then V (f) is a complex submanifold of Pn.

1.3 Dolbeault Cohomology

Let M be a complex manifold of dimension n and for k ≥ 1, let Vk T C,∗M be the vector bundle of differential k-forms on M. Then the splitting T C,∗M = T 1,0M ⊕ T 0,1M from Proposition 1.2.1 induces a splitting of Vk T C,∗M: ^ M ^ ^ ^ ^  k T C,∗M = p,q T C,∗M where p,q T C,∗M := p T 1,0M ⊗ q T 0,1M . p+q=k

Vp,q C,∗ Each graded piece T M has basis {dzi1 ∧· · ·∧dzip ∧dz¯j1 ∧· · ·∧dz¯jq | 1 ≤ i1, . . . , ip, j1, . . . , jq ≤ n} which we can write more compactly as {dzI ∧ dz¯J : |I| = p, |J| = q} where I,J ⊂ N and

dzI = dzi1 ∧ · · · ∧ dzip for I = {i1, . . . , ip} (likewise for dz¯J ). Passing to global sections of this vector bundle gives a type decomposition on the vector space of differential k-forms:  ^  M  ^  M ΩkM := Γ M, k T C,∗M = Γ M, p,q T C,∗M =: Ωp,qM. p+q=k p+q=k We say a differential k-form ω ∈ Ωp,qM ⊆ ΩkM has type (p, q). Write X ω = fIJ dzI ∧ dz¯J |I|=p,|J|=q using the same indexing notation as above, we have X X X ¯ dω = dfIJ ∧ dzI ∧ dz¯J = ∂fIJ ∧ dzI ∧ dz¯J + ∂fIJ ∧ dzI ∧ dz¯J . |I|=p,|J|=q |I|=p,|J|=q |I|=p,|J|=q In the last expression, the first term is a differential form of type (p + 1, q) and the second term is of type (p, q + 1). Put X ¯ X ¯ ∂ω = ∂fIJ ∧ dzI ∧ dz¯J and ∂ω = ∂fIJ ∧ dzI ∧ dz¯J . |I|=p,|J|=q |I|=p,|J|=q

Then dω = ∂ω + ∂ω¯ with ∂ω ∈ Ωp+1,qM and ∂ω¯ ∈ Ωp,q+1M. Observe that 0 = d2 = ∂2 + ∂∂¯+ ∂∂¯ + ∂¯2 and each piece lands in a different direct summand of Ωk+1M, so we must have ∂2 = ∂¯2 = 0 and ∂∂¯ = −∂∂¯ . This proves:

6 1.3 Dolbeault Cohomology 1 Complex Manifolds

Lemma 1.3.1. For any complex manifold M, there are C-linear operators ∂ :Ωp,qM → Ωp+1,qM and ∂¯ :Ωp,qM → Ωp,q+1M

which satisfy ∂2 = ∂¯2 = 0 and ∂∂¯ = −∂∂¯ .

Definition. The Dolbeault cohomology of a complex manifold M is the bigraded vector space M H•,•(M) = Hp,q(M) where Hp,q(M) := Hq(Ωp,•(M), ∂¯). p,q≥0 Remark. Taking the differential ∂ on Ω•,q(M) yields an isomorphic version of H•,•(M).

Definition. A differential form on M of type (p, 0) is called a holomorphic p-form.

A holomorphic differential form on M is locally of the form X ω = fI dzI |I|=p where the fI are holomorphic functions. Remark. The grading Ωk(M) = L Ωp,q(M) together with the decomposition of the differ- ential d = ∂ + ∂¯ gives Ω•(M) the structure of a double complex of vector spaces. There is an associated spectral sequence, called the Hodge-de Rham spectral sequence, whose E2-page p,q • p,q p,q is given by H (M) and which converges to HdR(M). In particular, if h = dimC H (M) k and bk = dimC HdR(M) is the kth Betti number of M, then

X p,q h ≥ bk p+q=k for all k ≥ 0. In the case when M is a K¨ahlermanifold (see Section 1.4), then the Hodge-de P p,q Rham spectral sequence collapses on the second page, in which case p+q=k h = bk for all k. Moreover, when M embeds holomorphically into Cn for some n ≥ 1, then M has the cohomology of an affine complex variety, so Hp,q(M) = 0 for all q > 0 and there is an p ∼ p •,0 isomorphism HdR(M) = H (H (M), ∂) for all p ≥ 0.

Example 1.3.2. Let E be a complex torus given by C/Λ for a lattice Λ ⊆ C. We may write Λ = Z ⊕ Zτ for some τ = a + bi ∈ C such that b > 0. Then E carries the structure of a complex curve (1-manifold) called an elliptic curve. Using the above techniques, one may show that

1,0 ∼ 2 0,1 ∼ 2 H (E) = Chdzi = SpanC{1, τ} ⊂ C and H (E) = Chdz¯i = SpanC{1, τ¯} ⊂ C . 2 ∼ 2 So different choices of τ determine different subspaces of H (E; C) = C and therefore different complex structures on E. In fact, the set of all elliptic curves up to isomorphism may be identified with the set of all τ ∈ C with b > 0.

7 1.4 K¨ahler Manifolds 1 Complex Manifolds

Example 1.3.3. For n ≥ 1, complex projective n-space CP n has a distinguished differential form known as the Fubini-Study form, which can be defined as follows. Recall that CP n has the following cohomology groups: ( k n C, k = 0, 2,..., 2n H (CP ; C) = 0, otherwise.

Let z ∈ CP n be a point with a neighborhood U having holomorphic map Z : U → Cn+1 r{0} n+1 n 2 n which is a section of π : C r {0} → CP . Define ωFS ∈ Ω (CP ) locally about z by i ω = ∂∂¯log |Z|2 FS 2π

n+1 where | · | is the usual Euclidean norm on C . Then ωFS is a (1, 1)-form and is real, ¯ meaningω ¯FS = ωFS. Moreover, dωFS = ∂ωFS = ∂ωFS = 0, so ωFS is a closed 2-form and 1,1 n defines a class [ωFS] ∈ H (CP ). On the other hand, one can show that ωFS is Poincar´e n n−1 n dual to the class of a hyperplane H ⊆ CP representing CP ,→ CP , so ωFS is not exact 1,1 n 2 n 2,0 n 1,1 n and therefore [ωFS] generates H (CP ). Since H (CP ; C) = H (CP ) ⊕ H (CP ) ⊕ H0,2(CP n), it follows that 2 n 1,1 n ∼ 2,0 n 0,2 n H (CP ; C) = H (CP ) = C and H (CP ) = H (CP ) = 0.

• n ∼ n+1 Recall that as a ring, H (CP ; C) = C[x]/(x ) where x is a generator in degree 2. The k k n above shows we can take x = [ωFS], so that [ωFS] generates H (CP ; C) for each 1 ≤ k ≤ n. k • n However, since ωFS is a (k, k)-form, this means the type decomposition of H (CP ; C) is precisely ( p,q n C, p = q H (CP ) = 0, p 6= q.

1.4 K¨ahlerManifolds

Recall that a Riemannian metric on a smooth manifold M is a smooth assignment g of an inner product on TxM for each x ∈ M. This can alternatively be characterized as a tensor field g ∈ Γ(M,T ∗M ⊗ T ∗M) that restricts to a symmetric, positive definite form at each x ∈ M. If M is a complex manifold, then there is an R-linear endomorphism J : TM −→ TM

2 satisfying J = −id. Explicitly, on TxM this can be written Jv = iv. Definition. A Riemannian metric g on a complex manifold M is compatible with the complex structure if g(Jv, Jw) = g(v, w) for all x ∈ M and v, w ∈ TxM. Proposition 1.4.1. Every complex manifold admits a compatible metric.

8 1.4 K¨ahler Manifolds 1 Complex Manifolds

Fix a complex Riemannian manifold (M, g) with g a compatible metric. Extending J to T CM defines a C-bilinear form

C C gC : Tx M ⊗ Tx M −→ C

0 0 for each x ∈ M such that if v, w ∈ TxM, then gC(v, w) = 0. This is because TxM may be 0 written TxM = {X − iJX | X ∈ TxM}. Definition. For a complex Riemannian manifold M with compatible metric g, the associ- 0 0 ated Hermitian form on M is the assignment h of an R-bilinear map h : TxM ⊗TxM → C 0 to each x ∈ M, defined by h(v, w) = gC(v, w¯) for all v, w ∈ TxM. Lemma 1.4.2. Let (M, g) be a complex manifold with compatible metric and associated Hermitian form h. Then

0 (a) For all v, w ∈ TxM, h(v, w) = h(w, v). In particular, h(v, v) ∈ R.

0 (b) For all v ∈ TxM, h(v, v) ≥ 0 with equality if and only if v = 0. 0 ¯ (c) For all v, w ∈ TxM and λ ∈ C, h(λv, w) = λh(v, w) = h(v, λw). Definition. A (1, 1)-form ω ∈ Ω1,1(M) is a positive form if ω¯ = ω and for all nonzero 0 v ∈ TxM, −iω(v, v¯) > 0. P The reason for this terminology is that any real (1, 1)-form ω = fjk dzj ∧dz¯k determines P a Hermitian form h = −i fjk dzj ⊗ dz¯k and h is positive definite (and thus defines a metric on M) if and only if ω is a positive form.

Definition. To a complex manifold M with compatible metric g, we associated a (1, 1)-form ω defined by ω(X,Y ) = −g(X,JY ) for X,Y ∈ TxM. Then M is a K¨ahlermanifold if ω is a closed form, and in this case g is called a K¨ahler metric on M.

n Example 1.4.3. On CP , the Fubini-Study form ωFS is closed and of the form ω(X,Y ) = −g(X,JY ) where g is the standard metric inherited from CP n+1. Thus CP n is a K¨ahler manifold.

Example 1.4.4. On Cn, the usual inner product may be expressed as a tensor field

n n X 1 X g = (dx ⊗ dx + dy ⊗ dy ) = (dz ⊗ dz¯ + dz¯ ⊗ dz ) j j j j 2 j j j j j=1 j=1

0 n where zj = xj + iyj. The corresponding Hermitian form on T C is given by  ∂ ∂   ∂ ∂  1 h , = g , = δjk ∂zj ∂zk ∂zj ∂z¯k 2

9 1.4 K¨ahler Manifolds 1 Complex Manifolds

so the ∂ are mutually Hermitian-orthogonal of length √1 . Meanwhile, the associated 2-form ∂zj 2 for this metric is determined by

 ∂ ∂   ∂ ∂  ω , = −g , i = 0 ∂zj ∂zk ∂zj ∂zk  ∂ ∂   ∂ ∂  ω , = −g , i = 0 ∂z¯j ∂z¯k ∂z¯j ∂z¯k  ∂ ∂   ∂ ∂  i ω , = −g , i = δjk. ∂zj ∂z¯k ∂zj ∂z¯k 2 So ω has the form n n i X X ω = dz ∧ dz¯ = dx ∧ dy 2 j j j j j=1 j=1 as expected.

Example 1.4.5. Any Riemann surface (complex 1-manifold) is automatically a K¨ahlerman- ifold because any 2-form is closed.

Example 1.4.6. Let M be a K¨ahlermanifold of complex dimension n, x ∈ M and choose ∗ an orthonormal basis {ε1, . . . , ε2n} of Tx M with respect to a basis {e1, . . . , e2n} of TxM. We may assume J : TxM → TxM acts by Je2j−1 = e2j for each 1 ≤ j ≤ n. Set ϕj = ε2j−1 + iε2j. 1,0 n Then {ϕ1, . . . , ϕn} is a basis for Tx M. The calculation on C in Example 1.4.4 then shows that the associated (1, 1)-form ω on M locally has the form

n n X i X ω = ε ∧ ε = ϕ ∧ ϕ¯ . 2j−1 2j 2 j j j=1 j=1

n A short calculation shows that ω = n! ε1 ∧ · · · ∧ ω2n = n! volM , where volM is the volume form on M with respect to g. In particular this is easy to see from the calculation at the end of Example 1.4.4 for Cn. Proposition 1.4.7. Let (M, g) be a K¨ahlermanifold and N ⊆ M any submanifold. Then restricting g to N makes N a K¨ahlermanifold.

Proof. Let i : N,→ M be the subspace inclusion and write ωM and ωN for the associated ∗ (1, 1)-forms on M and N, respectively. Then ωN = i ωM and since d commutes with pullback, we get dωN = 0. Therefore N is K¨ahler. Corollary 1.4.8. On a complex manifold M of dimension n with Hermitian inner product h 1 n and associated (1, 1)-form ω, the volume form on M is given by volM = n! ω . In particular, for any complex submanifold N ⊆ M of dimension d, Z 1 d volM (N) = ω . d! N Corollary 1.4.9. If (M, g) is a K¨ahlermanifold, then

10 1.4 K¨ahler Manifolds 1 Complex Manifolds

(1) The volume of any closed complex submanifold N ⊆ M depends only on the homology class [N] ∈ H•(M). (2) A closed complex submanifold N ⊆ M has [N] 6= 0. Corollary 1.4.10. If (M, g) is a compact K¨ahlermanifold then for each 0 ≤ k ≤ n, [ωk] 6= 0 and in particular H2k(M) 6= 0. Proof. If [ωk] = 0 then ωk = dα for some α ∈ Ω2k−1(M). In this case, we have Z Z n! vol(M) = ωn = ωk ∧ ωn−k M M Z Z = dα ∧ ωn−k = (dα ∧ ωn−k + α ∧ dωn−k) since ωn−k is closed M M Z = d(α ∧ ωn−k) by the Leibniz rule M Z = α ∧ ωn−k = 0 by Stokes’ theorem. ∂M But this implies vol(M) = 0 which is impossible. Theorem 1.4.11 (Hodge Decomposition). If M is a compact K¨ahlermanifold, then for each k ≥ 0 there is an isomorphism

k ∼ M p,q HdR(M) = H (M). p+q=k Proof. (Sketch) Let ∆ be the Laplace-Beltrami operator on M, defined by

∆ : Ω•(M) −→ Ω•(M) ω 7−→ dd∗ω + d∗dω

where d∗ :Ωk(M) → Ωk−1(M) is the adjoint of the exterior derivative on M. Explicitly, d∗ is uniquely determined by hdα, βi = hα, d∗βi for all α, β ∈ Ω•(M). A differential k-form ω on M is called harmonic if ∆ω = 0; this is equivalent to dω = d∗ω = 0, which is easy to prove using the inner product definition of d∗. Thus a harmonic form ω ∈ Ωk(M) defines a k k cohomology class [ω] ∈ HdR(M). Let H (M) be the vector space of all harmonic k-forms on M. The Hodge theorem says that the natural inclusion

k k H (M) ,−→ HdR(M) is in fact an isomorphism for all k ≥ 0, so every de Rham cohomology class has a unique harmonic representative. On the other hand, since M is K¨ahler,its metric is compatible with the complex structure on M in a way that ensures ∆ preserves the type decomposition of differential k-forms k L p,q Ω (M) = p+q=k Ω (M), and therefore this type decomposition is also compatible with the ‘anti-holomorphic Laplace operator’

¯¯∗ ¯∗ ¯ • • ∆∂¯ = ∂∂ + ∂ ∂ :Ω (M) → Ω (M).

11 1.4 K¨ahler Manifolds 1 Complex Manifolds

Hence the Hodge theorem, suitably applied to ∆∂¯, says that every Dolbeault cohomology class has a unique ∆∂¯-harmonic representative, so there are isomorphisms k ∼ k M p,q ∼ M p,q HdR(M) = H (M) = H (M) = H (M). p+q=k p+q=k

Example 1.4.12. Let Σ = Σg be a Riemann surface of genus g, which is K¨ahlerby Exam- ple 1.4.5. Note that for any p or q > 1, Hp,q(Σ) = 0 since the complex dimension of Σ is 1. Then the Hodge decomposition for Σ is: ∼ 2 1,1 C = HdR(Σ) = H (Σ) 2g ∼ 1 1,0 0,1 ∼ g g 1,0 ∼ 0,1 C = HdR(Σ) = H (Σ) ⊕ H (Σ) = C ⊕ C since H (Σ) = H (Σ) ∼ 0 0,0 C = HdR(Σ) = H (Σ). This information can be encoded in a Hodge diamond: 1 g g 1

p,q ∼ q,p p,q p,q For an arbitrary K¨ahlermanifold M, H (M) = H (M) so if we set h = dimC H (M), then hp,q = hq,p for all p, q ≥ 0. Thus the Hodge diamond for any K¨ahlermanifold is always symmetric about the vertical axis. In fact, Poincar´eduality ensures that the diamond is also symmetric about the horizontal line k = dim M/2.

Example 1.4.13. Let Σg and Σh be Riemann surfaces of genera g and h, respectively, and set M = Σg × Σh. It is easy to check M is also K¨ahler.By the K¨unneththeorem, we have

2 2,0 1,1 0,2 HdR(M) = H (M) ⊕ H (M) ⊕ H (M) 1,0 1,0 1,1 0,0 1,0 0,1 =( H (Σg) ⊗ H (Σh)) ⊕ (H (Σg) ⊗ H (Σh)) ⊕ (H (Σg) ⊗ H (Σh)) 0,1 1,0 0,0 1,1 0,1 0,1 ⊕ (H (Σg) ⊗ H (Σh)) ⊕ (H (Σg) ⊗ H (Σh)) ⊕ (H (Σg) ⊗ H (Σh)) ∼ g h g h g g h = (C ⊗ C ) ⊕ C ⊕ (C ⊗ C ) ⊕ (C ⊗ C) ⊕ C ⊕ (C ⊗ C ). Similarly,

1 1,0 0,1 HdR(M) = H (M) ⊕ H (M) 1,0 0,0 0,0 1,0 =( H (Σg) ⊗ H (Σh)) ⊕ (H (Σg) ⊗ H (Σh)) 0,1 0,0 0,0 0,1 ⊕ (H (Σg) ⊗ H (Σh)) ⊕ (H (Σh) ⊗ H (Σh)) ∼ g h g h = C ⊕ C ⊕ C ⊕ C . Applying the symmetry described above, we get the following Hodge diamond for M: 1 g + h g + h gh 2(gh + 1) gh g + h g + h 1

12 1.5 Analytic Varieties 1 Complex Manifolds

Remark. The duality Hp,q(M) ∼= Hn−p,n−q(M) inducing the symmetry of the Hodge diamond for a K¨ahlermanifold is known as Serre duality in algebraic geometry.

1.5 Analytic Varieties

Definition. Let M be a complex manifold. An analytic variety in M is a subset V ⊆ M given locally on an open subset U ⊆ M by the vanishing locus

V ∩ U = V (f1, . . . , fk) of some holomorphic functions f1, . . . , fk : U → C. Example 1.5.1. An analytic hypersurface in M is an analytic variety which is locally defined by a single holomorphic function f : U → C. Such varieties have complex codimension 1 in M.

Example 1.5.2. Any complex submanifold N (say of dimension n) of M (of dimension m) is an analytic subvariety of M, as N is locally defined by the vanishing of the last m − n coordinates in a complex chart.

From complex analysis, we know that if f : C → C is a holomorphic function, then it is analytic, meaning in an open disk D about z0 ∈ C, f is equal to its Taylor series expansion: 1 Z f(w) f(z) = dw. 2πi ∂D w − z

n In several complex variables, a similar argument says that if U ⊆ C is open then f(z1, . . . , zn): U → C is holomorphic if and only if f is locally of the form

X k1 kn f = ak1,...,kn (z1 − u1) ··· (zn − un) in a neighborhood of (u1, . . . , un) and this power series converges on that neighborhood. One useful consequence of this fact is that if f, g : U → C are holomorphic, U is connected and f and g agree on some open subset of U, then f ≡ g on all of U. (This is sometimes called the principle of analytic continuation.)

Definition. A Weierstrass polynomial on an open set U ⊆ Cn is a holomorphic function g : U → C of the form

d d−1 g(z1, . . . , zn−1, w) = w + g1(z1, . . . , zn−1)w + ... + gd(z1, . . . , zn−1)

n−1 for some holomorphic functions g1, . . . , gd : C → C. More generally, a Weierstrass polynomial on a complex manifold M is one that is locally given by a Weierstrass polynomial on the complex charts of M.

13 1.5 Analytic Varieties 1 Complex Manifolds

We have the following analogue of the fundamental theorem of calculus for complex functions of several variables.

Theorem 1.5.3 (Weierstrass Preparation). Let f be a holomorphic function on a neighbor- n n−1 hood of 0 in C = Cz × Cw, where z = (z1, . . . , zn−1), and suppose f(0,..., 0, w) 6≡ 0. Then on a (possibly smaller) neighborhood of 0, one can write f = gh for a Weierstrass polynomial g and a holomorphic function h such that h(0) 6= 0.

Example 1.5.4. Let V = V (f) be a hypersurface in Cn. After a coordinate change, we may assume f satisfies f(0,..., 0, w) 6≡ 0. Then by the Weierstrass preparation theorem, f = gh in a neighborhood of 0, for g a Weierstrass polynomial and h(0) 6= 0, so V = V (g). Thus we may always assume hypersurfaces are (locally) cut out by a Weierstrass polynomial. For each z = (z1, . . . , zn−1), the zero set of g(z, w) as a function of w is the collection of discrete roots of a polynomial, so other than where there are multiple roots, each of these roots are in fact holomorphic functions in the multivariable z. Thus the projection (z, w) 7→ z exhibits V as a branched cover of (an open set of) Cn−1 with branch locus precisely equal to

n−1 {z ∈ C | g(z, w) has a multiple root} = V (disc(g)) where disc(g) is the discriminant of g. Therefore the branch locus of a hypersurface is itself an analytic variety. For example, take n = 2 and g(z, w) = w2 − z. Then the projection V → C, (z, w) 7→ z2 gives the classic degree 2 branched cover of C with branch locus {0}.

Real coordinates of V = V (w2 − z)

Theorem 1.5.5 (Extension). Let D be a disk around 0 ∈ Cn and f : D → C be a holomor- phic function such that f(0) = 0. Suppose h : D r V (f) → C is holomorphic and bounded. Then h extends uniquely to a holomorphic function h˜ : D → C.

For a complex manifold M and a point P ∈ M, let OP = OM,P be the local ring of the sheaf of holomorphic functions at P (i.e. the germs of holomorphic functions defined at P ). Then the Extension Theorem implies that each OP is an integral domain. What’s more, OP is a UFD:

Proposition 1.5.6. For each P ∈ M, the local ring OM,P is a UFD.

Proof. (Sketch) Since the question is local, we may assume M = Cn and P = 0. We induct on n. For f ∈ OCn,0, we can use the Weierstrass preparation theorem (1.5.3) to write

14 1.6 Divisors, Line Bundles and Sections 1 Complex Manifolds

f = gh for some Weierstrass polynomial g ∈ OCn−1,0[w] and some holomorphic h ∈ OCn,0 × with h(0) 6= 0, i.e. h ∈ O n . By induction, O n−1 is a UFD and by Gauss’s Lemma, so C ,0 C ,0 is OCn−1,0[w], so g is a product of irreducibles uniquely up to a unit. Since h is a unit, this gives an irreducible factorization of f uniquely up to a unit as well.

Theorem 1.5.7 (Division Algorithm). Let f, g ∈ OM,P where g is a Weierstrass polynomial of degree k. Then f = gh+r for some h, r ∈ OM,P with r a Weierstrass polynomial of degree strictly less than k.

Proof. Follows a similar proof as for the polynomial division algorithm.

Corollary 1.5.8. If g ∈ OM,P is irreducible and h ∈ OM,P such that h ≡ 0 on V (g), then g | h.

These results show that for any P ∈ M, OM,P more or less behaves like a polynomial ring. This makes the connection to algebraic geometry, with its foundations based in the algebra of polynomial rings, even more explicit.

1.6 Divisors, Line Bundles and Sections

The notion of a divisor from algebraic geometry can also be defined in the context of analytic varieties. Consider the set of all hypersurfaces V ⊆ M, i.e. those analytic varieties defined locally by V (f) for some holomorphic function f. (The following definitions are the same for any analytic subvariety of M, but we will not need them here.)

Definition. For a hypersurface V ⊆ M, a point P ∈ V is called a smooth point (or nonsingular point) of V if on a neighborhood U ⊆ M of P intersecting V nontrivially, the holomorphic differential map 0 ∂P f : TP M −→ C

is nonzero. Otherwise, P is called a singular point of V . Write Vsm and Vsing for the set of smooth and singular points, respectively, of V .

Proposition 1.6.1. For any hypersurface V ⊆ M, Vsing is a proper analytic subvariety of V .

Proof. Locally, if V is defined by a holomorphic function f then Vsing is defined by the vanishing of the partial derivatives ∂f ,..., ∂f . ∂z1 ∂zn Definition. A hypersurface in M is irreducible if it cannot be written as the union of two hypersurfaces in M.

Proposition 1.6.2. Let V be a hypersurface in M. Then

(1) V is irreducible if and only if Vsm is connected.

(2) If V1,...,Vm are the connected components of Vsm, then their closures V 1,..., V m are the irreducible components of V .

15 1.6 Divisors, Line Bundles and Sections 1 Complex Manifolds

Corollary 1.6.3. Every hypersurface V ⊆ M has a unique decomposition

V = V1 ∪ · · · ∪ Vm

where Vj are irreducible hypersurfaces. Definition. A divisor on a complex manifold M is a locally finite linear combination of irreducible hypersurfaces in M, X D = nV V, nV ∈ Z. V ⊆M Let Div(M) be the abelian group of divisors on M under addition of coefficients: X X X mV V + nV V = (mV + nV )V. V ⊆M V ⊆M V ⊆M

A divisor D is called effective if nV ≥ 0 for all irreducible hypersurfaces V ⊆ M. Definition. Let g be a holomorphic function on a neighborhood U ⊆ M of a point P ∈ V . The order of vanishing of g at P is

k ordP (g) = max{k ≥ 0 | g = f h for h ∈ OM,P } where V (f) = V ∩ U.

Lemma 1.6.4. For a hypersurface V ⊆ M and g ∈ OM (U), the order of vanishing ordP (g) is independent of the point P ∈ V ∩ U, that is, P 7→ ordP (g) is locally constant.

As a result, the order of vanishing ordV (g) of g along any hypersurface V ⊆ M is well- defined. Example 1.6.5. Let Σ be a Riemann surface. Then a divisor on Σ is just a formal linear combination of points of Σ: X D = nP P. P ∈X Thus Div(Σ) is the free abelian group on the points of Σ. Example 1.6.6. The above situation is atypical, as many complex manifolds have no divisors at all. However, projective manifolds, i.e. those admitting complex embeddings M,→ CP n, contain many hypersurfaces. For example, if V ⊆ CP n is a hypersurface then M ∩ V is a hypersurface in M. Examples of such hypersurfaces in CP n can be found by taking the vanishing set of any homogeneous polynomial f ∈ C[z0, . . . , zn]. Recall that a meromorphic function on M is an almost-everywhere defined function f g which is locally of the form f = h where g and h are holomorphic and h is not identically the zero function. Such an f is specificed by an open cover {Uα} of M and holomorphic functions gα, hα ∈ OM (Uα) such that hα 6≡ 0 and on Uα ∩ Uβ,

gαhβ|Uα∩Uβ = gβhα|Uα∩Uβ .

By Theorem 1.5.7 we may choose gα, hα relatively prime in OM (Uα).

16 1.6 Divisors, Line Bundles and Sections 1 Complex Manifolds

Example 1.6.7. Let M = CP 2 and consider the meromorphic function f : CP 2 → C defined by 2 2 2 −z0 + z1 + z2 f(z0, z1, z2) = . z1z2 Then f defines a divisor (f):

2 2 2 (f) = V (−z0 + z1 + z2) − V (z1z2).

2 2 2 Here, V (−z0 + z1 + z2) is called the divisor of zeroes of f sometimes written (f)0, while V (z1z2) is called the divisor of poles and is denoted (f)∞. Further, in this example V (z1z2) 2 is reducible (it’s the union of the z0z2- and z0z1-planes in CP ) so we have

2 2 2 (f) = V (−z0 + z1 + z2) − V (z1) − V (z2).

Definition. The principal divisor of a meromorphic function f : M → C is the divisor X (f) = ordV (f)V V ⊆M

where ord (f) := ord (g ) − ord (h ) for a local expression f = gα on some open U ⊆ M V V α V α hα α with Uα ∩ V 6= ∅. Write X X (f)0 = ordV (gα) and (f)∞ = ordV (hα) V ⊆M V ⊆M

for the divisor of zeroes and divisor of poles of f, so that (f) = (f)0 − (f)∞. As in algebraic geometry, there is an important connection between divisors and line P bundles which we exhibit now for complex manifolds. Observe that any divisor D = nV V can be described locally as a principal divisor of a meromorphic function: if V is locally defined by V (gα,v) for each Uα, then on Uα, D can be represented by (fα) where

Y nV fα = gα,V . V ∩Uα6=∅

These local expressions must satisfy the following condition on Uα ∩ Uβ:

fα × (fα) = (fβ) ⇐⇒ ∈ OM (Uα ∩ Uβ) . fβ

× fα (Here, OM (U) denotes the nonvanishing holomorphic functions on U.) Set gαβ = . Notice fβ that if gαβ = 1 on all overlaps, then the fα glue together to define a global meromorphic function f such that D = (f). However, in general we have gαβgβγgγα = 1 on all triple × overlaps Uα ∩ Uβ ∩ Uγ. Then the maps gαβ : Uα ∩ Uβ → C are the transition maps of a line bundle on M, denoted OM (D) or just O(D) if the underlying complex manifold is clear. Definition. A holomorphic vector bundle on a complex manifold M is a holomorphic map π : E → M which is a complex vector bundle such that the transition functions are holomorphic.

17 1.6 Divisors, Line Bundles and Sections 1 Complex Manifolds

Lemma 1.6.8. For any divisor D ∈ Div(M), O(D) is a holomorphic line bundle.

Definition. The set of isomorphism classes of holomorphic line bundles on M, which is a group under ⊗, is called the Picard group of M, written Pic(M).

Theorem 1.6.9. For any complex manifold M, the assignment

Φ : Div(M) −→ Pic(M) D 7−→ O(D)

is a group homomorphism.

Proof. From the above description of O(D) using transition functions, it is clear that O(D1 + ∼ D2) = O(D1) ⊗ O(D2) for any divisors D1,D2 ∈ Div(M).

Definition. Two divisors D1,D2 ∈ Div(M) are called linearly equivalent, denoted D1 ∼ D2, if there exists a nonzero meromorphic function f on M such that D1 = D2 + (f). ∼ Lemma 1.6.10. Let D1,D2 ∈ Div(M). Then O(D1) = O(D2) if and only if D1 ∼ D2.

Proof. The line bundles O(D1) and O(D2) are isomorphic if and only if there is an open × 2 cover {Uα} of M and a collection of holomorphic functions hα ∈ O(Uα) such that gαβ ◦ 1 i hβ = hα ◦ gαβ on Uα ∩ Uβ, where gαβ are the transition functions defining O(Di), i = 1, 2. i i fα i By the above, gαβ = f i where fα are the meromorphic functions locally defining Di. So ∼ β O(D1) = O(D2) if and only if

2 2 fα fβ 1 = 1 on Uα ∩ Uβ. fα ◦ hα fβ ◦ hβ

n 2 o fα This latter condition says that the collection 1 defines a global meromorphic function fα◦hα f on M such that

X 2 1 D2 − D1 = (ordV (fα) − ordV (fα))V V ⊆M X f 2  = ord α V V f 1 V ⊆M α  2  X fα = ordV 1 V since hα is holomorphic f ◦ hα V ⊆M α = (f).

This completes the proof. Let M(M) denote the abelian group of meromorphic functions on M, under pointwise addition.

18 1.6 Divisors, Line Bundles and Sections 1 Complex Manifolds

Corollary 1.6.11. There is an exact sequence of abelian groups

M(M) → Div(M) −→Φ Pic(M).

Moreover, if M is compact then this extends to an exact sequence

× Φ 0 → C → M(M) → Div(M) −→ Pic(M)

× where C → M(M) is the map α 7→ cα, the constant function on α. Example 1.6.12. Let M = CP n and consider the hyperplane H = V (f) where f is a homogeneous linear polynomial. Notice that different choices of f give hyperplanes with isomorphic line bundles (since the ratio of any two homogeneous linear polynomials is a global meromorphic on CP n), so we might call L = O(H) the hyperplane line bundle on n n CP . For now, assume f = z0. Consider the open cover {Uj} of CP , where

n Uj = {[z0, . . . , zn] ∈ CP | zj 6= 0}.

On this cover, H is defined by a collection of functions fj : Uj → C given by ( 1, j = 0 fj([z0, . . . , zn]) = z0 , j 6= 0. zj These determine the following transition maps of O(H):

fj zk gjk = = on Uj ∩ Uk. fk zj

On the other hand, there is a distinguished line bundle E → CP n defined by

n n n+1 E = {(`, v) | ` ∈ CP , v ∈ `} ⊂ CP × C , called the tautological line bundle. Over the same open cover {Uj}, E is trivialized by

−1 ϕj : Uj −→ E|Uj    z0 zn ([z0, . . . , zn], λ) 7−→ [z0, . . . , zn], λ ,..., 1,..., zj zj  λ  = [z0, . . . , zn], (z0, . . . , zn) . zj Then the transition maps for E are:   −1 λ (ϕj ◦ ϕk )([z0, . . . , zn], λ) = ϕj [z0, . . . , zn], (z0, . . . , zn) zk   λzj = [z0, . . . , zn], zk

E zj i.e. g = , the inverse of the transition map on Uj ∩ Uk for O(H). This proves that jk zk E ∼= O(H)∗ = O(−H).

19 1.6 Divisors, Line Bundles and Sections 1 Complex Manifolds

Definition. For a holomorphic vector bundle π : E → M, a holomorphic section (resp. meromorphic section) of π is a holomorphic (resp. meromorphic) function s : M → E such that π ◦ s = idM . The group of all holomorphic (resp. meromorphic) sections of π is denoted Γh(M, π) or Γh(M,E) (resp. Γ(M, π) or Γ(M,E)).

Locally, a holomorphic section s ∈ Γh(M, π) is determined by sections sα : Uα → E|Uα such that sα = gαβsβ on each overlap Uα ∩ Uβ. P Example 1.6.13. Suppose D = nV V ∈ Div(M) is a divisor with local defining functions fα fα on Uα. By definition, the transition maps of O(D) are gαβ = on Uα ∩ Uβ so the fα glue fβ together to give a global section fD of O(D). If D is an effective divisor, fD is a holomorphic section of O(D), but in general fD need only be meromorphic. Conversely, if L → M is a line bundle and s : M → L is a nonvanishing meromorphic section, there is a natural divisor associated to the pair (L, s), namely X (s) = ordV (s)V. V ⊆M

Theorem 1.6.14. For any complex manifold M, there is a one-to-one correspondence

Div(M) ←→ {(L, s) | L ∈ Pic(M), s ∈ Γ(M,L) is nonvanishing}

D 7−→ (O(D), fD) (s) 7−→ (L, s).

This restricts to a one-to-one correspondence

eff Div (M) ←→ {(L, s) | L ∈ Pic(M), s ∈ Γh(M,L) is nonvanishing}

where Diveff(M) denotes the subset of effective divisors on M.

Corollary 1.6.15. The image of Φ : Div(M) → Pic(M) is precisely the isomorphism classes of line bundles on M admitting nonzero meromorphic sections.

For a divisor D ∈ Div(M), define a vector space

L(D) = {f ∈ M(M) | D + (f) is effective},

sometimes called the Riemann-Roch space for D.

Proposition 1.6.16. Let L = O(D) be the line bundle defined by a divisor D ∈ Div(M) and let sD be a meromorphic section of L defined by a set of local defining functions fα for D. Then there is a one-to-one correspondence

L(D) ←→ Γh(M,L)

f 7−→ fsD.

20 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Example 1.6.17. For M = CP n and L = O(H) the hyperplane bundle, this correspondence gives

n ∼ n Γ(CP , O(H)) = L(H) = {f ∈ M(CP ) | f has a pole of order ≤ 1 along H}.

Suppose H = V (z0). Then these functions all have the form

p(z0, . . . , zn) f(z0, . . . , zn) = z0

n+1 n ∼ n+1 where p is a homogeneous linear polynomial on C , so Γ(CP , O(H)) = C . Similarly, for any d ≥ 2, Γ(CP n, O(H)⊗d) = Γ(CP n, O(dH)) can be identified with the space of homogeneous polynomials p(z0, . . . , zn) of degree d, so

n ∼ d n+1 ∗ Γ(CP , O(dH)) = Sym (C ) .

1.7 Cohomology and Chern Classes

p Recall that for a space X, the Cechˇ cohomology groups Hˇ (U,F ) for a sheaf F on X, with respect to an open cover U = {Ui}, are defined as the cohomology of the complex

p Y C (U,F ) := F (Ui0,...,ip )

i0,...,ip

with differential

d : Cp(U,F ) −→ Cp+1(U,F ) p+1 ! X k α 7−→ (−1) αi0,...,ik−1,ik+1,...,ip+1 |Ui0,...,ip+1 k=0

where Ui0,...,ip denotes the intersection Ui0 ∩ · · · ∩ Uip . For a refinement V of U, there is a map Cp(V,F ) → Cp(U,F ) defined by restriction which induces a map on cohomology: p p Hˇ (V,F ) → Hˇ (U,F ). Write

p p Hˇ (X,F ) := lim Hˇ (U,F ). −→ In addition, there is always a homomorphism

p Hˇ (U,F ) −→ Hp(X,F ) where Hp(X, −) denotes the (derived functor) sheaf cohomology of X.

0 Example 1.7.1. For any sheaf F on X and any open cover U of X, Hˇ (U,F ) = F (X) = 1 H0(X,F ). In addition, Hˇ (X,F ) → H1(X,F ) is always an isomorphism.

More generally:

21 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Theorem 1.7.2 (Leray). If F is a sheaf on X which is acyclic for a cover U = {Ui}, i.e. Hp(U ,F | ) = 0 for all p > 0 and i , . . . , i , then the map i0,...,ip Ui0,...,ip 0 p p Hˇ (U,F ) −→ Hp(X,F ) is an isomorphism for all p ≥ 0.

Example 1.7.3. If A is an abelian group and AX denotes the constant sheaf on X with ˇ p ∼ p ∼ p p coefficients in A, then H (U,AX ) = H (X,AX ) = H (X; A), where H (X; −) denotes singular cohomology with coefficients. In the future we will just denote this constant sheaf by A. ˇ 0 ∼ Example 1.7.4. If M is a connected complex manifold, H (M, OM ) = OM (M) = C. Example 1.7.5. If M is a complex manifold, for each open U ⊆ M let M(U) be the vector space of meromorphic functions on U. Then U 7→ M(U) defines a sheaf on M and H0(M, M) = M(M) by the above. Moreover, there is an injective morphism of sheaves × OM ,→ M and we have ∼ 0 × ∼ 1 × Div(M) = H (M, M/O ) and Pic(M) = H (M, OM ). ∼ 1 × Explicitly, the isomorphism Pic(M) = H (M, OM ) is given by L 7→ [(gαβ)] where (gαβ) is the 1-cocycle given by the transition functions gαβ defining L (see Section 1.6). Recall from Corollary 1.6.11 that for any complex manifold M, there is an exact sequence M(M) → Div(M) → Pic(M). This in fact coincides with the long exact sequence in sheaf cohomology coming from the short exact sequence of sheaves

× × 0 → OM → M → M/OM → 0. Theorem 1.7.6 (De Rham). For any manifold M, there is an isomorphism • ∼ • HdR(M) = H (M, R) between the de Rham cohomology and constant sheaf cohomology of M. As a result, the singular cohomology of any real manifold can be computed by de Rham cohomology. Dolbeault’s theorem is an analogous statement for complex manifolds. Theorem 1.7.7 (Dolbeault). For any complex manifold M and every p, q ≥ 0, there is an isomorphism Hp,q(M) ∼= Hq(M, Ωp). Consider the short exact sequence of sheaves

exp × 0 → Z −→OM −−→OM → 0 where Z is the constant sheaf and exp denotes the exponential map f 7→ e2πif . This induces a long exact sequence in sheaf cohomology:

1 1 × δ 2 · · · → H (M, OM ) → H (M, OM ) −→ H (M, Z) → · · · 1 × Again note that H (M, OM ) = Pic(M).

22 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Definition. The (first) Chern class of a holomorphic line bundle L on M is the cohomology 2 1 × class c1(L) := δ([L]) ∈ H (M, Z), where [L] denotes the class in H (M, OM ) corresponding to L.

The fact that the connecting map δ in a long exact sequence is a homomorphism implies the following basic facts about Chern classes.

Lemma 1.7.8. Let L be a holomorphic line bundle on a complex manifold M. Then

(a) If L is trivial, then c1(L) = 0.

0 0 0 (b) For any other holomorphic line bundle L on M, c1(L ⊗ L ) = c1(L) + c1(L ). In ∗ particular, c1(L ) = −c1(L).

∗ ∗ (c) For any holomorphic map f : N → M between complex manifolds, c1(f L) = f c1(L). The same arguments define the first Chern class of any smooth (i.e. topological) line bundle on M. Explicitly, replacing OM with the sheaf A of smooth functions on M, there is also a short exact sequence of sheaves

exp × 0 → Z −→A −−→A → 0. The corresponding long exact sequence is

1 1 × δ 2 · · · → H (M, A) → H (M, A ) −→ H (M, Z) → · · · Then H1(M, A×) may be identified with the set of isomorphism classes of smooth line bundles 2 on M and for any such bundle L, we take c1(L) to be the class δ([L]) ∈ H (M, Z).

Proposition 1.7.9. For any holomorphic line bundle L → M, c1(L) only depends on the underlying smooth line bundle of L and this is classified by its first Chern class.

Proof. The morphism of sheaves OM → A fits into a commutative diagram

exp × 0 Z OM OM 0

id exp 0 Z A A× 0 Taking cohomology yields another commutative diagram

1 × δ 2 H (M, OM ) H (M, Z)

id ∼ 0 = H1(M, A) H1(M, A×) H2(M, Z) H2(M, A)

23 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Then since A is acyclic, H1(M, A) = H2(M, A) = 0 so the connecting homomorphism in the bottom row is an isomorphism. Both statements follow immediately.

Example 1.7.10. Let Σ = Σg be a Riemann surface of genus g with structure sheaf O = OΣ. 1 ∼ 0,1 ∼ g 2 ∼ By Dolbeault’s theorem and Example 1.4.12, H (Σ, O) = H (Σ) = C and H (Σ, O) = 0,2 1 × 1 ∼ 2g 2 ∼ H (Σ) = 0. Moreover, H (Σ, O ) = Pic(Σ), H (Σ, Z) = Z and H (Σ, Z) = Z since Σ is a real 2-manifold, so the long exact sequence coming from the exponential sequence on Σ becomes 2g g 0 → Z → C → Pic(Σ) → Z → 0. ∼ g 2g 1 1 ∼ g 2g Hence Pic(Σ) = Z × (C /Z ). The second factor, H (Σ, O)/H (Σ, Z) = C /Z , is called the Jacobian of Σ, written J(Σ). Topologically, J(Σ) is homeomorphic to a torus of genus g. For a line bundle L over Σ, pairing c1(L) with the fundamental class [Σ] ∈ H2(Σ; Z) defines an integer called the degree of L:

deg(L) = hc1(L), [Σ]i. P Definition. Let Σ be a Riemann surface and D = nP P a divisor on Σ. The degree of P D is the integer deg(D) = nP . Lemma 1.7.11. For any divisor D ∈ Div(Σ), deg(D) = deg(O(D)). Proposition 1.7.12. If L ∈ Pic(Σ) such that deg(L) < 0 then L admits no nonzero holo- morphic sections. Proof. We saw in Example 1.6.13 that if L is a line bundle over Σ which admits a holomorphic section s, then L = O(D) where D = (s) is the effective divisor defined by s. Hence deg(L) = deg(s) ≥ 0 since s is holomorphic. Let M be a complex manifold and recall from Proposition 1.6.16 that there is an iso- ∼ 0 morphism L(D) = H (M, O(D)) given by multiplication by sD, where sD is a meromorphic section of L defined by a collection of local defining functions for D. More generally, let E → M be a vector bundle and write E(D) = {meromorphic sections s : M → E | D + (s) ≥ 0}. Then the same argument proves: Proposition 1.7.13. If D ∈ Div(M) is a divisor and E → M is a vector bundle, then there is an isomorphism E(D) −→∼ H0(M,E ⊗ O(D))

f 7−→ f ⊗ sD. Definition. The holomorphic Euler characteristic of a vector bundle E → M is the integer ∞ X i i χh(M,E) = (−1) h (E) i=0 where hi(E) := dim Hi(M,E). The holomorphic Euler characteristic of a complex manifold M is the holomorphic Euler characteristic of its structure sheaf,

χh(M) = χh(M, OM ).

24 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Proposition 1.7.14. For any complex manifold M,

∞ X i 0,i χh(M) = (−1) h (M) i=0 where hp,q(M) = dim Hp,q(M). Proof. Apply Dolbeault’s theorem. Suppose D = V is a smooth, irreducible hypersurface in M and consider the restriction map E → E|V . Lemma 1.7.15. For an irreducible hypersurface V ⊆ M and any vector bundle E → M, the map E → E|V is a surjective morphism of sheaves fitting into a short exact sequence

0 → E ⊗ O(−V ) → E → E|V → 0. Therefore there is a long exact sequence

0 0 0 1 0 → H (M,E ⊗ O(−V )) → H (M,E) → H (V,E|V ) → H (M,E ⊗ O(−V )) → · · · Corollary 1.7.16. If V ⊆ M is an irreducible hypersurface, then

χh(M,E) = χh(M,E ⊗ O(−V )) + χh(V,E|V ).

Corollary 1.7.17. Let Σ = Σg be a Riemann surface of genus g and L → Σ a line bundle. Then for any point P ∈ Σ, χh(Σ,L) = χh(Σ,L ⊗ O(−P )) + 1. 0 Proof. For any P ∈ Σ, χh(P,L|P ) = h (L|P ) is the dimension of the fibre of L over P , which is 1.

Corollary 1.7.18. For any Riemann surface Σ of genus g, χh(Σ) = 1 − g. 0,0 0,1 Proof. Applying Example 1.4.12, we get χh(Σ) = h (Σ) − h (Σ) = 1 − g. Corollary 1.7.19 (Riemann-Roch Theorem). Suppose Σ is a Riemann surface of genus g and D is a divisor on Σ. Then

χh(Σ, O(D)) = deg(D) − g + 1.

Alternatively, this may be written χh(Σ,L) = deg(L) − g + 1 for any line bundle L ∈ Pic(Σ). We might want to know more about the dimensions h0(L) and h1(L) for a given line bundle, but it turns out that these are not determined solely by the topology of L → Σ. When M is compact, there is a deep result in sheaf theory which tells us that h1(L) may be computed as h0 of a different line bundle. Theorem 1.7.20 (Serre Duality). Suppose M is a compact complex manifold of dimension n and E → M is a vector bunde. Then there is an isomorphism of C-vector spaces p ∼ n−p ∗ ∗ H (M,E) = H (M,E ⊗ O(KM ))

∗ where (−) denotes vector space dual and KM is a distinguished divisor on M which is defined up to linear equivalence.

25 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Here’s a sketch of the proof; along the way we will define KM ∈ Div(M). For any vector bundle E → M, define the vector spaces  ^   ^  Ωk(M,E) := Γ M, kT ∗M ⊗ E and Ωp,q(M,E) := Γ M, p,qT ∗M ⊗ E .

Then we have a decomposition M Ωk(M,E) ∼= Ωp,q(M,E). p+q=k

An element of Ωp,q(M,E) can be thought of as a ‘vector-valued differential form of type (p, q), with values in E’. Equivalently, sections of Vk T C,∗M ⊗ E may be identified with alternating multilinear maps TM ×TM → E and such maps have a natural type decomposition agreeing with the above. Locally, ω ∈ Ωp,q(M,E) is of the form

X I J ω = ωIJ dz ∧ dz¯ ⊗ σ |I|=p,|J|=q

for some σ ∈ Γ(M,E). We define a ‘star operator’ ? :Ωp,q(M,E) → Ωn−p,n−q(M,E∗)∗ as follows. Assume E and M are endowed with Hermitian metrics which are compatible with the map E → M. When ∼ r E = C is a trivial bundle, ? is uniquely defined by the formula

α ∧ ?β = hα, βi · d volM

p,q for any α, β ∈ Ω (M) and volM a Hermitian volume form on M. Locally, given a Hermitian 1,0 0,1 n-frame ε1, . . . , εn for T M with dual basisε ¯1,..., ε¯n for T M, the operator ? acts by

?(ε ∧ · · · ∧ ε ∧ ε¯ ∧ · · · ∧ ε¯ ) = ±ε 0 ∧ · · · ∧ ε 0 ∧ ε¯ ∧ · · · ∧ ε¯ 0 i1 ip j1 jq i1 in−p j1 jn−q

0 0 0 0 where {i1, . . . , ip, i1, . . . , in−q} = {1, . . . , n}, {j1, . . . , jq, j1, . . . , jn−q} = {1, . . . , n} and ± is the product of signs of these permutations of (1, . . . , n). For example, when n = 3,

?(ε1 ∧ ε2) = ε3 ∧ ε¯1 ∧ ε¯2 ∧ ε¯3,?(ε2 ∧ ε¯1) = −ε1 ∧ ε3 ∧ ε¯2 ∧ ε¯3, and so on.

For nontrivial E, this local definition extends to the whole bundle using the Hermitian metric on E. Now by the theory of harmonic functions (see Section 1.4 and the proof of Hodge de- composition), we may identify Hp,q(M,E) with the space Hp,q(M,E) of harmonic functions p,q E with values in E. Explicitly, α ∈ H (M,E) if and only if ∆∂¯ α = 0, where E ¯¯∗ ¯∗ ¯ • • ∆∂¯ = ∂∂ + ∂ ∂ :Ω (M,E) −→ Ω (M,E)

• as in the proof of Hodge decomposition. Let ∆∂¯ be the Laplace operator on Ω (M).

E Lemma 1.7.21. For any vector bundle E → M, ∆∂¯ ? = ?∆∂¯. Therefore ? descends to an operator ? : Hp,q(M,E) → Hn−p,n−q(M,E∗)∗.

26 1.7 Cohomology and Chern Classes 1 Complex Manifolds

Lemma 1.7.22. For all p, q ≥ 0, ? : Hp,q(M,E) → Hp,q(M,E∗) is an isomorphism.

This gives us a string of isomorphisms

Hp(M,E) ∼= H0,p(M,E) by Dolbeault’s theorem ∼= H0,p(M,E) by the proof of Hodge decomposition ∼= Hn,n−p(M,E∗)∗ by the Lemma  ^ ∗ ∼= Hn,n−p(M,E∗)∗ ∼= Hn−p M, nT ∗M ⊗ E∗ .

It remains to interpret VnT ∗M ⊗ E∗. Note that the bundle VnT ∗M is a line bundle, so it is isomorphic to O(KM ) for some divisor KM . This completes the sketch of Serre duality.

Definition. For a compact complex manifold M of dimension n, any divisor KM such that Vn ∗ ∼ T M = O(KM ) is called a canonical divisor on M. Example 1.7.23. When Σ is a Riemann surface and L → Σ is a line bundle, Serre duality says that 1 ∼ 0 ∗ H (Σ,L) = H (Σ,L ⊗ O(KΣ)). Corollary 1.7.24 (Riemann-Roch Theorem, Second Version). Suppose Σ is a Riemann surface of genus g and L is a line bundle on Σ. Then

0 0 ∗ h (L) − h (L ⊗ O(KΣ)) = deg(L) − g + 1.

Alternatively, if D is a divisor on Σ then

0 0 h (D) − h (KΣ − D) = deg(D) − g + 1.

Corollary 1.7.25. If KΣ is a canonical divisor on a Riemann surface Σ of genus g, then deg(KΣ) = 2g − 2.

Proof. Apply Riemann-Roch with D = KΣ.

Note that deg(KΣ) = 2g − 2 is precisely −χ(Σ) where χ denotes the topological Euler characteristic. To see this directly, let e(Σ) = e(T Σ) be the Euler class of Σ and observe ∗ that O(KΣ) = T Σ, so we have

∗ −χ(Σ) = −he(T Σ), [Σ]i = h−c1(T Σ), [Σ]i = hc1(T Σ), [Σ]i = hc1(KΣ), [Σ]i = deg(KΣ).

Corollary 1.7.26. If D is a divisor on a Riemann surface Σ of genus g and deg(D) > 2g−2, then h0(D) = deg(D) − g + 1. Likewise, if L is a line bundle on Σ with deg(L) > 2g − 2 then h0(Σ,L) = deg(L) − g + 1.

Proof. If deg(D) > 2g − 2 then deg(KΣ − D) = 2g − 2 − deg(D) < 0 so Proposition 1.7.12 0 shows that h (KΣ − D) = 0.

27 1.7 Cohomology and Chern Classes 1 Complex Manifolds

This shows that if L is a line bundle on a Riemann surface of genus g (the same statement will hold for divisors) such that deg(L) < 0 or deg(L) > 2g − 2, then h0(L) is topologically determined. However for line bundles with 0 ≤ deg(L) ≤ 2g − 2, this is not the case in general. What are holomorphic sections good for?, you might ask. Well, suppose L is a holomor- phic line bundle on a complex manifold M and dim H0(M,L) = N + 1 for some N ≥ 0. Let 0 {s0, . . . , sN } be a basis for H (M,L). Then there is a point x ∈ M for which not all sj(x) are zero. In a local trivialization of L near x, the tuple (s0(x), . . . , sN (x)) defines a point in N+1 C r {0} and on a different trivialization, this tuple is well-defined up to gαβ(x) where N gαβ is a transition function defining L. In other words, the point [s0(x), . . . , sN (x)] ∈ CP is well-defined. Definition. A complete linear system for a line bundle L → M is the collection |L| of effective divisors corresponding to H0(M,L), i.e. if L ∼= O(D) for a divisor D ∈ Div(M), then 0 0 ∼ 0 ∼ |L| = {D ∈ Div(M) | D ∼ D} = P(H (M,L)) = P(L(D)). A linear system for L is a collection |W | of effective divisors corresponding to a linear sub- space W ⊆ H0(M,L). The dimension of a linear system is the dimension of the projective subspace of P(L(D)) to which it corresponds. In particular, note that dim |W | = dim W − 1. We call a linear system of dimension 1, i.e. when h0(L) = 2, a pencil (of divisors) for L. Definition. The base locus of a linear system |W | for L → M is the analytic subvariety

B|W | = {x ∈ M | s(x) = 0 for all s ∈ W } ⊆ M.

If B|W | = ∅, we say |W | is basepoint free. 0 The important point here is that if B|L| = ∅, then choosing a basis for H (M,L) produces N a well-defined holomorphic map ϕ|L| : M → CP . Moreover, a different choice of basis changes ϕ|L| by a projective transformation. Let’s consider the case of a Riemann surface Σ of genus g. Let L ∈ Pic(Σ) be a line N bundle. Then a natural question to ask is: when does ϕ|L| give an embedding Σ ,→ CP ? 0 First note that B|L| = ∅ if and only if for all P ∈ Σ, there is a section s ∈ H (Σ,L) such 0 0 that s(P ) 6= 0. Equivalently, we are asking if the natural map H (Σ,L) → H (P,L|P ) is surjective for all P . Consider the short exact sequence of sheaves

0 → IP → L → L|P → 0.

Here, IP is the sheaf of sections of L which vanish at P . Explicitly, IP = L ⊗ O(−P ) and the long exact sequence in cohomology from this reads:

0 0 1 H (Σ,L) → H (P,L|P ) → H (Σ,L ⊗ O(−P )). 1 ∼ 0 ∗ ∗ By Serre duality, H (Σ,L ⊗ O(−P )) = H (Σ,L ⊗ O(KΣ − P )) , so surjectivity of the 0 ∗ restriction map is equivalent to having H (Σ,L ⊗ O(KΣ − P )) = 0. If deg(L) > 2g − 1, then ∗ deg(L ⊗ O(KΣ − P )) = − deg(L) + (2g − 2) − 1 < 0

28 1.7 Cohomology and Chern Classes 1 Complex Manifolds

and this cohomology group will vanish by Proposition 1.7.12. Therefore if deg(L) > 2g − 1, N |L| is basepoint free and ϕ|L| defines a holomorphic map Σ → CP . The next question is when this map is an embedding. This happens precisely when, for all P 6= Q in Σ, the N+1 vectors (s0(P ), . . . , sN (P )) and (s0(Q), . . . , sN (Q)) are linearly independent in C . Let 0 0 AP := H (Σ,IP ) ⊂ H (Σ,L) be the subspace of sections of L vanishing at P . Note that AP has codimension 1 in H0(Σ,L). Then P 6= Q map to distinct points in CP N if and only if there is a section s ∈ AP such that s(Q) 6= 0. Consider the sequence of sheaves

0 → IP,Q → L → L|{P,Q} → 0

where IP,Q now denotes the sheaf of sections of L vanishing at P and Q simultaneously. 0 ∼ 0 0 Also note that H ({P,Q},L|{P,Q}) = H (P,L|P ) ⊕ H (Q, L|Q) = L(P ) ⊕ L(Q). Then ϕ|L| is one-to-one if and only if H0(Σ,L) → L(P ) ⊕ L(Q) is onto and to show this condition, it 1 suffices to prove H (Σ,IP,Q) = 0. By Serre duality,

1 1 ∼ 0 ∗ H (Σ,IP,Q) = H (Σ,L ⊗ O(−(P + Q))) = H (Σ,L ⊗ O(KΣ + P + Q)).

Thus we see that if deg(L) > 2g, deg(KΣ +P +Q)−deg(L) = 2g −deg(L) < 0 which implies 1 H (Σ,IP,Q) = 0. N Finally, we might ask when ϕ|L| is an immersion, i.e. when dP ϕ|L| : TP Σ → Tϕ|L|(P )CP ∗ ∗ N is an injective linear map for all P ∈ Σ. Dualizing, this is the same as dP ϕ : T P → |L| ϕ|L|(P )C ∗ 0 TP Σ being surjective. For a given P ∈ Σ, we may choose a basis {s0, . . . , sN } for H (Σ,L) with s0(P ) 6= 0 and s1(P ) = ··· = sN (P ) = 0. Then in a neighborhood of P , ϕ|L| is given by N N sj ϕ|L|(Q) = (t1(Q), . . . , tN (Q)) ∈ C ⊂ CP where tj = . s0 ∗
Then the differential is given by (dtj) so the adjoint is the row vector dP ϕ|L| = dt1 ··· dtN . ∗ ∗ 0 Thus dP ϕ|L| is surjective precisely when for any α ∈ TP Σ, there exist a section s ∈ H (Σ,IP ) such that ds = α. Consider the exact sequence of sheaves

d ∗ 0 → L ⊗ O(−2P ) −→ L −→ (T Σ ⊗ L)|P → 0

where d sends a section s to ds. In the long exact sequence, we have

0 0 ∗ 1 · · · → H (Σ,L) → H (P, (T Σ ⊗ L)|P ) → H (Σ,L ⊗ O(−2P )) → · · ·

so we are interested in when H1(Σ,L ⊗ O(−2P )) = 0. Again by Serre’s duality and a dimension argument, this vanishing occurs when deg(L) > 2g. Therefore we have proven:

Theorem 1.7.27. Every compact Riemann surface Σ admits a holomorphic embedding Σ ,→ CP N for some N ≥ 1. Proof. Let g be the genus of Σ and take L → Σ to be a line bundle with degree deg(L) > 2g. Then the above argument shows that the complete linear system |L| is basepoint free and N 0 hence ϕ|L| :Σ → CP is defined, where h (L) = N + 1. Moreover, since deg(L) > 2g, the above also shwos ϕ|L| is an embedding.

29 1.8 Connections on Vector Bundles 1 Complex Manifolds

Remark. It is a deep theorem that every analytic submanifold of CP N is algebraic, so as a consequence we see that every Riemann surface may be realized as a complex subvariety of some projective space.

We have seen that for a line bundle L on a Riemann surface Σ, the conditions that ϕ|L| is well-defined and an embedding are related to the vanishing of H1(Σ, −) for various line bundles built from L. Since dim Σ = 1, there is a nice topological criterion for the vanishing of such cohomology groups: the degree of those line bundles need only be sufficiently positive (this is essentially because irreducible divisors are merely points). This nice situation is special to the dimension 1 case. If M is a complex manifold of dimension greater than 1, there still exists a criterion for H1(M,L) to vanish for a particular line bundle L → M, but it is geometric and not merely topological in nature. To introduce this criterion, called positivity of a line bundle, we first discuss the theory of connections on topological vector bundles.

1.8 Connections on Vector Bundles

30