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Home , Chern class, Line bundle

Topic: First Chern classes of K¨ahler Mitchell Faulk Last updated: April 23, 2016

We study the first of various K¨ahlermanifolds. We only consider two sources of examples: Riemann surfaces and projective submanifolds. We conclude by establishing a relationship between the negativity of a and the non-existence of holomorphic sections.

Contents

1 K¨ahlermanifolds 1

2 Holomorphic line bundles 4

3 Positive line bundles 7

4 First Chern class 9

5 11

6 Riemann surfaces 12

7 Subvarieties of 14

8 Connections on vector bundles 17

9 Curvature 21

10 First Chern class and sections 22

1 K¨ahlermanifolds

We follow somewhat closely the notation of [GH]. Throughout, M denotes a complex of complex dimension n. This means in particular, there is n an open cover Uα of M together with continuous maps ϕα : Uα → C mapping Uα homeomorphically onto an open subset of Cn in such a way that the transition maps

−1 ϕα ◦ ϕβ : ϕβ(Uα ∩ Uβ) → ϕα(Uα ∩ Uβ) are biholomorphisms between open subsets of Cn.

1 Example 1.1. For example, the space Cn is a of dimension n with the usual global coordinates.

n n+1 Example 1.2. Let CP denote the set of complex lines through the origin in C . For a point (Z0,...,Zn) n+1 n in C \{0}, we let [Z0,...,Zn] denote the line it determines. If Uj denotes the subset of CP given by

Uj = {[Z0,...,Zn]: Zj 6= 0}

then we have complex coordinates on Uj given by

n ϕj : Uj → C ! Z0 Zcj Zn [Z0,...,Zn] 7→ ,..., ,..., . Zj Zj Zj

Example 1.3. Suppose that f1, . . . , fk ∈ C[x0, . . . , xn] are homogeneous polynomials and V is the well- n defined subset of CP given by

V = {[Z] = [Z0,...,Zn]: f(Z) = 0}.

n If V enjoys the structure of a smooth submanifold of CP , then V is a complex submanifold by the implicit function theorem [GH]. We call such a complex submanifold a projective submanifold.

Example 1.4. If M is a complex manifold, then there is an endomorphism J : TM → TM of the of M defined in the following way. If (z1, . . . , zn) are local complex coordinates on M, write zj = √ xj + −1yj to obtain local real coordinates on M. Then let J be the unique linear map satisfying

 ∂  ∂  ∂  ∂ J = J = − . ∂xj ∂yj ∂yj ∂xj

It follows that J does not depend on the choice of local coordinates, and hence defines a global section of End(TM). Moreover, it is apparent that J 2 = −id. Thus, if we extend J to an endomorphism of the complexified tangent bundle TMC = TM ⊗ C, then we obtain a decomposition

1,0 0,1 TMC = T M ⊕ T M where T 1,0M is the +i eigenspace for J and T 0,1M is the −i eigenspace for J. We call T 1,0M the holo- morphic tangent bundle, and it enjoys the structure of a complex manifold with local coordinates given by {zj, ∂/∂zj} where ∂ 1  ∂ √ ∂  = − −1 . ∂zj 2 ∂xj ∂yj

2 1 ∗ ∗ The complexified Ω M = T MC = T M ⊗ C decomposes similarly into

Ω1M = Ω1,0M ⊕ Ω0,1M

where Ω1,0M = (T 0,1M)⊥ and Ω0,1M = (T 1,0M)⊥. We call Ω1,0M the holomorphic cotangent bundle, and it enjoys the structure of a complex manifold with local coordinates {zj, dzj} where

√ dzj = dxj + −1dyj.

Let g be a Riemannian metric on M. Suppose that g is hermitian in the sense that g(JX,JY ) = g(X,Y )

for each pair of vector fields X,Y on M. Extend g to a metric on the complexified tangent bundle TMC. One can readily check that g is completely determined by the functions

 ∂ ∂  g¯ = g , kj ∂z¯k ∂zj and in fact a local expression for g is j k g = gkj¯ dz ⊗ dz¯ .

Define a (1, 1)-form ω by the rule ω(X,Y ) = g(JX,Y ).

Then ω is compatible with J in the sense that ω(JX,JY ) = ω(X,Y ) and a local expression for ω is given by √ j k ω = −1gkj¯ dz ∧ dz¯ .

Definition 1.5. We say a hermitian metric g on M is K¨ahler if the corresponding (1, 1)-form ω is closed, that is, dω = 0.

Note 1.6. A closed (1, 1)-form ω can determine a K¨ahlermetric in the following way. Suppose that ω is compatible with J in the sense that ω(JX,JY ) = ω(X,Y ). If the (0, 2)-tensor defined by

g(X,Y ) = ω(X,JY )

is symmetric and positive definite, then the metric g is K¨ahler. √ n P j j Example 1.7. The complex manifold C with the metric determined by ω = −1 j dz ∧ dz¯ is K¨ahler.

n Example 1.8. We describe a (1, 1)-form on CP called the Fubini-Study metric ωFS. If U is an open n n subset of CP and Z : U → Cn+1 \0 is a local holomorphic section of the projection map π : Cn+1 \0 → CP ,

3 we let ωFS be the (1, 1)-form with local expression

√ −1∂∂¯log kZk2 .

This does not depend on the choice of holomorphic section Z because if Z0 is another such section, then

Z0 = fZ for some local holomorphic C-valued function f, and we find that

√ √ √ −1∂∂¯log kZ0k2 = −1∂∂¯log kZk2 + −1∂∂¯log kfk2 ,

√ ¯ 2 but the last expression −1∂∂ log kfk is zero since f is holomorphic. It follows that ωFS exists globally. n Now the group U(n + 1) acts transitively on CP and preserves the form ωFS. Thus to check whether the corresponding symmetric bilinear form is positive definite, it suffices to check at one point, such as

[1, 0,..., 0]. In the coordinate chart U0, the form ωFS has local expression

√ ¯ 2 ωFS = −1∂∂ log(1 + |z| )

√ P j j where z = (z1, . . . , zn) are the coordinates on U0. At the point z = 0, this is −1 j dz ∧ dz¯ . Hence the corresponding hermitian matrix at this point is the identity matrix, which is positive definite.

Example 1.9. If V is a complex submanifold of a K¨ahlermanifold (M, ω), then the induced metric on V is K¨ahleras well, since the differential d commutes with the pullback i∗ corresponding to the inclusion i : V → M.

2 Holomorphic line bundles

Definition 2.1. Let M be a complex manifold. A holomorphic line bundle over M consists of a family

{Lp}p∈M of one-dimensional complex vector spaces together with the structure of a complex manifold on the disjoint union L = ∪p∈M Lp in such a way that the natural projection map π : L → M is holomorphic and such that for each point p ∈ M, there is an open neighborhood U of p in M and a biholomorphism −1 ϕ : π (U) → U × C which is linear on each fiber Lp for p ∈ U and which commutes with the projection maps π = proj1 ◦ ϕ, where proj1 : U × C → U is the projection onto the first factor.

Definition 2.2. A smooth section s : M → L is called holomorphic if s is holomorphic as a map of complex manifolds. We let H0(M,L) denote the space of holomorphic sections.

Note 2.3. If L → M is a holomorphic line bundle, then there is an open cover Uα of M together with −1 biholomorphisms ϕα : π (Uα) → Uα × C which are linear on the fibers and which commute with the pro- jection maps. Thus, on the intersections Uα ∩Uβ, we obtain holomorphic maps called transition functions

4 ∗ gαβ : Uα ∩ Uβ → GL(1, C) = C satisfying

−1 ϕβ ◦ ϕα (p, v) = (p, gβα(p)v)

for p ∈ Uα ∩ Uβ and v ∈ C. The collection {gαβ} satisfies the cocycle relations

 gαβgβγ = gαγ

gαα = 1,

and hence determines an element of the Cechˇ group H1(M, O∗). Conversely, it can be shown

that any 1-cocyle {gαβ} determines a holomorphic line bundle, and two such cocycles determine isomorphic line bundles if and only if they differ by an exact 1-cycle. Thus, we obtain an identification of the group of isomorphism classes of holomorphic line bundles with H1(M, O∗).

Definition 2.4. If L is a holomorphic line bundle over M, then there is a holomorphic line bundle L−1 over −1 M which is dual to L in the sense that each fiber Lp is the dual to Lp. If gαβ are transition −1 −1 functions for L, then transition functions for L are given by gαβ .

Note 2.5. Suppose L has transition functions gαβ with respect to an open cover Uα of M with trivializations

ϕα. If s is a holomorphic section, then the function ϕα ◦ s : Uα → Uα × C can be expressed as ϕα ◦ s(p) =

(p, sα(p)) for some sα : Uα → C. Moreover, from the definitions of the transitions

functions gαβ, we find that sβ = gβαsα on the overlap Uα ∩Uβ. Conversely, given a collection of holomorphic

functions sα : Uα → C satisfying sβ = gβαsα on the overlaps, we obtain a holomorphic section of L.

n Example 2.6. We describe a line bundle O(−1) on CP in the following manner: let O(−1) be the sub- n bundle of the trivial bundle CP × Cn+1 whose fiber over the point determined by the line ` is the line ` n itself. If Uj denotes the chart on CP given by

Uj = {[Z0,...,Zn]: Zj 6= 0},

then it can be shown that transition functions for the line bundle O(−1) relative to these charts are

Zj gjk([Z0,...,Zn]) = . Zk

We let O(1) denote the dual to O(−1). Moreover, by taking tensor powers, we obtain holomorphic line bundles O(`) for each ` in Z. One can show that the transition functions for O(`) satisfy

 ` (`) Zk gjk ([Z0,...,Zn]) = . Zj

n It can be shown that every holomorphic line bundle on CP is of the form O(`) for some `, but a proof of

5 this is beyond the scope of these notes. n The following characterization of the space H0(CP , O(`)) of global sections is not too difficult to obtain.

n • For ` < 0, we have H0(CP , O(`)) = 0.

0 n • For ` > 0, the space H (CP , O(`)) can be identified with the space of homogeneous polynomials of

degree ` in the n + 1 variables Z0,...,Zn.

Indeed, let us consider the case when n = 1 and ` = 1. In this case, a global section of O(1) consists of two

holomorphic functions s0 : U0 → C and s1 : U1 → C satisfying

  Z0 s1 = s0. Z1

A holomorphic function s1 : U1 → C has a expansion

 2 Z0 Z0 s1 = a0 + a1 + a2 + ··· . Z1 Z1

We can also write  2 Z1 Z1 s0 = b0 + b1 + b2 + ··· . Z0 Z0

The relation between s0 and s1 implies that

 2  2 Z0 Z1 Z1 Z0 Z0 b0 + b1 + b2 + b3 + ··· = a0 + a1 + a2 + ··· . Z1 Z0 Z0 Z1 Z1

We thus find that b0 = a1 and b1 = a0, and all other aj, bk are zero. If we set f = a1Z0 + a0Z1, then we find

that s1 = f/Z0 and s0 = f/Z1. Conversely, any such f determines a holomorphic section of O(1).

n 1,0 Example 2.7. If M has dimension n, the top exterior power KM := Λ Ω M of the holomorphic cotangent bundle is a holomorphic line bundle on M called the of M. Locally a trivialization of 1 n 1 n KM is given by dz ∧ · · · ∧ dz where (z , . . . , z ) are complex coordinates on M.

n Example 2.8. We claim that the canonical bundle of CP is isomorphic to O(−n − 1).

In the chart U0, we use the coordinates

Zj zj = , j = 1, . . . , n. Z0

n n A local frame for KCP in these coordinates is given by dz1 ∧ · · · ∧ dzn, which gives a trivialization of KCP

on the chart U0. For i 6= 0, we use the coordinates

0 Zj zj = 0 , j 6= i. Zi

6 n A local frame for KCP in these coordinates is given by

i 0 0 0 (−1) dz0 ∧ · · · dzci ∧ · · · ∧ dzn,

n which gives a trivialization of KCP on Ui. 0 The coordinates zj on the chart Ui satisfy

0 zj zj = 0 , j 6= i z0 1 zi = 0 . z0

This implies that

0 0 0 dzj zjdz0 dzj = 0 − 0 2 , j 6= i z0 (z0) 0 dz0 dzi = − 0 2 . (z0)

It follows that dz1 ∧ · · · ∧ dzn has local expression in the chart Ui as

i 1 0 0 0 dz1 ∧ · · · ∧ dzn = (−1) 0 n+1 dz0 ∧ · · · ∧ dzci ∧ · · · ∧ dzn (z0)  −n−1 i Z0 0 0 0 = (−1) dz0 ∧ · · · ∧ dzci ∧ · · · ∧ dzn Zi

n Thus the transition functions for KCP relative to the cover Uj satisfy

 −n−1 Z0 gi0([Z0,...,Zn]) = . Zi

One can show that more generally, the transition functions satisfy

 −n−1 Zk gjk([Z0,...,Zn]) = . Zj

n That is, KCP is isomorphic to O(−n − 1).

3 Positive line bundles

Definition 3.1. A hermitian metric h on a holomorphic line bundle L consists of a family {hp}p∈M of hermitian inner products, each hp an inner product on Lp, which is smooth in the sense that for each pair of local smooth sections s1, s2 of L, the local C-valued function p 7→ hp(s1(p), s2(p)) is smooth. We sometimes

7 write the metric as

hs1, s2ih = h(s1, s2).

2 We often write kskh = hs, sih.

Definition 3.2. Let L be a holomorphic line bundle with hermitian metric h. The curvature of (L, h) is the (1, 1)-form on M with local expression

√ ¯ 2 − −1∂∂ log kskh

for a local non-vanishing holomorphic section s of L.

Remark 3.3. The curvature is well-defined for the following reason. If s0 is another non-vanishing holomor- 0 0 2 2 2 phic section, then there is a local holomorphic function f such that s = fs, and hence ks kh = |f| kskh. We then find that

√ √ √ √ ¯ 0 2 ¯ 2 ¯ ¯ ¯ − −1∂∂ log ks kh = − −1∂∂ log kskh − −1∂∂ log f − −1∂∂ log f.

Since f is holomorphic, so is log(f) and the second term on the right-hand side is therefore zero. Since f¯ is anti-holomorphic, so is log(f¯), and hence the third term is

√ √ − −1∂∂¯log f¯ = −1∂∂¯ log f¯ = 0.

Thus √ √ − −1∂∂¯log h(s0) = − −1∂∂¯log h(s),

meaning that the curvature is independent of the choice of the non-vanishing holomorphic section s.

n Example 3.4. Let O(−1) denote the tautological line bundle over CP . Recall that we can view O(−1) n as the sub-bundle of the trivial bundle CP × Cn+1 whose fiber over the point determined by the line ` is the line ` itself. The standard Euclidean metric k·k on the trivial bundle induces a metric on O(−1). The curvature of this metric is defined to be √ − −1∂∂¯log kZk2 where Z is a local non-vanishing holomorphic section of O(−1). It follows immediately that the curvature n is given by −ωFS where ωFS is the Fubini-Study K¨ahlerform on CP .

Definition 3.5. We say that a (1, 1)-form α on M is positive if the 2-tensor defined by (X,Y ) 7→ α(X,JY ) is positive definite. A negative (1, 1)-form is defined in an analogous way.

Example 3.6. Say that a (1, 1)-form α has local expression

√ j k α = −1αkj¯ dz ∧ dz¯ .

8 Then α is positive if and only if αkj¯ defines a positive definite matrix. For example, the Fubini-Study form n ωFS on CP is a positive (1, 1)-form.

Definition 3.7. A holomorphic line bundle L is called positive if it admits a hermitian metric whose corresponding curvature form is positive. The notion of negative line bundle is defined in an analogous way.

n Example 3.8. Example 3.4 shows that the line bundle O(−1) over CP admits a hermitian metric whose corresponding curvature form is negative. Hence O(−1) is a negative line bundle. If O(1) denotes the dual to O(−1), then the curvature form corresponding to the metric dual to the

above metric on O(−1) is given by ωFS, and hence is positive. Thus we find that O(1) is a positive line bundle.

Moreover, the curvature of O(`) with the induced metric is given by `·ωFS. It follows that the line bundle

n O(`) is positive if and only if ` is a positive integer. In particular, the canonical bundle KCP 'O(−n − 1) is negative.

4 First Chern class

Definition 4.1. Let L be a holomorphic line bundle. The first Chern class c1(L) of L is the cohomology class determined by the (1, 1)-form with local expression √ − −1 ∂∂¯log ksk2 2π h

for some hermitian metric h on L and some local holomorphic non-vanishing section s of L.

Remark 4.2. The first Chern class does not depend on the choice of hermitian metric for the following 0 2 2 reason. If h is another hermitian metric on L, then kskh0 = λ kskh for some positive local function λ. We compute that √ √ √ − −1 − −1 − −1 ∂∂¯log ksk2 = ∂∂¯log ksk2 − ∂∂¯log λ. 2π h0 2π h 2π

But the second term is exact as ∂∂¯log λ = (∂ + ∂¯)∂¯log λ = d(∂¯log λ).

2 Remark 4.3. One can show that c1(L) is an integral cohomology class, that is, c1(L) ∈ H (M, Z). But a proof of this fact is beyond the scope of these notes. More specifically, there is the following interpretation of the first Chern class. The of sheaves over M exp ∗ 0 → Z ,→ OM −−→OM → 0 gives rise to a long exact sequence in cohomology, and the first connecting

1 ∗ 2 c1 : H (M, OM ) → H (M, Z)

9 1 ∗ is the map taking a line bundle to its first Chern class. Here we identify H (M, OM ) with the group of

isomorphism classes of holomorphic line bundles over M. From this viewpoint, it is clear that c1 satisfies

0 0 c1(L ⊗ L ) ' c1(L) + c1(L )

and ∗ c1(L ) = −c1(L)

where L∗ denotes the dual to L.

The following lemma is a key tool in complex .

Lemma 4.4 (∂∂¯-Lemma). Let M be a compact, K¨ahlermanifold. If ω and η are two real (1, 1)-forms representing the same cohomology class, then there is a smooth function f : M → R such that

√ η = ω + −1∂∂f.¯

It is clear that if such a function f exists, then η and ω represent the same cohomology class. However, the crux of the matter is that there is a converse type of statement showing the existence of such an f.A standard proof of the ∂∂¯-lemma involves defining adjoints ∂∗, ∂¯∗ and using Hodge theory.

0 Corollary 4.5. If η is a real (1, 1)-form representing the cohomology class c1(L), then there is a metric h on L such that 2πη is the curvature of h0.

Proof. Say that c1(L) is represented by √ − −1 ω = ∂∂¯log ksk2 2π h

for a metric h on L. By the ∂∂¯-lemma, there is a smooth function f : M → R such that

√ η = ω + −1∂∂f.¯

Let h0 be the metric determined by h0 = e−2πf h. Then note that the curvature of h0 is

√ √ √ ¯ 2 ¯ 2 ¯ − −1∂∂ log kskh0 = − −1∂∂ log kskh + 2π −1∂∂f √ = 2πω + 2π −1∂∂f¯

= 2πη.

This completes the proof.

Definition 4.6. Say that the first Chern class c1(L) is positive if c1(L) is represented by a positive (1, 1)- form.

10 Lemma 4.7. A line bundle L is positive if and only if c1(L) is positive.

Proof. If L is positive, then there is a hermitian metric h on L whose curvature form is positive, and hence

c1(L) is represented by a positive (1, 1)-form.

Conversely, if c1(L) is represented by a positive (1, 1)-form, then the previous lemma shows that there is a metric h on L whose curvature is this positive (1, 1)-form, and hence L is positive.

Definition 4.8. For a complex manifold M, we define the first Chern class of M by the rule

−1 c1(M) = c1(KM )

−1 where KM denotes the anti-canonical bundle, that is, the line bundle dual to the canonical bundle KM .

n n Example 4.9. The canonical bundle of C is trivial, and hence the first Chern class c1(C ) is zero. This −1 −1 −1 follows from the isomorphism K n ' K n and then the observation c1(K n ) = c1(K n ) = −c1(K n ). C C C C C

−1 Example 4.10. We have seen that K n 'O(−n − 1), from which we find K n 'O(n + 1). Example 3.8 CP CP

shows that there is a metric on O(n + 1) whose curvature form is (n + 1)ωFS. It follows that the first Chern n −1 class c1( ) = c1(K n ) = c1(O(n + 1)) is positive. CP CP

5 Ricci curvature

Let (M, g) be a K¨ahlermanifold of dimension n. Viewing g as a metric on the holomorphic tangent bundle −1 n T1,0M, we obtain a metric det(g) on the anti-canonical bundle KM = Λ T1,0M.

−1 Definition 5.1. The Ricci curvature of (M, g) is the curvature of det(g) on KM :

√ Ric(g) = − −1∂∂¯log det(g).

If we are given a K¨ahlerform ω on M, we sometimes write Ric(ω) to denote Ric(g), where g is the corre- sponding K¨ahlermetric.

Definition 5.2. A K¨ahlermanifold (M, ω) is called K¨ahler-Einstein if there is a constant k such that Ric(ω) = kω.

n Theorem 5.3. The K¨ahlermanifold (CP , ωFS) is K¨ahler-Einsteinwith Ric(ωFS) = (n + 1)ωFS.

Proof. Remember that on the coordinate chart U0, the Fubini-Study form has local expression

√ ¯ 2 ωFS(z1, . . . , zn) = −1∂∂ log(1 + |z| ) P  P ! √ P dz ∧ dz¯ z¯idzi ∧ zjz¯j = −1 i i i − i j . 1 + |z|2 (1 + |z|2)2

11 It follows that on U0, the corresponding metric g has local expression

2 δjk¯(1 + |z| ) − zjz¯k g ¯ = . jk (1 + |z|2)2

2 The matrix zjz¯k is symmetric of rank one with nonzero eigenvalue given by its trace |z| . Thus, under a suitable linear change of coordinates, we find that g has local expression

1 g = (1 + |z|2)I − diag(|z|2, 0,..., 0) (1 + |z|2)2 n 1 = diag(1, 1 + |z|2,..., 1 + |z|2). (1 + |z|2)2

It follows that det(g) has local expression

(1 + |z|2)n−1 1 det(g) = = . (1 + |z|2)2n (1 + |z|2)n+1

The Ricci form of ωFS is defined by

√ ¯ Ric(ωFS) = − −1∂∂ log det(g).

Using our expression of det(g) above, we find that Ric(ωFS) has local expression in U0 as

√ ¯ 2 Ric(ωFS) = (n + 1) −1∂∂ log(1 + |z| ) = (n + 1)ωFS.

A similar computation shows the equality Ric(ωFS) = (n + 1)ωFS in any coordinate chart Uj.

6 Riemann surfaces

n So far, we have only computed the first Chern class of Cn and CP . The goal of this section is to understand the first Chern class of a Riemann surface in terms of its genus. Let M be a compact, connected Riemann surface. The [M] is a generator of the

homology group H2(M, Z) ' Z. So the linear functional given by pairing with [M] describes an isomorphism H2(M, Z) ' Z.

Definition 6.1. Define the degree of a line bundle L to be the integer hc1(L), [M]i.

2 In other words, the degree of L is equal to c1(L) under the isomorphism H (M, Z) ' Z. To proceed further, we need the following classical result.

Theorem 6.2 (Gauss-Bonnet). Let M be a compact Riemann surface with Gaussian curvature k and Φ. Then Z k · Φ = 2πχ(M) = 2π(2 − 2g) M

12 where χ(M) denotes the Euler characteristic of M and g denotes the genus of M.

This result is somewhat surprising, since the left-hand side seems to depend upon the choice of Riemannian metric, while the right-hand side is expressed purely in terms of topological invariants of M.

Theorem 6.3. Let M be a compact Riemann surface with Gaussian curvature K and Ricci curvature Ric. Then Ric = K · Φ.

Proof. Say that the metric is h2dz ⊗ dz¯. We then compute

√ Ric = − −1∂∂¯log(h2) √ = −2 −1∂∂¯log(h) √  ∂2  = −2 −1 log(h) dz ∧ dz¯ ∂z∂z¯ 1√ = − −1∆ log(h)dz ∧ dz¯ 2 ∆ log(h) 1√  = −1h2dz ∧ dz¯ h2 2 = k · Φ,

where −∆ log(h) k = h2

is the usual Gaussian curvature and 1 √ Φ = −1h2dz ∧ dz¯ 2!

is the volume form corresponding to the metric.

Corollary 6.4. For a compact, Riemann surface M of genus g, we have

deg(KM ) = −χ(M) = 2g − 2.

Hence

• If g > 1, then c1(M) < 0.

• If g = 1, then c1(M) = 0.

• If g = 0, then c1(M) > 0.

Proof. The first Chern class of M satisfies

−1 1 c1(M) = c1(T1,0M) = c1(KM ) = [ 2π Ric].

13 −1 By Gauss-Bonnet, the degree of KM is

Z −1 1 deg(KM ) = Ric = χ(M) = 2 − 2g. 2π M

The result follows.

7 Subvarieties of projective space

To conclude our list of examples, we study the first Chern class c1(V ) of a smooth projective submanifold n V ⊂ CP . It turns out that we can understand the first Chern class in terms of the degrees of the polynomials defining V .

Definition 7.1. For a complex submanifold V ⊂ M, define the normal bundle NV to be the quotient

0 → T1,0V → T1,0M|V → NV → 0

∗ of the holomorphic tangent bundle T1,0M|V restricted to V . The conormal bundle NV is the dual to the normal bundle. Note that the conormal bundle consists of all cotangent vectors to M that vanish on the holomorphic tangent bundle T1,0V .

Lemma 7.2. For a complex submanifold V ⊂ M of codimension 1, the canonical bundle KV satisfies

KV = (KM |V ) ⊗ NV

Proof. We have an exact sequence of the form

0 → TV → TM|V → NV → 0.

Taking duals, we get an exact sequence

−1 −1 −1 0 → NV → TM|V → TV → 0.

Using the properties of the determinant, we conclude that

−1 KM |V ' KV ⊗ NV , as desired.

Lemma 7.3. Let L → M be a holomorphic line bundle with holomorphic global section s such that V =

14 s−1(0) ⊂ M is a complex submanifold. Then

NV = L|V .

−1 Proof. We show that the line bundle NV ⊗ L|V is trivial by constructing a global non-vanishing section.

Say that the transition functions of L are gαβ with respect to some open cover Uα with trivializations ϕα,

and let sα : Uα → C be the corresponding local holomorphic functions determining the section s. Because

s is constant along V , the functions dsα vanish along the tangent bundle TV . Moreover, we note that on

(Uα ∩ Uβ) ∩ V , we have

dsα = d(gαβsβ) = (dgαβ)sβ + gαβ(dsβ) = gαβ(dsβ).

−1 It follows that the collection dsα assembles to form a global section of NV ⊗ L|V . Moreover, because V is

smooth everywhere, none of the dsα ever vanish, and this global section is nonvanishing, as desired.

n Corollary 7.4. Suppose that V ⊂ CP is a projective hypersurface of degree d > 0, that is, V is the zero locus of a global section of O(d). Then

KV 'O(d − n − 1)|V .

Hence,

• If d > n + 1, then the first Chern class of V is positive c1(V ) < 0.

• If d < n + 1, then the first Chern class of V is negative c1(V ) > 0.

• If d = n + 1, then the first Chern class of V is zero c1(V ) = 0.

Proof. Because V is the divisor of zeros of a global section of O(d), the previous lemma shows that NV =

O(d)|V . The gives

KV ' (O(−n − 1) ⊗ O(d))|V

'O(d − n − 1)|V .

The rest of the claims now follow from the definition of the sign of the first Chern class.

Lemma 7.5. More generally, suppose that V ⊂ M is a complex submanifold of codimension r. Then we have Vr KV ' KM |V ⊗ NV

15 Proof. Again we have an exact sequence

−1 −1 −1 0 → NV → TM|V → TV → 0.

−1 Again using properties of the determinant, because NV is of rank r, we have

Vr −1 KM |V ' KV ⊗ NV .

Rearranging completes the proof.

Lemma 7.6. If d1, . . . , dr are integers, then

Vr (O(d1) ⊕ · · · ⊕ O(dr)) = O(d1 + ··· + dr).

Proof. We use induction on r. The claim is obvious when r = 1. For r > 1, we have an exact sequence of the form

0 → O(dr) → O(d1) ⊕ · · · ⊕ O(dr) → O(d1) ⊕ · · · ⊕ O(dr−1) → 0.

Using properties of the determinant and the inductive hypothesis, we conclude that

Vr Vr−1 V1 (O(d1) ⊕ · · · ⊕ O(dr)) = (O(d1) ⊕ · · · ⊕ O(dr−1)) ⊗ O(dr)

= O(d1 + ··· + dr−1) ⊗ O(dr)

= O(d1 + ··· + dr).

This completes the inductive step, and the proof.

n Corollary 7.7. Suppose that V ⊂ CP is a projective submanifold of codimension r determined by the

transverse intersection of r hypersurfaces of degrees d1, . . . , dr. If d = d1 + ··· + dr, then

KV 'O(d − n − 1)|V .

Hence,

• If d > n + 1, then the first Chern class of V is positive c1(V ) < 0.

• If d < n + 1, then the first Chern class of V is negative c1(V ) > 0.

• If d = n + 1, then the first Chern class of V is zero c1(V ) = 0.

Proof. We first claim that the normal bundle to V is (O(d1) ⊕ · · · ⊕ O(dr))|V . If Vj denotes the hypersurface of degree dj, then the normal bundle to Vj is O(dj)|Vj . Since TV ⊂ TVj for each j, we get natural surjective

16 maps NV → NVj for each j and hence a natural map

NV → NV1 ⊕ · · · ⊕ NVr 'O(d1) ⊕ · · · ⊕ O(dr),

which we claim is an isomorphism. Indeed at each point, the fibers of both are r dimensional, so it suffices to show that the map is injective. Because the intersection is smooth, we have

r \ TpV = TpVj. j=1

If a vector in the fiber of NV over p mapped to zero in the fiber of NV1 ⊕ · · · ⊕ NVr , this would mean that the vector belongs to each TpVj, which would imply that the vector belongs to TpV , and hence is zero in the

fiber of NV . Now the adjunction formula implies that

K ' Kn| ⊗ Λr(O(d ) ⊕ · · · ⊕ O(d )| ) V P V 1 r V

'O(−n − 1)|V ⊗ O(d1 + ··· + dr)|V

'O(d − n − 1)|V .

The claims follow.

8 Connections on vector bundles

Definition 8.1. A complex of rank r over M consists of a family {Ep}p∈M of r- dimensional complex vector spaces together with the structure of a smooth manifold on the disjoint union

E = ∪p∈M Ep in such a way that the projection map π : E → M is smooth and for each point p ∈ M, there is a neighborhood U of p in M and a diffeomorphism ϕ : π−1(U) → U × Cr which is linear on the fibers and which commutes with the projection maps to U.

0 Definition 8.2. A connection on a complex vector bundle E → M is a complex linear map D :ΩM (E) → 1 ΩM (E) satisfying Leibniz rule D(fs) = df ⊗ s + f(Ds)

∞ 0 for f ∈ C (M, C) and s ∈ ΩM (E).

Note 8.3. If (z1, . . . , zn) are complex coordinates for M, then for each k, a connection D gives rise to a 0 0 C-linear map Dk :ΩM (E) → ΩM (E) defined by

 ∂  D s = (Ds) . k ∂zk

17 Moreover, the map Dk satisfies the Leibniz rule in the sense that

∂f D (fs) = s + f(D s) k ∂zk k

∞ 0 0 for each f ∈ C (M, C). We obtain similar mappings Dk¯ :ΩM (E) → ΩM (E) which satisfy analogous Leibniz rule relations.

Conversely, if we are given such Dk and Dk¯ satisfying Leibniz rule, then we obtain a connection D on E defined by k k Ds = dz ⊗ (Dks) + dz¯ ⊗ (Dk¯s).

Note 8.4. The decomposition Ω1M = Ω1,0M ⊕ Ω0,1M implies that any connection D on E decomposes as D = D0 + D00 where

0 0 1,0 D :ΩM (E) → ΩM (E) 00 0 0,1 D :ΩM (E) → ΩM (E).

Definition 8.5. A hermitian metric h on a complex vector bundle E consists of a family {hp}p∈M of

hermitian inner products, each hp an inner product on Ep, which is smooth in the sense that for each pair of

local smooth sections s1, s2 of E, the local C-valued function p 7→ hp(s1(p), s2(p)) is smooth. We sometimes write the metric as

hs1, s2ih = h(s1, s2).

Note 8.6. If h is a hermitian metric on E, then we obtain a bundle morphism h : E → E−1 defined in the

following way. For a section s of E, we let h(s): E → C denote the bundle map defined by

h(s)(t) = hs, t¯ih.

In this way, we can view h as a section of the bundle Hom(E,E−1).

Definition 8.7. A connection D on a hermitian vector bundle (E, h) is called unitary if

∂kh(s1, s2) = h(Dks1, s2) + h(s1,Dk¯s2)

∂k¯h(s1, s2) = h(Dk¯s1, s2) + h(s1,Dk¯s2)

for each pair of sections s1, s2 of E.

Definition 8.8. If D is a connection on E, then we obtain a connection on Hom(E,E−1) defined in the

18 following way. For sections s, t of E and a section h of Hom(E,E−1), we require

∂k(h(s)(t)) = (Dkh)(s)(t) + h(Dks)(t) + h(s)(Dkt)

∂k¯(h(s)(t)) = (Dk¯h)(s)(t) + h(Dk¯s)(t) + h(s)(Dk¯t).

Lemma 8.9. A connection D on a hermitian vector bundle (E, h) is unitary if and only if Dh = 0, where we view h as a section of End(E).

Proof. The statement Dh = 0 is equivalent to Dkh = Dk¯h = 0 for each k. According to the above definition, this is equivalent to

∂k(h(s)(t)) = h(Dks)(t) + h(s)(Dkt) with a similar equality for k¯. But according to the definition of h, this is equivalent to

¯ ¯ ¯ ¯ hs, tih = hDks, tih + hs, Dktih = hDks, tih + hs, Dk¯tih, with a similar expression for k¯. This is clearly equivalent to the requirement that D is unitary.

Definition 8.10. A complex vector bundle E → M is called holomorphic if E admits the structure of a complex manifold in such a way that the projection map π : E → M is holomorphic and for each point p ∈ M, there is a neighborhood U of p in M and a biholomorphism ϕ : π−1(U) → U × Cr which is linear on the fibers and which commutes with the projection maps.

Definition 8.11. A section s : M → E of a holomorphic vector bundle E is called holomorphic if s is holomorphic as a map of complex manifolds. We let H0(M,E) denote the space of holomorphic sections of E.

¯ p,q p,q+1 Definition 8.12. If E is holomorphic, then there is a natural map ∂ :ΩM (E) → ΩM (E) described in p,q the following way. For a section s ∈ ΩM (E), write

α s = s ⊗ eα

α where s are local smooth (p, q) forms and eα is a local holomorphic frame for E. Then set

¯ ¯ α ∂s = (∂s ) ⊗ eα.

This definition does not depend on the representation of s, because if fα is another holomorphic frame in α α α which s = t fα, then there are local holomorphic functions gβ on M such that fβ = gβ eα, and hence

¯ β ¯ β α ¯ β α ¯ α (∂t ) ⊗ fβ = (∂t ) ⊗ gβ eα = ∂(t gβ ) ⊗ eα = (∂s ) ⊗ eα.

19 Definition 8.13. A connection D = D0 + D00 on a holomorphic vector bundle E is said to be compatible with the holomorphic structure if D00 = ∂¯.

Theorem 8.14. If E is a holomorphic vector bundle with hermitian metric h, then there is a unique connection D on E which satisfies the following two properties.

(i) D is unitary

(ii) D is compatible with the holomorphic structure.

Proof. Assume that D is a connection on E satisfying (i) and (ii). Let {eα} be a local holomorphic frame for E and write β Deα = θαeβ

β for some local matrix of 1-forms θα. Also define local smooth functions hαβ by

hαβ = h(eα, hβ).

β To say that D satisfies (ii) implies that θα are of type (1, 0). The fact that D is unitary implies that

γ γ dhαβ = θαhγβ + θβhαγ .

Decomposing into types, we find that

γ ∂hαβ = θαhγβ ¯ γ ∂hαβ = θβhαγ .

Writing this in terms of matrices, we find that

∂h = θh

∂h¯ = hθ∗.

This implies that θ = (∂h)h−1, hence the connection D is uniquely determined. On the other hand, such a connection D can be shown to exist by defining θ locally by θ = (∂h)h−1.

Definition 8.15. The unique connection which is compatible with the metric and the holomorphic structure is called the Chern connection on E, and we often denote this connection by ∇.

Note 8.16. A hermitian metric g on TM determines a Chern connection on the holomorphic tangent bundle T 1,0M. One can show that the metric is K¨ahlerif and only if the Chern connection agrees with the Levi-Civita connection on T 1,0M.

20 Note 8.17. If D1 is a connection on E1 and D2 is a connection on E2, then the map D described by

D = D1 ⊗ idE2 + idE1 ⊗ D2

is a connection on E1 ⊗ E2.

Definition 8.18. If E is a holomorphic vector bundle and M is a K¨ahlermanifold, then we obtain a connection on the bundle TM 1,0 ⊗ E by using the Chern connection on both factors. We denote this connection by ∇ when no confusion will arise.

9 Curvature

p p+1 Definition 9.1. We can extend any connection D on E to a C-linear map D :ΩM (E) → ΩM (E) by forcing Leibniz rule D(ψ ⊗ s) = dψ ⊗ s + (−1)pψ ⊗ (Ds)

p 0 2 for ψ ∈ Ω M and s ∈ ΩM (E). The curvature of the connection D is then the composition FD = D : 0 2 ΩM (E) → ΩM (E).

∞ 2 Lemma 9.2. The curvature FD is C (M)-linear and hence corresponds to a section FD ∈ ΩM (End(E)).

Proof. This is a simple computation using Leibniz rule. If f ∈ C∞(M) and s is a smooth section of E, then

FD(fs) = D(df ⊗ s + f ⊗ (Ds))

= ddf ⊗ s − df ⊗ (Ds) + df ⊗ (Ds) + f ⊗ (FDs)

= f(FDs) where we used the fact that d2 = 0.

Lemma 9.3. If L is a holomorphic line bundle with hermitian metric h, then the curvature F of the Chern connection has local expression 2 F`k¯ = −∂`¯∂k log kskh where s is a holomorphic non-vanishing section of L.

Proof. Let s be a holomorphic non-vanishing section. The curvature of the Chern connection is of type (1, 1) and satisfies F s = ∇0∇00 + ∇00∇0s = ∇00∇0s

00 ¯ where we used the fact that s is holomorphic and hence ∇ s = ∂s = 0. Say that ∇ks = Aks for some

21 00 0 smooth functions Ak. Then note that ∇ ∇ s has local expression

00 0 ` k ∇ ∇ s = dz¯ ∧ ∇`¯(dz (∇ks))

` k = (dz¯ ∧ dz )∇`¯(Aks)

` k = (dz¯ ∧ dz )(∂`¯Ak)s.

where again we used that s is holomorphic. It follows that F`k¯ has local expression F`k¯ = −∂`¯Aks. However, because ∇ is unitary, we know that

2 2 ∂k kskh = h∇ks, si + hs, ∇k¯si = Ak kskh .

It follows that 2 ∂k kskh 2 Ak = 2 = ∂k log kskh . kskh The claim now follows.

10 First Chern class and sections

We conclude by discussing the implications of a line bundle being negative. Essentially, the result is that such a line bundle has no holomorphic sections at all.

Theorem 10.1. If L is a holomorphic line bundle over a compact K¨ahlermanifold M such that c1(L) < 0, then there are no nonzero global holomorphic sections of L.

We require a lemma about integration by parts on a K¨ahlermanifold M.

Lemma 10.2. Let h be a metric on L. Then for each holomorphic section s of L, we have

Z Z k`¯ k`¯ g h∇`¯∇ks, sih dV = − g h∇ks, ∇`sih dV. M M

Proof. Let s be a holomorphic section of L. Let g :Ω1,0M → T 0,1M denote the isomorphism determined by the K¨ahlermetric g and let h : L → L−1 denote the isomorphism determined by the metric h. Define a vector field X by the rule X = g(h(∇0s)(¯s))

Note that X has local expression ¯ ∂ X = X` ∂z¯`

where `¯ k`¯ X = g h(∇ks)(¯s).

22 Since ∇ is compatible with both g and h, note that

0 0 ∇m¯ X = g(h(∇m¯ ∇ s)(¯s)) + g(h(∇ s)(∇m¯ s¯)).

We find that ∇m¯ X has local expression

 ¯ ¯  ∂ ∇ X = gk`h∇ ∇ s, si + gk`h∇ s, ∇ si m¯ m¯ k h k m h ∂z¯`

The divergence is then `¯ k`¯ k`¯ (∇`¯X) = g h∇`¯∇ks, sih + g h∇ks, ∇`sih

The result now follows from Stoke’s theorem:

Z Z (divX)dV = d(ιX dV ) = 0. M M

We now prove Theorem 10.1.

Proof. Because c1(L) < 0, there is a hermitian metric on L whose curvature form

2 Fk`¯ = −∂k∂`¯ log kskh

is negative definite. The Chern connection satisfies

∇k∇`¯ = ∇`¯∇k + Fk`¯.

If s is a holomorphic section, then

k`¯ 0 = g h(∇k∇`¯s)s

k`¯ = g h(∇`¯∇k + Fk`¯s)s

k`¯ k`¯ 2 = g h(∇`¯∇k)s + g Fk`¯|s|h

k`¯ 2 6 g h(∇`¯∇k)s − c|s|h.

k`¯ for some constant c > 0, because Fk`¯ is negative definite and g is positive definite. Integrating this over M and using the previous lemma gives

Z Z Z Z k`¯ 2 2 2 0 6 − g h(∇ks)∇`s dV − c |s|hdV = − |∇s|g⊗hdV − c |s|hdV. M M M M

2 This implies that |s|h = 0 and we conclude that s = 0 from the positivity of h.

23 2 Lemma 10.3. Let s be a holomorphic section of a line bundle such that ∇s = 0. Then the norm kskh is constant.

Proof. We compute that

∂khs, sih = h∇ks, si + hs, ∇k¯si = 0.

The result follows.

Proposition 10.4. Let L be a holomorphic line bundle over a compact, K¨ahlermanifold M with c1(L) = 0. If L is not the trivial bundle, then H0(M,L) = 0.

Proof. The 2-form which is identically zero is a class representing 2πc1(L). By Corollary 4.5, there is a metric h on L whose corresponding curvature form is 0. It now follows from the proof of the previous proposition that for a holomorphic section s we have

k`¯ k`¯ 0 = hg ∇k∇`¯s, si = hg ∇`¯∇ks, si.

Integrating over M and using integration by parts gives that

Z 2 0 = − |∇s|g⊗h dV. M

Hence the norm of s is constant. If the norm of s is nonzero, then s defines a global non-vanishing section and hence L is trivial. Otherwise, the norm of s is zero, and s is the zero section.

References

[GH] P. Griffiths and J. Harris, Principles of , Wiley, New York (1978).

24