Chapter VI Kodaira’s Projective Embedding Theorem 217 1. Hodge Manifolds 217
2. The Canonical Connection and Curvature of a Hermitian Holomorphic Vector Bundle
for a unitary frame; i.e., θ is to be skew-Hermitian. In each Uα we can choose the trivial skew-Hermitian matrix of the formθα=0; i.e.,θ (fα)=0.
If we make a change of frame in Uα, then we see that we require that (1.23) θ (fαg)=g−1dg+0
by Lemma 1.6(a). Therefore, define θ (fαg) by (1.23), and noting that h(fαg)=tgh(f )g¯ =tgg, we obtain¯
dh(fαg)=d(tg¯·g)
=dtg¯·g+tg¯·dg
=dtg(¯ tg)¯ −1·tg¯·g+tg¯·g·g−1·dg
=tθ (f¯ αg)h(fαg)+h(fαg)θ (fαg),
which verifies the compatibility. Let{ϕα}be a partition of unity subordinate to {Uα} and let Dα be the connection in E|Uα defined by
(Dαξ )(fα)=dξ(fα).
Dα is defined with respect to other frames overUαby formula (1.23) and is compatible with the Hermitian metric on E|Uα, by construction. Then we let D =
αϕαDα, which is a well-defined (first-order partial-differential) operator
D:E(X, E)−→E1(X, E).
Moreover, D is compatible with the metric h on E since Dξ,η + ξ, Dη =
α
ϕα[Dαξ,η + ξ, Dαη]
=
α
ϕαdξ,η =dξ,η.
Q.E.D.
Remark: It is clear by the construction in the proof of Proposition 1.11 that a connection compatible with a metric is by no means unique because of the various choices made along the way. In the holomorphic category, we shall obtain a unique connection satisfying an additional restriction on the type of θ.
2. The Canonical Connection and Curvature of a Hermitian Holomorphic
Suppose then that we have a connection on E
D:E(X, E)−→E1(X, E)=E1,0(X, E)⊕E0,1(X, E).
Then D splits naturally intoD=D′+D′′, where D′:E(X, E)−→E1,0(X, E) D′′:E(X, E)−→E0,1(X, E).
Theorem 2.1: Ifh is a Hermitian metric on a holomorphic vector bundle E −→ X, then h induces canonically a connection, D(h), on E which satisfies, for W an open set in X,
(a) For ξ,η∈E(W, E)
dξ,η = Dξ,η + ξ, Dη; i.e., D iscompatible with the metric h.
(b) If ξ ∈O(W, E), i.e., is a holomorphic section of E, thenD′′ξ =0.
Proof: First, we point out that (b) is equivalent to the fact that the connection matrix θ (f ) is of type (1,0) for a holomorphic frame f. This follows, since for ξ ∈O(W, E) and f a holomorphic frame, we have
Dξ(f )=(d+θ (f ))ξ(f )
=(∂+θ(1,0)(f ))ξ(f )+(¯∂+θ(0,1)(f ))ξ(f ), where θ=θ(1,0)+θ(0,1) is the natural decomposition. Therefore
D′ξ(f )=(∂+θ(1,0)(f ))ξ(f ) and
D′′ξ(f )=(¯∂+θ(0,1)(f ))ξ(f ).
But ∂ξ(f )¯ =0 since ξ and f are holomorphic. Thus D′′ξ(f )=θ(0,1)(f )ξ(f ).
Suppose now that we have a connectionDsatisfying (a) and (b). Then let f =(e1, . . . , er)be a holomorphic frame overU ⊂X andθ the associated connection matrix. Since D is compatible with the metric h, we have, by (1.22),
dh=hθ+tθ h.¯
Since, in addition, D satisfies (b), we have seen that θ is of type (1, 0).
Thus, by examining types we see that
∂h=hθ
and ∂h¯ =tθ h,¯
from which it follows that
(2.1) θ=h−1∂h.
We can then define θ by (2.1). Such a connection matrix clearly satisfies (a) and (b). Moreover, if f′=f g is another holomorphic frame, we have
h(f g)=tgh(f )g,¯ so that
h−1(f g)=g−1h(f )−1[tg¯]−1 and
gθ (f g)=g[h−1(f g)∂h(f g)]
=h(f )−1[tg¯]−1∂[tgh(f )g¯ ]
=h(f )−1[tg¯]−1[tg∂h(f )g¯ +tgh(f )∂g¯ +∂tgh(f )g¯ ]. But g is a holomorphic change of frame, from which it follows that
∂tg¯ = ¯∂tg=0 and ∂g=dg.
Thus
gθ (f g)=h(f )−1∂h(f )g+dg
=θ (f )g+dg.
Recalling Lemma 1.6(a), we see that this is the necessary transformation formula for θ to define a global connection.
Q.E.D.
This theorem gives a simple formula for the canonical connection in terms of the metric h; namely,
(2.2) θ (f )=h(f )−1∂h(f )
for a holomorphic frame f. Moreover, D = D′+D′′ has the following representation with respect to a holomorphic frame f:
D′=∂+θ (f ) D′′= ¯∂.
(2.3)
Thus we have the following proposition.
Proposition 2.2: Let D be the canonical connection of a Hermitian holo- morphic vector bundle E −→X, with Hermitian metric h. Let θ (f ) and (f )be the connection and curvature matrices defined by D with respect to a holomorphic frame f. Then
(a) θ (f )is of type (1, 0), and ∂θ (f )= −θ (f )∧θ (f ).
(b) (f )= ¯∂θ (f ), and (f )is of type (1, 1).
(c) ∂(f )¯ =0, and ∂(f )= [(f ), θ (f )].
Proof: Let h=h(f ),θ=θ (f ), and=(f ). Then we first note that θ is of type (1, 0) by (2.2). Then by using
∂h−1= −h−1·∂h·h−1
∂2=0,
we see that
∂θ=∂(h−1∂h)= −h−1·∂h·h−1∧∂h
= −(h−1∂h)∧(h−1∂h)= −θ∧θ,
which gives us part (a). Part (b) is a simple computation, namely, =dθ+θ∧θ=∂θ+θ∧θ+ ¯∂θ
= ¯∂θ, by using part (a). Part (c) then follows from
¯
∂= ¯∂2θ=0 and Proposition 1.10.
Q.E.D.
Let E −→X be a holomorphic vector bundle and let f =(e1, . . . , er) be a frame for E defined near a point p ∈ X. Choose local coordinates z=(z1, . . . , zn) near p so that p is given by z=0. Then we can write
f (z)=(e1(z), . . . , er(z))
to denote the dependence on the variable znearz=0. Suppose that his a Hermitian metric onE→Xand thatf (z)is the above frame. Then we write
h(z)=h(f (z))
near z=0. The next lemma tells us that we may always find a local frame f near p such that h(f (z)) has a very nice form. Let (z)=(f (z)).
Lemma 2.3: There exists a holomorphic frame f such that (a) h(z)=I+O(|z|2).
(b) (0)= ¯∂∂h(0).
Proof: Suppose that (a) holds. Then it follows that h−1(z)=I+O(|z|2), from which we see that
(z)= ¯∂∂h(z)+O(|z|), and hence (b) follows.
To show (a), we shall make two changes of frame. First we note thath(0) is a positive definite Hermitian matrix, and thus there exists a nonsingular matrix g∈GL(r,C)such that
g∗h(0)g=I,
where for any matrix M we let M∗=tM. The matrix¯ g induces a change of frame f −→ ˜f =f ·g, and we see that
¯
h(z)=h(f (z))˜ =h(f g)
=g∗hg
¯
h(z)=I+O(|z|).
(2.4)
Assume now that h(z) is given satisfying (2.4). We want to consider a change of frame of the form
g=I+A(z), where A(z)=(
jAjρσzj)is a matrix of linear holomorphic functions ofz.
Since A(0)=0, this change of frame will preserve (2.4). By choosing A(z) such that
(2.5) h(z)˜ =g(z)∗h(z)g(z)=I+O(|z|2),
we will have proved (a). But (2.5) is equivalent, by Taylor’s theorem, to the vanishing of the first derivatives of (2.5) at z=0; i.e., dh(0)˜ =0. Thus we compute
dh(z)˜ =dh(z)+dA∗(z)·h(z) +h(z)dA(z)+O(|z|).
Therefore
dh(0)˜ =∂h(0)+dA(0)+ ¯∂h(0)+dA∗(0).
Suppose that we let
(2.6) Aiρσ =−∂hρσ(0)
∂zi
, Then we see that
dA(0)= −∂h(0), which implies that
dA∗(0)= −¯∂h(0).
Then the choice of A(z)given by (2.6), depending on the derivatives of the metric h, ensures that (2.5) holds.
Q.E.D.
This lemma allows us to compute the curvature at a particular point without having to compute the inverse of the local representation for the metric, provided that we have the right frame.
We want to give one principal example concerning the computation of connections and curvatures. Further examples of specific Hermitian metrics on tangent bundles are found in Chap. VI, where we shall discuss Kähler manifolds. In Sec. 4 we shall look at the special case of line bundles in more detail.
Example 2.4: Let Ur,n −→ Gr,n be the universal bundle over the Grassmannian manifold Gr,n (Example 1.2.6). We see that a frame f = (e1, . . . , er)for Ur,n−→Gr,n consists of an open set U ⊂Gr,n and smooth functions
ej :U −→Cn,
so thate1∧. . .∧er =0. Thusf =(e1, . . . , er)can be thought of as ann×r matrix with coefficients being smooth functions in U and whose columns are the vectors {ej}, and the matrix f is of maximal rank at each point z∈U. A holomorphic frame will simply have holomorphic coefficients. We define a metric on Ur,n by letting
(2.7) h(f )=tf f¯
for any framef forUr,n. This metric results from consideringUr,n⊂Gr,n×Cn and restricting the standard Hermitian metric onCnto the fibres ofUr,n−→
Gr,n. First we note that h(f ) is positive definite since (recall that f has maximal rank)
tzh(f )z¯ =t(f z)(f z)= |f z|2>0 if z=0.
Moreover, if g is a change of frame, then we compute that h(f g)=t(f g)(f g)=tg¯tf f g¯ =tgh(f )g,¯
so that (1.7) is satisfied, and thus we see that hdefined by (2.7) on frames gives a well-defined Hermitian metric onUr,n, since the frame representation transforms correctly.
We can now compute the canonical connection and curvature for Ur,n with respect to this natural metric. If f is any holomorphic frame forUr,n, then by (2.2) and Proposition 2.2, we see that
θ (f )=h−1(f )∂h(f ) (f )= ¯∂(h−1(f )∂h(f )).
We obtain, letting θ =θ (f ), etc., as before,
(2.8) =h−1·tdf ∧df−h−1·tdf ·f ·h−1∧tf¯·df,
where h−1 = [tff¯ ]−1. In the case r =1 (projective space), we can obtain a more explicit formula. If ϕ ∈ [Ep(W )]n, ψ ∈ [Eq(W )]n, for W an open subset of Cn, we set
ϕ, ψ =(−1)pq tψ¯ ∧ϕ,
which generalizes the usual Hermitian inner product on vectors inCn [note that this is compatible with (1.19), where we have the usual inner product on E=W×Cn given by u, v =tvu, u, v¯ ∈Cn]. Then the curvature form for U1,n becomes
(2.9) (f )= −f, fdf, df − df, f ∧ f, df f, f2 ,
where f is a holomorphic frame for U1,n. If we choosef to be of the form
f =
⎡
⎢⎢
⎢⎢
⎣ ξ1
·
·
· ξn
⎤
⎥⎥
⎥⎥
⎦ ,
where ξj ∈O(U ) and
|ξj|2= |f|2=0, then
df =
⎡
⎢⎢
⎢⎢
⎣ dξ1
·
·
· dξn
⎤
⎥⎥
⎥⎥
⎦ ,
tdf =(dξ1, . . . , dξn)
=(dξ¯1, . . . , dξ¯n), and we obtain
(2.10) (f )= −
|f|2 n i=1
dξi∧dξ¯i− n i,j=1
¯
ξiξjdξi∧dξ¯j
|f|4 .
Recall that the functions ξ1, . . . , ξn are functions of the local coordinates on G1,n=Pn−1, and that, in particular, (f ) is a well-defined 2-form on U ⊂Pn−1. Alternatively, we can think of(ξ1, . . . , ξn)as being homogeneous coordinates for Pn−1, and by the homogeneity of (2.9), we see that the expression in (2.9) induces a well-defined 2-form on all of Pn−1, which agrees with the 2-form onU mentioned above. We shall see this differential form again when we study Kähler metrics in Chap. V.
Returning to the general case ofUr,n→Gr,n, we have seen in Lemma 2.3 that, by a proper choice of holomorphic frame forUr,n and a proper choice of local holomorphic coordinates near some fixed point, we can find a very simple expression for the curvature. We shall now see an example of this.
Let
f0= Ir
0
be a frame for Ur,n at the point x0∈Gr,n defined by x0=
Ir 0 ,
where denotes the span of the columns of the frame matrix inside, which is a subspace of Cn and thus a point in Gr,n. Letting
Bǫ = {Z∈Mn−r,r ∼=C(n−r)r: |Z|< ǫ}, the mapping
Bǫ−→Gr,n given by
Z−→
Ir Z
is a coordinate system for Gr,n near x0, with the property that x0 corre- sponds to Z = 0. There is a natural action of GL(n,C) on Gr,n given by left multiplication of frames (i.e., left multiplication of homogeneous coordinates). Namely, if f ∈Mn,r, x= f, and u∈GL(n,C), then set
u(x)= u·f.
Moreover, U (n), the unitary group, is transitive on Gr,n under this action (a well-known fact of linear algebra).
Therefore if y0 is any point in Gr,n and y0=u(x0) for a unitary matrix u, then the mapping
Z−→u I
Z
gives local coordinates at y0 ∈ Gr,n. The metric at y0 has the form, with respect to the frame
f (z)=u I
Z , h(Z)=h(f (Z))=
u
I Z
∗ u
I Z
=I+Z∗Z
=I+O(|Z|2),
which is the form occurring in Lemma 2.3(a) (note that the dependence on u disappears completely). Thus we see that
(y0)=(0)= ¯∂∂(I+Z∗Z)(0) (y0)=dZ∗ ∧dZ(0)
(2.11)
which is the same for all points of Gr,n with respect to these particular systems of local coordinates. We shall use this expression for the curvature to compute certain Chern classes of this vector bundle in the next section.