Differential Geometry of Complex Vector Bundles

by Shoshichi Kobayashi

Overview

Holomorphic vector bundles have become objects of interest not only to algebraic and differential geometers and complex analysts but also to low dimensional topologists and mathematical physicists working on gauge theory. This book, which grew out of the author's lectures and seminars in Berkeley and Japan, is written for researchers and graduate students in these various fields of mathematics.

Originally published in 1987.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9780691632643
Publisher:	Princeton University Press
Publication date:	04/19/2016
Series:	Princeton Legacy Library , #793
Pages:	318
Product dimensions:	7.10(w) x 10.10(h) x 0.90(d)

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Differential Geometry of Complex Vector Bundles

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By Shoshichi Kobayashi

PRINCETON UNIVERSITY PRESS

Copyright © 1987 The Mathematical Society of Japan
All rights reserved.
ISBN: 978-0-691-08467-1

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CHAPTER 1

Connections in vector bundles

Although our primary interest lies in holomorphic vector bundles, we begin this chapter with the study of connections in differentiable complex vector bundles. In order to discuss moduli of holomorphic vector bundles, it is essential to start with differentiable complex vector bundles. In discussing Chern classes it is also necessary to consider the category of differentiable complex vector bundles rather than the category of holomorphic vector bundles which is too small and too rigid.

Most of the results in this chapter are fairly standard and should be well known to geometers. They form a basis for the subsequent chapters. As general references on connections, we mention Kobayashi-Nomizu [1] and Chern [1].

§ 1 Connections in complex vector bundles (over real manifolds)

Let M be an n-dimensional real C∞ manifold and E a C∞ complex vector bundle of rank (=fibre dimension) r over M. We make use of the following notations:

AP=the space of C∞ complex p-forms over M,

AP(E)=the space of C∞ complex p-forms over M with values in E.

A connection D in E is a homomorphism

D: A0(E) -> A1(E)

over C such that

(1.1) D(fσ) = σdf + f · Dσ for f [member of] A0, σ [member of] A0 (E),

Lets =(s1, ···, sr) be a local frame field of E over an open set U [subset] M, i.e.,

i) si [member of] A0(E|υ) i=1, ···, r,

ii) (s1 (x), ···, sr(x)) is a basis of Ex for each x [member of] U.

Then given a connection D, we can write

(1.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We call the matrix 1-form ω = (ωji) the connection form of D with respect to the frame field s. Considering s = (s1, ···, sr) as a row vector, we can rewrite (1.2) in matrix notations as follows:

(1.2)' Ds = s · ω.

If [xi] = [summation][xi]i si, [xi]i [member of] A0|U, is an arbitrary section of E over U, then (1.1) and (1.2) imply

(1.3) D[xi] = [summation]si(d[xi]i + [summation]ωij[xi]j).

Considering [xi] = i([xi]1, ···, [xi]r)as a column vector, we may rewrite (1.3) as follows:

(1.3)' D[xi] = d[xi] + ω[xi].

We call D]xi] the covariant derivative of [xi].

Evaluating D on a tangent vector X of M at x, we obtain an element of the fibre Ex denoted by

(1.4) Dx[xi] = (D[xi])(X) [member of] Ex.

We call Dx[xi] the covariant derivative of [xi] in the direction of X.

A section [xi] is said to be parallel if D]xi] = 0. If c = c(t), 0 ≤ t ≤ a, is a curve in M, a section [xi] defined along c is said to be parallel along c if

(1.5) Dc'(t)[xi] = 0 for 0 ≤ t ≤ a,

where c'(t) denotes the velocity vector of c at c(t). In terms of the local frame field s, (1.5) can be written as a system of ordinary differential equations

(1.5)' d[xi]i/dt + [summation] ωij(c'(t))[xi]J = 0.

If [xi]0 is an element of the initial fibre Ec(0). it extends uniquely to a parallel section [xi] along c, called the parallel displacement of [xi]0along c. This is a matter of solving the system of ordinary differential equations (1.5)' with initial condition [xi]0. If the initial point and the end point of c coincide so that x0 = c(0) = c(a), then the parallel displacement along c induces a linear transformation of the fibre Ex0. The set of endomorphisms of Ex0 thus obtained from all closed curves c starting at x0 forms a group, called the holonomy group of the connection D with reference point x0.

We shall now study how the connection form [oemga] changes when we change the local frame field s. Let s' = (s'1, ···, s'r) be another local frame field over U. It is related to s by

(1.6) s = s' · a,

where a: U -> GL(r; C) is a matrix-valued function on U. Let [oemga]'=(ω'ij) be the connection form of D with respect to s'. Then

(1.7) ω = a-1 ω'a + a-1 da.

In fact,

sω= Ds = D(s'a) =(Ds')a + s'da = s'ω'a + s'da = s(a-1ω'a + a-1da).

We extend a connection D: A0 (E) ->A1(E) to a C-linear map

(1.8) D: Ap(E) -> Ap+1(E), p ≥ 0,

by setting

(1.9) D(σ · φ) =(Dσ) [conjunction] φ + σ · dφ for σ [member of] A0(E), φ [member of]AP.

Using this extended D, we define the curvature R of D to be

(1.10) R = D [??] D: A0(E) -> A2(E).

Then R is A0-linear. In fact, if f [member of] A0 and σ [member of] A0(E), then

D2 (fσ) = D(σdf + f · Dσ) = Dσ [conjunction] df + df [conjunction] Dσ + fD2σ = fD2σ.

Hence, R is a 2-form on M with values in End(E). Using matrix notations of (1.2)', the curvature form Ω of D with respect to the frame field s is defined by

(1.11) sΩ = D2s.

Then

(1.12) Ω = dω+ω[conjunction]ω.

In fact,

sΩ = D(sω) = Ds [conjunction] ω + sdω = s(ω [conjunction] ω + dω).

Exterior differentiation of (1.12) gives the Bianchi identity:

(1.13) dΩ = Ω [conjunction] ω - ω [conjunction] Ω.

If ω' is the connection form of D relative to another frame field s'= sa-1 as in (1.6) and (1.7), the corresponding curvature form Ω' is related to Ω by

(1.14) Ω = a-1Ω'a.

In fact,

sΩ=D2s=D2(s'a)= D(Ds'a +s' da)=D2s'a - Ds' [conjunction] da + Ds' [conjunction] da = s'Ω'a=sa-1Ω'a.

Let {U, V, ···} be an open cover of M with a local frame field sυ on each U. If U [intersection] V ≠ [empty set], then

(1.l5) sU = sVgVU on U [intersection] V,

where gVU: U [intersection] V -> GL(r; C) is a C∞ mapping, called a transition function. Given a connection D in E, let ωU be the connection form on U with respect to sU. Then (1.7) means

(1.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conversely, given a system of gl(r; C)-valued l-forms ωU on U satisfying (1.16), we obtain a connection D in E having {ωU} as connection forms.

If ΩU is the curvature form of D relative to sU, then (1.14) means

(1.17) ΩU = g-1VUΩVgVU on U [intersection] V.

It is sometimes more convenient to consider a connection in E as a connection in the associated principal GL(r; C)-bundle P. In general, let G be a Lie group and g its Lie algebra identified with the tangent space TeG at the identity element e of G. Let P be a principal G-bundle over M. Let {U} be an open cover of M with local sections {sU} of P. Let {gVU} be the family of transition functions defined by {(U, sU)}; gVU: U [intersection] V ->G is defined by

sU(x) = sV(x)gVU(x) x [member of] U [intersection] V.

A connection in P is given by a family of g-valued l-forms ωU on U satisfying (1.16).

A connection in P induces a connection in every bundle associated to P. In particular, a connection in a principal GL(r; C)-bundle P induces a connection in the vector bundle EP = P × ρCN associated to P by any representation ρ: GL(r; C)->GL(N; C).

For further details on connections in principal bundles, see Kobayashi-Nomizu [1].

§ 2 Flat bundles and flat connections

Let E be a C∞ complex vector bundle over a real manifold M as in the preceding section. A flat structure in E is given by an open cover {U, sU} with local frame fields such that the transition functions {gVU} (see (1.15)) are all constant matrices in GL(r; C). A vector bundle with a flat structure is said to be flat. On the other hand, a connection D in a vector bundle E is said to be flat if its curvature R vanishes.

A flat vector bundle E admits a natural flat connection D; namely, if the flat structure is given by {U, sU}, then D is defined by

(2.1) DsU=O.

Since the transition functions {gVU} are all constants, the condition DsU = 0 and Dsv=0 are compatible on U [intersection] V and the connection D is well defined. We note that if ωU is the connection form of D relative to sU, then (2.1) is equivalent to

(2.1 )' ωU=0.

From (2.1) (or (2.1 )'), it follows that the curvature of D vanishes.

Conversely, a vector bundle E with a flat connection D admits a natural flat structure {U, sU}. To construct a local frame field sU satisfying (2.1), we start with an arbitrary local frame field s' on U and try to find a function a: U -> GL(r; C) such that sU = s'a satisfies (2.1 )'. Let ω' be the connection form of D relative to s'. By (1. 7), the condition ωU = 0 is equivalent to

(2.2) a-1ω'a + a-1da = ωU = 0.

This is a system of differential equations where ω' is given and a is the unknown. Multiplying (2.2) by a and differentiating the resulting equation

ω'a + da = 0,

we obtain the integrability condition

0=(dω')a-ω' [conjunction] da = (dω')a+(ω' [conjunction] ω')a=Ω'a.

So the integrability condition is precisely vanishing of the curvature Ω' = 0.

In general, if s and s' are two local parallel frame fields, i.e., if Ds = Ds' = 0, then s=s'a for some constant matrix a [member of] GL(r; C) since the connection forms ω and ω' vanish in (1. 7). Hence, if DsU = DsV = 0, the gVU is a constant matrix. This proves that a flat connection D gives rise to a flat structure {U, sU}.

Let E be a vector bundle with a flat connection D. Let x0 be a point of M and π1 the fundamental group of M with reference point x0. Since the connection is flat, the parallel displacement along a closed curve c starting at x0 depends only on the homotopy class of c. So the parallel displacement gives rise to a representation

(2.3) ρ: π1 -> GL(r; C).

The image of ρ is the holonomy group of D defined in the preceding section.

Conversely, given a representation (2.3), we can construct a flat vector bundle E by setting

(2.4) E = [??] × ρCr.

where [??] is the universal covering of M and [??] × ρCr denotes the quotient of [??] × C by the action of π1 given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(We are considering π1 as the covering transformation group acting on [??]). It is easy to see that E carries a natural flat structure coming from the product structure of [??] × Cr. The vector bundle defined by (2.4) is said to be defined by the representation ρ. In summary, we have established the following

(2.5) PropositionFor a complex vector bundle E of rank r over M, the following three conditions are equivalent:

(1) E is a flat vector bundle,

(2) E admits a flat connection D,

(3) E is defined by a representation ρ: π1 ->GL(r; C).

A connection in a vector bundle E may be considered as a connection in the associated principal GL(r; C)-bundle. More generally, let G be a Lie group and P a principal G-bundle over M. Let {U} be an open cover of M with local sections {sU} of P. Let {gVU} be the family of transition functions defined by {(U, sU)}; each gVU is a C∞ maps from U [intersection] V into G. A flat structure in P is given by {(U, sU)} such that {gVU} are all constant maps. A connection in P is said to be flat if its curvature vanishes identically.

Let [??] be the universal covering space of M; it is considered as a principal π1 - bundle over M, where π1 is the fundamental group of M acting on [??] as the group of covering transformations. Given a homomorphism ρ: π1 ->G, we obtain a principal G-bundle P=[??]×ρG by "enlarging" the structure group from π1 to G. Then P inherits a flat structure from the natural flat structure of the product bundle [??] × G over [??]. The following proposition is a straightforward generalization of (2.5).

(2.6) PropositionFor a principal G-bundle P over M, the following three conditions are equivalent:

(1) P admits a fiat structure,

(2) P admits a fiat connection,

(3) P is defined by a representation ρ: π1 ->G.

(Continues...)

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Excerpted from Differential Geometry of Complex Vector Bundles by Shoshichi Kobayashi. Copyright © 1987 The Mathematical Society of Japan. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents

• FrontMatter, pg. i
• Preface, pg. vii
• Contents, pg. ix
• Chapter I. Connections in Vector Bundles, pg. 1
• Chapter II. Chern Classes, pg. 30
• Chapter III. Vanishing Theorems, pg. 49
• Chapter IV Einstein-Hermitian vector bundles, pg. 98
• Chapter V. Stable vector bundles, pg. 133
• Chapter VI. Existence of approximate Einstein-Hermitian structures, pg. 193
• Chapter VII. Moduli spaces of vector bundles, pg. 237
• Bibliography, pg. 291
• Index, pg. 299
• Notations, pg. 303