TRANSACTIONSOF THE
AMERICANMATHEMATICALSOCIETY
Volume 193, 1974
ALMOSTCOMPLEXSTRUCTURESON
COMPLEXPROJECTIVE SPACES
BY
ALAN THOMAS
ABSTRACT. In this paper we classify
complex
projective
Introduction.
plex structure
the almost complex
as roots of a certain
polynomial
structures
on a
equation.
A real oriented vector bundle E is said to admit an almost com-
if there is a complex vector bundle
vector bundles.
class
space
An almost complex structure
of such bundles
F.
F such that £-F
on F is a (complex)
When X is a 27?-dimensional
is its tangent
bundle,
an almost complex structure
plex structure
on X, and X together
almost complex manifold.
manifold and TX
on TX is called
an almost com-
with an almost complex structure
A special
with complex tangent bundle Jx.
oriented
as oriented
isomorphism
is called
an
case of this is when X is a complex manifold
Since
(jX)R = TX, X always admits an almost
complex structure.
In this paper we study the generic
spaces,
and classify
mial equation.
their almost complex structures
which elements
of almost complex structures.
well in other cases
2-planes
ciently
these
will appear
later.
Lnip)
of these,
of the integers
is the lens space
etc.) and the details
For example,
one can easily
(«x, •. •, am) H* (aj,..
p ' Z
—» Z be projection
defined
of main theorems.
as an abelian
sions
[7]
used to solve this problem apply
one almost complex structure.
and statement
polyno-
of Hirzebruch
to Uin), S ", G2(C") = Grassmannian
bundle of the complex structure
1. Definitions
copies
(for example
products
admits precisely
plex tangent
as roots of a certain
of cohomology arise as Chern classes
The techniques
in C", S2m + l x Lnip)-wheie
large, various
cases
S m*
the complex projective
In this context we give an answer to a question
posed in 1954 concerning
equally
complex manifolds,
group.
of
and p is suffi-
and calculations
show that
S "
in
x
(In fact this is the comby Calabi
and Eckmann [9].)
Let Zm be the product of 77?
We consider
Zm C Zm+1 by the inclu-
•, am, 0), and put Z°°= IJ
, Z™. For each 77?,let
onto the 777thcoordinate.
Received by the editors December 13, 1972 and, in revised form, July 9, 1973.
AMS(MOS)subject classifications (1970). Primary 53C15; Secondary 32C10, 55B15.
Key words and phrases.
Almost complex,
vector bundle,
complex projective
space.
Copyright© 1974,AmericanMathematicalSociety
123
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124
ALANTHOMAS
Let r e H (CPm)
CP7"-1
be the cohomology
via the natural
with generators
orientation
t, t , • • •, tm.
corresponding
The abelian
For convenience
with Z"1 by means of the isomorphism
Definition.
class
of CPm.
of notation
aj< + a2r
The kúi Newton function
to the submanifold
group H^CP™)
we identify
+ • • • + a tm \-, dj,
is free
this group
a2, • • •, a ).
sfe: Z°°—»Z is defined inductively
by
(Í) Sj = pj,
(ii) sk(a)= p A¿)s k_A.a) - p2ia)sk_2ia)
(-l)kkpk(a).
+ • •• + i-l)k~xpk_Aa)stia)
+
We notice that sA.a) is a polynomial in pÀa), ■■■, pkia), the first k coordinates of a.
Inductively
(i)
we define
a homomorphism
¡k: Z
—* Z by
/j = identity,
(ii) /¿dj,--.,
Theorem
over CPn
A.
afe)= fk_li<*2'" *» aP -^
The n-ple
iff fAs^
...,
For » a positive
a = (a,, • ■•, a)
is the Chern vector
s.) - 0 mod ¿! /or
integer
- l)/jfc_x^«!»• • • » a¿_i)>
1 < k <<» wèere
of a vector bundle
s. = s{ia).
let
7(72)= r - largest
integer < An,
r in) = r = smallest
integer
>An,
uin)= u = 0 if « is even,
= 1 if » s 3 mod 4,
= 2 if « = 1 mod 4.
Define
f:
Zr —» Z[[z]] (the ring of formal power series
over the integers)
by
the equation
ruv •••, x,)=(1+ù*+hi+d- D!¿")"v n (^Yfe
and let '/"777d „1 .. •, x - ,) be the coefficient
of r1", that is
f%cv.. .,*,)« 1/^,...,*/.
777
We note that /"(x 1? • • •, x ,) is a polynomial
Theorem
B.
plex structures
dj,
There is a 1-1 correspondence
on CP"
n + 1. Moreover,
and the set of solutions
the almost
complex
• • •, a^,) has ith Chern class
is precisely
/"dj,...,
Inductively
of degree less than or equal to m.
the set of almost
corresponding
/"(x.,
after the nth term.
a homomorphism
g, : Z
—» Z by
(i) gj = identity,
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com-
• . •, x /) =
to the solution
/?(a j, • ••, a ,) so that the 'total
a /) truncated
we define
structure
between
of the equation
Chern class'
ALMOSTCOMPLEXSTRUCTURESON COMPLEXPROJECTIVE SPACES
(ii) gk(<*ii•" >ak) = gfe_, («2»*" ' ai) ~(k - I) gk-i(ai>'"
Theorem C.
plex structure
125
» ak-J'
The n-ple a = (a,, • • •, a ) is the Chern vector of an almost com-
on CP" iff
(1) a. = 7?+■1 mod 2;
(2) an = n + 1;
(3) gkisv sy..-,
s2k_l) = 0 mod2ii2k- IVX 2<k<r;
(4) if n is odd, g ,(«,,•••,
s ) = 0 mod u(nl);
(5) s,. = « + 1 for 1 < k < r;
where
s. = s .(a).
I\S
2. Preliminaries.
We recall briefly from [2] the functors
l\j
G and H and their rela-
tion with K-theory. The reader is referred to this paper for explicit and further details.
For X a finite connected
whose elements
based
CW-complex,
G(X) is the abelian
group
are formal expressions
1 + a. +••• + «. +•••
where a. £ H '(X)
with addition © defined by
(1 +íZj + ■••) ©(1 + bx + ...)=
where c. = a. + a._ybyr
1 + cj + •••
a._yb2 + •••+ bf
If £ is a C"-bundle over X with z'th Chern class
"total Chern class* of F
c.(E) £ H2i(X) then the
7(E) = l + cl(E) + ... + cn(E)
is an element
of G(X), and since if F is a Cm-bundle
c (F), we see that
c induces
a natural
over X c(E®F)
= c(£)©
homomorphism
c : K(X) -» G(X).
Then
c is an isomorphism
modulo finite groups.
(In fact a much stronger
is true [2].)
The image of c is the subgroup of 'vectors'
Chern class
of a vector bundle.
.
Let H(X) = ÍP W "(X) as an abelian
graded
result
which occur as total
group.
We give H(X) the structure
ring with product denoted by juxtaposition
of a
defined in terms of cup product
by the formula
(m + n)\
xJ* =:—TT-T x U*
m » imOinl) m
There is a natural
homomorphism
,
where
,,2i/„\
x. £ H'(X).
o: G(X) —» H(X) which is defined
in terms of
Newton polynomials with the property that if a £ H2(X) then oil + a) = (a, a2, • • •, ef).
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126
ALAN THOMAS
We define
s : K(X) —*H(X) to be the composite
morphism.
s = oc, which is a ring homo-
If F is a complex vector bundle over X, then the z'th term s. of
s(E —dim E) is related
to the Chern classes
s. - clsj_l
Both o and s are isomorphisms
c. by the formula
+ ... + (-l)kkck
= 0.
modulo finite groups.
3. Proof of Theorem A. We now consider
the case
X = CP".
Let H be the
line bundle over CP" defined by
/7 = 1(V, v) eCPnxCn+1:
where we think of CP"
v £V\
as the set of 1-dimensional
subspaces
of Cn+ . Let
= H —1 € K(CP") and let 77R be the image of 77under the 'realification'
77
map
R: K(X) -> KOiX).
The following
theorem
is well known.
(See, for example,
[3].)
'V-
3.1 Theorem. (1) K(CP") is generated by 77,77 ,...,
single relation
rf+
rj" subject to the
=0.
(2) KOiCP") is generated by powers of r¡R subject
(i) if n is even, (77R)r+1 = 0;
to the relations
(ii) if n ml mod 4, 2(77R)r+1
= 0 and ÍVrY'*1 = °,
(iii) if n = 3 mod 4, dR)r+1 = 0.
In particular,
mod 4.
If
K(CP")
is free abelian,
« = 1 mod 4, then
KO(CP")
and a copy of Z2, so has a unique
3.2
Lemma.
Proof.
finitely
Since
generated,
and so is KO(CP")
is the direct
element
G(CP")
is a free abelian
G(CP")
is isomorphic
so it suffices
provided
» 4 1
sum of a free abelian
of order 2, namely
group
(77R)r+ .
group of rank ».
to K(CPn)
modulo finite groups,
to show it is torsion
free.
Suppose
it is
a £ G(CP")
and k £ Z such that ka = 0. Writing a = 1 + af + higher terms where af £ H2riCPn) is
nonzero, we see that
ka = 1 + ka + higher terms
so that
k = 0 since
H2riCPn)
is torsion
free.
It follows that the maps
c : KiCP")- G(CP"),
are all monomorphisms.
a: G(CP")- W(CP"), 7: KiCP")-+ HiCP")
We identify
G(CP")
with Z" by means of the bijection
1 + ají + • • • + anr" h» dj,
• • •, a).
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ALMOSTCOMPLEXSTRUCTURESON COMPLEXPROJECTIVE SPACES
Let C = Im c, and S = Im s.
\a £ Z" such that
Then C can be identified
(s j(a), s2(a),...,
with the set of elements
s (a)) £ S}.
For k £ Z, let vk = Hk _ 1 e K(CP"). Then c (V;t)-Ufae
s(v,)
= U, 4 ,...,
127
/é"). Since i> • w. = v.
G(CPn) and
, - v. - v¡, and n = Vj, one can
show inductively that
T4-».-(ÎH-*GH-,--<-«4-'(/_i)''.
and hence
K(CP")
is generated
by f,»«..»f
. So 5 consists
of all elements
of
the form
öjs(vj)
i.e. those elements
+ a2s(v2)
+ • • • + ans(vn)
where a¿ e Z,
ib., • • •, b ) £ Z" such that the system of simultaneous
equa-
tions
öj + 2«2 + 3«3 + • • • + nan = b^,
äj + 2 a2 + 3 a} + ... + n an = b2,
■
a1 + 2na2 + 3nai + ---+nnan
has a solution
= bn
with a. integers.
3.3 Lemma, (i) fij, j2, j\ • • •, /') = ;!/(/ - i)l.
(ii)
The augmented
lent over the integers
matrix of the above system
to the matrix
M defined
of equations
is row equiva-
by
M..= 0 ifi>j,
= ///» 72>• • • » ñ
= /.(¿j,
Proof.
f°r i < i < ">
•••, b/> for /=72+l.
Part (i) is proved by induction
on i, starting
with /j(/)
= / = /'!/(/ - 1)!.
Then
/I+1(/>• • •, /*♦*)= ///2, • • •, ;i+1) - «y//, • ■• »/"')= (; - OffU • • • 7/')
using the fact that /. is a homomorphism,
and the result
follows.
Part (ii) is proved using (i) and by applying to the augmented
row operations
Since
which inductively
define
matrix those
/..
M.. = il by the lemma, we see that
S= \(b,, •••, b ): f.ib,, •.., b) = 0 modt! for 1 < i < tt.1
which completes
the proof of Theorem
A.
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128
ALAN THOMAS
For X a CW-complex, let Vect iX) be the set of (isomorphism
C-bundles
Vect
over X. If dim X = 2», then Vectn(X)
(X), then by stability
uniquely
trivial
properties
up to isomorphism
C"-bundle.
the natural
is an abelian
there is a C"-bundle,
classes
group.
of)
If E, F e
say E • F determined
such that F © F = (F • F) © n, where n denotes
The function
(F, F) t-» E • F defines
the
the group structure,
and
map
Vectn(X) —K(X),
E v-*E-n
is an isomorphism.
If E £ Vect 77(CP"), then ciE) e C, and this defines a function /:
Vect (CP")
'71
—»C. Conversely, if a £ C then there is a C"-bundle E with /(E) = a. If E' is a
second such C-bundle, then c (E - E' ) = 0 so E - E' = 0 in K(CP") and by stability,
E = E
and the isomorphism
class
of E is well defined.
This defines
a
function g: C —*Vectn(CP") such that fg = 1.
3.4 Lemma. The maps
f:
Vect 77(CP") -» C and g:
C — Vect 71(CP") are in1
'
°
verse
isomorphisms
(of abelian
This lemma together
is a C-bundle
bundle
problem
E
over CP"
over
CP"+S
to determine
groups).
with Theorem A classifies
we say that E extends
which restricts
necessary
to extend over CP"+Ä.
to E.
and sufficient
The following
C"-bundles
over CP"+S
over CP". If E
if there is a C-
It is an interesting
conditions
is a solution
and unsolved
for a Cr-bundle
in the simplest
case
over CP"
ir = »,
s= 1).
3.5
Proposition.
A C-bundle
over
CP"
extends
over
CPn+1
iff
f «(*!#•••»
* 77+1,) = 0modd+l)!
1
'77+1
where s 7. = s 7Xc), and c = ic.iE),
cAE), • • • , c 72(E), O).
14
Proof.
Necessity is an easy corollary
bundle over CP"+1 with c.ÍF)=CiÍE)
esis
of Theorem A. Let F be the C"+
and Lemma 3.4, there is such a bundle.
by stability,
has a nonzero
Over CP"
we can write
and then by Lemma 3.4 G = E by examining
section
over CP"
tion extends
to a nonzero
dimensional
bundle over CPn+
4. Almost complex
over CP"+
. So F=
, and E* restricted
structures.
F'©
to CP"
If X is a 2«-dimensional
e: KiX) = Vect ft (X) — H2"(X) > Z
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F S G © 1
Chern classes.
and since the Euler class
section
-
fot 1 < i < n, and cn+1(F)= 0. By hypothSo F
of F is zero this sec1 for some E' an nis E.
oriented
manifold,
let
ALMOSTCOMPLEXSTRUCTURESON COMPLEXPROJECTIVE SPACES
be the function which associates
nth Chern class,
4.1
elements
complex vector bundle its
that is, its Euler class.
Proposition.
correspondence
to an «-dimenáional
129
// X is a 2n-dimensional
between
the set of almost
oriented manifold, there is a 1-1
complex
structures
on X and the set of
x £ K(X) satisfying
(1) eix) = e(X) = Euler number of X;
(2) R(x) = TX - 2n in KOiX), where R: K(X)-> KO(X)is the realification homomorphism, and TX is the tangent
Proof.
If A is an almost complex structure then RÍA -n) = TX-2n
= c (A)= Euler class
n
Vect^Of)
of TX = Euler class
be the corresponding
TX, whence
element.
AR © 1 = TXffilso
which is the obstruction
4.2
bundle of X.
Corollary.
(1) H2'(X)
of X. Conversely
*
and eiA - n)
if x £ K(X), let A €
Then RÍA - n) = TX - 2n, so ^r^s^
AR = TX since they have the same Euler class,
to removing the trivial
// X is a 2n-dimensional
has only torsion relatively
line bundle.
oriented manifold such that
prime to (j - l)\,
(2) KO(X) has no 2-torsion,
then there is a 1-1 correspondence
between
almost
elements E £ Vect
Of) such that
_n
(a) c2.iE © E) = p¿(X) = itb Pontrjagin class,
complex structures
on X and
l<i<n,
(b) e(E)=e(X).
Proof. If x £ £(X),then R(X)= TX - 2n iff CRix) = C(TX)- 2n (where
C: KOiX) —♦ ¡(.(X) is the complexification
homomorphism)
since
KOiX) has no
2-torsion and RC is multiplication by 2. So RÍE ~n)= TX - 2n iff E © E - 2n
= C(TX)~ 2n, that is iff E © E and C(TX) are stably isomorphic. By a result
of Peterson
[4] under the conditions
bly isomorphic
A particular
example
4.3 Proposition.
(1) m/j,...
of the hypothesis,
iff they have the same Chern classes,
,wr,
of Corollary
4.2 is X = CP"
2 vector bundles
and the result
are sta-
follows.
for n 4 1 mod 4.
The kernel of R:£(CPn) -» KO(CP") is freely generated by
n even;
(2) w j, • • • , w >rf, n =. 3 mod 4;
(3) «7j ,• •• ,wf, 27/", « = 1 mod 4 where wk = Hk - H~k.
Proof.
This is an easy consequence
of the theorems
of Fujii on the structure
of KO(CP") ([5], [6]) and the fact that
(77—rj)(r] + rj) = w,
which is easily
j + (linear
proved by induction
starting
combination
of w., •••,
with k = 0.
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w,)
130
ALANTHOMAS
Let J denote the complex tangent bundle of the complex manifold CP" and
let T denote the real tangent bundle of the differentiable
manifold
CP".
Then
JR « T, and so if x £ K(CP"), Rix) = T - 2n iff
x =(J-
for some integers
») + ajWj
+ ••• + afwf+
uar,7]n
a j, « • • , a ,a ». For such an x,
(l + kt\
yi-kt)
cd) - (i+ty+Hi+d- i)!/")"v n
lsfesr
= /"dj,
• ••, ar»)
and so cd) = nth Chern class
= /^dj,-
that any almost complex structure
-• ,ar> ). So using
determines
/£(*!> "•»
Conversely,
if dj,-
so that Rix) = T and eix)=
For an element
x =(J-
iff sd)
equations
= su
+ ua ,77"
« + 1 so by Proposition
4.1 x determines
x e K(CP")
from Proposition
4.3 by methods
for some a¿
+ • • • + a siw ) + ua is(qn)
fot some a. iff the n
(l < z < «)
aj^siwy)
have a solution
+ • • • + ars{wj
with a¿ integers.
+ ar,uskrf)
= six)
= s2.(j-n)=»+
- s.CJ-
a)
Since
s(uzt) = 2d, 0, k3,0,k\0,
we see that s2.d)
similar
is of the form
n) + a.z^j + • • • + awf + uar,r¡n
- «) + ajsdj)
an almost
the proof of Theorem B.
The proof of Theorem C now follows
to §3.
of the equation
let
») + a ,z¿/. + • ■• + aw
This completes
4.1 we see
*r») = « + 1.
• • , a () is a solution,
x = ( J-
complex structure.
a solution
Proposition
...)
1. The equations
for i odd are
2dj + 2a2 + • • • + rar) = sA.x) -in + l),
2ia1 + 23a2 + • • • + r'a) = s}d) - (» + l),
2dt + 22r_1a2 + • • • + rtr-la)
= s2r_,d)
and if n is odd
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- (» + 1)
ALMOSTCOMPLEXSTRUCTURESON COMPLEXPROJECTIVE SPACES
2(«j + 2"a2 + .. • + rna) - uinDa^ = sn(x) -in
131
+ l).
Theorem C now follows from the following lemma.
4.4 Lemma, (i) g.(j, ;3,. • • , j2i~ l) = (/ + i - 1) !/ (/ - ¿)l.
(ii) The augmented
over the integers
matrix of the above system
to the matrix N defined
of equations
is row equivalent
by
N.. = 0 if i > ],
= 2g//,73,---,/2t'1)
fori<i<r,
= g is,, s,, • • •, s. . .)
&t- V
3'
for 2<i,
2«-l
= Sj-(w+l)
'
^ -
for im 1, jmf
and j = r +l\
+1
. .
> where s. = s.(x),
"
and if n is odd
/V..
ii = 0
if Ki<r,
— —
= - 2«(n!),
jmr't
'
i m j m r.
The proof of this lemma is similar to the proof of Lemma 3 with the additional
observation
that
gi(sl - (n + 1), s¡ - (n + 1), . • •, s2z_i - (« + 1))
= gzUj, •••, s2;_ ,_)-§,.(« + 1, 72+ 1, •••, n + 1)
- S/Sl' *' ' » s2i- P
= s, —(n + l)
for ' £ 2»
for 2 = 1.
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manifolds,
Advanced
North-Holland, Amsterdam 1968.
structures
on eight-
Studies
in
MR 40 #4972.
and ten-dimensional
manifolds,
MR41 #9283.
DEPARTMENT OF MATHEMATICS,UNIVERSITY COLLEGE OF SWANSEA,SINGLETON
PARK, SWANSEA,WALES
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