Complexes of Partial Differential Equations (Mathematical Monograph) 0300018878, 9780300018875

Citation preview

(

\

ALDO ANDREOTTI ,i

.

Complexes of Partial Differential Operators

I

I ,I

Yale Mathematical

Monographs

I I

Editorial Committee Nathan Jacobson Shizuo Kakutani William S. Massey George D. Mostow

6

Complexes

of Partial Differential Operators

by ALDO ANDREOTTI //1

New Haven and London, Yale University Press, 1975

Contents

Published with assistance from the foundation established in memory of Amasa Stone Mather of the Class of 1907, Yale College. Copyright © 1975 by Yale University. All rights reserved. This book may not be reproduced, in whole or in part, in any form (except by reviewers for the public press), without written permission from the publishers. Library of Congress catalog card number: 75-8440 International standard book number: 0-300-01887-8 Set in Times Roman type. Printed in the United States of America by Eastern Press, Inc. Published in Great Britain. Europe. and Africa by Yale University Press. Ltd .• London. Distributed in Latin America by Kaiman & Polon, Inc .• New York City; in Australasia and Southeast Asia by John Wiley & Sons Australasia Pty. Ltd .• Sydney; in India by VBS Publishers' Distributors Pvt., Ltd .• Delhi; in Japan by John Weatherhill. Inc., Tokyo.

Preface I. Elementary and Levi Convexity 2. On the Hans Lewy Problem

II

3. Complexes of Differential Operators with Constant Coefficients

19

4. Boundary Value Problems for General Complexes of Differential Operators 5. Complexes of Differential Operators with Variable Coefficients Bibliography

27 38

Preface

The lectures that comprise this monograph were greatly inspired by the work of Malgrange on operators with constant coefficients. I have presented a summary of some joint work with H. Grauert, C. D. Hill, S. Lojasiewicz, B. Mackichan, and M. Nacinovich (in the third chapter). The aim is to carry the classical results of the Cauchy-Riemann equations to systems of linear partial differential equations. I am not a professional "analyst". Therefore I beg indulgence from the reader for any lack of elegance, but I have tried to be clear. A.A. Corvallis. Oregon 1975

.,

1. Elementary and Levi Convexity

1. Elementary Convexity a) We consider R" as an affine space. Let n be an open set in R" and let cp; n ---+ R be a Coo function. At a point a E n we consider the Taylor expansion of cp cp(x)

= q>(a)

+ ~ox, cp(a)(xi- ail + j~ox,x, cp(a)(xj- aj)(xj - aj) + 0( II x - a 113).

The quadratic form H(cp).(v) = ~O"'j

cp(a)vjvj

is called the Hessian of sp at the point a E n. An affine change of coordinates on R" acts on H(cp). as a linear change of coordinates v

---+

Av,

det A#-O

and therefore the number of positive and the number of negative eigenvalues of the Hessian at a does not-depend on the affine choice of coordinates in R". A function cp on n is called q-conuex if at each point a En the number of strictly positive eigenvalues is ~ n - q. This notion is an affine notion. Let a E n be a point where (dcp). = O. Then any differentiable change of coordinates near a acts on H(cp). as the linear change of coordinates v

---+

J(a)v

where J(a) is the jacobian matrix of that change of variables. Therefore at a critical point of cp (i.e. where do = 0) the signature Hessian

is independent

of the choice of differentiable

of the

local coordina tes.

b) Let n be an open set in R" with a smooth boundary. By this we mean that there exists a Coo function q>; R" ---+ R with the following properties; i) n = {x E Rnlcp(x) < O}, ii) at every point Xo E an = James K. Whittemore

n - n, (do),

Lectures. May 1974.

#- O.

Elementary and Levi Convexity

2

This means that near any point Xo EOn, ({J can be taken among a set of local differentiable coordinates so that, locally, near Zo is diffeomorphic to a half space {({J ~ OJ. The boundary on of n is given by the equation {({J = O} and the tangent space at Xo to on has the equation

n

La,; Let us consider

the Hessian

La,." H«({J)I(v)T,,(OOI ::

1 La.,

({J(Xo) Vi =

of

({J

O.

restricted

d.

d({J =

d. (h

d({J

there exists a disk of dimension

affinely imbedded

h(xo)

dh. IjI) = h d. dljl

+

> O. 2 dh. dl/t

H«({J)IT,,(00) = h(o)H(I/t)ITx

+

I/t d. dh.

.

(00)

a

+

q(xo) ~ n -

1.

As an exercise one can show that there exists a disk of dimension affinely imbedded in RO: r : OP ------> R"

p(xo) ;;:, n -

I - k

=

{x E nl({J(x)


c , such that 0'" ::> K2 and set O2 = co~nected component of K2 in 0', . Continuing in this way we get the conclusion. Then we get the following , Corollary: (a) Any a-convex domain in the general sense in R" is elementary convex.

and has no torsion.

Proof It is not restrictive to assume that we have on n a q-convex, proper function cp: Q ---+ R having only nondegenerate critical points, because we can "approximate" as well as we like with first and second derivatives the given function exhibiting the q-convexity, with one having this property. At a critical point, the number of negative eigenvalues of H(cp)cannot exceed in number n - (n - q) = q. Thus the index i(p) of each critical point p is ,,;; q. Hence Q has the homotopy type of a cellular complex with cells all of dimension :0::; q. Consequently Hj(Q, Z) = a if i > q, and also for the same set of i's, Hj(n, K) = a where K is any field. We have also by the universal coetTicent theorem

0=

Hq+ 1(0, K) = (Hq+ 1(0, Z) ® K) EB Tor (Hq(O, Z), K).

Thus Hq(Q, Z) cannot have torsion. NOTE: Hj(Q, R) = Horn (Hj(O, Z), R) conjecture above.

=

2, Another proof of the theorem will be given later.

a if i > q, as expected by the

Elementary and Levi Convexity

6

Levi Convexity

7

I

2. Levi Convexity a) Let n be an open set in C", cp: n R a Coo function. Using complex coordinates Zl' ... , z, and theirs complex conjugates, the Taylor expansion of cp at a point a E n has the form cp(zl

=

cp(a)

La, cp(a)(z. Lo z.zp~(a)(z. tLai i cp(a)(z. Lo,;6cp(aHz.-

+t

+

+ The hermitian

+

• p

quadratic

- a.) + LOi cp(a)(z. - a.) a.Hz,6 - ap)" a.Hz,6 - ap) a.)(z,6 - a,6) + O(llz - aI13).

v

is hermitian form in (n - 1) variables whose signature is independent of the choice of local hulomorphic coordinates near Zo and of the choice of the defining function cpo If p(zo) is the number of strictly positive eigenvalues of .2"(CP)T" Po) and q(xo) the number of strictly negative eigenvalues of 2'(cp)I a '''.:ve have T 'A

=

---->

Moreover

LOz; ./1 cp(a) v, ifl'

+

q(xo) ~ n - 1.

there exists an analytic disk of dimension r : DP

near a acts on 2'(cp).

with a

(DP = {t

E

CPILltil2

< I},

r

holomorphic

+

0(1Iz'113)

(so that 2'(CP)a = H(cp).). b) Let n be an open set in en with a smooth boundary and a defining function cpoAt each point Zo E an one can consider the analytic tangent space

T~,(an):

which is the maximal linear complex subspace contained in the real tangent space. As in the case of elementary convexity we now have the following statement. At every boundarv point Xo E on and for any choice of the defining function

O.

And more generally one has the following T lieorem: (see [2]), If" a is a q-conuex open set in en (in the general sense), 1/)('/1

[or all i

H i(r2, (n) = 0

Theorem: Let a be a O-convex open set in en with a smooth boundary. Then open set of holomorphy. Proof (ex) Let {x.} c n be a divergent sequence. If {x.} is divergent in en we can find a polynomial p with s1;lplp(x,)1 = 00. We can therefore assume that {x.] is bounded in en and, passing to a subsequence, that 1i!TI x , = Zo E aa. Let a = {z E enlcp(z) < O}, where cP is a defining function for a. We can find a coordinate neighborhood U of Zo such that, on U,

a is an

cp(z) = Re f

+ .2':(cp)zo(z) + 0(1IzIl3), where f is holomorphic.

Thus {z E UIf(z) = O} (l n = I), as the Levi form is assumed to be positive definite at zo, provided U is taken sufficiently small. Also replacing tp by eC~- 1 with c > 0 large, we may assume that !t'(cp)zo is positive definite in U (provided U is sufficiently small). Let Q: U ---+ R be a Coo function with the properties

> q.

o
of ~ we have 0---+

r(XI, ygq) ---+

exact with complexes (y) Apply We get, as sequence

HO(XI

U

X2, .F)

...

F(X, ygq) EElr(X2, ygq) ---+

...

is a flabby resolution

r(XI

(l

X2, ygq)

------>

0

obvious definition of the maps. Tnis gives an exact sequence of whose cohomology sequence is the Mayer- Vietoris sequence. the Mayer-Vietoris sequence to 11 = a u V and to the sheaf (I. H 1(11, (!) = 0 by the theorem of Cart an and Serre, a short exact

Elementary and Levi Convexity

10

----.-----------------In particular I/f is holomorphic functions h nand h, such that

hn

-

in

h,

n

= I/f

1\

V and we can find holomorphic

on

2. On the Hans Lewy Problem

n 1\ V

i.e. hn = I/f This shows

n

that

li!1llhn(x,.)1 =

+

h..

t. Introduction

00.

Remark. The theorem can be greatly generalized to show that an open set in C" with a smooth boundary satisfying the condition of Levi's theorem

is actually an open set of holomorphy (solution of Levi problem). First one shows that Levi's condition implies that n is O-convex in the general sense. Then one realizes that n is the union of an increasing sequence of O-convex open sets, i.e. of open sets of holomorphy. Then, by a theorem of Behnke and Stein, itself must be an open set of holomorphy. Note that to show that a O-convex open set in the general sense is an open set of holomorphy is an immediate consequence of the theorem of Cartan and Serre for coherent sheaves (apply the theorem to the exact sequence of sheaves a -----+ " -----+ (!) --+ TTCx,.--+ a where" is the sheaf of germs of holomorphic functions vanishing on the points of a divergent sequence {x.} en; f is a coherent sheaf). b) By a tube-open-set in en = R" EB iR" we mean an open set n of the form

n

n=

t» x iR",

where co is an open set in R". The following theorem is elementary and due to Bochner and Martin [8]. Theorem: Let 0 = co x iR" be a connected tube in Co. n is an open set of holomorpliy if and only if w is convex (cf. Horrnander [14], p. 41). As a corollary one can deduce a proof of Hadamard's theorem. Indeed let w c R" be open connected with smooth boundary and a-convex. Then 0 = w x iRn is an open set of C" with a smooth boundary and (as one easily verifies) Levi O-convex. By the solution of the Levi problem, n is an open set of holomorphy. By the theorem of Bochner and Martin we then must have w convex, as we wanted.

n

an

a) Let be an open set in R" with a smooth boundary and let us denote by i: --+ Q the injection map. _ Let us denote by 0"'(0) the space of Coo differential forms of degree r on 0 (i.e. with coefficients having continuous partial derivatives .of every order up to the boundary of 0). Let us denote by ','(00) the similar space for the

an

manifold 00. We have a natural

r'"

map i* : 0"(')(0)

--+

0"(')(00)

which is compatible with the operators of exterior differentiation in 00 (d,'n i* = i*do) and thus one obtains a natural map I.)

H'(O, R)

--+

H'(aO,

in nand

R) 'V r.

More generally given an open set U c R" and a Coo function (1: U --+ R, setting = {x E Ule(x) ,,;; O} and assuming de i' a on the POInts where e = 0, we get for the boundary of 0 in U a smooth hypersurface and a natural map (*) in cohomology. . This map is however locally trivial in the sense that we can find for every point x E a small connected neighborhood U(xo) such that when we replace 00 by Q 1\ U(xo) and by 1\ U(xo), the map (*) reduces to o --+ 0 if r > 0 and R ~ R if r = O. b) When we consider the analogous situation in en with respect to the operator (1 of exterior differentiation with respect to anti-holornorpb ic coordinates, the situation is much richer and not so trivial. This IS what we will try to explain in the sequel. Note that now has no complex structure and thus the previous argument breaks down from the start.

n

on

on

an

an

on

2. The Boundary Complex Although the situation for a complex manifold,

can be described without any substantial change we will restrict our attention to open sets in en. II

12

On the Hans Lewy Problem

Let U be an open set in C" and let II : U U+ US

= =

=

°

---+

R be a Coofunction. We define ~ O} ~ O}

{zEUlll(Z) {zEUlll(Z) {zEUle(z)

= O}

and we will assume that dQ of- on S so that S is a smooth hypersurface, On U we can consider the Dolbeault complex Ch(U)

=

{CO(U) ~

CI(U) ~

C2(U) ~

... },

where C'(U) denotes the space of Coo exterior form of type (0, s) on U and usual exterior differentiation with respect to anti-holomorphic coordinates. Analogously one defines the complexes C*(U+) and C*(U-). Define

o the

.I"'(U)

=

{cp E CS(U)lcp = ea.

+ oe

1\

p, a. E C"(U), P E C"-I(U)}.

We have oJf'(U) c Jf' + I(U) so that J*(U) = ,JlJ'(U) is a subcomplex of C*(U) (and indeed a "differential ideal"). Similarly, one defines the complexes Jf*(U±). Finally one can consider the quotient complex Q*(S) = {Qo(S)

---+

QI(S) ~

Q2(S) ~

... }

The Boundary Complex

NOTE: Hf(U) = Hf(U, (I)), but this is no longer true for U± (unless one defines on U ± a sheaf (I) ± with some special behavior on S). Remark. We have JO(U) = eCO(U) so that QO(S) ~ tS'(S). Then if f E QO(S) is the restriction of a holomorphic function on U to S, we must have Os r = 0 (tangential Cauchy-Riemann equations). This system of equations can be also described by

dll\ dz ,

Jf*(U)

---+

1\

Q*(S)

---+

O.

Note that the quotient complex Q*(S) is concentrated on S and that actually, locally,

II ;:

H*(U) = the cohomology of C*(U) H *(U ±) = the cohomology of C*(U i) H*(S) = the cohomology of Q*(S).

= 0

x4

(x/ + x/)

-

=

z'l) -

1;(Z2 -

IZ111.

Then S = {e = O} is the product of the paraboloid x4 = x/ + x/ in R3 and the X3 - axis . At each point of C2, dZ, and Oll = - 1;dz,1 - z,dz', can be taken as a basis for (0, I)-forms. Thus we get QO(S) ~ tS'(S) QI(S) ~ tS'(S) 1\ dz', Ql(S) = 0 so that tS'(S)

1\

dz,

----+

O}.

To compute explicitly Os we have to do the following (according to the definitions). Given u E tS'(S) choose an extension ii, Cooon Cl. Evaluate aii and "restrict" the result to S, i.e. computing modulo on the points of S. We can select ii independent of x4 so that

oe

(tS'(S) = Coofunctions on S) as one verifies taking a basis for (0, 1) forms, containing QQ. Moreover, either by direct verification or by use of Peetre's theorem, one verifies that the operators Os are differential operators. It may be remarked that one can also define the quotient complex by using the complexes C*(Ui) and J*(U±), but the resulting complex is always the same. We can thus consider four types (at least) of cohomology groups

dznls

1\

os.

Q*(S) ;: {tS'(S) ~ C*(U) ---

..

where I is any Coo extension of f to U. Similar interpretations hold for the other operators Example. Let U = C2, ZI = XI + iX2' Z2 = X3 + iX4' and take

defined by the exact sequence 0----+

13

"Iu-

a

(aii -

__

az'1

-

2'lZ

aii)

lax3

d-Z

-

2'l-aQ aii

1\

dZI'

"I

az'l

I

and "I

usu

=

(

au OZI -

2' au) IZIax3

In conclusion, the complex Q*(S) is isomorphic to tS'(R3) ~ where L ZI

=

XI

- iz1-l- is the Lewy operator (on R3 ~ S when we select aZI uX3 i Xl and X3as coordinates).

=~ +

tS'(R3) ------+ 0

14

On the Hans Lewy Problem

(*)

a) A Coo function f E $(U) is called flat on S if it vanishes on S with all its partial derivatives. If ff(U, S) is the space of flat functions on S, then 6"(U)/ff(U, S) = CarS) is the space of COO"Whitney functions" on S. We set = {cp E

I the

C(U)

iL

ff*(U)

C*(U)

--+

--+

C*(S)

--+

O.

Note that one has the exact sequence: (I) 0 -+ Jf*(U)

ff,U)

-+ C*(S) -+ Q*(S)

--+

0

.

This will enable us to relate the colomology of C*(S) to H*(S). One has the following lemma. Lemma. For any choice of V and S the sequence

-L

-L

0--> yo(U) .l: yl(U) y2(U) yO(U) .~ I(U) y2(U) is an exact sequence. Proof (a) Let u E yo(U) and assume that u is flat on S. Then u E Jfo(U) implies u = Q al for some al E CO(U) and the assumption 0 u flat on S implies

oQa Thus al

=

QIX2for some a2

E

I

=

=

5

=

fk for k

Now let f E .P(U), s ~ 1, thus f = ea one can construct 131 E CS-I(U) with

PI Is

=

+

oe " p. Using the above remark

+ oQ " 13I

Qa I

Q(al

=

(131)

-

+ iQ2f32 + {Q3/33)

f - 0(Q/31

=

=

m -

00

cp=E-Q-fJ I

m

+

= 0,

0, so that Yz

so that

= QCl.3

+

Q3(1X-3 i/33)'

Proceeding in this way we get a formal power series in

Q

1

m

m

I Qk

k

13k)

=

Qm +

I

I'm

+

I'

Using the remark at the beginning we can construct a g e CS-I(U) such that

aQ" I = iJ'cp aQ" I = ~ k!

0, I, 2, . ..

+ oQ " /31' one realizes that

Qal

Set ')'2 = a2 - i0f32' By the assumption oQ " 1'215 oQ " 133, with 133 verifying (*). Then

akg

=

=

= 1,2, ....

and set 1'1 = al - 0131' By the assumption we get oQ " 1'1 Is 1'1 = ea2 + oQ " 132, and we may assume 132 verifies (*). Then f - O(Qpl + ie2f32) = Q2(a2 - i0f32)'

OonS

Pis

=

f - o(ef3d

f - 0(7 Q2 a2• But then

for k

with 131 verifying condition (*). Let us now assume that 0 f is flat on S. Write

0 on S.

implies '1.2 = ('IX) for some a.l E CO(u) so that u = Q3aJ' In this way one show that u = QklXk. ak E CO(U) for every k, thus u E Y(U). (fJ) One makes use, to treat the general case, of the following fact. Given 011 S a sequence fo, fl' f2, ... of coo functions there exists a Coo function F 011 V such that

VI

f

with 13mverifying (*) so that

=

0

=

Setting f = (f - 0 Q " 131) + oQ " 131 every f E Y"(U) can be written as

°

CO(U), hence u

OQI "a2

akF

5

coefficients of qJ are flat on S}.

We have Oy' (U) c y' + I(U) so that ff*(U) = ff' (U) is a subcomplex of C*(U) and also of Y*(U). One can then con;ider the quotient complex C*(S) defined by the exact sequence

o -->

a f31 I V k

3. Mayer-Vietoris Sequence

s :(U)

15

Mayer- Vietoris Sequence

s

13k + I for k = 0, J, 2,

s

Set v = Qg then Q v E y,-I(U) and f - OV is flat on S. b) One can now prove that we do have always all exact S('(/"eI1CI' (M (/.\'('/'Vietoris sequence) 0--+ ---+

. Proof

o

HO(U) ---+ HO(U+) EDHO(U-) ---+ HO(S) --+ HI(U) ---+ HI(U+) EDHI(U-) ---+ HI(S) ---+

We have (with obvious notations) a short exact sequence ---+

C*(U)

---+

C*(U +) EDC*(U -)

---+

C*(S)

---+

0

from which we get a Mayer- Vietoris sequence with H*(S) replaced

by

16

On the Hans Lewy Problem

H*(C*(S), c). But by virtue of the Lemma and exact sequence (I) we do have H*(C*(S), (1) ~ H*(S). c) One can replace in all considerations developed above C'" functions with Ceo functions with compact support. This gives the Mayer-Vietoris sequence with compact support

o

---+ ---+

H~(U) H~(U)

---+ ---+

H~(U+) ffi H~(U-) H~(U+) ffi H~(U-)

---+ ---+

H~(S) H~(S)

---+

HO(U-)

By Serre duality

-----+

HO(S) -----> H~(cn).

H~(C", (9) ~ H"-'

(C", (9)

= O.

This theorem is duc to Bochner, Fichera, Martinelli, and Severi (see [7], [10], [2l]). NOTE: The same argument can be applied to obtain the same theorem on any Stein manifold (or on any (n-2)-complete manifold). b) Equations without solutions. Let us go back to the example above which gave us the Hans Lewy equation. Let a E S and let U be any domain of holomorphy containing a. As H'(U) = 0 = H2(U), we get in particular HI(S n U) ~ HI(U+) ffi H'(U-). Let U be a coordinate patch, for instance, and let U+ denote the convex piece with U- denotes the concave piece. We make use of the following. Lemma. Let n be an open subset ofC2 - {O} containing a closed haifsphere L = Consider the

0 - closed,

{lz,12 + IZ212 =

O' frO) =

~J (2m)

f ({) dZI dZ2 (Martinelli formula).

s

•

(fl) Assume that LC.e. = O.Thus J1 == LC.({), = (1", on n. From the Martinelli formula applied to S = {L 1 Zj 12 = s] we get L c ~ O'f(O) 'a!

a) Extension theorems. Let U = C" and let S be a closed compact hypersurface such that C n - S consists of two connected components. Let U+ be the unbounded piece and U- the piece bounded by S. If n ~ 2 then any Coo function f on S satisfying the compatibility conditi~ns csf = 0 is the trace on S of a function Ceo on U" and holomorphic on U-. Proof By the Mayer-Vietoris sequence with compact supports we get an ex act seq uence ----->

e.

They define cohomology classes E H 1(0, (9) which are linearly independent. Proof (a) One uses the following remark. If S is any sphere centered at the origin in some open set U 3 0 and if f is holomorphic on U then

---+

4. Applications

o

17

Applications

For small positive a we integral over S n {Re z, The first integral, if a is Thus by Stokes' formula

J s f J1 dZI dz

= ~21.)

2•

m

can break up the integral on the right side as an > - a} and an integral over S r. {Re z, < - a}. small, is extended over a surface contained in n. we can write it as

J f", dZI dZ

2

(l{S

r'I

1 Re

We can find a closed rectangle

1.1

> - /)'fl

Q containing

a(S n {Re z, > - a})

u (S n {Re ZI < - a})

but not containing a small closed ball B centered at the origin. Given any holomorphic function g on a neighborhood of B, we can find a sequence of entire holomorphic functions I,such that f, ---+ g uniformly on B, I, --+ 0 uniformly on Q, because B u Q is Runge in C2. But then, by the previous remark, we deduce that L

c.,a. 0' g(O) 1

=

0

for every g holomorphic in B. This implies c. = 0 V iX. Corollary: dim., H'(U-) = 00 (for any coordinate ball U centered at aJ. It follows that we can find a fundamental sequence co., v = O. J. 2. of neighborhoods of any point a E R3 and for each v a C" [unction f, E 6"(wo) such that the equation

e, Re ZI ~ O}.

(0, l)forms

has no solution u in w,.. We can improve on this statement, showing that we can find f E tS"(wo) such

18

On the Hans Lewy Problem

that the above equation has no solution u in any smaller to showing that the H. Lewy complex on R3 tff ~

tff ----+

WV'

This amounts

3. Complexes of Differential Operators with Constant Coefficients

0,

where tff denotes the sheaf of germs of coo functions, does not admit the Poincare lemma in dimension 1 near any point a E R3. Proof Set w = Wo and let us consider the following diagram tff(w) ~

Ii,

E,.

tff(wv)

1. Stating the Problem

jL ------>

tff(wv)

where E,. = {(a, (3) E tff(w) x tff(w)lr,a = Lf3} and where r , denotes the restriction map. Each space tff(w,) has a nautral structure of a Frechet space and the maps r, and L are continuous. Therefore E, has the structure of a Frechet space and i, = pr,,(W) is also continuous. By the Banach theorem, either i,.(E,.) = tff(w) or i,(E,) is a set of first category. The first possibility is ruled out by the previous remark. Thus /" (E ) is of first category and therefore also ,!;hi,. (E,). It follows then that one can find f E ~(())o) - ,VII/" (E,). Th is answers our question. Concludinq remark. The example studied in this chapter is due to H. Lewy (see [18J). The methods we have employed help to decide whether or not the complex Q*(S) associated with the tangential Cauchy-Riemann equations presents a phenomenon similar to that considered by Lewy, i.e. the absence of the Poincare lemma. This is what one could call the Lewy problem. Exercise. What information gives you the Mayer- Vietoris sequence in the example we have discussed for cohomology in dimension zero?

Let 0 be an open set in R" and let tff(O) be the space of Coofunctions on 0, '@(O) the subspace of tff(O) of those functions with compact support in O. We set tff5(0) = tff(O) x ... x tff(O) s-tirnes and similarly for .@S(O). Given a matrix A

=

(a ij

(~)) 1 " I ~

~l' ...

~n'

i " p j ~ q

with polynomial entries -inthe variables

a~. =

replacing ~ j with the symbol

D, we get a differential operator

I

A(D). We can consider A(D) as a map (0)

tffq(O)

A(D)

I

I,'P(O).

One is led to consider the system of partial differential equations (1) A(D)u

=

f

for f E

tffP(O)

and

Remark 1. If(1) is solvable and if QW matrix such that (2)

Q(D) A(D)

u =

E tffq(O).

(Q1(~), ... Qp(~)) is a polynomial

= 0, then necessarily Q(D)f = 0.

o.«: Remark Z, If (I) is solvable

and if Q( () ~ (

:

o,«:

>"

pol yn ornial rn atrix

such that (3) A(D) Q(D) == 0, then for any v E tff(O), u

+

Q(D) v is also a solution of (I).

Obviously the solutions Q(~) of (2) form a module over the ring q> of poly19

Complexes with Constant Coefficients

20

nomials and thus we can find a finite basis. This will give an operator such that (4)

1,"(12)

gq(n)

~

A(Dl,

6'P(12) ~

0--+ !!JP'

qD)

6"(12)

We thus have to consider the problem of inserting a differential operator AWl into a complex of differential operators having a Poincare lemma. 2. Hilbert Theorem (Forward Resolution) a) A satisfactory answer to the problem stated above is provided by some theorems of Ehrenpreis, L¢jasiewicz (for an essential step) and Malgrange (for the most general form of the statements). We limit ourselves to stating basic theorems of this theory. Theorem A: Let n be an open convex set ill R". The necessary and sufficient condition for the system (1) to have a solution u E ,g'q(D)for a given f E ,g'P(D) is that :for any polynomial vector Q(~) such that Q(~) A«) == 0 we should have also Q(D)f = O. From this one easily deduces the following. Corollary: The necessary and sufficient condition for a sequence of differential operators

to be exact on any open convex set 0 c R" is that the sequence of :?J-homomorphisms

(.31' = C[~l'

theorem: Let &

C[ ~I'

=

... ,

~nJ. Given

a &-homomorphism

!!JP'

one can find an exact sequence of &-homomorphisms

is a com plcx .. as exact as possi ble."

be on exact sequence

Hilbert's

&p, ~

,g'Q(!1) A(D), ,g'P(!1) 8(0), ,g"(!1)

is a complex "as exact as possible." Similarly for the solution Q(~) of(3) which will give another operator such that (5)

B(D)

21

Hilbert Theorem (Forward Resolution)

... ,~"]).

The sufficiency of the cond ition is a direct conseq uence of Theorem A. The necessity should have additional argument. However, we will omit it as only the sufficiency part is of practial interest. b) By virtue of the above corollary the problem to continue the differential operator A(O) to the right with a complex of differential operators with constant coefficients is reduced to a purely algebraic problem, and the solution is provided by a classical theorem of Hilbert that we state in the following form.

.9P,.,

~

--+

...

:?JP'

~

.9P,

~

.9P,

~

of length d ~ n (d = 2 ifn = 1). Taking thus for IXIthe homomorphism given by the matrix tA(~) = 'Ao(~) (so that Po = p, PI = q) and representing the homomorphisms ai' i > I, by polynomial matrices tAi(~)' we then obtain a complex of differential operators ,g'P'(!1) Ao(Ol, ,g'P'(!1) A,(O), ,g'P'(12)__

...

AiDl,

I,r'(n) ~

0

which is exact whenever 0 is open and convex. c) One may ask what the situation will be if we replace the space ,g'(D) with the space ~(!1) of functions with compact support. Again from the theory of division of distributions one can derive the following. Proposition. Let 0 be open and convex in R". The necessary and slIfliciel1t condition for the sequence of differential operators

to be exact is that the sequence of :?J-homomorphisms &Q

AW,

&p

8(~),

:?J'

be an exact sequence. In particular from the Hilbert resolution for which we have IXjrepresented by a matrix Bj(~)' one derives the complex o __

~p,(D)

8d(0), ~P,.,(!1) Bd_,(£?) ...

which is exact on convex open sets. This situation is actually the "dual" as Bj(D) = 'Aj(D).

82(D), gp,(D)

situation

8,(0),0}P.(D)

of the one treated above

3. Backward Resolutions Let us consider the operator

o

--+ .9P, ~

where IXI = 'AI(~)' PI

!!JP'"

=

A(O) and the corresponding

--+

p, Po

=

...

--+

.'j'P'

glP"

~

q, and where N

=

Hilbert resol ut ion ---->

N

Coker (XI'

---+

0

22

Generic Koszul Complex

Complexes with Constant Coefficients

23

If it is at all possible to extend the corresponding complex of differential operators on the left (keeping the Poincare lemma on convex sets) ...

->

gP-'(il)

__

tS'Po(il) A(D4 tS'P'(il) __

...

__

tS'P'(il)

->

we realize that it is necessary and sufficient that N be imbeddable in a finitely generated free module f/JP'. This is equivalent to saying that N is free of torsion: T(N)

=

(XI = (xz, xt, yz, yt),

0,

0

--+

y

0

,

I

~ _ ~ _ ~

and where N = Coker (XIis such that T(N) = N. Correspondingly one has the complex of differential operators (1)

tS'(Q) ~

tS'4(Q) ~

02

%t

oxoz 02

oxot 02

oyoz

given by

02

oyot A3(0)

= NI and so forth. One arrives

Theorem: The necessary and sufficient condition that the finitely eP-module N be included in an exact sequence of length k --+

= (OIoy, -

%t,

-

0

->

o

-%z

a/oy

o

-%x

o

o

%t

o

a/ay

o o

o

a/ax, a/oz).

Complex (1) is exact on open convex sets. Note that the kernel of A I(0) is the set of functions g : il --+ C of the form g = a(x, y)

~;merated

+ b(z, t)

with a, b, Cooin their respective variables. f/JP-.

of stable maps is that i) (or k = 1, r(N) = 0 ii) 'tor k = 2. r(N) = 0 and N = N*'" (i.e. N be reflexive) iii) for k > 2, N = N** and Exti(N*, f/J) = Of or 1 ~ j ~ k - 2. This theorem can be found in a slightly different form in Palamodov [24]. Therefore the theorems of Hilbert and Palamodov provide all the necessary information to include A(O) in the longest possible complex of differential operators with constant coefficients keeping the Poincare lemma. Example: Let n = 4, f/J = C[x, y, z, t]. One has the maximal Hilbert resolution

where (Xiis given by the matrix

tS'(il)

tS'4(Q) ~

where

f/JI

(stable imbedding). We can then repeat the argument for f/JI/N at the following

0 0

(

where T(N) denotes the torsion module of N. If rfN) = 0 then we do have several ways of imbedding N into a free ':1'-module. The best (for reasons that will be left obscure) is to consider N* = Hom(N, eP), and to take a basis (XI ... (X1 of N* and consider the imbedding N

(X2=

t

-z

4. Generic Koszul Complex

,

,I

a) It has been said that "in natural phenomena one docs not ask what happens in all cases but what happens in the majority of cases."( I) Accepting this point of view one can be satisfied to have an explicit resolution of a generic polynomial homomorphism f/J' ~

represented by a matrix A(~)

=

(a ii W)

f/Jr 1 e i " r . j ~ s

I E;

1. "In naturalibus non quaeritur Aquinas, Summa Theologica.

quid semper fiat sed quid in pluribus accidat."

Thomas

Complexes with Constant Coefficients

24

As the matrix is generic, it will have maximal rank (over the field of rational functions in the variables O. We will first assume r ~ s. We want to make precise the notion for A(e) to be generic and to state for generic A the corresponding Hilbert resolution. b) Let.lt, x ,(C) = Home (C', C') be the space of matrices of type r x s with entries in C. The group GL(r, C) x GL(s, C) operates on .It, x ,(C) by (a, p, M)

----+

aMp(a

E

GL(r, C), M

E

vI(, x .(C),

P E GL(s,

Jo

U

JI

u J,.

U ...

.?4~ = space of homogeneous polynomials of degree k in the indeterminates y I> ••• , y, and with coefficients in .9Ih. Thus an element E .?4~has an expression

P

P

=

r

1.1 - k To each row of the matrix

P•...•

CPI

C)).

The space . ;{{r x s(C) gets partitioned into orbits according to the action of this group, one for each value of the rank of M. Denoting by J" the orbit where the rank equals r - e (i.e. drops bye units) we thus have .It,x,(C)=

25

Generic Koszul Complex

CPr=

By exterior

multiplication

,

A we associate

=

arl

dt,

with

CPi

(P., ...• , E .s;Ih).

the exterior

+ ... +

dt,

all

+ ... +

al•

ars

l-forrn

dt,

dt,.

we get a morphism

d( ~ Finally,

y;'

YI·' ...

'

.91(+

I.

if we introduce

One easily verifies that

J

Q

=

Jp u J, + I

U

u J,

....

is an algebraic irreducible variety of codimension e(s - r + e). Given a matrix A(e) of type r x s with polynomial entries one can consider the map a:

en

-----+

vI(,x,(C)

We define a map ---+

the map &i" ~

We can identify

,.",h + I

\l

,.",h

""k

""k-I·

&i" with the map "

/\ 'Pr,

N

dr+1

------>

~

E

O}, U- = {x E {x E UlQ(X) = O}

UI(!(x) ~ =

and assume d (! #- 0 on S. Let Ej be a sequence of differentiable Ej = U x cPr will be the trivial bundle

.wl

(s,tO)' ~

~::::~-2 ~

U+ = {x

S

that one could also write as

0------> (do)'

Our purpose is now to extend to general complexes of differential operators (with constant or variable coefficients) the considerations developed in chapter 2 about the system of Cauchy-Riemann equations. Again we restrict our attention tok", although all considerations can be carried over to any differentiable manifold. Let U be an open set in R" and (! : U ------> R be a Coo function; we set as usual.

fJJs

qJ' ~

given by (u.; ... , u.) -----> With similar argument setting D = L Yj ({Ji'

1. Preliminaries

the following ~r+2

~

exact

sequence,

We assume operators.

~ Each

where N is the co-kernel of the last map D. This could be called the generic co-Koszul complex, from the previous one by application of the function

as this was obtained

Horn, ( . , f!J).

oj,

locally,

==

{is'°(U) ~

is expressed

of spaces

is'l(U)

~

by a matrix oj

==

O},

vector bundles over U. In general of rank Pj' We set

= space of Coo sections

that we do have a complex

is'*(U)

0

qu, Ej)

UIQ(x) :(

of Ej over U.

is' j(U)

and partial

is'2(U) ~ of partial

(L a,(x)hk D'),

...

differential

differntial

}. operators

~

h s; r I ~ k ~ Sj

with coefficients a.(x)hk Coo on U. By assumption we have oj+l

Dj = O.

If ffj(U, Ui) == {s E is'j(U)ls == 0 on Ui}, we can define gj(U±) = gj(U)/ ffj(U, Ui) and we obtain in this way two other complexes 6"*(U ±). 27

Boundary Value Problems

Examples. I) One has the classical de Rham and Oolbeault complexes. More generally one has all complexes that come from a Koszul complex. For instance, if( ... , en' In particular GgJ is a Noetherian ring. It follows then that the ring gJ itself is a Noetherian ring (as left and right module over itself). Given a matrix A(x, D) of type PI x Po with entries in qJ we can associate to it a qJ-homomorphism

by sending each vector v(x, D) E gJ P', regarded as a horizontal vector v(x. 0) (vl(x, D), ... , vp, (x, D)) into the (horizontal) vector v(x, 0) A(x, 0). In this way we see that the set of integrability conditions of Alx, 0) is the ~-module kernel of the gJ-homomorphism A. As ~~, is Noetherian, that gJ-module is finitely generated and therefore we get an exact sequence

D} c!'(n),

Continuing in this way one obtains a resolution going backward from the ~-module N.•. = Coker A. It can be shown that the resolution may be chosen of length d ~ n, (d = 2,dn = 1): (1) 0

---+ ---+

38

gJk = gJ,

=

ii) if u is a solution of the considered equation, then, for any co-integrability condition S(x. 0), u + S(x, 0) v is also a solution for any v. To find all integrability and co-integrability conditions is to put the operator E inside a complex c!'(n) ~

V

a(a b) = a(a) arb),

(integrability conditions).

ii) Find all operators S(x, D) such that

E(x, D) u(x)

c: gJ2 c: ... , gJj gJk c: gJj + k'

Therefore the associated graded ring G~ is defined and, as

Q(x, D) E(x, D) == 0

E(x, D) S(x, D)

[DI' ... , Dn],

be ring of partial differential operators with coefficients in . Clearly ~ is not a commutative ring, because for 0: E , [D, 0:] ¥- 0 in general, for all j's. If a E ~, a = L a.(x) D· and for some 10:1 = k, a. ¥- 0; k is called the ,., " k order of the operator a and ,., ~ k a. = u{ (a) the symbol of a. If we set ~k = {a E ~I order of a ~ k}, we get a filtration of ~ gJo = c: ~l

one is led naturally to the following questions. i) Find all operators Q(x, D) such that

be the ring of formal power series centered

gJP, ~ ~p,-, ~ '" ~p, N.•. ---+ O(d ~ n)(see [6]).

~

gJP' ~

qJ"'"

From this we obtain a complex of differential operators on formal power series

Complexes with Variable Coefficients

40

is exact (~,. Similarly the research of the co-integrability conditions leads to the problem of imbedding NA in a finitely generated free !'i)-module and to choose the imbedding wisely so as to be able to extend the resolution (1) to the right as much as possible. This will provide us with the formulations of the analogous problems solved for operators with constant coefficients by Hilbert's and Palamodov's theorems. b) Let us assume that (1) is an exact sequence (and possibly also extended to the right as much as possible). We can ask the following question: Is then the corresponding sequence (2) an exact sequence (i.e. is the formal Poincare lemma true)? In general the answer is no. Exumpte. In I variable x the complex

O.

as follows in the case of a nondegenerate

Xju

W(U+)

(the other

groups

are zero).

H

1\ HO(S)

(n odd) HP(U+)

(the other groups

are zero).

1 W(S) ~ W(U+)

By the argument used in the second degenerate Levi forms one has =

= 0

(a~j) E .g'(0) )

with the following properties: (IX) The vector fields X J are linearly independent ofO. (P) The system (1) is in involution, i.e.

(y) The vector fields Xl' ... , X" ~I' (thus

0 ..; I ..;

n) at each point

'.Xj ,; IYj

EB

W(U-)

U-

dimcHP(U+)

.-1'

(k:j

E

over C at each

point

.g'(0) ).

... , ~I are linearly

independent

of t».

Two local structures (XI' ... , XI) (YI, ... , YI) on two open sets nl, O2 in Rn+1 will be considered equivalent if there exists a diffeomorphism t : 01 --+ O2 such that

W(S)

O en is an imbedding, then on t», the functions ,*Zl' ... , ,*zn (z, being the holomorphic coordinates in en) give solutions of (1). Moreover as ,(w) is generic, the condition on the rank must be satisfied. This follows from the fact if we set

n

a

I

uZj

a

n+1

== L

a vector y = L a, :;-

Ct.s

s=

I

~is

holomorphic

uXs

and tangent to r(w) if and

47

Abstract Local Cauchy-Riemann Structures

dim KerA

+ dim

KerA ='rank(L,[) = = dim Ker A + dim Ker A - dim Ker A = n + n - (n - I) = n + I.

!1 Ker

A

Once this is proved it is a question of applying the criterion given before to verify that ,(w) is a generic submanifold of en. c) That the given system (1) admits solution other than the constants is by no means obvious. The problem of local imbeddability has a positive answer in the following instances. (ex) The system (1) has real analytic (complex valued) coefficients, (Pl I = n, in this case it reduces to the Newlander-Nirenberg theorem.'!' The proof of (c] is elementary, not so the proof of (Pl. In the other cases the answer is doubtful. As example of L. Nirenberg [23] shows that for I = 1, n = 3 the answer is negative. Indeed Nirenberg's example is the data of a complex vector field X on a neighborhood of the origin in R3 with the property that i) X, X, [X, ~] are linearly independent at the origin, ii) every solution u of Xu = 0 in a neighborhood of the origin must be constant in some neighborhood of the origin. Now the system Xu = 0 defines a Cauchy-Riemann hood of the origin in R3 of type (2, 1).(2)

structure in a neighbor-

only if a = JCt., JCt. = O. Sufficiency: Let z, = CPj(x1, ... , Xn+ I)' 1 ~ i ~ n be solutions of (1) satisfying the rank conditions on w. Consider the map

r :w given by x ---+ (CPl(X), ... , CPn(x)). Indeed it is enough to show that ran k

ci(CPl'

.

...

en

--->

We claim that,

is a local imbedding. f,

, CPn (PI' ... , (Pn)

o(xl' , .. ,xn+l)

= n

+'1

.

Let (L, [) denote this matrix. The columns of L form a basis of the space {IJE en+1

iAI] = O}

=

Ker A,

where A

The columns of (L, L) thus form a basis of Ker A Ker A

!1 Ker

n - I. Thus

A

=

Ker

(1) and thus,

=

(a~i}).

+ Ker A in

by assumption

e +'. n

:f I, More generally one has a positive answer to the problem of imbedding in this particular case, The system (I) satisfies in addition to conditions (a), (P~ (1') also the condition

Now

(y), has dimension

(0)

See [22]. 2. Counterexamples equation of Nirenberg

[XJ' X.] = I:cj.(x)X,

+

I:dj.X.

cj., dj.

E