Annales de l’institut Fourier (AIF)

Barlet, D

Maire, H.-M.

Asymptotique des intégrales-fibres

Baouendi, M. S.

Rothschild, L. P.

ANNALES DE L’INSTITUT FOURIER
M. S. BAOUENDI
L. P. ROTHSCHILD
Harmonic functions satisfying weighted sign
conditions on the boundary
Annales de l’institut Fourier, tome 43, no 5 (1993), p. 1311-1318
<http://www.numdam.org/item?id=AIF_1993__43_5_1311_0>
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Ann. Inst. Fourier, Grenoble
43, 5 (1993), 1311-1318
HARMONIC FUNCTIONS SATISFYING WEIGHTED
SIGN CONDITIONS ON THE BOUNDARY
by M.S. BAOUENDI and L.P. ROTHSCHILD
0. Introduction.
In this paper we prove a local unique continuation theorem for
harmonic functions defined in an open set of the half space, R^. These
results involve a local boundary sign condition on the product of the
function and a homogeneous polynomial. Classical results in this direction
go back to the global theorems of G. Giraud and E. Hopf, concerning
the nonvanishing of the normal derivative at a boundary point where an
extremum is reached. (See e.g. Miranda [8] and the references given there.)
For X e W1 , we write X = (x,y), with x e R71"1 and y C R and
denote by R!j. the half space {y > 0}. Let 0 be an open neighborhood
of 0 in R71 and put ^ = R^ H 0 and V = R71-1 H 0. We shall assume
that Q. is connected. If u is a continuous function in f^, we say that u is
flat at 0 if for every positive integer N there is a constant CN such that
KX)| < C^W.
We now state our main results.
THEOREM 1. — Let u € (7°(Q) be harmonic in fl, and satisfy the
following conditions:
The authors were partially supported by National Science Foundation Grant DMS
9203973.
Key words : Unique continuation - Harmonic functions - Hopf Lemma.
A.M.S. Classification : 35J30 - 31B25 - 30B40.
1312 M.S. BAOUENDI and L.P. ROTHSCHILD
(1) There exists P(x), a homogeneous polynomial inn—1 variables such
that P{x)u{x, 0) >, 0 for x € V.
(2) For every positive integer N , the function \x\~Nu{x,0) is integrable
inV.
(3) For every multi-index a in n variables with |a| < d, where d is the
degree of P(x), the function y i-> (<9^-n)(0, y) is Sat at y = 0.
Then u(x^0) = 0 in a neighborhood ofO in V, and hence u extends as a
real analytic function in a neighborhood ofO in R"'.
For applications, a more precise version of Theorem 1 will be useful.
THEOREM 1'. — The conclusion of Theorem 1 still holds if condition
(3) is replaced by the weaker condition:
(35) y ^ Of^-^V^^^)! is Qat at y = 0, where Q is the
\dx dy/ \ y / la;==o
homogeneous polynomial of degree d in n variables satisfying P(x) =w^x^)-
Note that for an arbitrary polynomial P(x) the existence of the
polynomial Q(X) in condition (3') follows from Proposition 1.4 below. We
now state some consequences of this theorem.
COROLLARY 1. — Let u 6 (7°(n) be harmonic in fl, and satisfy the
following conditions:
(i) There exists P(x), a homogeneous polynomial inn—1 variables such
that P{x)u(x, 0) ^ 0 for x € V.
(ii) For every multi-index a in n variables with \a\ < d, where d is the
degree ofP(x), 9^u{X) is flat at 0.
Then u(X) = 0 in ̂ .
In particular, for n = 2, we have the following corollaries:
COROLLARY 2. — Assume n = 2. Let u € C°(fl.) be harmonic in fl,
and flat at 0. Then u = 0 in Q if one of the following conditions holds:
(a) u(x^ 0) does not change sign in V.
r\
(b) xu(x^ 0) does not change sign in V and y i—^ ^-(0, y) is flat at y = 0 .
(JX
WEIGHTED SIGN CONDITIONS FOR HARMONIC FUNCTIONS 1313
(c) xu(x,Q) does not change sign in V and y \—> v(0,y) is flat at y = 0,
where v(x, y) is a harmonic conjugate ofu near 0 in f^.
The following is an immediate consequence of Corollary 2.
COROLLARY 3. — Assume n = 2. Let h be a holomorphic function
in ^, flat at 0. Jf u = 5R/i is continuous in f2 and the function V 3 x ^
n(rc, 0) does not change sign, except perhaps at 0, then /i ̂  0.
Theorem 1 was first proved for the case P(x) = 1 by the authors in [5],
both for open sets in the half space and also for open sets in the ball. It is
likely that unique continuation results with weighted sign conditions similar
to condition (1) of Theorem 1 can be found for more general open sets with
analytic boundaries. Unique continuation for holomorphic functions with
nonnegative real part on the boundary was given by the authors in [4], and
was used to prove a generalized Schwarz reflection principle for holomorphic
functions mapping the real line into a C1 totally real manifold or into a real
analytic set. Earlier results in this direction were obtained by the authors
jointly with S. Alinhac [3], as well as by Bell-Lempert [6], Alexander [I],
and Huang-Krantz [7]. We should also mention here more recent work
of Alexander [2], in which the sign boundary condition for holomorphic
functions is weakened in a different direction.
1. Some properties of the Poisson integral in a half space.
Let f(x) be a continuous function with compact support in R71"1. We
denote by w(X) == Pf(X) the solution of the Dirichlet problem Aw = 0,
w{x,0) = f(x}^x € R71"1, given by the Poisson integral,
(1.1) w(x, y) = -Pf(x, y) = c» f^ ̂  ^ ̂ p)^ dx'^ ̂  € R^
where Cn = ——^— (see e.g. [10]). We start with the following proposition:
PROPOSITION 1.2. — Let f C C^QR71"1) with compact support
and w(X) given by (1.1). Assume that for every positive integer N ,
Ircl-^/Cr) C L^R71-1). Then for every a e N71, the function y ̂  9^w{0,y)
isinC°°(R^) .
Proof. — For y > 0, we may differentiate (1.1) under the integral
sign and then set x = 0. Then 9^w(0, y) is a finite sum of terms of the
1314 M.S. BAOUENDI and L.P. ROTHSCHILD
form
R(x^y)f{x1)^(1.3) /
JR"-17^ i (2/2+IW5
where .R(a/, y) is a polynomial in n variables, and k > 1. Since [^"^/(a-) e
L^R71"1) for every TV, we may use the dominated convergence theorem
to conclude that the function given by (1.3) is infinitely differentiable for
yeR. D
We shall also need the following:
PROPOSITION 1.4. — Let Pd^) be the space of all real valued
polynomials in n variables homogeneous of degree d. Then the linear
mapping ^ : ̂ (R71) -^ ̂ (IT) given by
(1.5) P(X) = ^(Q)(X) = ixr^o^) (—),
is a vector space isomorphism.
Proof. — It is easily checked that if Q{X) e ^(lET), then the right
hand side of (1.5) is again a homogeneous polynomial of degree d. To prove
the proposition, it suffices to show that the mapping <I> is injective. For
this, assume that Q e Pd(M71) satisfies
(L6) ^Xp^)'0' ^"\w.
Let T be the tempered distribution on y1 given by the finite part of ——.
Then its Fourier transform T is of the form t = ci log |$| +02, where ci and
C2 are constants. (See e.g. [9].) On the other hand, from (1.6) we obtain
(1.7) < )̂̂ ,
where S is a distribution in R71 supported at the origin. Taking the Fourier
transform of (1.7) and using the form of t we obtain
(1-8) 0(0(^1 log |$|+C2)=A(0,
with R{^) a polynomial in n variables. Since ci ^ 0, (1.8) can hold only if
Q = 0. This completes the proof of Proposition 1.4. D
PROPOSITION 1.9. — Let f and w be as in Proposition 1.2 and P
and Q be homogeneous polynomials of degree d related by Proposition 1.4.
Suppose that P(x, y) is independent ofy. Then for every integer j > 0 the
following holds:
f1-1" m^) ("?) I...= '-1''̂  /..-.
WEIGHTED SIGN CONDITIONS FOR HARMONIC FUNCTIONS 1315
with M, = ( ^(n/2+d)(n/2+d+l)...(n/2+d+j-l)
Proof. — We shall apply the operator Q ( — — ) to both sides of (1.1)
\uX /
for X C R!jl. By Proposition 1.4 and the fact that P(x^y) is independent
of t/, we obtain
(1.11)
Q 9 /w(x,y)\ _ [ P{x' - x)f{x') ^ ,
v{9x'9y){~T~)~ ^An^+k-^l2)71/2^ 5 ^)€K+•
Define v(y) by
(L12) ^(^C^L.^-
By Proposition 1.2, v(i/) is smooth for y > 0, up to y == 0, hence admits a
Taylor expansion at 0. We claim that for every positive integer N we have
(1.13) ^)=Ea^2J+o(1^12N+2)-
j=o
where the Oj are given by
(1.14) a,=(-l)^M,^^^<to.
Indeed, we have, for every integer N > 0 and for k > 0,
(1.15)
(2/2 + M2^
_ V^. , ( f c ) ( f c + l ) . . . ( f c + J - l ) ^
2-/ ^ .» |a;|2j+2fc
j=o J 1 '
^ (fc)(fc + 1) . . . (fc + N) / I-1 (1 - T)^ ^V2^2+ v l ) ——AT»——vyo w^m^ ^ '
By substituting (1.15) into (1.11) with x replaced by x — x ' and with
k = n/2 + d, we obtain the desired expansion (1.13) for v(y), by putting
x = 0, provided we show that the integral(U6) L. r ̂ ^^'p(xmx)dTdx
is bounded independently of y > 0. The latter follows from the fact
that the integrand in (1.16) is dominated by the function \P(x)f(x)\
j^j-(n+2d+2N+2)^ |̂ch ig integrable by the assumption on /. The proof
of Proposition 1.9 is complete. D
1316 M.S. BAOUENDI and L.P. ROTHSCHILD
Let n and V be as in the introduction, and u € C°(Q,) and harmonic
in Q. Let \ € C§°(V) with ^(.r) = 1 near 0 and \{x) > 0, and put
f(x) = \(x)u(x^0). We denote by w(X) the solution of the Dirichlet
problem Aw = 0, w(rc,0) = f(x),x € R17"1, given by the Poisson integral
(1.1). We write
(2.1) u{X)=(u-w){X)+w(X), Xe^ .
Note first that the function u — w is harmonic in f!, continuous in fl, and
vanishes in a neighborhood of 0 in V, by the choice of the cut-off function
^. Hence by the classical local regularity of the Dirichlet problem in the
real analytic setting, u — w extends to be real analytic in a neighborhood
of 0 in V. Using also Proposition 1.2, we may conclude the following:
PROPOSITION 2.2. — Let u € C°{fl.) be harmonic in fl,, and assume
that for every positive integer N , the function [^["^^(rc.O) is integrable
in V. Then if 77 > 0 is sufficiently small, for every a € N71, the function
y^O^u^isinC00^}).
Similarly, from Proposition 1.9 and (2.1), using again the real analyt-
icity of u — w near the origin, we have:
PROPOSITION 2.3. — Let u be as in Proposition 2.2. IfP and Q
are two homogeneous polynomials in n variables related by Proposition 1.4
with P independent of y , there exists C > 0 such that for every integer
j > 0 the following holds:
(2.4)
(^N^)(^)L.«-(- /̂̂1 fft3n( 9 V^WM t 1V. M { PQcM^OL, ^^+iW)^^Qxj^-rnx^ ^^jy ^dx ^c '
with Mj as in Proposition 1.9.
We may now complete the proof of Theorem 1. Let P(x) be the
polynomial (in n — 1 variables) given in assumption (1) of Theorem 1.
By regarding P as a polynomial in X = (x^y), but independent of
y^ and applying Proposition 1.4, one has a unique polynomial Q(x^y)^
homogeneous of degree d satisfying (1.5). Put
^ ^(^C^L
WEIGHTED SIGN CONDITIONS FOR HARMONIC FUNCTIONS 1317
By Proposition 2.2, g € ^^([O,^]) for 77 > 0 and sufficiently small. Since
by assumption (37) of Theorem 1', ^Q/) is flat at 0, all its derivatives must
vanish at 0. Hence we conclude from Proposition 2.3, noting that Mj >_ 1,
(26) c fp^u^dx<C^l
^•OJ n Jv {x^2^^ '
As in [5], we reason now by contradiction. If u(x^0) does not vanish in
any neighborhood of 0, then since P(x)u(x^ 0) > 0 by assumption (1) of
Theorem 1, for every positive e sufficiently small we would have
(2.7) / P(x)u(x,0)dx > 0.
J\x\<e
On the other hand it follows from (2.6) that we have for every e > 0
sufficiently small
(2-8) ——bj / P(x)u{x,0)dx < C^\ j > 0.
t J\x\<e
Taking the jth root of both sides of (2.8), making use of (2.7), and letting
j go to infinity we obtain 1 < Ce2. Since e can be taken arbitrarily
small, we reach a contradiction, which proves the vanishing of u(x^ 0) in
a neighborhood of 0 in R71""1. The real analyticity of u in a neighborhood
of W1 then follows from the classical real analyticity up to the boundary of
the Dirichlet problem. This completes the proof of Theorem 1'. D
Proof of Theorem 1. — We note that Proposition 1.4 guarantees the
existence of the polynomial Q(X) of condition (3') of Theorem 1'. It is then
clear that (3') implies (3); hence Theorem 1 follows from Theorem 1'. D
Proof of Corollary 1. — It is not hard to see that conditions (i)
and (ii) of the corollary imply conditions (1), (2), (3) of Theorem 1.
Therefore, by applying the theorem, we conclude that u(x, 0) vanishes in
a neighborhood of 0 in V, and hence ^extends to be real analytic in a
neighborhood of 0. This, together with the flatness of n, i.e. condition (ii),
implies that u must vanish identically in the connected set f^. D
Proof of Corollary 2.
(a) Take P(X) = Q(X) = ± 1 and apply Corollary 1 to u.
(b) Take P(X) =±x and hence Q(X) = qp x / 2 in Theorem 1' to conclude
that u{x^ 0) = 0 for a* in a neighborhood of 0. The desired conclusion then
follows from the flatness of u as in the proof of Corollary 1.
(c) Take P(X) = ±x and hence Q(X) = =p x /2 . In order to apply Theorem
QuV , we must show that T—(O, y) is flat at y = 0. If v is a harmonic conjugatedx
1318 M.S. BAOUENDI and L.P. ROTHSCHILD
of n, by the Cauchy-Riemann equations, we have —(0,2/) = —(0,z/) for
ox dy
y > 0 sufficiently small. Since by Proposition 2.2, j—(Q^y) is smooth up toox
y = 0, the same is true for —(O,^/). The assumption of flatness of v at 0
implies that —(O,^/) (and hence also —-(0,2/)) is also flat. The rest of the
dy ox
proof is as in (b).
Remark 2.9. — Corollary 3 follows from Corollary 2 parts (a) and
(c). However, we note that if the condition u = St/i G G°(Q) is replaced by
the stronger condition h € C°(fl.), then the result can be proved using only
Corollary 2 part (a). Indeed under this stronger condition if u(x, 0) changes
sign at 0, i.e. xu(x^ 0) does not change sign, then we can apply part (a) to
the harmonic function ?Sizh(z) (with z = x + iy).
BIBLIOGRAPHY
[1] H. ALEXANDER, Boundary behavior of certain holomorphic maps, Michigan Math.
J., 38 (1991), 117-128.
[2] H. ALEXANDER, A weak Hopf Lemma for holomorphic mappings, preprint.
[3] S. ALINHAC, M.S. BAOUENDI, L.P. ROTHSCHILD, Unique continuation and
regularity at the boundary for holomorphic functions, Duke J. Math., 61 (1990),
635-653.
[4] M.S. BAOUENDi and L.P. ROTHSCHILD, Unique continuation and a Schwarz
reflection principle for analytic sets, Comm. P.D.E., 18 (1993), 1961-1970.
[5] M.S. BAOUENDi and L.P. ROTHSCHILD, A local Hopf lemma and unique continu-
ation for harmonic functions, Duke J. Math., Inter. Research Notices, 71 (1993),
245-251.
[6] S. BELL and L. LEMPERT, A C°° Schwarz reflection principle in one and several
complex variables, J. Dm. Geom., 32 (1990), 889-915.
[7] S. HUANG and S G. KRANTZ, A unique continuation problem for holomorphic
mappings, Comm. P.D.E., 18 (1993), 241-263.
[8] C. MIRANDA, Partial differential equations of elliptic type, Ergeb.Math. Grenzgeb.
(n.F.), 2, Springer-Verlag, Berlin ,1970.
[9] L. SCHWARTZ, Theorie des distributions, Hermann, Paris, 1966.
[10] E.M. STEIN, Singular integrals and differentiability properties of functions, Prince-
ton University Press, Princeton NJ, 1970.
M.S. BAOUENDI and L.P. ROCHSCHILD,
Department of Mathematics-0112
University of California
SAN DIEGO, LA JOLLA CA 92093-0112 (USA).


Harmonic functions satisfying weighted sign conditions on the boundary

Pham, Frédéric

Principe de Huygens et trajectoires classiques complexes ou Balian et Bloch vingt ans après

Xu, Quanhua

ANNALES DE L’INSTITUT FOURIER
QUANHUA XU
Erratum : Notes on interpolation of Hardy spaces
Annales de l’institut Fourier, tome 43, no 2 (1993), p. 569
<http://www.numdam.org/item?id=AIF_1993__43_2_569_0>
© Annales de l’institut Fourier, 1993, tous droits réservés.
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(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
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nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
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ERRATUM
"NOTES ON INTERPOLATION OF HARDY SPACES9
par Quanhua XU
Article paru dans Ie tome 42 (1992), fascicule 4, pp. 875-889
This is to correct an error in my paper "Notes on interpolation of
Hardy spaces" in Ann. Institut Fourier, 42-4 (1992), 875-889.
The error lies in the passage from line 8 to line 9 in page 886. It can
be corrected by modifying the definitions of Vk and Uk in page 884; Vk and
Uk are now changed as follows
fci-i
Vk = ^ 2^/2m(e,)
i/p.,} ,
\J=k-l )
fei-1
Uk(s)= Y^ 2-"/2^.(s), se[0,l].
j=k-l
With this change, the a^'s in page 884 are all equal to 1; everything else
goes on in the same way with some necessary minor modifications, except
relations (16) and (18), which now read respectively as
i/p
\s)gk(s)uk(s) = { ̂  2^m(e,) \ =v^
U=fc-i /
/1 r( ) k(s) [ ( ,)}
Jo \^-i I
,̂
and
I' i^r72 < c2-^/2^.JT


Erratum : Notes on interpolation of Hardy spaces

Łojasiewicz, Stanislas

Sur la géométrie semi- et sous- analytique

Dillinger, H.

Delabaere, E.

Résurgence de Voros et périodes des courbes hyperelliptiques

Zarrabi, Mohamed

Contractions à spectre dénombrable et propriétés d'unicité des fermés dénombrables du cercle

Gouvêa, Fernando Q.

Mazur, Barry

On the characteristic power series of the U operator

Bolondi, Giorgio

Migliore, Jean-Claude

On curves with natural cohomology and their deficiency modules

Bialynicki-Birula, Andrzej

ANNALES DE L’INSTITUT FOURIER
ANDRZEJ BIALYNICKI-BIRULA
On induced actions of algebraic groups
Annales de l’institut Fourier, tome 43, no 2 (1993), p. 365-368
<http://www.numdam.org/item?id=AIF_1993__43_2_365_0>
© Annales de l’institut Fourier, 1993, tous droits réservés.
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(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
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tion commerciale ou impression systématique est constitutive d’une in-
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Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
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Ann. Inst. Fourier, Grenoble
43, 2 (1993), 365-368
ON INDUCED ACTIONS OF ALGEBRAIC GROUPS
by Andrzej BIALYNICKI-BIRULA^
Let If be a subgroup of an algebraic group G. Let Y be an algebraic
space with an action of H (shortly an algebraic If-space). The aim of this
note is to study some properties of G XH V, defined as a quotient of G x Y
by the action of H determined by h(g^ y) = (gh~1^ hy)^ for all h € H, g e G
and y G Y. Left translations by elements of G on G determine an action of
G on G XH H. The importance of the space G XH Y follows from the fact
that the map Y '—> G XH H , which to y G Y attaches the image of (1, y)
in G XH H solves the universal problem of jFf-equivariant morphisms of Y
into G-spaces. In analogy with the theory of modules and representations
we can say that the space G XH Y is induced from the H -space Y by the
group extension H C G and that the action of G on G XH Y is induced by
the action of H on Y. In applications the notion is used for constructing
a space with an action of G, when a space with an action of its subgroup
H is given. Properties of G XH Y in the case where Y is quasi-projective
were studied in the classical paper [Se]. Though results presented here are
perhaps predictable or even known, we hope that the paper will be useful
as a reference.
In order to make our arguments more lucid, we are going to start with
considering more general situations.
1. Let X and Y be two algebraic H -spaces. Then X XH Y is defined
as a quotient of the product X x Y by the action of H defined by
h(x^ y ) = (hx^ /M/), for h € H. In general, neither the meaning of the notion
(*) supported by Polish KBN Grant GR-87.
Key words : Principal fiber bundles - Actions of algebraic groups - Algebraic spaces.
A.M.S. Classification : 14L30.
366 ANDRZEJ BIALYNICKI-BIRULA
of quotient, nor its existence (when the meaning of the quotient has been
already fixed) is clear. In the note we consider only the case when H is
affine and X is a principal locally isotrivial ^-fibration in the category
of algebraic spaces. In this case we require the quotient X XH Y to be
an algebraic space, and the map X x Y — ^ X x H Y t o b e affine and a
geometric quotient in the sense of [GIT]. If X = G, where G is an affine
algebraic group containing H as its subgroup with an action of H by right
translations, then by [Se] the above assumptions concerning H and X are
satisfied.
THEOREM 1. — Let H, X, Y be as above. Moreover assume that
X is normal. Then S XH Y exists in the category of algebraic spaces. If,
moreover, X is an algebraic variety, Y is normal and can be covered by
H-invariant open quasi-projective subsets, then X XH Y is an algebraic
variety.
Proof. — The theorem will be proved in several steps.
1st step. Assume that the jy-fibration on X is trivial i.e. X = H x U,
where U is an algebraic space. Then X x Y = H x U x Y — > U x Y defined
by (/i, x, y) ̂  (^, h~'^y) satisfies desired conditions. Thus X XnY = UxY.
2nd step. Assume that the .H-fibration on X is isotrivial with the base
space U. Since X is normal, U is normal and then there exists a Galois
ramified cover Z —>• U such that X Xu Z is trivial. It follows from the
1st step that there exists {X XH Z) XH Y and by Deligne's theorem [K]
p. 183-4 there exists its quotient by the action of the Galois group (induced
by the action on Z) in the category of algebraic spaces. The quotient can
be identified with X XH Y. If moreover Y is normal quasi-projective and
U is affine, then (X Xjj Z) XH Y is normal quasi-projective. Because the
quotient of a normal quasi-projective variety by an action of a finite group
is quasi-projective, hence X Xjj Y is also quasi-projective.
3rd step. Assume that X and Y are covered by If-invariant open
subsets [Ui, i € J}, {Vj, j e J}, such that Uz XH Vj exist, for all z e I
and j € J (in the category of algebraic spaces). Then X XH Y also exists
(in the same category) and {Uz XH Vj} form an open covering of X XH Y.
Moreover if Uz XH Vj are quasi-projective, then X XH Y is an algebraic
variety. Proof of this step is obvious.
4th step. Now we consider the general case. Notice first that the base
space of the ff-fibration given on X can be covered by open subsets {Wfc},
ON INDUCED REPRESENTATIONS 367
k € JC, such that for every k e K^ the inverse image L ;̂ of IV/c in X, as a
principal U-fibration, is isotrivial. Then it follows from the 2nd step that,
for every k G K, U^ XH Y exists and from the 3rd step that X XH Y
exists in the category of algebraic spaces. Moreover, if Y can be covered
by H -invariant open quasi-projective subsets Vj, where j € J, then by the
second part of the 3rd step, we infer that X XH Y is an algebraic variety.D
COROLLARY 2. — Let Y be an algebraic space with an action of
an algebraic group H and let G be an affine algebraic group containing
H as its subgroup. Then G XH Y is an algebraic space with an action of
G induced by left translations on G. Moreover, if Y can be covered by
H-invariant quasi-projective open subsets, then G XH Y is an algebraic
variety.
THEOREM 3. — Let G be a connected affine algebraic group and
let H be its subgroup. Let Y be a normal algebraic space with an action
of H. Then G XH Y is an algebraic variety if and only if Y can be covered
by H-invariant quasi-projective open subsets.
Proof. — It follows from Corollary 2 that, if Y can be covered by
-H-invariant open quasi-projective subsets, then G XH Y is an algebraic
variety. Let us assume now that G XH Y is an algebraic variety. Since
G is connected, G XH Y by Sumihiro Theorem [Su] can be covered by
(^-invariant open quasi-projective subsets. Intersecting these subsets with
HxnY C GxnY we obtain an lif-invariant quasi-projective open covering
of H XH Y. Since Y c± H x n Y we obtain that Y can be covered by open
quasi-projective jFf-invariant subsets. D
It follows from the above results and Sumihiro Theorem that when-
ever H is connected, any induced G-space from an algebraic normal H-
variety is also an algebraic (normal) variety. However in case where H is
a finite subgroup of a connected algebraic group and Y is an algebraic H-
variety which can not be covered by ^-invariant open quasi-projective open
subsets, then the induced algebraic G-space is not an algebraic variety. For
example, if two element group Z^ acts on a normal algebraic variety Y in
such a way that a Zs-orbit is not contained in any affine open subset (see
[H] or Chap. 4§ 3 in [GIT] for an example), then for any connected affine
group G containing £2 as a subgroup, G x z^ Y is an algebraic space but
not an algebraic variety.
368 ANDRZEJ BIALYNICKI-BIRULA
BIBLIOGRAPHY
(H] H. HIRONAKA, An example of a non-kahlerian deformation, Ann. of Math., 75
(1962), 190-208.
[K] D. KNUTSON, Algebraic spaces, Lecture Notes in Mathematics, 203, Springer-
Verlag, 1971.
[GIT] D. MUMFORD, J. FOGARTY, Geometric Invariant Theory, 2nd edition, Ergeb.
Math. 36, Springer-Verlag, 1982.
[Se] J.-P. SERRE, Espaces fibres algebriques in Anneaux de Chow et Applications,
Seminaire Chevalley, E.N.S. Paris, 1958.
[Su] H. SUMIHIRO, Equivariant completions I, J. Math. Kyoto Univ., 14 (1974), 1-28.
Manuscrit recu Ie 23 avril 1992.
Andrzej BIALYNICKI-BIRULA,
Instytut Matematyki UW
Banacha 2
02097 Warszawa (Pologne).


On induced actions of algebraic groups

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