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Homologie des ensembles semi-pfaffiens

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Relative discrete series of line bundles over bounded symmetric domains

Amoroso, Francesco

Mignotte, Maurice

On the distribution on the roots of polynomials

Globevnik, Josip

Partial indices of analytic discs attached to lagrangian submanifolds of ${\mathbb {C}}^N$

Barbot, Thierry

Flots d'Anosov sur les variétés graphées au sens de Waldhausen

Félix, Yves

The center of a graded connected Lie algebra is a nice ideal

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Sur le genre arithmétique des courbes rationnelles

Berndtsson, Bo

ANNALES DE L’INSTITUT FOURIER
BO BERNDTSSON
The extension theorem of Ohsawa-Takegoshi and
the theorem of Donnelly-Fefferman
Annales de l’institut Fourier, tome 46, no 4 (1996), p. 1083-1094
<http://www.numdam.org/item?id=AIF_1996__46_4_1083_0>
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Ann. Inst. Fourier, Grenoble
46, 4 (1996), 1083-1094
THE EXTENSION THEOREM OF OHSAWA-TAKEGOSHI
AND THE THEOREM OF DONNELLY-FEFFERMAN
by Bo BERNDTSSON (*)
1. Introduction.
In [OT] Ohsawa and Takegoshi proved the following theorem.
THEOREM A. — Let V be a bounded and pseudoconvex domain in
C71, and let </? be plurisubharmonic in V. Let H be a hyperplane. Then, for
any holomorphic function, /, on V H H there is a holomorphic function F
in V such that F = / on H and
( iFpe-^ <C^( I/I
J-D J H
2e-^.
I '
Moreover, C-p depends only on the diameter ofV.
This theorem is particularly useful as there is no loss in the estimate
and the constant only depends on the size of P.
The original proof of Ohsawa-Takegoshi was based on a 9-theorem
involving complete Kahler metrics, inspired by a theorem of Donnelly and
Fefferman [DF]. Later, Manivel [M] proved, using related methods, a more
general version of Theorem A for sections of vector bundles.
Recently Siu [S] found a simpler proof of Theorem A which avoids the
use of general metrics and the somewhat complicated commutator identities
(*) Supported by the Goran Gustafsson foundation for research in Natural Sciences and
Medicine.
Key words: Plurisubharmonic - Pseudoconvex - 9.
Math. classification: 32F20.
1084 BO BERNDTSSON
used in the approach of Ohsawa-Takegoshi. Siu's proof uses only the
Hormander-Kohn-Morrey formalism for the 9-equation in domains in C^,
but the idea from [OT], [DF] of twisting the ^-complex by multiplication
with a function is still visible in Siu^s proof. Related ideas have also been
used recently by McNeal [McN] in his work on estimates for the Bergman
kernel.
The aim of this note is to give yet another proof of Theorem A. It
is not radically different from the methods of Siu and McNeal in as much
as the same crucial integral formula is still used. However, this formula
is derived and interpreted in a different, in our opinion more transparent
way. We also show how the same methods lead to a simple proof of a more
general version of the 9-theorem refereed to above, which also generalizes
a recent result of Diederich-Ohsawa [DO].
Consider more generally the problem of extending the function / from
a subvariety V = {h = 0}, where h is holomorphic in P. A well-known
scheme to solve such problems is to first construct a local extension, /.
Then one multiplies / with a cut-off function ^, and solves a 9-problem
Bn.f9-
h
Then F = f\ — hu solves the extension problem. To avoid using any
special property of the local extension /, Siu considers cut-off functions
with support in an 6-neighbourhood of V, and then lets e tend to zero. The
difficulty is that one needs an estimate independent of e, and for this the
usual Hormander estimate is not enough.
A variant of this scheme, which we will use here, is to consider the
9-problem
(1.1) Qu=f-9^
and then put F = hu. Clearly F is then holomorphic and
F=h(n-{)^f,
so -F = / on V since u — f / h is holomorphic, hence smooth. (This method
was introduced in [A].)
An immediate difficulty here is that the right hand side of (1.1) is not
a form with -^-coefficients but a current with measure coefficients. More
seriously, it is clearly impossible to even find a solution to (1.1) in L^. It
is however, possible to find a solution such that hu € L2 and we shall show
how the Z^-methods can be adapted to prove this directly.
OHSAWA-TAKEGOSHPS EXTENSION THEOREM 1085
The main tool we use is a differential identity (Lemma 2.1) from [Be2],
[Be3] (the one-dimensional case was used in [Bel]). This differential identity
implies immediately the integral formula used by Siu and McNeal. Indeed,
the two formulas are essentially equivalent, but the differential identity is in
our opinion more suggestive and might hopefully have further applications
(one is given in [Be3]).^
Finally, it's a pleasure to thank Jeff McNeal for stimulating discus-
sions on the topic of this paper.
2. The extension theorem.
We shall use the following two lemmas from [Be2] and [Be3]. For the
convenience of the reader proofs are given in an appendix.
LEMMA 2.1. — Let a = ^djdzj be a smooth (0,l)-form in a
domain in C71, and let (p be a smooth function. Then
^ Q^oT^3^6'^ = ~2Re^9'pa' ̂ ^ + W26^
+ E 11| I26"' - i^i26"' + E w^-'-
Here Q*p is the formal adjoint of the ^-operator in L^e"^), i.e.,
^a=-e^^(e-^,).
In the one-dimensional case the left hand side equals
AH^,
and the formula can be used to obtain pointwise control of a by integrating
against a fundamental solution of the Laplace operator. This was used in
[Bel] to obtain Z^-estimates for 9 in one variable. In higher dimensions one
can instead integrate against log|/i|, h holomorphic, and get an estimate
for the integral of a over the zero-variety of h. This is what leads to the
extension theorem. The integrated version of Lemma 2.1 is
LEMMA 2.2. — Let V = [p < 0} be a smoothly bounded domain
in C71 and let w be a smooth function on T>. Let a and (p be as in
(*) Note added 30/7/96 : Since this paper was written I have realized that Lemma 2.1
is essentially contained in an earlier paper by Siu, [S2].
1086 BO BERNDTSSON
Lemma 2.1 and assume a satisfies the 9-Neumann boundary conditions
on 9V, a ' Qp = 0. Then
j w^(PjWoike~lp - j ^w^ajOke-^ 4- j w^a^e-^
r v^i<9a..2 „ r v-^ .- ds+jwW^e~r+S^I'•""wrm
= 2 Re IwQQ^a • ae'^ + /w|9a|2e~</?.
In the proof of Theorem A one may assume that P is smoothly
bounded and strictly pseudoconvex, that (p is smooth on P, and that /
extends to a holomorphic function in a neighbourhood of P, as long as one
obtains a constant Cp that only depends on the diameter of P. We will
have use for a more or less standard lemma.
LEMMA 2.3. — Let g be a 9-closed (0, l)-current denned in a
neighbourhood ofV. Assume u is a L1 -function in V and
(2.1) I g ' a e - t f t = [ u't^ae-^
JT> J-D
for all smooth, 9-closed (0, l)-forms a on V, satisfying the 9-Neumann
boundary conditions. Then 9u = g in the sense of distributions.
Proof. — It is enough to prove that (2.1) holds for any smooth (0,1)-
form a with compact support in V. Decompose a = a1 +a2 where 9a1 = 0
and a2 -L Ker(9) in Z^e"^). It follows from the regularity of the 9-
Neumann problem that a1 and a2 are both smooth. Note that a2 ± Im 9
implies 9^a2 = 0 and that a2 satisfies the 9-Neumann boundary conditions.
Hence a1 satisfies (2.1) by hypothesis. Moreover,
/^.aY-^O
since g can be approximated by 9-closed (0, l)-forms in L2. Since (2.1)
holds for a1, it also must hold for a. D
LEMMA 2.4. — Let g, y, and V be as in the previous lemma, and
assume the inequality
|2
\/9'ae-- </W2^
OHSAWA-TAKEGOSHFS EXTENSION THEOREM 1087
where 1 / p , is an integrable positive function, holds for all 9-closed (0,1)-
forms a satisfying the 9-Neumann boundary conditions. Then there is a
solution, u, to the equation 9u = g such that
(M2/^-^ < c.
Proof. — By elementary Hilbert space theory there is a function v
such that
/ /* -=_g-v?g . ae"^ = y vQ^a——
for all a of the above type, satisfying
/H^.c.
Let u = v / t i . Clearly u satisfies the stated inequality; in particular u is in
L1 by the hypothesis on fi. By the previous lemma u solves 9u = g, so we
are done. D
Choose coordinates so that the hyperplane H in Theorem A is
{z\ z\ = 0}. Suppose V is included in the set where |zi| < 1, and let
w = -log i—.3.
7T \Z^2
Since w > 0 on P, Lemma 2.2 implies, if 9a = 0 and a satisfies the 9-
Neumann boundary conditions,
/ |ai|2e-^+ fw^a^e-^ <2Re IwQQ^a'ae-^
= 2 / w^a^e-^ - 2 Re / 9^a9w . ae-^.
Hence
(2.2) ^^ |,,|̂  i ̂ lo |̂S;o|'.- + ^ / W^-
Next we apply Lemma 2.2 once more, this time with w = Wi = 1 — \zi\2.
We then get
f^e-y ^ /(I - \Zl\2)\yya\2e-y+2 f\^a\\ai\\zi\e-y
^ y(i - iziî aiv-^ + 2 y i^ai2!^!^-^ + j y laiî
so
/>|Ql|2e-v<4/'|^Q|2e-v.
1088 BO BERNDTSSON
Using this in (2.2), together with the elementary inequality a;(log(l/a;) +
2) <, 2 for 0 < x < 1, we find
Ll01'26"^/10^^26"'^/^2^/^o'"1' " - ̂ J ^ N21^"1 ' ' T r Y l^al | 2
^fw^^fw^.
Now, let g = / • 9—, and assume
î
/ l/l2^ = 1.
Jzi=0
Then if a is a smooth 9-closed (0,l)-form satisfying the 9-Neumann
boundary conditions, we get
| L.ae-^Tr2! /> /aie
2 «i r |2y?
1 ^ ' Jzi=0
<^ I |al|26-^^47^/>|9;a|2e-.^1=0 J ^ ki I2
This is in principle the inequality we need, but there is a minor problem
arising from the fact that l/|^i|2 is not integrable. To remedy this, we
note that if we instead choose wi in the last part of the argument as
wi = 1 - \z\\26, where 6 < 1, we get instead the inequality
i/^'^L^3^/^'^
where Cg tends to 47T as 6 tends to 1. By Lemma 2.4 there is a solution us
to 9us = g satisfying
/^ll^e-^C,.
By the argument from the introduction Fs = z-^ug solves the extension
problem and clearly
/W^<C,.
An immediate passage to the limit gives us a solution F such that
yiFiv-^^TT,
so we have proved Theorem A, with CD = 47T if D is included in the set
where |^i| < 1 (Siu obtained CD = 647r/9(l + l^e)^). Note also that a
OHSAWA-TAKEGOSHPS EXTENSION THEOREM 1089
similar argument shows that there is a solution to the extension problem
satisfying
fWz^e-^^Ce
for any 6 > 0.
It is also worth mentioning that we can apply exactly the same ar-
gument with the coordinate function z\ replaced by a general holomorphic
function h bounded by 1 in P (just replace the choice of w and wi by
I/TT log \h\2 and 1 - \h\2 respectively). We then obtain a version of the more
general results obtained in [M] and [OhT]:
THEOREM 2.1. — Let V be a bounded and pseudoconvex domain
in C71, and let (p be plurisubharmonic in V. Let V = {z € P; h{z} = 0} be a
hypersurface denned by a holomorphic function bounded by 1 inT>. Then,
for any holomorphic function, f, on V there is a holomorphic function F
in T> such that F = / on V and
( [Fpe-^ < 47T / I/I'|2g-^ ^ A^- I 1 ^ | 2 .1-- ^ — -1-" / \j | Qn9 •
p Jv 1^1J-D 
o-^P
3. The <9-theorem.
It is clear that by choosing w in Lemma 2.2 to be any positive
plurisuperharmonic function, one gets estimates for the integrals of 9-closed
forms just like in the previous section. We shall now use this to give a simple
proof of the theorem of Donnelly and Fefferman.
Let M be a complex manifold of dimension n, equipped with a
complete Kahler metric ^.
Assume 0 has a global potential ^, so that Q. = i9Q^, and assume -0
satisfies the crucial condition that 9^ is uniformly bounded when measured
in the ^-metric. The theorem of Donnelly and Fefferman says that under
these assumptions we can solve the equation
9u=g
with an estimate
IMÎ  < C\\g\\^
for any 9-closed (p, g)-form g, provided p + q ^ n. The instance of this
theorem that is relevant to the previous discussion is when (p,g) = (n, 1),
1090 BO BERNDTSSON
and we shall concentrate on this case in the sequel. We shall also assume
that M = T) is a bounded pseudoconvex domain in C71, and we can then
identify (0, g)-forms with (n, g)-forms in the natural way. The theorem then
says that for any 9-closed (0, l)-form g = Yigjdzj we can solve Qu = g with
the estimate
(3.1) / H^C/^^^,
J-D Jv^V T>
where {^3k) = (V^fe)"1? and both integrals are taken with respect to
Lebesgue measure. We shall now show how the arguments of Section 2
lead to a simple proof of a more general statement.
The assumption ||9'0||Q <: C means that
(3.2) Wz) • a|2 < C2 Y^^(z)a,ak
for all z G T> and a G C". Rescaling we may, of course, assume C = 1, and
then (3.2) just says that e~^ is plurisuperharmonic. We shall now apply
Lemma 2.2 with
w = e-^
where 0 < 6 < 1.
Let a be a 9-closed (0, l)-form satisfying the 9-Neumann boundary
conditions, and assume (p is plurisubharmonic and smooth on P. We then
get from Lemma 2.2
6(1-6) f^^a^e-^ < f\a;a\2e-^+2 f\9^a\\a-9^e-^.
By the condition on ip this is dominated by
(1 + 26/{1 - 6)) f I^Q^e-^ + 6(1 - 6)/2 f ̂  ̂ a^e-^,
so we obtain
(3.3) f ̂  ̂ ma,e-^ < -^^ f ̂ 0-^
for any smooth 9-closed (0, l)-form a that satisfies the 9-Neumann bound-
ary conditions. Now let g be a 9-closed (0, l)-closed (0,1) form in Z> and
assume
J^^g^e-^^l.
Then, by (3.3)
/w-v|2 <——4—— /'ly/yl2^^-^f\9^a•Q.e~v\  — ——  I \o a \ e9 ae | - 6(6 - I)2 J ' v '
OHSAWA-TAKEGOSHPS EXTENSION THEOREM 1091
for all forms a of the type we are dealing with. By Lemma 2.4 this implies:
THEOREM 3.1. — Let V be a bounded pseudoconvex domain in C71
and let (p be plurisubharmonic in P. Let ̂  be plurisubharmonic and assume
-0 satisfies
Wz) . a|2 ^^j-k(z)a,ak
for allzeV and a e C71. Let 0 < 6 < 1. Then for any 9-closed (0, l)-form
g in T) there is a solution u to the equation 9u = g such that
/H2e-v+" < w^ /E^W-^.
In particular, choosing (p = 6^, we get the theorem of Donnelly and
Feffermann, even without assuming the metric to be complete. Somewhat
weaker results were previously obtained by Diederich-Ohsawa [DO] and
Diederich-Herbort [DH]. Their theorems have the same feature of allowing
plurisubharmonic weights with "the wrong sign" in the exponent, but
do not specify the full range of permitted values of ^, and still assumes
completeness of the metric. Note that already the example V = {z €
C; \z\ < 1} and
^=log^l\^
shows that Theorem 3.1 would be false with 6 = 1, and even with the
constant replaced by c/(6 — 1).
4. Appendix.
Here we shall give the proofs of Lemmas 2.1 and 2.2.
Proof of Lemma 2.1. — This is of course nothing but a direct
computation. In the proof we will use the notation 6j = e^Q/Qzje'^, and
also write 9k for Q/Qzjc. Note that
9/9zk(uve~(p) = (Sk^ve'^ + uQkve'^,
Q/Qz^uve-^) = (Qk^ve-^ + uSkve^,
and that 6j and 9k satisfy the commutator relations
[^•A]=<^fc-
1092 BO BERNDTSSON
Using these formulas we find
^^-(Q,Qfce-^) = (6j9ka,)ak€-y + (9^(9,0'^ + ajQ^e^
+(6^)(6k0k)e-^.
But
^ajQ^kake^ =-Y^ajQ^ae-^ = -a • 99*ae~'f',
since 9*a= —^6jCtj. Using the commutation relations we also see that
^(^•afcQ-jQafce-^ = -99*a • ae-^ + ̂ ^•fcQyafce-^.
Hence we obtain after summing
y-: V 92 (aj-Qfce-^) = -2Re95*a • ae-^ + ̂ afe-^
f ^ ^ OZjOZk
+ ̂  QkOijQjOike^ + ̂  (pjkajake^.
But __
^%a,^=^|^|2-|9a|2,
so we are done.
Proof of Lemma 2.2. — If we integrate Lemma 2.1 over D, we obtain
the fundamental identity used in the proof of Hormander's theorem. Here
we shall multiply by w > 0 before integrating. We must then evaluate
^^ l̂̂ 0^-
Stokes' theorem gives
1 = - I' ̂ Wj9k(a,ake-^ + t ^w9k(a,ake-^)p,dS/\9p\
= f Y^Wjk^ake-^ + f ̂ wBk^ake-^pjdS/^p}
+ f ^w,a,pkake^dSI\9p\.
J 9
Now introduce the notation a ' 9p = On, and assume that On = 0 on
the boundary. This clearly makes the last boundary integral disappear.
Moreover, and just like when w = 1, the first boundary term also simplifies
in this case. To see this we argue as in [H2] or [KF]. First o.n = 0 implies
^pjQk^ajOke-^) =^ak(9k0ij)pje~lp.
OHSAWA-TAKEGOSHFS EXTENSION THEOREM 1093
Then one makes the important observation that since On = 0 on the
boundary we can write ̂ c^jpj = pg near the boundary for some smooth
g. Therefore
^ak9k^ajpj=0
on the boundary, or in other words,
^ak(9k0ij)pj = -Yajakpj-k.
All in all we have then shown that
I = f^w^ake-f - f w^p^ake-^dS/\9p\.
Combining this with Lemma 2.1 we obtain Lemma 2.2.
BIBLIOGRAPHY
[A] E. AMAR, Extension de fonctions holomorphes et courants, Bull. Sciences Mathe-
matiques, 2eme serie, 107 (1983), 25-48.
[Bel] B. BERNDTSSON, Weighted estimates for 8 in domains in C, Duke Math J., 66
(1992), 239-255.
[Be2] B. BERNDTSSON, Some recent results on estimates for the ^-equation, in Contri-
butions to Complex Analysis and Analytic Geometry, Eds. H. Skoda and J-M
Trepreau, Vieweg, 1994.
[Be3] B. BERNDTSSON, Uniform estimates with weights for the ^-equation, preprint 1994,
to appear in J. Geom. Anal.
[DH] K. DIEDERICH and G HERBORT, Extension of holomorphic I^-functions with
weighted growth conditions, Nagoya Math. J., 126 (1992), 141-157.
[DOh] K. DIEDERICH and T. OHSAWA, An estimate for the Bergman distance on pseu-
doconvex domains, Annals of Math., 141 (1995), 181-190.
[H] L. HORMANDER, I^-estimates and existence theorems for the 5-equation, Acta
Math., 113 (1965).
[KF] J. J. KOHN and G. FOLLAND, The Neumann problem for the Cauchy-Riemann
complex.
[M] L. MANIVEL, Un theoreme de prolongement L2 pour des sections holomorphes
d'un fibre hermitien, Math. Z., 212 (1993), 107-122.
[McN] J. MCNEAL, On large values of Z^-holomorphic functions, Math. Res. Letters, 3
(1996), 247-260.
1094 BO BERNDTSSON
[OhT] T. OHSAWA and K. TAKEGOSHI, On the extension of L2 holomorphic functions,
Math. Z., 195 (1987), 197-204.
[S] Y-T SlU, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi,
preprint 1995.
[S2] Y-T SlU, Complex-Analyticity at harmonic maps, vanishing and Lefschetz theo-
rems, J. Diff. Geom., 17 (1982), 55-138.
Manuscrit recu Ie 22 janvier 1996,
accepte Ie 26 avril 1996.
Bo BERNDTSSON,
Chalmers University of Technology
Department of Mathematics
41296 Goteborg (Sweden).


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