Annales de l’institut Fourier (AIF)

Egilsson, Ágúst Sverrir

Linear hamiltonian circle actions that generate minimal Hilbert bases

König, Heinz

Upper envelopes of inner premeasures

Zamora, Alexis García

ANNALES DE L’INSTITUT FOURIER
ALEXIS GARCÍA ZAMORA
Sheaves associated to holomorphic first integrals
Annales de l’institut Fourier, tome 50, no 3 (2000), p. 909-919
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Ann. Inst. Fourier, Grenoble
50, 3 (2000), 909-919
SHEAVES ASSOCIATED TO
HOLOMORPHIC FIRST INTEGRALS
by Alexis GARCIA ZAMORA
0. Introduction.
Through this paper S will be a projective, smooth and irreducible
surface over C. A holomorphic foliation over S is an element T €
PH°(S,Os 0 C~1), where C is an invertible sheaf on S and Qs is the
tangent sheaf. Thus, to any foliation there is associated via the natural
isomorphism H°(S^ Q^(g)£~1) ^ Hom(£, 65), a morphism C —> Qs defined
up to multiplication by a non-zero complex number, in the sequel this
morphism will be denoted by F.
Thus, F is defined in a trivializing open set Ua by a vector field
9 , „ 9
•^cn — ^Q:"^——— ~^~ -^a'o
OZa.l OZaU oi,! ^Q',2
defined up to multiplication by a scalar.
Associated to F there exists a dual map ujy : C —> (9^)*, which is
given in local coordinates by the 1-holomorphic form dual to the previous
vector field.
The singular set of T is defined as the set of points where the
associated map of vector bundles F : L —> TS is not injective. This is
equivalent of saying that Sing T is the set of points where Ao; = B^ = 0. It is
Partially supported by CONACyT Grant 27962-E.
Key words'. Holomorphic foliations — First integrals.
Math. classification: 37F75 - 34A26 - 14H99.
910 ALEXIS GARCIA ZAMORA
more natural to speak about the singular subvariety of T which is defined as
the subvariety of S associated to the ideal sheaf Ty generated by {Aa, Ba}.
In this paper we work with foliations having isolated singularities, i.e, we
assume that the subvariety Sing^" is supported on a finite set of points.
Assume that F admits a holomorphic first integral, that is: there
exists a holomorphic map / : S —> P1 such that any irreducible component
of the fibers of / is a solution of F. We assume, in addition that the generic
fiber of / is irreducible.
Recall that an irreducible analytical curve C C S is a solution of
F if for each x € Co = C - Sing(^), we have F{C^} = T^Co, where
C,^ == (C 0 Os,x)/^s,x-> ls the vector fiber of C at x. In others words, C is
a solution of F if the restricted map Fc '' Cc —^ QS,C factorizes through
the natural inclusion Qc c->> ©5 lo-
in [7] Poincare studied the following problem: assume that F is a
foliation on S = P2 admitting a meromorphic first integral, that is, there
exists a rational map / : P2 —^ P1 (therefore defined away from a finite
number of points), such that the irreducible components of the fibers of /
are leaves of f', away from the indeterminancy locus of /. Then, find an
upper bound for the degree of the general fiber of /, depending of some
information given by F'. (For a more precise statement of the problem see
[7] and [9].)
Poincare ([7] or [9], remark following Example 2.3) noticed that for
foliations of degree m > 4 in P2, the previous problem can be solved finding
a bound for the geometric genus of the general fiber of /. Thus, in a more
general setting, its seems reasonable to study the following problem: given a
foliation F on an algebraic surface 5, admitting a holomorphic first integral,
find a bound for the genus of the general fiber of / in terms of information
depending of F'.
This is the problem that we study in the present paper. In particular
we prove the following result (Theorem 2.2):
Assume that for some integer n ̂  1, h°{S, Os{-nKs)) > 0. Iff is a
foliation on S admitting a holomorphic first integral, then
(2n - l)(g - 1) ^ h\S^ ̂ "'(-(n - l)Ks)) + h°(S^ C') + 1.
The class of surfaces satisfying the previous hypothesis includes all
the 0-dimensional Kodaira surfaces and the Del Pezzo surfaces. The bound
ANNALES DE L'lNSTITUT FOURIER
SHEAVES AND FIRST INTEGRALS 911
obtained in this case depends on the dimension of the cohomology groups
of the sheaves defining F', so this gives a solution to a question analogous
to the Poincare problem for above classes of surfaces.
In Section 3 we study the cohomology of C'~ in the case where S
is a rational surface. It is a case of interest because, given a foliation F
in P2 admitting a meromorphic first integral /, it is possible, after a finite
number of blowings-up, to obtain a rational surface S, and a foliation F '
on S admitting a holomorphic first integral /' whose fibers are birationally
equivalent to the fibers of / (see [8]). Moreover, the cohomology of the
sheaves defining F ' can be computed in terms of the cohomology of the
sheaves defining F and information on the blowings-up that gives rise to 5'.
Our results on Section 3 are based on the study of the direct image of
jC'~ under /. We prove that the two direct-image sheaves are locally free
and we give some partial information on the nature of the decomposition
of these sheaves as the direct sum of invertible sheaves.
Almost all the proofs in Sections 2 and 3 are based on some rela-
tionships between F, £'~1 and the canonical map df obtained in Section 1
(Proposition 1.1, Corollary 1.2 and Theorem 1.3). Part a) of Proposition 1.1
is probably well known, we have taken the formulation and proof from ([3]).
1. Foliations and first integrals.
PROPOSITION 1.1. — a) Let F : C —> Qs be a foliation on S. Then
there exists an exact sequence:
0 -^ C -^ Os ̂  C'~1 —— Osing^ —— 0.
b) I f f : S — > P 1 is a first integral of F, then we have the following
exact sequence:
0 —. C -^ Os -^ Os(2C) —> I —> 0,
or, in other words, C is the kernel of the natural map df : Os ~~^ /*Qp1-
Here C denotes a generic fiber off.
Proof. — a) Let {(^,^,1^0,2)} be a covering of S by coordinates
charts, with transition functions 0a^. Assume that C' is trivial on U^.
Assume that in this choice of coordinates the map o^~ is described locally
by means of
^Q == Aa(Za,l^a,2)dZa,l + B^ (^,1,^,2)^0,2.
TOME 50 (2000), FASCICULE 3
912 ALEXIS GARCIA ZAMORA
The local vector field
XQ. = - ^ a T ^ ^ a T t — — — 5
^Q,I <9^a,2
is a local generator of the kernel of ujy and has isolated singularities. Thus,
if Y is any vector field such that u}a(Y) = 0, then on Ua — Sing.77 we may
find a holomorphic function h such that Y = hX^^ this function h extends
to an element of Ou^i by Hartog's Theorem ([4]), since Sing^ has discrete
support. So, the kernel of o^- : Qs —> ^'~1 is an invertible sheaf generated
by X. On the other hand the image ofujr is the ideal generated by Ao, and
BQ, this proves the assertion, since Sing^ is the 0-dimensional subvariety
defined by Aa and Ba.
b) Consider the canonical map
df:Qs-^rQx.
call its kernel 65 j. The map df is locally defined by means of
^ ( v 9 ^Y 9 \ Y 9f ^Y 9fOJ 1 AQ l———— -r AQ: ,2-Q——— ) — AQ 1-^——— + AQ, 2-a———•\ a^i oz^) Qza^ az^2
Thus, we see that the kernel of df is the subsheaf of Os generated at
each local ring Op by the local vector field
J_ (_9J__9_ _ 9f 9 \
hp \ QZaft QZa.l 9Za,l 9^,2 ) p '
where hp is the greatest common divisor of (9——) and (^7 ) . This
proves that Qs/f ls an invertible sheaf. Moreover, the local generator of
Os,f nas isolated zeros.
Now, using the fact that / is a first integral of F it is easy to see that
there exists a map of sheaves C —^ Qs,f such that the diagram
0 —> Qsj —^ Q-s?
7
T /
c
commutes. Indeed, we first claim that if / : S —> X is a first integral of F
the composite map
c-^es^rex.
is the zero map. For this it is sufficient to check that the assertion is valid
in the vector stalk at each point x e S. Let x be a nonsingular point of T,
then there exists a unique solution C that passes through x, and we have
ANNALES DE L'lNSTITUT FOURIER
SHEAVES AND FIRST INTEGRALS 913
J^^Cx) = T^C. But df(TxC) = 0, since C is an irreducible component of a
fiber of /. On the other hand, if re is a singular point of T then ^(Cx) = 0
and the claim is trivial. As Os, f is the kernel of df, we obtain the desired
morphism.
Now, we need to prove that this map is surjective. This map is
surjective on S — Sing.F, because it is surjective on the vector stalks at
these points. So if Xy denotes the local expression of a generator of C C Os
and Xa is a local generator Os,f then we must have a relation hXy = Xa
on Ua — Sing^", where h is a holomorphic function, now, both Xy and Xa
has only a finite number of zeros, and thus h must be a nowhere vanishing
function on Ua (Hartog's Theorem). This means that h is an unity in 0(Ua)
and so C ̂  Os ,/• D
An easy corollary of part a) of the proposition is the following well
known fact:
COROLLARY 1.2. — C (g) C~1 = JCs'
Proof. — Outside the support of Osing^~ we have ICs~1 ̂  del Os ^
C <g) C'~1. The lemma follows as an application of Hartog's Theorem. D
For a slightly different proof of this fact and other general properties of
the sheaves C and /y~1 see [2]. The reader problably will find the notation
used in that reference more appropiate, here I have followed the notation
in [3], the primary source where I learned the subject.
Assuming that T admits a first integral, call ^^ ̂ ki riijFij the
divisor given by the preimage by / of the critical values of /, (here i runs
on the set of critical values and ki is the number of irreducible components
in the fiber corresponding to i).
THEOREM 1.3. — Let F be a foliation admitting a holomorphic first
integral f with generic fiber C, using the previous notation we have
C'-^Os (2C-^(^-1)^).
Proof. — Let us denote that J = O s / ^ ' Then we have an isomor-
phism J ^ C'~1 away from Sing.77.
Thus, the inclusion
0 —— J —— OsW,
TOME 50 (2000), FASCICULE 3
914 ALEXIS GARCIA ZAMORA
allows us to define a morphism C'-1 -^ Os(2C), outside Sing.F, which is
obviously injective. Now, we take a closer look at the structure of Z.
I is the cokernel of the map df. Note that the image of df is locally
generated, away from SingJ^, by the divisor E^(^ - 1)^-. We conclude
that, away from Sing^ we have an exact sequence:
0 —— £/-1 —— OsW -^ 0^(n.,-lW, —— 0.
£'~1 and Os(2C) being invertible and Smgf discrete this exact sequence
extends to one on S. Now, we use the fact that
Os(2C')o^.(^._^ ^ 0£.,(n.,-l)P.,,
([I], page 90, Lemma 8.1) to conclude that
£'-\-2C)^Os(-^(n^-l)F^.
^
Note that this isomorphism generalizes relationships a, page 39 and
6, page 41 in [7].
2. Bounding g in some particular cases.
In this section we study the cohomology of the sheaf C'~1 and other
related sheaves. As will be clear after the following discussion a good
understanding of these cohomology groups can be helpful for the study
of the Poincare Problem.
We start with the following observation:
PROPOSITION 2.1.— Let f : S -^ P1 be a holomorphic first integral
of the foliation F : C-.Qs.If g denotes the genus of the general fiber C
of f, then
9 ^ h1^^-1) + h^S^'-^-C)).
Proof. — Consider the standard exact sequence
(2.1) 0 —^ Os(-C) — Os — Oc — 0.
Its follows almost immediatly from the definition of the solutions of^
that C (g) Oc ^ QC- Thus, from Corollary 1.2 we obtain: C'~10 Oc ^ Oc.
If we apply the functor (g)/;'"1 to the exact sequence 2.1, we obtain
0 —— C'-\-C) -^ C1-1 —. Oc —— 0,
ANNALES DE L'lNSTITUT FOURIER
SHEAVES AND FIRST INTEGRALS 915
the result follows at once after considering the long exact sequence in
cohomology associated to the previous one. D
A similar argument allows us to solve the analogous to the Poincare
problem on some kinds of surfaces:
THEOREM 2.2. — Under the same hypothesis of Proposition 2.1,
assume, moreover, that h°{S, Os(-nKs)) > 0, for some positive integer n.
Then
(2n - l)(g - 1) ^ h\S^ C'~\-(n - l)Ks)) + h°(S, C') + 1.
Proof. — First of all, we observe that , C being a fiber of /, we
have Os (C) 0 Oc ^ Oc ([I], Lemma 8.1). Thus, applying the adjunction
formula we obtain Os(Ks) 0 Oc ^ Oc(Kc). Furthemore we obtain an
exact sequence:
(2.2)
0 —— C' \-(n - l)Ks -C) —— ̂ "'(-(n - l)Ks) —— Qc^-^ —— 0.
Now, it follows from the Riemann-Roch formula ([5], page 108) that
^^Qc^n-l))=(2n-l)(g-l).
Then we obtain, from 2.2, the following inequality:
(2n-l)(^-l)^^ l(^/: / - l(-(n-l)^))+/,2(^/: /~ l(-(n- l)Ks-C)).
But ^(^/^(-(n - l)Ks - C)) = h^S.C^nKs + C)), by Serre's
duality. On the other hand the global section of Os{-nKs) defines an
inclusion:
0 —— Os{nKs) —— Os,
from this we obtain
h°(S, C'(nKs + C)) ^ /i°(5, C\C)).
Finally we observe that the exact sequence 2.1, followed by tensor
multiplication by Os(C) leads to the exact sequence
0 —— C' —— C\C) —— Oc —— 0.
Thus, we have the inequality h°(S, C'{C)) ^ h°(S, C') +1. D
We note that the class of surfaces satisfying the hypothesis of the
theorem includes the surfaces of Kodaira dimension 0 (K3 and abelian
surface, among others) and Del Pezzo's surfaces. Unfortunately general
rational surfaces and surfaces of general type do not satisfy the hypothesis.
TOME 50 (2000), FASCICULE 3
916 ALEXIS GARCIA ZAMORA
3. Direct images of £'~1 when S is rational.
In this section we start with the study of the cohomology of H!~ in
the case where S is a rational surface. The results that we present here are
just a first step in this study, for the moment we have been not able to
obtain a more satisfactory description.
Thus, in this section we always assume that S is a rational surface.
We start by observing that h2(S,Os) = 0 and this implies, by Proposi-
tion 1.1 a), that h2^, C'~ ) = 0. Indeed, the sequence a) in Proposition 1.1
splits into two short exact sequences:
0 —>C —>Qs —^ J —^ 0,
and
0 — J — C'~1 — Osing^ — 0,
from this it is easy to conclude, taking long exact sequences in cohomology,
that h2(S, r~1) = 0. We denote by g, = h1 (E.^-l)^ Os,(n^-i)^).
Our main result is:
THEOREM 3.1. — The direct-images R0/^'1) and R1/^'1)
are locally free sheaves of rank 1 and g , respectively. Moreover
a) R°W1) ^ Opi(2 - k), where k = E^°(E,(^ - l)^.
OE^-I)^-)-
b) R^f^C1'1) ^ C^p^-l) 0s Opi C* Opi{l), where r,s,t are
integers satisfying r 4- s +1 = g and — r - ^ - t = g — ̂  ̂ .
Proof. — We use the results in [6], Section 5, following the notation
in the reference we call <7t, t € P1 to the fiber of / over t, if M is a coherent
sheaf Md wm denote the restriction of M to Ct.
Now, C'c} ̂  Oci for general t, this implies that /^(C^'^1) ^ 1 for
any t, for a similar reason we have that h°{Ct, C ' C t } ^ 1 ̂  any i" Then, for
any fiber Q of / we have C'c} ̂  Oc^ This implies that h°(Ct, C'c1) = 1
for any t € P1 and then we conclude that R0/^^'1) is locally free of rank
1. From it follows at once that J?1/ '̂'"1) is locally free of rank ^, since the
Euler characteristic of £'^ is locally constant on t and /^(C^,/^1) = g
for general t. This proves the first part of the theorem. In order to prove
parts a) and b) we need the following elementary lemma:
ANNALES DE L'lNSTITUT FOURIER
SHEAVES AND FIRST INTEGRALS 917
LEMMA 3.2. — Let S be a rational surface and f a proper holomor-
phic map from S onto P1. If the general fiber of f is an irreducible curve
of geometric genus g , then
a) R°UOs)^Op^
b) R^^Os^^^Op^-l).
Proof of 3.2. — We note that Osc, ^ Ocr Moreover, S being a
rational surface, we have h°{S, Os) = 1 and ^(S, Os) = h2^, Os) = 0.
Now, it follows from the Leray spectral sequence associated to / that
h°(?\R°f^Os))=^
and
h\P\R°f^Os)) =ho(P\Rlf^Os)) =h\P\Rlf^Os)) =0.
Finally, use Grothendieck's Theorem on locally free sheaves on P1
(any locally free sheaf on P1 splits in a direct sum of invertible sheaves([5],
page 129)) and the Riemann-Roch formula to get the result. D
Now we apply the direct image functor to the exact sequence deduced
in the proof of Theorem 1.3. The long exact sequence obtained in this way
splits into two short exact sequences:
(3.1)
0 — R°W1) —. R°MOs(2C)) —> ̂ °/*(0(^._^J — 0,
(3.2) o — R1/^-1) — R^^Os^C)) — ̂ V*(^(^._^J,
since .R1/ '̂"1) is locally free and R°f^(^(nij - l)^j) is supported on
a finite set of points.
From Lemma 3.2 and the projection formula we deduce that R°f^{0s
(2C)) =0pi{2). Thus,
deg^CC'-1) =2-A; ,
where k = ̂ (P1^0/*^^-!)^,)) = ^°(E(^ - l)^,OE(n,-i)^,).
Thus, part a) of the theorem follows.
b) From Lemma 3.2 and projection formula it follows that
^(0^(20)) ^e^eWl).
TOME 50 (2000), FASCICULE 3
918 ALEXIS GARCIA ZAMORA
This implies that in the decomposition J?1/^^'""1) ^ ©jLiOpi (a^),
we must have a^ ^ 1. Moreover, h2{S,Os) = 0 implies that ^{S^C1'1) =
hl(P\Rl(Cf~l))=0.
Now, as the cohomology of R1 f^C'~1 decomposes in the the direct
sum of the cohomology of Opi (a^), it is easy to deduce, using the Riemann-
Roch formula that a^ ^ —1. From this we conclude that
R1^-1) ̂  e^p^-i) e' Opi e' Opi(i).
It is clear that r-{-s-\-t = g , since ^/^(/y"1) has rank g. Lastly, note
that the exact sequence 3.2 implies that
g = degR^^OsW) = deg R1 W -1) + ̂ (P^A^n,-!)^)).
On the other hand,
^°(P\^(<^(n,-i)Fj) = Y,9i. since h\C^ 0(E(n,-i)F,)^) = 0
for any t G P1 ([6] page 53, Corollary 3). The proof of the theorem is
concluded. D
We would like to note some immediate consequences of the theorem.
First of all note that
COROLLARY 3.3. — r = /i2^,/:'"^-^)).
Proof.
r^PM^W^Xl)]-1),
by Serre duality we have
r = ̂ (P1,^1^-1^-!)) = /r^.r'-^-C)). D
COROLLARY 3.4. — Let F and S be like in Theorem 3.1, if
h°(S^ C'~1) > 0, then f has at most two non-reduced fibers.
Proof. — Its follows at once from part a) of the theorem.
COROLLARY 3.5. — Under the same hypothesis of Theorem 3.1
2g - 2 - Y^g, = /i°(P1, R1^-1)).
ANNALES DE L'lNSTITUT FOURIER
SHEAVES AND FIRST INTEGRALS 919
Proof. — Use Riemann-Roch formula and the fact that
/^(P1,^1/^-1))^:
h\¥\R\Cf-l))=deg(RlW~l))^^nk(Rlf^Cf~l)).
The corollary follows at once from this relationship and part b) of the
theorem. D
BIBLIOGRAPHY
[1] W. BARTH, C. PETERS, and A. VAN DE VEN, Compact Complex Surfaces, Springer
Verlag, 1984.
[2] M. BRUNELLA, Feuilletages holomorphes sur les surfaces complexes compactes,
Ann. scient. EC. Norm. Sup., 30 (1997), 569-594.
[3] X. GOMEZ-MONT, and R. VILA, On Meromorphic Integrals of Holomorphic
Foliations in Surfaces, Unpublished.
[4] P. GRIFFITHS, and J. HARRIS, Principles of Algebraic Geometry, John Wiley &
Sons, 1978.
[5] G. KEMPF, Algebraic Varieties, Cambridge University Press, 1993.
[6] D. MUMFORD, Abelian Varieties, Oxford University Press, 1970.
[7] H. POINCARE, Sur 1'integration algebrique des equations differentielles du primer
ordre, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191.
[8] A. SEIDENNBERG, Reduction of Singularities of the Differential Equation Ady —
Bdx, Am. Journal of Math., (1968), 248-269.
[9] A.G. ZAMORA, Foliations on Algebraic Surfaces having a Rational First Integral,
Public. Mat de la Universita Aut. de Barcelona, 41 (1997), 357-373.
Manuscrit recu Ie 15 mars 1999,
accepte Ie 2 novembre 1999.
Alexis GARCIA ZAMORA,
IMATE-Unidad Morelia
A.P. 61-3
C.P. 58089 Morelia, Mich (Mexique).
amgarcia@j upiter. ecu. umich. mx
TOME 50 (2000), FASCICULE 3


Sheaves associated to holomorphic first integrals

Mathieu, David

Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations

Buchdahl, Nicholas

ANNALES DE L’INSTITUT FOURIER
NICHOLAS BUCHDAHL
A Nakai-Moishezon criterion for non-Kähler surfaces
Annales de l’institut Fourier, tome 50, no 5 (2000), p. 1533-1538
<http://www.numdam.org/item?id=AIF_2000__50_5_1533_0>
© Annales de l’institut Fourier, 2000, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(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-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
50, 5 (2000), 1533-1538
A NAKAI-MOISHEZON CRITERION
FOR NON-KAHLER SURFACES
by Nicholas BUCHDAHL
0. Introduction.
In Corollary 15 of [B], the classical Nakai-Moishezon criterion for a
compact complex surface X was generalised to yield a characterization of
the set of classes in ^^(X) which can be represented by a Kahler form,
a result obtained independently by Lamari [L]. Under the assumption that
&i(X) is even, this result was further generalised in Theorem 16 of [B] to
the case of <99-closed modulo 99-exact (l,l)-forms. The purpose of this
paper is to demonstrate that the assumption on b]_(X) can be dropped
entirely. Namely, the following will be proved:
THEOREM. — Let X be a compact complex surface equipped with
a positive 99-closed (1,1)-form uj and let (p be a smooth real 99-closed
(1, l)-form satisfying f ^ ^ p / \ ( p > 0 , f ^ ( p / \ u ; > 0 and f^(p > 0 for every
irreducible effective divisor D C X with D • D < 0. Then there is a smooth
function g on X such that y -h i99g is positive.
Theorem 16 of [B] differs from this only in that it assumes &i(X)
is even and that f^ ^ > 0 for every effective divisor D C X; however,
this inequality must hold for any effective divisor D with D ' D >_ 0 by
Proposition 5 of that paper.
Keywords: Complex surface — Kahler metric — Nakai-Moishezon criterion.
Math. classification: 32J15 - 53C55 - 32C30.
1534 NICHOLAS BUCHDAHL
Acknowledgement. — This first version of this paper was written
during the second half of 1999 when the author was a visitor at PInstitut
des Hautes Etudes Scientifiques. The author wishes to express his deep
gratitude to IHES for its hospitality.
1. Proof of the theorem.
Let X be a compact complex surface. Since the theorem has already
been proved in the case of even first Betti number, it will be assumed
henceforth that h{X) is odd. The same notation as in [B] is employed
throughout, so A^'9 denotes the sheaf of germs of smooth (p, ^)-forms on
X, with A^'^X) denoting the global sections. A <9<9-closed positive (1,1)-
form LO e A ^ (X) is chosen once and for all, its existence guaranteed by
Gauduchon's theorem [G].
For any / e A^pC) there is a function g e A°'°(X), unique up to
the addition of a constant, such that ^ A (/ + g " ) is a constant multiple
of a;2 where g " := i99g. Since &i(X) is odd, the proof of Lemma 8 in [B]
implies that there is a unique form 0-0 € A^X) with the properties that
it is d-exact and satisfies uj A 0-0 = uj2. The harmonic representative of a
closed (1, l)-form / on X satisfying uj A / = cuj2 for some constant c is
then f-cao. This form is anti-self-dual with respect to uj, a manifestation
of the fact that the intersection form on H2{X,R) restricted to H}^{X) is
negative definite ([BPV], IV 2.13).
For a holomorphic line bundle L on X, there is a unique hermitian
metric on L such that the corresponding hermitian connection has cur-
vature fL satisfying uj A /L = Const-^2. If s C F(X,0(L)) is non-zero
and E is the associated effective divisor ^(O), the equation of currents
27T E = i f-i99\og \s\2 holds by the Poincare-Lelong theorem ([GH]). The-
refore f^ ̂  = ̂  f^ fL A ip for any smooth <9<9-closed (1, l)-form (p. When
the divisor E is given without reference to L, the notation fa will be used
to denote /L for L = 0{E).
A real divisor on X is by definition a finite formal sum of the form
D = Z^ ^iDi where Di C X is an irreducible effective divisor on X and ^
is a real number; D is effective if ^ > 0 for all z, in which case the usual
notation D >_ 0 is employed; similarly, D > E iff D — E ^ 0. As for integral
divisors, the notation fr) is used to denote ̂  ̂ /D,.
The intersection form on ^(X.IR) is denoted by the dot product
symbol. Thus E ' E is the self-intersection number of an effective divisor
ANNALES DE L'lNSTITUT FOURIER
A NAKAI-MOISHEZON CRITERION 1535
E in X, realised by the integral —4^- f^ SE A fa- The notation extends
by M-linearity to all real divisors, and is further extended to denote the
pairing between 99-closed (1, l)-fbrms: y? • -0 := f^ (p A ^ for 99-c\osed
(p, ̂  G A^X). If ip == ifi) for some real divisor D, the notation (p - D may
also be used in place of ^y • ifr>-
LEMMA 1. — Let E C X be an effective integral divisor such that
E ' E = 0. Then for any £ > 0 there is a smooth function g such that
zfp + g " > -^.
Proof. — If there is no smooth function g on X such that ife + Q" +
euj is positive in a neighbourhood of E, the Hahn-Banach Theorem implies
the existence of a current T and a constant c such that /^(z/£;4-a^+(//) ^ c
for every smooth function g and T(^) > c for every smooth 2-form '0 whose
(1, l)-component is positive in a neighbourhood of E.
It follows immediately that T is a (1, l)-current, that 99T = 0, that c
must be non-positive, that T{i/E + ̂ ) <, c, that T(^) ^ 0 for any smooth
(1, l)-form ^ which is positive in a neighbourhood of E and finally that
the support of T must be contained in E. By Lemma 32 of [HL], it follows
that T == ^^ /^£z where hi is a non-negative constant and £'1, £"2,... are
the irreducible components of E. Since E ' E =0 and ^i (X) is odd, [E] = 0
in H2^, M). Hence Er E == 0 for all z, and this gives a contradiction since
then c ̂  T(z/£; + euj} = T(euj) > c.
It can therefore be supposed that E is the zero set of a section s
of a holomorphic line bundle L which has a hermitian connection whose
curvature form / satisfies if > —euj in an open neighbourhood U of E. After
rescaling s if necessary, it can be assumed that {x € X | \s(x)\ <^ 1} C U.
Let \ be a smooth convex increasing function on R such that 0 <
\'(t} < 1 for all t, with \(t) = t for t ^ 0 and with ^(t) = -1 for t < -1.
Then z^(x(log H2)) = x'(log H2) z991og H2 + x"(log |5|2) z9(log |5|2) A
B(log |5|2) ^ x'(log H2) V, so z/ - ̂ 9(x(log |^|2)) ^ (1 - x'(log |^|2)) if ̂
—euj, as required. D
Remark. — The above proof also works in some cases when b\ {X) is
even. For example, if E is irreducible (with E • E = 0), or if every effective
divisor on X has non-negative self-intersection.
LEMMA 2. — Suppose ^ € A^(X) satisfies 99^ = 0, ^ • ^ = 0,
^ • uj > 0 and ^ • D > 0 for every effective divisor D C X. Then for any
TOME 50 (2000), FASCICULE 5
1536 NICHOLAS BUCHDAHL
e > 0 there is a smooth function g such that ^ + g " >. —EUJ.
Proof. — By Lemma 7 of [B], -0 can be approximated arbitrarily
closely in L2 norm by forms of the kind p — g " where p is smooth and
positive and g is smooth. Following exactly the same argument as used in
the proof of Theorem 11 of [B], a sequence of smooth functions Qn and
smooth positive (1, l)-forms pn can be found such that ||'0 + g^ — pn^L2^)
is converging to 0 and gn is converging in L1 to define an almost-positive
closed (1, l)-current P = g^ > —'0. Applying the same arguments as in
the proofs of Theorems 11 and 16 in [B] shows that for any given e > 0
there is a real effective divisor De and a smooth function ge such that
—if De + Q'l > ~^ ~ euj' The construction of De is such that it can be
assumed that De' >_ De for e' < e and the coefficient of an irreducible
component common to both De and De' is the same in both.
Now take a sequence of positive numbers e converging monotonically
to 0. Since \e := CLJ -}- ^ — ifoe + Q'i ls positive, 0 < \e ' Xe ==
e2 uj •a;+47r2 De ' De +2euj -t^ — ^KEUJ - De — 27r^ - D e . The hypotheses on
0 and negativity of the intersection form restricted to H ^ (X) therefore
imply that the cohomology classes [De} € H2(X, M) are uniformly bounded.
After passing to a subsequence if necessary, the corresponding sequence of
harmonic representatives can be assumed to converge smoothly. Moreover,
the inequality 0 < uj'\e =£^'^-\-^'^—^^'De implies that the increasing
sequence of non-negative numbers {c<; • De} is bounded above and hence
converges. Therefore the sequence of forms {foe} converges smoothly to a
closed (1, l)-form /p satisfying f^ • f^ = 0 = ̂  - fr> and uj A ify = cuJ1 for
some constant c >, 0. Since [ify} = 0 in ^(X.IR) it follows if^ = cao.
If c == 0, it follows from the fact that {uj • De} is non-negative and
increasing that uj-De = 0 for all e\ in this case De = 0 for all e and therefore
^ + g'e >. —^ as required.
If c > 0, the identity ^ • 0-0 == 0 and Proposition 5 of [B] imply that
^j-\-g" is a non-negative multiple of (TO for some smooth function g . If there
is a non-zero integral effective divisor E on X such that E ' E == 0, since
[ao} = 0 in ^(X, R) it follows that (TO ' E = 0 and by Proposition 5 of [B]
again, that (TO is a positive multiple of %/£;; in this case, the desired result
follows from Lemma 1. If X has algebraic dimension 1, it is well-known
that X is an elliptic surface ([BPV], VI 4.1) and therefore such a divisor E
exists.
If X has algebraic dimension 0, then by [BPV], IV 6.2, there are only
ANNALES DE L'lNSTITUT FOURIER
A NAKAI-MOISHEZON CRITERION 1537
finitely many irreducible curves on X so that for e sufficiently small, the
real divisors D^ are independent of e. Hence /p = fr> for some genuine
real effective divisor D on X satisfying D ' D = 0. By Lemma 4 in §3.5 of
Ch. V of [Bou], the symmetric negative semi-definite intersection matrix
M associated with the irreducible components of a connected component
of D has a 1-dimensional kernel, and the entries in a generating vector v all
have the same sign. Since v must be a multiple of a column of the cofactor
matrix of M, after multiplying by a real constant it has positive integer
entries. This implies that there is an effective non-zero integral divisor E
on X with E • E = 0, so the desired result follows from the previous para-
graph. D
The proof of the main theorem can now be completed. Let ip e
A^pC) be a 99-closed form satisfying the hypotheses of the theorem.
By the proof of Theorem 14 of [B], there is a form u € A°^(X) such that
(p := (p 4- Qu + 9u is positive; (the hypothesis that &i(X) be even in that
theorem is used only in the final sentence of the proof).
By Proposition 5 of [B], ( p ' y is strictly positive. Let to be the smaller
solution of the equation (y? — toy) ' ((/? — to(p) =0, and set ^ := y — toy.
Since ((p— t(p) •((/?— t(p) > 0 for t satisfying 0 <, t < to, the sign oiuj'((p—tf))
cannot change for such t so uj-^ > 0. Since (<^ — ( p ) ' ((p — (p) = —2[|<9u||2 < 0,
it follows that to <: 1 and therefore for any effective divisor E C X,
^ ' E = (1 - t o ) y ' E > 0 .
The form -0 therefore satisfies the hypotheses of Lemma 2. Applying
that lemma, given e > 0 there is a smooth function ge such that ip + g^ >
—EUJ^ so if e is chosen so small that t^—euj > 0, it follows that ^4-fi^ > 0,
as required. D
Remark. — The methods of this paper show that if (p e A^X)
satisfies the hypotheses of the theorem except for the condition that J^ (p
be positive for every effective E C X with negative self-intersection, there
is an effective real divisor D on X such that y — ifo ls i99- homologous to
a positive form.
BIBLIOGRAPHY
[BPV] W. BARTH, C. PETERS and A. VAN DE VEN, Compact Complex Surfaces, Berlin-
Heidelberg-New York, Springer, 1984.
TOME 50 (2000), FASCICULE 5
1538 NICHOLAS BUCHDAHL
[Bou] N. BOURBAKI, Groupes et Algebres de Lie, Ch. 4,5,6, "Elements de mathemati-
ques" Fasc. XXXIV, Paris, Hermann, 1968.
[B] N. P. BUCHDAHL, On compact Kahler surfaces, Ann. Inst. Fourier, 49-1 (1999),
287-302.
[G] P. GAUDUCHON, Le theoreme de Pexcentricite nulle, C. R. Acad. Sci. Paris, 285
(1977), 387-390.
[GH] P. A. GRIFFITHS and J. HARRIS, Principles of Algebraic Geometry, New York,
Wiley, 1978.
[HL] R. HARVEY and H. B. LAWSON, An intrinsic characterisation of Kahler manifolds,
Invent. Math, 74 (1983), 169-198.
[L] A. LAMARI, Le cone kahlerien d'une surface, J. Math. Pures Appl., 78 (1999),
249-263.
Manuscrit recu le 15 novembre 1999,
revise le 16 mars 2000,
accepte le 27 avril 2000.
Nicholas BUCHDAHL,
University of Adelaide
Department of Pure Mathematics
Adelaide 5005 (Australia).
nbuchdah@maths.adelaide.edu.au
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