ON THE KÄHLER RANK OF COMPACT COMPLEX
SURFACES
MATEI TOMA
Abstract. Harvey and Lawson introduced the Kähler rank and com-
puted it in connection to the cone of positive exact currents of bidimen-
sion (1, 1) for many classes of compact complex surfaces. In this paper
we extend these computations to the only further known class of sur-
faces not considered by them, that of Kato surfaces. Our main tool is
the reduction to the dynamics of associated holomorphic contractions
(C2, 0) → (C2, 0).
Harvey et Lawson ont introduit et calculé le rang de Kähler en re-
lation avec le cône des courants positifs fermés de bidimension (1, 1)
pour beaucoup de classes de surfaces complexes compactes. Dans ce
travail nous étendons ces calculs à la seule classe de surfaces connues
et qui n’avait pas été considérée par eux, celle des surfaces de Kato.
Notre outil principal est la réduction à la dynamique des contractions
holomorphes (C2, 0) → (C2, 0) associées.
1. Introduction
In [8] Harvey and Lawson give a characterisation of Kählerianity for compact
complex surfaces in terms of existence (or rather non-existence) of closed
positive currents which are (1, 1)-components of a boundary. The authors
also investigate and describe the cones formed by such currents for many
types of non-Kähler surfaces: elliptic, Hopf, Inoue. Later Lamari proved
that every non-Kähler surface admits non-trivial positive d-exact currents
of bidimension (1, 1); cf. [10]. In order to estimate the degree of non-
Kählerianity of a compact complex surface smooth positive d-exact currents
are considered in [8] and the Kähler rank is defined as follows: the Kähler
rank is two if the surface admits some Kähler metric, one if it admits some
positive d-exact (1, 1)-form with some supplementary property and zero in
Date: October 30, 2018.
I wish to thank the Max-Planck-Institut für Mathematik in Bonn and the University
of Osnabrück for their hospitality and for financial support during the preparation of
this paper. Furthemore I thank Charles Favre and Karl Oeljeklaus for useful discussions
and the referee for his remarks which improved the presentation of the paper.
Keywords: compact complex surface, global spherical shell, closed positive current, iter-
ation of polynomial maps.
Mots-clé: surface complexe compacte, coquille sphérique globale, courant positif fermé,
itération des applications polynômiales
AMS Classification (2000): 32J15, 32H50 .
http://arxiv.org/abs/0704.1195v2
2 MATEI TOMA
the remaining case; see the more precise Definition 2. Unfortunately it is
not clear whether the Kähler rank is a bimeromorphic invariant.
We consider in this paper instead a bimeromorphic invariant which we call
the modified Kähler rank and which we define to be two if the surface is
Kähler, one if the cone of positive d-exact currents of bidimension (1, 1) is
larger than a half-line and zero if this cone is a half-line. The two notions
agree in the cases considered in [8]. Our main result is the computation
of the cones of positive exact (1, 1)-currents for Kato surfaces. These are
surfaces whose minimal models have positive second Betti number and ad-
mit a global spherical shell (see Definition 9) and are the only “known”
compact complex surfaces not considered in [8]. (We refer the reader to [1]
for the general theory of compact complex surfaces.) It turns out that the
modified Kähler rank does not coincide with the Kähler rank in general. In
order to perform our computations we reduce ourselves to the investigation
of plurisubharmonic functions with a certain invariance property with re-
spect to polynomial automorphisms of C2 associated to Kato surfaces. As
a corollary we obtain
Theorem 1. (a) Every positive d-exact (1, 1)-current on a Kato surface,
and more generally on any “known” non-Kählerian compact complex sur-
face, is a foliated current for some holomorphic foliation of the surface.
(b) All positive d-closed (1, 1)-currents on the “known” non-Kählerian com-
pact complex surfaces excepting on parabolic Inoue surfaces are foliated cur-
rents for some holomorphic foliations.
See Section 3 for definitions.
2. The Kähler rank
We let X be a compact complex surface and denote by Pbdy = Pbdy(X),
P∞bdy = P
bdy(X) the cones of positive currents of bidimension (1, 1) which
are boundaries (i.e. are d-exact), respectively smooth such currents. These
objects have been first studied in [8]. In [10] it was shown that X is Kähler
if and only if Pbdy(X) is trivial. It is easy to see that on a non-Kähler
surface every positive d-closed differential form of type (1, 1) is d-exact, cf.
[10]. The following definition of Harvey and Lawson gives a measure of
non-Kählerianity by looking at the positive closed differential(1, 1)-forms
on X . It is roughly speaking the largest generic rank of a positive closed
(1, 1)-form on X .
Definition 2. Let B(X) = {x ∈ X | ∃φ ∈ P∞bdy(X) φx 6= 0}. The Kähler
rank of X is defined to be two if X admits a Kähler metric. If X is non-
Kähler the Kähler rank is set to be one when B(X) contains a non-trivial
Zariski open subset of X and zero otherwise.
It is not known whether the Kähler rank is a bimeromorphic invariant.
We propose also the following:
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 3
Definition 3. The modified Kähler rank of X is defined to be two when
X admits no non-trivial positive exact current of bidimension (1, 1), zero
when it admits exactly one such current up to a multiplicative constant and
one otherwise.
One can show easily that the modified Kähler rank is a bimeromorphic
invariant by taking push-down and pull-back of currents through blowing
up maps. See the proof of Proposition 7 for a more precise description.
For elliptic surfaces, primary Hopf surfaces and Inoue surfaces one sees that
the Kähler rank and the modified Kähler rank coincide using the precise
description of Pbdy given in [8] in these cases.
Proposition 4. Let T be a positive exact current of bidimension (1, 1) on
the compact complex surface X. Then there is a representation ρ : π1(X) →
(R,+) and a plurisubharmonic function u on the universal cover X̃ of X
such that T = ddcu and u ◦ g = u+ ρ(g) for all g ∈ π1(X).
The function u can be chosen to be smooth if T is smooth.
Proof. One has b1 = dimCH
1(X,C) = dimCH
1(X,OX)+dimCH
0(X,Ω1X) =
h0,1 + h1,0, cf [1]. Denoting the sheaf of closed differential (1, 0)-forms by
dOX and looking at the long exact cohomology sequence of
0 → CX → OX → dOX → 0
one gets an exact sequence
0 → H0(dOX) → H
1(CX) → H
1(OX).
By the above equality follows now the surjectivity of the natural map
H1(CX) → H
1(OX). This is given by mapping a de Rham cohomology class
[β] of a differential form β = β1,0+β0,1 onto the Dolbeault cohomology class
of β0,1.
Let now T be a positive exact current of bidimension (1, 1) on X . Then T =
dS with S = S1,0+ S0,1, S1,0, S0,1 currents of order zero and bidegree (1, 0)
and (0, 1) respectively, and S1,0 = S̄0,1. Since ∂̄S0,1 = 0, S0,1 represents a
cohomology class in H1(OX) and let β = β
1,0 + β0,1 be a closed differential
form with [β0,1] = [S0,1] in H1(OX) and U a current of degree 0 on X with
S0,1 = β0,1 + ∂̄U . The lift β̃ of β to the universal cover X̃ is d-exact and
let f be a smooth function on X̃ with df = β. In particular ∂̄f = β0,1.
This implies S0,1 = ∂̄(f + U) and T = dS = d(∂̄(f + U) + ∂(f̄ + Ū)) =
i∂∂̄(2ℑm(f + U)).
Moreover for g ∈ π1(X) we have d(f ◦ g − f) = 0 hence f ◦ g − f must be
constant. Set ρ(g) = 2ℑm(f ◦g−f). The current 2ℑm(f +U) is associated
to a plurisubharmonic function u on X̃ . Since u − 2ℑmf = 2ℑmU comes
from X we see that u has the desired automorphy behaviour with respect
to the action of π1(X).
It is clear that u can be chosen to be smooth when T is smooth. �
4 MATEI TOMA
Definition 5. We say that an effective reduced divisor C = C1 + ... + Cn
on X is a cycle of rational curves if n ≥ 1, C1,..., Cn are rational curves
and either n = 1 and C1 has a node or n > 1, all components C1,..., Cn are
smooth and the dual graph of C is cyclic.
Corollary 6. For a compact complex surface X with a cycle of rational
curves C and b1(X) = 1 the Kähler rank is zero.
Proof. Under the above hypotheses the natural map Z ∼= π1(C) → π1(X)
is an isomorphism by a Theorem of Nakamura, [13]. Let g be a generator
of π1(X). If the Kähler rank of X were one, we would get a smooth non-
constant plurisubharmonic function u on the universal cover X̃ satisfying
u ◦ g = u+ c
for some constant c ∈ R by Proposition 4. The inverse image C̃ of the cycle
of rational curves C of X is an infinite chain of rational curves on X̃ . Since
u is smooth and constant on each link of this chain we would get c = 0
hence u would descend to X . But here u must be constant contradicting
our assumptions. �
Proposition 7. The modified Kähler rank is a bimeromorphic invariant.
Proof. It is known that for smooth surfaces the property of being Kähler is
invariant under bimeromorphic transformations.
Let now X be non-Kähler and X ′ the blown-up surface at a point x ∈ X .
Any plurisubharmonic function u on the universal cover X̃ may be lifted
to X̃ ′. Conversely let u′ be a plurisubharmonic function on X̃ ′. It will be
constant on the exceptional divisors of X̃ ′ coming from X ′ → X . Thus u′ is
the pull-back of a plurisubharmonic function on X̃ , which satisfies the same
invariance condition as u with respect to the action of π1(X) ∼= π1(X
This gives a bijective correspondence between Pbdy(X) and Pbdy(X
′), hence
the invariance of the modified Kähler rank. �
From the above argument it also follows that a counterexample to the
bimeromorphic invariance of the Kähler rank could only be given by some
non-elliptic non-Kähler minimal surface X of Kähler rank zero admitting
a continuous plurisubharmonic function u on the universal cover X̃ which
is smooth outside a discrete subset of X̃ and exhibits the automorphy be-
haviour from Proposition 4. From [8] we gather that this is not the case of
Hopf surfaces or Inoue surfaces and as we shall see it is not going to be the
case of Kato surfaces either.
Finally let us mention the following description of the cone P (X) of positive
d-closed (1, 1)-currents:
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 5
Remark 8. Let C1, ..., Cn be the irreducible curves of negative self-
intersection on a non-Kählerian surface X and [C1], ..., [Cn] the corre-
sponding currents of integration. Then
P (X) = Pbdy(X) +
R≥0[Ci].
Proof. Let us denote by Fj the irreducible curves ofX of zero self-intersection.
Since X is non-algebraic it will admit no curve of positive self-intersection.
Thus the Siu decomposition of a positive closed current T takes the form:
ai[Ci] +
bj [Fj ] +R,
where ai, bj ∈ R and R is a positive current whose Lelong level sets are
finite. By [11] Prop. 4.3 and its proof R is nef. Since X is non-Kählerian it
follows that R must be exact, cf. [11] Thm. 7.1. The integration currents
[Fj] are also exact, since F
j = 0 and the intersection form on H
(X) is
negative definite for a non-Kählerian surface. �
3. Kato surfaces
We recall here some facts on Kato surfaces which will be needed in the
sequel. We refer the reader to [9], [3], [4], [5], [6] for more details.
Definition 9. A global spherical shell on a compact complex surface X is
the image Σ of the sphere S3 by a holomorphic embedding of a neighbourhood
of S3 from C2 into X such that X \ Σ is connected. We call X a Kato
surface if X admits a global spherical shell and the second Betti number of
a minimal model of X is positive.
The other minimal surfaces admitting a global spherical shell are the Hopf
surfaces. Their second Betti number is zero.
The notion of global spherical shell and a construction method for sur-
faces with global spherical shells were introduced by Ma. Kato in [9]. An
important analytic object associated to this construction method was intro-
duced and studied by Dloussky in [3]. It is the germ of a holomorphic map
(C2, 0) → (C2, 0). We shortly recall the facts now.
Take the unit ball B in C2 around the origin and blow-up the origin. Choose
a point P1 on the exceptional curve C1 thus obtained and blow it up again.
Continue by blowing up a point on the last created exceptional curve. Af-
ter n blow-ups one considers the blowing down map π : B′ → B. The
exceptional divisor on B′ is a tree of n smooth rational curves. The only
(−1)-curve among them is the last created curve Cn. Choose a point Pn on
Cn, a biholomorphic map σ : B̄ → σ(B̄) onto a small compact neighbour-
hood of Pn in B
′ and glue the two components of the boundary of B′ \σ(B)
by means of σ ◦π. In this way one obtains a minimal compact complex sur-
face X . One can show that the image of S3 through σ is a global spherical
shell on X and that b2(X) = n. Thus X is a minimal Kato surface.
6 MATEI TOMA
The images of the exceptional curves are the only (compact) rational curves
on X . They form an effective reduced divisor which we denote by D. De-
pending on the structure of D one subdivides the class of minimal Kato
surfaces into:
(1) Enoki surfaces, when D is a cycle of rational curves and D is homo-
logically trivial,
(2) intermediate surfaces, when D consists of a cycle of rational curves
and of at least one further rational curve attached to the cycle,
(3) Inoue-Hirzebruch surfaces, when D consists of one or two cycles of
rational curves and D is not homologically trivial.
In particular Kato surfaces admit cycles of rational curves and therefore
their Kähler rank is zero.
In the case of Enoki surfaces a further curve might appear. In such a case
the curve will be elliptic and the surface is called a parabolic Inoue surface.
By definition the Dloussky germ associated to X is the germ of the map
f := π ◦ σ : B → B around the origin. It can be shown that the conjugacy
class of this germ determines the isomorphy class of X . We shall denote
by X(f) a Kato surface associated to such a germ of holomorphic map
f : (C2, 0) → (C2, 0). One can relate certain analytic objects on X to germs
of objects on (C2, 0) which are invariant under f as follows.
One recovers first the universal cover X̃ ofX from the above construction by
considering an infinite number of copies (Ai)i∈Z of B̄
′ \ σ(B) and by gluing
for all i ∈ Z the pseudoconvex component of the border ∂Ai−1 of Ai−1 to
the pseudoconcave component of ∂Ai by means of σ ◦ π again. If one glues
in this way only the copies (Ai)i≤0 and then caps the pseudoconcave end of
A0 by a copy of B using σ, one obtains a non-compact complex surface X̂ ,
a holomorphic map p : X̃ → X̂ , a (−1)-curve Ĉ on X̂ and a point Ô on Ĉ
such that p extends the identity map of ∪i≤0Ai, Ĉ is the isomorphic image
of a rational curve C of X̃ through p and p restricts to an isomorphism
X̃ \ p−1(Ô) → X̂ \ {Ô}. In fact p−1(Ô) is the union of the infinitely many
rational curves appearing after C on X̃ in the “order of creation”, cf. [3],
Prop. 3.4. Thus p : X̃ → X̂ can be seen as a blowing down of the infinitely
many exceptional curves in p−1(Ô). The generator g of π1(X) mapping Ai
to Ai+1 induces a holomorphic map ĝ : X̂ → X̂ with p ◦ g = ĝ ◦ p. One sees
that Ô is fixed by ĝ, that ĝ(Ĉ) = Ô, and that the germ of ĝ at Ô is the
same as the germ of π ◦ σ : B → B at the origin.
Let now u be any plurisubharmonic function on X̃ . The restriction of u
to X̂ \ {Ô} extends to a plurisubharmonic function û on X̂ , [7]. It is clear
that û ◦ ĝ = û − c in case u ◦ g = u − c for some c ∈ R. Conversely one
gets a plurisubharmonic function u = û ◦ p starting from û. In fact, since
the germ of ĝ around Ô is contracting, it is enough to have only a germ of
û around Ô satisfying û ◦ ĝ = û − c in order to recover u on X̃ with the
property u ◦ g = u− c.
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 7
According to [3], [4], [5], [6] we get the following three normal forms for
representatives of conjugacy classes of Dloussky germs:
(1) in the case of Enoki surfaces
f(z, w) = (αz, wzs +Q(z)),
where α ∈ ∆∗ = ∆ \ {0}, s ≥ 1 and Q is a complex polynomial of
degree at most s and with Q(0) = 0; we have denoted by ∆ the unit
disc in C;
(2) in the case of intermediate surfaces
f(z, w) = (zp, λwzs +Q(z)),
where p ≥ 2, s ≥ 1, λ ∈ C∗ and Q(z) =
m=1 amz
m + az
p−1 is
a complex polynomial with gcd{p, m | am 6= 0} = 1 and a = 0 if
(p− 1) ∤ s or λ 6= 1;
(3) in the case of Inoue-Hirzebruch surfaces
f(z, w) = (zawb, zcwd),
where the matrix
is a product of b2(X) matrices of the form
with at least one factor of the first kind.
In the rest of this paper we shall determine the germs of plurisubharmonic
functions around the origin of C2 satisfying u ◦ f = u − c for some fixed
c ∈ R>0 and each type of germ f as above.
4. The main results
Theorem 10. (a) Let f : C2 → C2,
f(z, w) = (αz, wzs +Q(z)),
with α ∈ ∆∗, s ≥ 1 and Q is a complex polynomial of degree at most s
and with Q(0) = 0. Let u : C2 → [−∞,∞[, u(z, w) = log |z|. Then up
to some additive constant, u is the only plurisubharmonic function on C2
which satisfies u ◦ f = u+ log |α|.
(b) On an Enoki surface the integration current on the cycle of rational
curves is the only positive exact current of bidimension (1, 1) up to multi-
plicative constants.
Proof. We start by proving part (b) of the theorem.
Let C = C1 + ... + Cn be the cycle of rational curves on the Enoki surface
X . Since C is homologically trivial the current of integration [C] along C is
a positive d-exact current of bidimension (1, 1). We denote by E the elliptic
curve on X in case it exists.
Let T be an arbitrary positive exact current of bidimension (1, 1) on X and
mCi := inf{ν(T, x) |x ∈ Ci}, mC = minmCi , mE its generic Lelong numbers
8 MATEI TOMA
along Ci, C and E respectively. We denote by χA the characteristic function
of a subset A of X .
The Siu decomposition of T has the form
T = χET + χCT + χX\(C∪E)T = mE [E] +
mCi [Ci] + χX\(C∪E)T =
= mE [E] +
(mCi −mC)[Ci] +mC [C] + χX\(C∪E)T,
see [2] 6.18, 3.2.4. Since C is the only homologically trivial effective reduced
non-trivial divisor X we get as in Remark 8 that χX\(C∪E)T is exact and
T = mC [C] + χX\(C∪E)T.
Hence we may replace T by χX\(C∪E)T which is positive, d-exact, has van-
ishing generic Lelong number along C and is the trivial extension toX of the
restriction of T to X \(C∪E). After normalisation we obtain a correspond-
ing plurisubharmonic function v on C2 satisfying v ◦ f = v + log |α|. We
denote the current ddcv on C2 by T again. Consequently we have f ∗T = T .
Moreover, since C corresponds to the axis A = {z = 0} ⊂ C2, the generic
Lelong number mA of T along A must vanish. Since f(A) = {0} and in
general ν(f ∗T, x) ≥ ν(T, f(x)) (cf. [12]) it follows that ν(T, 0) = 0.
The differential form dz defines a holomorphic foliation on C2 which is
invariant by f , since f ∗(dz) = αdz.
Claim: T is a foliation current for this foliation.
This means that for any test function φ ∈ C∞c (C
2) one has T (φdz∧dz̄) = 0.
As before T is the trivial extension to C2 of its restriction to C2 \ A. It is
therefore enough to check that T (φdz ∧ dz̄) = 0 for test functions φ with
support in C2 \ A.
We have
|T (φdz ∧ dz̄)| = |f ∗T (φdz ∧ dz̄)| = |T ((f−1)∗(φdz ∧ dz̄))| =
= |T (
φ ◦ f−1
dz ∧ dz̄)| ≤
max |φ|
2σT (f(Suppφ)),
where σT = T ∧
(dz ∧ dz̄ + dw ∧ dw̄) denotes the trace measure of T .
Iterating we obtain
(1) |T (φdz ∧ dz̄)| ≤
max |φ|
|α|2n
2σT (f
n(Supp φ)),
for any n ∈ N.
We now need to estimate how large fn(B(0, R)) is. Take C1 = max{
|Q(z)|
| z ∈
∆̄}. We denote by P (R1, R2) the bidisc of radii R1, R2 centered at the origin
of C2. Then
f(P (1, R2) ⊂ P (|α|, R2 + C1),
f 2(P (1, R2) ⊂ P (|α|
2, |α|(R2 + C1) + |α|C1) ⊂
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 9
⊂ P (|α|2, |α|(R2 + C1) + (
1− |α|
− 1)C1),
f 3(P (1, R2) ⊂ P (|α|
3, |α|3(R2 + C1) +
1− |α|
|C1) ⊂
⊂ P (|α|3, |α|3(R2 + C1) + (
1− |α|
− 1)C1),
f 4(P (1, R2) ⊂ P (|α|
4, |α|6(R2 + C1) +
1− |α|
|C1) ⊂
⊂ P (|α|4, |α|6(R2 + C1) + (
1− |α|
− 1)C1),
and further
fn(P (1, R2) ⊂ P (|α|
n, |α|
n(n−1)
2 (R2 + C1) +
|α|n−1
1− |α|
|C1).
From this and from (1) it follows that there is a constant C2 such that
|T (φdz ∧ dz̄)| ≤ C2max |φ|
σT (B(0, |α|
|α|2n
for all n sufficiently large. But
σT (B(0, |α|
|α|2n
= πν(T, 0) = 0,
and the claim follows.
Using the claim it can be easily shown that T is invariant under translations
in the w-direction, hence T = pr∗1S, where S is a positive current on C
of dimension zero and pr1 : C
2 → C denotes the first projection. Set
f1 : C → C, f1(z) = αz. Then pr1 ◦ f = f1 ◦ pr1, hence f
1 (S) = S. The
current S is of the form µidz ∧ dz̄, where µ is a positive measure on C.
Denote by ∆(r) the disc of radius r around the origin of C. The invariance
property of S implies
µ(∆(r)) = µ(∆(|α|r)) = µ(∆(|α|nr))
for all n ∈ N and r ∈ R>0. This entails that S is supported at the origin of
C and hence that T = 0, since T is not carried by A.
We now turn to part (a) of the theorem. Let v be a plurisubharmonic
function on C2 satisfying v ◦ f = f + log |α|. By part (b) of the theorem we
see that ddcv ≤ ddcu or ddcu ≤ ddcv. But then u− v or v − u would give a
plurisubharmonic function on X which has to be constant. �
Notation For α ∈ R>0 we denote by Kα the set of continuous α-periodic
functions ψ : R → R which fulfil the inequality
−ψ′′ + ψ′ + 1 ≥ 0
in generalised sense.
10 MATEI TOMA
Notice that Kα is infinite dimensional: for any α-periodic smooth function
φ : R → R and for any small enough factor ǫ ∈ R>0 one has ǫφ ∈ Kα.
Theorem 11. (a) Let f : ∆× C → ∆× C,
f(z, w) = (zp, λwzs +Q(z)),
where p ≥ 2, s ≥ 1, λ ∈ C∗ and Q(z) =
m=1 amz
m + az
p−1 is a complex
polynomial with gcd{p, m | am 6= 0} = 1 and a = 0 if (p − 1) ∤ s or
λ 6= 1. The plurisubharmonic functions u : ∆×C → [−∞,∞[ which satisfy
u ◦ f = u− log p are precisely the functions of the form
u(z, w) = − log(− log |z|)− ψ(log(− log |z|))
for ψ ∈ Klog p.
(b) The cone of positive exact currents of bidimension (1, 1) on the inter-
mediate surface X(f) corresponds bijectively to the cone of currents of the
form cddcu on ∆×C for u as above and c ∈ R>0. In particular, the modified
Kähler rank of X(f) is one.
Proof. There are no homologically trivial divisors on X(f) this time (ex-
cepting 0 of course). As in the case of Enoki surfaces we reduce ourselves
to the investigation of closed positive currents T on ∆ × C with vanishing
Lelong number at the origin and satisfying f ∗T = T . Moreover we may
again suppose that T is the extension to ∆×C of its restriction to ∆∗ ×C.
The differential form dz defines a holomorphic foliation on ∆× C which is
invariant under f .
We start again by showing that T is a foliation current for this foliation.
For this it is enough to check that the measure
idz ∧ dz̄ ∧ T
vanishes on ∆∗ × C. We use the invariance of T by f again.
Take 0 < r1 < r2 < 1, r
′ > 0, D := (∆(r2) \ ∆(r1)) × ∆(r
′), D′ :=
f−n(fn(D)) and A(r1, r2) = A(r1, r2, r
′) := (idz ∧ dz̄ ∧ T )(D) the measure
of the set D. We have of course D′ ⊃ D. Since f : ∆∗ × C → ∆∗ × C is p
to 1 we get
(idz ∧ dz̄ ∧ T )(fn(D)) = pn(i|z|2(p
n−1)dz ∧ dz̄ ∧ T )(D′) ≥
≥ pn(i|z|2p
dz ∧ dz̄ ∧ T )(D′) ≥ pnr
1 A(r1, r2).
Hence
(2) A(r1, r2) ≤
n i(∂∂̄(|z|2 + |w|2p) ∧ T )(fn(D))
As before we need to estimate the width of fn(D). Let C1 = max{
|Q(z)|
| z ∈
∆̄} and C2 = max(1, |λ|). For 0 < r < 1, 0 < r
′ we see that
f(P (r, r′)) ⊂ P (rp, rC2(r
)) ⊂ P (rp, rC2(r
(1− r)C2
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 11
f 2(P (r, r′)) ⊂ P (rp
, rp+1C22r
rp−1C22C1
(1− r)C2
) ⊂ P (rp
, rp−1C22(r
(1− r)C2
f 3(P (r, r′)) ⊂ P (rp
p−1C22(r
(1− r)C2
⊂ P (rp
2+p−1C32r
2−1C32C1
(1− r)C2
For sufficiently large n we thus obtain
fn(P (r, r′)) ⊂ {|z|2 + |w|2p < 2C1r
which in combination with the inequality (2) gives
A(r1, r2) ≤
n 2i(∂∂̄(|z|2 + |w|2p) ∧ T )({|z|2 + |w|2p < 2C1r
But the factor
2i(∂∂̄(|z|2 + |w|2p) ∧ T )({|z|2 + |w|2p < 2C1r
converges to 4C1ν(T, φ, 0), where ν(T, φ, 0) is the Lelong number with re-
spect to φ = log(|z|2 + |w|2p). This Lelong number vanishes by the Com-
parison Theorem for Lelong numbers [2], since the usual Lelong number
vanishes. So for any ǫ > 0 we can find some N ∈ N such that
A(r1, r2) ≤
for all n ≥ N . Moreover this inequality holds also for smaller r1 and r2.
δ = (
and look at the division (r1, δr1, δ
2r1, ..., δ
2pnr1 = r2) of the interval [r1, r2].
We have seen that A(δir1, δ
i+1r1) ≤
ǫ = 1
ǫ hence A(r1, r2) =
i=1A(δ
i−1r1, δ
ir1) ≤ 2
ǫ, which proves that T is a foliation current.
As in the proof of Theorem 10, T is invariant under translations in the w-
direction, hence T = pr∗1S, where S is a positive current on ∆ of dimension
zero and pr1 : ∆ × C → ∆ is the first projection. On the other hand
T = ddcu, where u is a plurisubharmonic function on ∆ × C with u ◦ f =
u− log p. The restriction of u to a general leaf {z = const.} of our foliation
is a harmonic function in w. We shall show that ∂u
= 0, the case of ∂u
being similar.
Since T is a foliation current one has that ∂
∂z̄∂w
= 0 as distributions.
In particular ∂u
= 0 is invariant by translations in the z-directions. The
invariance condition by translations in w implies that
(3) λzs
(f(z, w)) =
(z, w).
12 MATEI TOMA
Take now a small non-zero solution z′ of the equation
(4) w = λzsw +Q(z)
for fixed small w, w 6= 0. Using the invariance in the z-direction together
with (3) and (4) we obtain
(z, w) =
(z′, w) = λ(z′)s
((z′)p, w) = λ(z′)s
(z, w),
hence ∂u
(z, w) = 0 for small w. But for large w the vanishing holds as well
since we can iterate on (3).
Thus u is constant on the leaves {z = const.} and therefore descends to a
subharmonic function u on ∆ fulfilling the condition u(zp) = u(z) − log p
for all z ∈ ∆.
Take now z ∈ ∆ and a pn-th root of unity θ. The invariance condition
implies u(z) = u(zp
) + n log p = u(θz) hence each value u(z) is attained
on a dense subset of the circle {|z| = const.}. By semi-continuity it follows
that u depends only on r = |z|.
Let v = u|[0,1[. By the maximum principle and the invariance condition on
u one sees that v is strictly increasing. Using this, the semi-continuity of u
and the definition of subharmonicity (cf. [2] 1.4.13.b) one infers that v is
continuous. The Laplace operator takes the form ∂
in polar
coordinates (r, φ). We thus reduce ourselves to the search of continuous
functions v : [0, 1[→ [−∞,∞[ which satisfy
(5) v(rp) = v(r)− log p
(6) v′′(r) +
v′(r) ≥ 0.
Let h :] − ∞,∞] → [0, 1[, h(t) = exp(− exp t) and ψ = −v ◦ h − h. The
conditions (5) and (6) translate into
ψ(t+ log p) = ψ(t),
−ψ′′ + ψ′ + 1 ≥ 0
and thus ψ ∈ Klog p and the theorem is proved. �
For the case of Inoue-Hirzebruch surfaces we need some preparations. Let
f : C2 → C2, f(z, w) = (zawb, zcwd), and denote by A the matrix
We suppose that A is a product of matrices of the form
with at least one factor of the first kind. We have det(A) = ±1 and by
Lemma 2.2 from [4], trace(A) > 2 unless A =
or A =
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 13
any case one sees that the eigenvalues λ1, λ2 of A
t are real, irrational and one
of them is larger than 1. We set λ = λ1 > 1. Let (α, β) = (α1, β1), (α2, β2)
be eigenvectors associated to λ1 and λ2 respectively. An easy computation
shows that αβ = α1β1 > 0, α2β2 < 0. We choose α, β, α2 ∈ R>0 and set
φ = φ1 = α log |z|+ β log |w|, φ2 = α2 log |z|+ β2 log |w|,
U := {φ < 0} ⊂ C2.
Then φi ◦ f = λiφi for i = 1, 2.
Now we can state
Theorem 12. (a) For A, λ, α, β as above let φ = α log |z| + β log |w|,
U = φ−1([−∞, 0[), f : U → U ,
f(z, w) = (zawb, zcwd).
Then the plurisubharmonic functions u on U which satisfy u ◦ f = u− log λ
are precisely the functions of the form
u = − log(−φ)− ψ(log(−φ)),
for ψ ∈ Klog p.
(b) The cone of positive exact currents of bidimension (1, 1) on the Inoue-
Hirzebruch surface X(f) corresponds bijectively to the cone of currents of
the form cddcu on U for u as above and c ∈ R>0. In particular, the modified
Kähler rank of X(f) is one.
Proof. As in the proof of Theorem 11 we reduce ourselves to the investi-
gation of currents T on U of the form T = ddcu with u plurisubharmonic
on U and such that u ◦ f = u − log λ and ν(T, 0) = 0. Moreover we may
suppose that T is the extension to U of its restriction to U ∩ (C∗ × C∗).
There is again a holomorphic foliation on U , this time singular, which will
be shown to be compatible with T . On U∩(C∗×C∗) this foliation is induced
by the smooth positive (1, 1)-form ddcv where v := − log(−φ). As before it
is enough to show that the measure ddcv∧T is zero on any compact subset
of U ∩ (C∗ × C∗).
For a better visualisation of the present situation we introduce the map
E : C × C → C∗ × C∗, E(ζ, ω) = (exp ζ, expω). We denote the linear
automorphism induced by A on C2 again by A and get the relations
f ◦ E = E ◦ A,
φ ◦ E(ζ, ω) = αiℜeζ + βiℜeω, i = 1, 2.
In particular E−1(U) = {αℜeζ + βℜeω < 0}.
We also see that U is covered by compact subsets of the formD = D(c1, δ, c2) :=
φ−11 ([−c1δ,−c1]) ∩ φ
2 ([−c2, c2]) for c1, c2 ∈ R>0, δ ∈ [1,∞[. Since both T
and ddcv are invariant by f we have
A(c1, δ) = A(c1, δ, c2) := (dd
cv ∧ T )(D(c1, δ, c2)) =
= (ddcv ∧ T )fn(D(c1, δ, c2))) = (dd
cv ∧ T )(D(λnc1, δ, λ
−nc2)).
We need to estimate ddcv on fn(D). On D we have
14 MATEI TOMA
−c1δ ≤ α1 log |z|+ β1 log |w| ≤ −c1,
−c2 ≤ α2 log |z| + β2 log |w| ≤ c2,
hence
c1δβ2 − c2β1
α2β1 − α1β2
) ≤ |z| ≤ exp(
c1β2 + c2β1
α2β1 − α1β2
−c1δα2 − c2α1
α2β1 − α1β2
) ≤ |w| ≤ exp(
−c1α2 + c2α1
α2β1 − α1β2
Therefore we get on fn(D)
λnc1δβ2 − λ
−nc2β1
α2β1 − α1β2
) ≤ |z| ≤ exp(
λnc1β2 + λ
−nc2β1
α2β1 − α1β2
−λnc1δα2 − λ
−nc2α1
α2β1 − α1β2
) ≤ |w| ≤ exp(
−λnc1α2 + λ
−nc2α1
α2β1 − α1β2
hence fn(D) ⊂ B(0, rn), where
r2n = exp(2
λnc1β2 + λ
−nc2β1
α2β1 − α1β2
) + exp(2
−λnc1α2 + λ
−nc2α1
α2β1 − α1β2
Now on D again
i∂∂̄v =
i∂φ ∧ ∂̄φ
i(α dz
+ β dw
) ∧ (α dz̄
+ β dw̄
min{|z|2, |w|2 | (z, w) ∈ D}
i(dz ∧ dz̄ + dw ∧ dw̄)
where C1 is a constant not depending on D. This implies
A(c1, δ) ≤
min{φ2(z, w)|z|2, φ2(z, w)|w|2 | (z, w) ∈ fn(D)}
σT (B(0, r
As in the previous cases
σT (B(0,r
converges to zero, therefore for any
ǫ > 0 we get
C1σT (B(0,r
< ǫ as soon as n is large enough. Setting
2c1 max{α1,−β2}
α2β1−α1β2
+ 1 we get by our estimates r2n ≤ exp(−λ
nC2) and
min{|z|2, |w|2 | (z, w) ∈ fn(D)} ≥ exp(−λnC2δ), hence
A(c1, δ) ≤
exp(λnC2(δ − 1))
Set m = ⌊λ2n⌋ and consider the division (1, exp 1
, exp 2
, ..., exp m
= e) of
the interval [1, e]. For any subinterval our estimates hold with eC2 instead
of C2, hence
A(c1, e, c2) =
A(c1 exp
, exp
, c2) ≤
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 15
exp(λnC2(exp
− 1))
≤ exp(C2e
exp 1
and the last term converges to exp(C2e)
showing that A(c1, e, c2) = 0.
As in the proof of Theorem 11 we next check that u is constant on the leaves
of the foliation given by α dz
+ β dw
on U ∩ (C∗ × C∗).
Indeed, for V := u ◦ E one has V ◦ A = V − log λ on {αℜeζ + βℜeω < 0}
and ddcV is a foliation current for the foliation given by αdζ + βdω.
We consider a linear change of coordinates which diagonalizes A on C2
leading to A(ξ, τ) = (λξ, λ−1τ). Keeping the notation V for V after this
coordinate change we notice that V restricted to the leaves {ξ = const.} is
harmonic and that ∂V
are invariant by translations in the ξ-direction.
As in the case of intermediate surfaces we get
(ξ, τ).
Iterating this relation and using the continuity of ∂V
on {ξ = const.} we
obtain
τ) = λn
(ξ, τ)
which forces ∂V
to vanish. Similarly ∂V
= 0. Thus V is constant on the
leaves {ξ = const.}.
Returning now to the coordinates (ζ, ω) on C2 we shall show that V only
depends on ℜeζ and ℜeω. The function V is doubly periodic in ℑmζ ,
ℑmω since V = u ◦ E. Fix ℜeζ and ℜeω and look at a leaf {αζ + βω =
const.}. Under this restriction the imaginary parts must satisfy some rela-
tion αℑmζ + βℑmω = const. which describes a dense subset in the “torus
of the imaginary parts” R2/(2πZ)2 since α/β is irrational. On this subset V
is constant and by semi-continuity it must be constant on the whole torus.
Switching once more to coordinates (ξ, τ) we see that V depends on ℜeξ
alone. We have reduced ourselves in this way to the search of continuous
functions v : R<0 → R subject to the conditions
v(λt) = v(t)− log λ, ∀t ∈ R<0,
v′′ ≥ 0.
Remark that the continuity of v can be deduced as in Theorem 11 after first
restricting u to the line {z = w}.
Take now h : R → R<0, h(s) = − exp(s) and ψ = −v ◦ h − h. Then the
conditions on v translate into ψ ∈ Klog λ. Thus u = − log(−φ)−ψ(log(−φ))
and the proof is finished. �
Finally, the first part of Theorem 1 is a direct consequence of the description
of Pbdy(X) given in [8] and in the above Theorems. For the second part it
suffices to apply Remark 8 and to notice that the only case of a curve on a
“known” non-Kählerian surface, which is not invariant under a holomorphic
foliation, is that of the elliptic curve on a parabolic Inoue surface.
16 MATEI TOMA
References
[1] Barth W., Hulek K., Peters C., Van de Ven A.: Compact complex sur-
faces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin, Springer-Verlag,
(2004).
[2] Demailly J.P.: Complex analytic and algebraic geometry,
http://www-fourier.ujf-grenoble.fr/∼demailly/books.html.
[3] Dloussky G.: Structure des surfaces de Kato, Mémoire de la S.M.F 112.n◦14
(1984).
[4] Dloussky G.: Sur la classification des germes d’applications holomorphes con-
tractantes, Math. Ann. 280, (1988), 649-661.
[5] Dloussky G., Oeljeklaus K.: Vector fields and foliations on surfaces of class
VII0, Ann. Inst. Fourier 49, (1999), 1503-1545.
[6] Favre, Ch.: Classification of 2-dimensional contracting rigid germs, Jour. Math.
Pures Appl. 79 (2000), 475-514.
[7] Grauert, H., Remmert, R.: Plurisubharmonische Funktionen in komplexen
Räumen, Math. Z. 65 (1956), 175-194.
[8] Harvey, R., Lawson, H. B.: An intrinsic characterization of Khler manifolds,
Invent. Math. 74 (1983), 169-198.
[9] Kato Ma.: Compact complex manifolds containing “global spherical shells”, Pro-
ceedings of the Int. Symp. Alg. Geometry, Kyoto 1977, Kinokuniya Book Store,
Tokyo 1978.
[10] Lamari, A.: Courants kählériens et surfaces compactes, Ann. Inst. Fourier 49
(1999), 263-285.
[11] Lamari, A.: Le cône Kählérien d’une surface, J. Math. Pures Appl. 78 (1999),
249-263.
[12] Méo, M.: Image inverse d’un courant positif fermé par une application analytique
surjective, C. R. Acad. Sci. Paris Sr. I Math. 322 (1996), 1141–1144.
[13] Nakamura I.: On surfaces of class VII0 with curves II, Tohôku Math. Jour. 42,
(1990), 475-516.
Institut de Mathématiques Elie Cartan, Nancy-Université, B.P. 239, 54506
Vandoeuvre-lès-Nancy Cedex, France and Institute of Mathematics of the
Romanian Academy.
E-mail address : toma@iecn.u-nancy.fr
URL: http://www.iecn.u-nancy.fr/∼toma/
http://www-fourier.ujf-grenoble.fr/~demailly/books.html
	1. Introduction
	2. The Kähler rank
	3. Kato surfaces
	4. The main results
	References

  Harvey and Lawson introduced the Kaehler rank and computed it in connection
to the cone of positive exact currents of bidimension (1,1) for many classes of
compact complex surfaces. In this paper we extend these computations to the
only further known class of surfaces not considered by them, that of Kato
surfaces. Our main tool is the reduction to the dynamics of associated
holomorphic contractions.

Introduction
In [8] Harvey and Lawson give a characterisation of Kählerianity for compact
complex surfaces in terms of existence (or rather non-existence) of closed
positive currents which are (1, 1)-components of a boundary. The authors
also investigate and describe the cones formed by such currents for many
types of non-Kähler surfaces: elliptic, Hopf, Inoue. Later Lamari proved
that every non-Kähler surface admits non-trivial positive d-exact currents
of bidimension (1, 1); cf. [10]. In order to estimate the degree of non-
Kählerianity of a compact complex surface smooth positive d-exact currents
are considered in [8] and the Kähler rank is defined as follows: the Kähler
rank is two if the surface admits some Kähler metric, one if it admits some
positive d-exact (1, 1)-form with some supplementary property and zero in
Date: October 30, 2018.
I wish to thank the Max-Planck-Institut für Mathematik in Bonn and the University
of Osnabrück for their hospitality and for financial support during the preparation of
this paper. Furthemore I thank Charles Favre and Karl Oeljeklaus for useful discussions
and the referee for his remarks which improved the presentation of the paper.
Keywords: compact complex surface, global spherical shell, closed positive current, iter-
ation of polynomial maps.
Mots-clé: surface complexe compacte, coquille sphérique globale, courant positif fermé,
itération des applications polynômiales
AMS Classification (2000): 32J15, 32H50 .
http://arxiv.org/abs/0704.1195v2
2 MATEI TOMA
the remaining case; see the more precise Definition 2. Unfortunately it is
not clear whether the Kähler rank is a bimeromorphic invariant.
We consider in this paper instead a bimeromorphic invariant which we call
the modified Kähler rank and which we define to be two if the surface is
Kähler, one if the cone of positive d-exact currents of bidimension (1, 1) is
larger than a half-line and zero if this cone is a half-line. The two notions
agree in the cases considered in [8]. Our main result is the computation
of the cones of positive exact (1, 1)-currents for Kato surfaces. These are
surfaces whose minimal models have positive second Betti number and ad-
mit a global spherical shell (see Definition 9) and are the only “known”
compact complex surfaces not considered in [8]. (We refer the reader to [1]
for the general theory of compact complex surfaces.) It turns out that the
modified Kähler rank does not coincide with the Kähler rank in general. In
order to perform our computations we reduce ourselves to the investigation
of plurisubharmonic functions with a certain invariance property with re-
spect to polynomial automorphisms of C2 associated to Kato surfaces. As
a corollary we obtain
Theorem 1. (a) Every positive d-exact (1, 1)-current on a Kato surface,
and more generally on any “known” non-Kählerian compact complex sur-
face, is a foliated current for some holomorphic foliation of the surface.
(b) All positive d-closed (1, 1)-currents on the “known” non-Kählerian com-
pact complex surfaces excepting on parabolic Inoue surfaces are foliated cur-
rents for some holomorphic foliations.
See Section 3 for definitions.
2. The Kähler rank
We let X be a compact complex surface and denote by Pbdy = Pbdy(X),
P∞bdy = P
bdy(X) the cones of positive currents of bidimension (1, 1) which
are boundaries (i.e. are d-exact), respectively smooth such currents. These
objects have been first studied in [8]. In [10] it was shown that X is Kähler
if and only if Pbdy(X) is trivial. It is easy to see that on a non-Kähler
surface every positive d-closed differential form of type (1, 1) is d-exact, cf.
[10]. The following definition of Harvey and Lawson gives a measure of
non-Kählerianity by looking at the positive closed differential(1, 1)-forms
on X . It is roughly speaking the largest generic rank of a positive closed
(1, 1)-form on X .
Definition 2. Let B(X) = {x ∈ X | ∃φ ∈ P∞bdy(X) φx 6= 0}. The Kähler
rank of X is defined to be two if X admits a Kähler metric. If X is non-
Kähler the Kähler rank is set to be one when B(X) contains a non-trivial
Zariski open subset of X and zero otherwise.
It is not known whether the Kähler rank is a bimeromorphic invariant.
We propose also the following:
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 3
Definition 3. The modified Kähler rank of X is defined to be two when
X admits no non-trivial positive exact current of bidimension (1, 1), zero
when it admits exactly one such current up to a multiplicative constant and
one otherwise.
One can show easily that the modified Kähler rank is a bimeromorphic
invariant by taking push-down and pull-back of currents through blowing
up maps. See the proof of Proposition 7 for a more precise description.
For elliptic surfaces, primary Hopf surfaces and Inoue surfaces one sees that
the Kähler rank and the modified Kähler rank coincide using the precise
description of Pbdy given in [8] in these cases.
Proposition 4. Let T be a positive exact current of bidimension (1, 1) on
the compact complex surface X. Then there is a representation ρ : π1(X) →
(R,+) and a plurisubharmonic function u on the universal cover X̃ of X
such that T = ddcu and u ◦ g = u+ ρ(g) for all g ∈ π1(X).
The function u can be chosen to be smooth if T is smooth.
Proof. One has b1 = dimCH
1(X,C) = dimCH
1(X,OX)+dimCH
0(X,Ω1X) =
h0,1 + h1,0, cf [1]. Denoting the sheaf of closed differential (1, 0)-forms by
dOX and looking at the long exact cohomology sequence of
0 → CX → OX → dOX → 0
one gets an exact sequence
0 → H0(dOX) → H
1(CX) → H
1(OX).
By the above equality follows now the surjectivity of the natural map
H1(CX) → H
1(OX). This is given by mapping a de Rham cohomology class
[β] of a differential form β = β1,0+β0,1 onto the Dolbeault cohomology class
of β0,1.
Let now T be a positive exact current of bidimension (1, 1) on X . Then T =
dS with S = S1,0+ S0,1, S1,0, S0,1 currents of order zero and bidegree (1, 0)
and (0, 1) respectively, and S1,0 = S̄0,1. Since ∂̄S0,1 = 0, S0,1 represents a
cohomology class in H1(OX) and let β = β
1,0 + β0,1 be a closed differential
form with [β0,1] = [S0,1] in H1(OX) and U a current of degree 0 on X with
S0,1 = β0,1 + ∂̄U . The lift β̃ of β to the universal cover X̃ is d-exact and
let f be a smooth function on X̃ with df = β. In particular ∂̄f = β0,1.
This implies S0,1 = ∂̄(f + U) and T = dS = d(∂̄(f + U) + ∂(f̄ + Ū)) =
i∂∂̄(2ℑm(f + U)).
Moreover for g ∈ π1(X) we have d(f ◦ g − f) = 0 hence f ◦ g − f must be
constant. Set ρ(g) = 2ℑm(f ◦g−f). The current 2ℑm(f +U) is associated
to a plurisubharmonic function u on X̃ . Since u − 2ℑmf = 2ℑmU comes
from X we see that u has the desired automorphy behaviour with respect
to the action of π1(X).
It is clear that u can be chosen to be smooth when T is smooth. �
4 MATEI TOMA
Definition 5. We say that an effective reduced divisor C = C1 + ... + Cn
on X is a cycle of rational curves if n ≥ 1, C1,..., Cn are rational curves
and either n = 1 and C1 has a node or n > 1, all components C1,..., Cn are
smooth and the dual graph of C is cyclic.
Corollary 6. For a compact complex surface X with a cycle of rational
curves C and b1(X) = 1 the Kähler rank is zero.
Proof. Under the above hypotheses the natural map Z ∼= π1(C) → π1(X)
is an isomorphism by a Theorem of Nakamura, [13]. Let g be a generator
of π1(X). If the Kähler rank of X were one, we would get a smooth non-
constant plurisubharmonic function u on the universal cover X̃ satisfying
u ◦ g = u+ c
for some constant c ∈ R by Proposition 4. The inverse image C̃ of the cycle
of rational curves C of X is an infinite chain of rational curves on X̃ . Since
u is smooth and constant on each link of this chain we would get c = 0
hence u would descend to X . But here u must be constant contradicting
our assumptions. �
Proposition 7. The modified Kähler rank is a bimeromorphic invariant.
Proof. It is known that for smooth surfaces the property of being Kähler is
invariant under bimeromorphic transformations.
Let now X be non-Kähler and X ′ the blown-up surface at a point x ∈ X .
Any plurisubharmonic function u on the universal cover X̃ may be lifted
to X̃ ′. Conversely let u′ be a plurisubharmonic function on X̃ ′. It will be
constant on the exceptional divisors of X̃ ′ coming from X ′ → X . Thus u′ is
the pull-back of a plurisubharmonic function on X̃ , which satisfies the same
invariance condition as u with respect to the action of π1(X) ∼= π1(X
This gives a bijective correspondence between Pbdy(X) and Pbdy(X
′), hence
the invariance of the modified Kähler rank. �
From the above argument it also follows that a counterexample to the
bimeromorphic invariance of the Kähler rank could only be given by some
non-elliptic non-Kähler minimal surface X of Kähler rank zero admitting
a continuous plurisubharmonic function u on the universal cover X̃ which
is smooth outside a discrete subset of X̃ and exhibits the automorphy be-
haviour from Proposition 4. From [8] we gather that this is not the case of
Hopf surfaces or Inoue surfaces and as we shall see it is not going to be the
case of Kato surfaces either.
Finally let us mention the following description of the cone P (X) of positive
d-closed (1, 1)-currents:
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 5
Remark 8. Let C1, ..., Cn be the irreducible curves of negative self-
intersection on a non-Kählerian surface X and [C1], ..., [Cn] the corre-
sponding currents of integration. Then
P (X) = Pbdy(X) +
R≥0[Ci].
Proof. Let us denote by Fj the irreducible curves ofX of zero self-intersection.
Since X is non-algebraic it will admit no curve of positive self-intersection.
Thus the Siu decomposition of a positive closed current T takes the form:
ai[Ci] +
bj [Fj ] +R,
where ai, bj ∈ R and R is a positive current whose Lelong level sets are
finite. By [11] Prop. 4.3 and its proof R is nef. Since X is non-Kählerian it
follows that R must be exact, cf. [11] Thm. 7.1. The integration currents
[Fj] are also exact, since F
j = 0 and the intersection form on H
(X) is
negative definite for a non-Kählerian surface. �
3. Kato surfaces
We recall here some facts on Kato surfaces which will be needed in the
sequel. We refer the reader to [9], [3], [4], [5], [6] for more details.
Definition 9. A global spherical shell on a compact complex surface X is
the image Σ of the sphere S3 by a holomorphic embedding of a neighbourhood
of S3 from C2 into X such that X \ Σ is connected. We call X a Kato
surface if X admits a global spherical shell and the second Betti number of
a minimal model of X is positive.
The other minimal surfaces admitting a global spherical shell are the Hopf
surfaces. Their second Betti number is zero.
The notion of global spherical shell and a construction method for sur-
faces with global spherical shells were introduced by Ma. Kato in [9]. An
important analytic object associated to this construction method was intro-
duced and studied by Dloussky in [3]. It is the germ of a holomorphic map
(C2, 0) → (C2, 0). We shortly recall the facts now.
Take the unit ball B in C2 around the origin and blow-up the origin. Choose
a point P1 on the exceptional curve C1 thus obtained and blow it up again.
Continue by blowing up a point on the last created exceptional curve. Af-
ter n blow-ups one considers the blowing down map π : B′ → B. The
exceptional divisor on B′ is a tree of n smooth rational curves. The only
(−1)-curve among them is the last created curve Cn. Choose a point Pn on
Cn, a biholomorphic map σ : B̄ → σ(B̄) onto a small compact neighbour-
hood of Pn in B
′ and glue the two components of the boundary of B′ \σ(B)
by means of σ ◦π. In this way one obtains a minimal compact complex sur-
face X . One can show that the image of S3 through σ is a global spherical
shell on X and that b2(X) = n. Thus X is a minimal Kato surface.
6 MATEI TOMA
The images of the exceptional curves are the only (compact) rational curves
on X . They form an effective reduced divisor which we denote by D. De-
pending on the structure of D one subdivides the class of minimal Kato
surfaces into:
(1) Enoki surfaces, when D is a cycle of rational curves and D is homo-
logically trivial,
(2) intermediate surfaces, when D consists of a cycle of rational curves
and of at least one further rational curve attached to the cycle,
(3) Inoue-Hirzebruch surfaces, when D consists of one or two cycles of
rational curves and D is not homologically trivial.
In particular Kato surfaces admit cycles of rational curves and therefore
their Kähler rank is zero.
In the case of Enoki surfaces a further curve might appear. In such a case
the curve will be elliptic and the surface is called a parabolic Inoue surface.
By definition the Dloussky germ associated to X is the germ of the map
f := π ◦ σ : B → B around the origin. It can be shown that the conjugacy
class of this germ determines the isomorphy class of X . We shall denote
by X(f) a Kato surface associated to such a germ of holomorphic map
f : (C2, 0) → (C2, 0). One can relate certain analytic objects on X to germs
of objects on (C2, 0) which are invariant under f as follows.
One recovers first the universal cover X̃ ofX from the above construction by
considering an infinite number of copies (Ai)i∈Z of B̄
′ \ σ(B) and by gluing
for all i ∈ Z the pseudoconvex component of the border ∂Ai−1 of Ai−1 to
the pseudoconcave component of ∂Ai by means of σ ◦ π again. If one glues
in this way only the copies (Ai)i≤0 and then caps the pseudoconcave end of
A0 by a copy of B using σ, one obtains a non-compact complex surface X̂ ,
a holomorphic map p : X̃ → X̂ , a (−1)-curve Ĉ on X̂ and a point Ô on Ĉ
such that p extends the identity map of ∪i≤0Ai, Ĉ is the isomorphic image
of a rational curve C of X̃ through p and p restricts to an isomorphism
X̃ \ p−1(Ô) → X̂ \ {Ô}. In fact p−1(Ô) is the union of the infinitely many
rational curves appearing after C on X̃ in the “order of creation”, cf. [3],
Prop. 3.4. Thus p : X̃ → X̂ can be seen as a blowing down of the infinitely
many exceptional curves in p−1(Ô). The generator g of π1(X) mapping Ai
to Ai+1 induces a holomorphic map ĝ : X̂ → X̂ with p ◦ g = ĝ ◦ p. One sees
that Ô is fixed by ĝ, that ĝ(Ĉ) = Ô, and that the germ of ĝ at Ô is the
same as the germ of π ◦ σ : B → B at the origin.
Let now u be any plurisubharmonic function on X̃ . The restriction of u
to X̂ \ {Ô} extends to a plurisubharmonic function û on X̂ , [7]. It is clear
that û ◦ ĝ = û − c in case u ◦ g = u − c for some c ∈ R. Conversely one
gets a plurisubharmonic function u = û ◦ p starting from û. In fact, since
the germ of ĝ around Ô is contracting, it is enough to have only a germ of
û around Ô satisfying û ◦ ĝ = û − c in order to recover u on X̃ with the
property u ◦ g = u− c.
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 7
According to [3], [4], [5], [6] we get the following three normal forms for
representatives of conjugacy classes of Dloussky germs:
(1) in the case of Enoki surfaces
f(z, w) = (αz, wzs +Q(z)),
where α ∈ ∆∗ = ∆ \ {0}, s ≥ 1 and Q is a complex polynomial of
degree at most s and with Q(0) = 0; we have denoted by ∆ the unit
disc in C;
(2) in the case of intermediate surfaces
f(z, w) = (zp, λwzs +Q(z)),
where p ≥ 2, s ≥ 1, λ ∈ C∗ and Q(z) =
m=1 amz
m + az
p−1 is
a complex polynomial with gcd{p, m | am 6= 0} = 1 and a = 0 if
(p− 1) ∤ s or λ 6= 1;
(3) in the case of Inoue-Hirzebruch surfaces
f(z, w) = (zawb, zcwd),
where the matrix
is a product of b2(X) matrices of the form
with at least one factor of the first kind.
In the rest of this paper we shall determine the germs of plurisubharmonic
functions around the origin of C2 satisfying u ◦ f = u − c for some fixed
c ∈ R>0 and each type of germ f as above.
4. The main results
Theorem 10. (a) Let f : C2 → C2,
f(z, w) = (αz, wzs +Q(z)),
with α ∈ ∆∗, s ≥ 1 and Q is a complex polynomial of degree at most s
and with Q(0) = 0. Let u : C2 → [−∞,∞[, u(z, w) = log |z|. Then up
to some additive constant, u is the only plurisubharmonic function on C2
which satisfies u ◦ f = u+ log |α|.
(b) On an Enoki surface the integration current on the cycle of rational
curves is the only positive exact current of bidimension (1, 1) up to multi-
plicative constants.
Proof. We start by proving part (b) of the theorem.
Let C = C1 + ... + Cn be the cycle of rational curves on the Enoki surface
X . Since C is homologically trivial the current of integration [C] along C is
a positive d-exact current of bidimension (1, 1). We denote by E the elliptic
curve on X in case it exists.
Let T be an arbitrary positive exact current of bidimension (1, 1) on X and
mCi := inf{ν(T, x) |x ∈ Ci}, mC = minmCi , mE its generic Lelong numbers
8 MATEI TOMA
along Ci, C and E respectively. We denote by χA the characteristic function
of a subset A of X .
The Siu decomposition of T has the form
T = χET + χCT + χX\(C∪E)T = mE [E] +
mCi [Ci] + χX\(C∪E)T =
= mE [E] +
(mCi −mC)[Ci] +mC [C] + χX\(C∪E)T,
see [2] 6.18, 3.2.4. Since C is the only homologically trivial effective reduced
non-trivial divisor X we get as in Remark 8 that χX\(C∪E)T is exact and
T = mC [C] + χX\(C∪E)T.
Hence we may replace T by χX\(C∪E)T which is positive, d-exact, has van-
ishing generic Lelong number along C and is the trivial extension toX of the
restriction of T to X \(C∪E). After normalisation we obtain a correspond-
ing plurisubharmonic function v on C2 satisfying v ◦ f = v + log |α|. We
denote the current ddcv on C2 by T again. Consequently we have f ∗T = T .
Moreover, since C corresponds to the axis A = {z = 0} ⊂ C2, the generic
Lelong number mA of T along A must vanish. Since f(A) = {0} and in
general ν(f ∗T, x) ≥ ν(T, f(x)) (cf. [12]) it follows that ν(T, 0) = 0.
The differential form dz defines a holomorphic foliation on C2 which is
invariant by f , since f ∗(dz) = αdz.
Claim: T is a foliation current for this foliation.
This means that for any test function φ ∈ C∞c (C
2) one has T (φdz∧dz̄) = 0.
As before T is the trivial extension to C2 of its restriction to C2 \ A. It is
therefore enough to check that T (φdz ∧ dz̄) = 0 for test functions φ with
support in C2 \ A.
We have
|T (φdz ∧ dz̄)| = |f ∗T (φdz ∧ dz̄)| = |T ((f−1)∗(φdz ∧ dz̄))| =
= |T (
φ ◦ f−1
dz ∧ dz̄)| ≤
max |φ|
2σT (f(Suppφ)),
where σT = T ∧
(dz ∧ dz̄ + dw ∧ dw̄) denotes the trace measure of T .
Iterating we obtain
(1) |T (φdz ∧ dz̄)| ≤
max |φ|
|α|2n
2σT (f
n(Supp φ)),
for any n ∈ N.
We now need to estimate how large fn(B(0, R)) is. Take C1 = max{
|Q(z)|
| z ∈
∆̄}. We denote by P (R1, R2) the bidisc of radii R1, R2 centered at the origin
of C2. Then
f(P (1, R2) ⊂ P (|α|, R2 + C1),
f 2(P (1, R2) ⊂ P (|α|
2, |α|(R2 + C1) + |α|C1) ⊂
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 9
⊂ P (|α|2, |α|(R2 + C1) + (
1− |α|
− 1)C1),
f 3(P (1, R2) ⊂ P (|α|
3, |α|3(R2 + C1) +
1− |α|
|C1) ⊂
⊂ P (|α|3, |α|3(R2 + C1) + (
1− |α|
− 1)C1),
f 4(P (1, R2) ⊂ P (|α|
4, |α|6(R2 + C1) +
1− |α|
|C1) ⊂
⊂ P (|α|4, |α|6(R2 + C1) + (
1− |α|
− 1)C1),
and further
fn(P (1, R2) ⊂ P (|α|
n, |α|
n(n−1)
2 (R2 + C1) +
|α|n−1
1− |α|
|C1).
From this and from (1) it follows that there is a constant C2 such that
|T (φdz ∧ dz̄)| ≤ C2max |φ|
σT (B(0, |α|
|α|2n
for all n sufficiently large. But
σT (B(0, |α|
|α|2n
= πν(T, 0) = 0,
and the claim follows.
Using the claim it can be easily shown that T is invariant under translations
in the w-direction, hence T = pr∗1S, where S is a positive current on C
of dimension zero and pr1 : C
2 → C denotes the first projection. Set
f1 : C → C, f1(z) = αz. Then pr1 ◦ f = f1 ◦ pr1, hence f
1 (S) = S. The
current S is of the form µidz ∧ dz̄, where µ is a positive measure on C.
Denote by ∆(r) the disc of radius r around the origin of C. The invariance
property of S implies
µ(∆(r)) = µ(∆(|α|r)) = µ(∆(|α|nr))
for all n ∈ N and r ∈ R>0. This entails that S is supported at the origin of
C and hence that T = 0, since T is not carried by A.
We now turn to part (a) of the theorem. Let v be a plurisubharmonic
function on C2 satisfying v ◦ f = f + log |α|. By part (b) of the theorem we
see that ddcv ≤ ddcu or ddcu ≤ ddcv. But then u− v or v − u would give a
plurisubharmonic function on X which has to be constant. �
Notation For α ∈ R>0 we denote by Kα the set of continuous α-periodic
functions ψ : R → R which fulfil the inequality
−ψ′′ + ψ′ + 1 ≥ 0
in generalised sense.
10 MATEI TOMA
Notice that Kα is infinite dimensional: for any α-periodic smooth function
φ : R → R and for any small enough factor ǫ ∈ R>0 one has ǫφ ∈ Kα.
Theorem 11. (a) Let f : ∆× C → ∆× C,
f(z, w) = (zp, λwzs +Q(z)),
where p ≥ 2, s ≥ 1, λ ∈ C∗ and Q(z) =
m=1 amz
m + az
p−1 is a complex
polynomial with gcd{p, m | am 6= 0} = 1 and a = 0 if (p − 1) ∤ s or
λ 6= 1. The plurisubharmonic functions u : ∆×C → [−∞,∞[ which satisfy
u ◦ f = u− log p are precisely the functions of the form
u(z, w) = − log(− log |z|)− ψ(log(− log |z|))
for ψ ∈ Klog p.
(b) The cone of positive exact currents of bidimension (1, 1) on the inter-
mediate surface X(f) corresponds bijectively to the cone of currents of the
form cddcu on ∆×C for u as above and c ∈ R>0. In particular, the modified
Kähler rank of X(f) is one.
Proof. There are no homologically trivial divisors on X(f) this time (ex-
cepting 0 of course). As in the case of Enoki surfaces we reduce ourselves
to the investigation of closed positive currents T on ∆ × C with vanishing
Lelong number at the origin and satisfying f ∗T = T . Moreover we may
again suppose that T is the extension to ∆×C of its restriction to ∆∗ ×C.
The differential form dz defines a holomorphic foliation on ∆× C which is
invariant under f .
We start again by showing that T is a foliation current for this foliation.
For this it is enough to check that the measure
idz ∧ dz̄ ∧ T
vanishes on ∆∗ × C. We use the invariance of T by f again.
Take 0 < r1 < r2 < 1, r
′ > 0, D := (∆(r2) \ ∆(r1)) × ∆(r
′), D′ :=
f−n(fn(D)) and A(r1, r2) = A(r1, r2, r
′) := (idz ∧ dz̄ ∧ T )(D) the measure
of the set D. We have of course D′ ⊃ D. Since f : ∆∗ × C → ∆∗ × C is p
to 1 we get
(idz ∧ dz̄ ∧ T )(fn(D)) = pn(i|z|2(p
n−1)dz ∧ dz̄ ∧ T )(D′) ≥
≥ pn(i|z|2p
dz ∧ dz̄ ∧ T )(D′) ≥ pnr
1 A(r1, r2).
Hence
(2) A(r1, r2) ≤
n i(∂∂̄(|z|2 + |w|2p) ∧ T )(fn(D))
As before we need to estimate the width of fn(D). Let C1 = max{
|Q(z)|
| z ∈
∆̄} and C2 = max(1, |λ|). For 0 < r < 1, 0 < r
′ we see that
f(P (r, r′)) ⊂ P (rp, rC2(r
)) ⊂ P (rp, rC2(r
(1− r)C2
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 11
f 2(P (r, r′)) ⊂ P (rp
, rp+1C22r
rp−1C22C1
(1− r)C2
) ⊂ P (rp
, rp−1C22(r
(1− r)C2
f 3(P (r, r′)) ⊂ P (rp
p−1C22(r
(1− r)C2
⊂ P (rp
2+p−1C32r
2−1C32C1
(1− r)C2
For sufficiently large n we thus obtain
fn(P (r, r′)) ⊂ {|z|2 + |w|2p < 2C1r
which in combination with the inequality (2) gives
A(r1, r2) ≤
n 2i(∂∂̄(|z|2 + |w|2p) ∧ T )({|z|2 + |w|2p < 2C1r
But the factor
2i(∂∂̄(|z|2 + |w|2p) ∧ T )({|z|2 + |w|2p < 2C1r
converges to 4C1ν(T, φ, 0), where ν(T, φ, 0) is the Lelong number with re-
spect to φ = log(|z|2 + |w|2p). This Lelong number vanishes by the Com-
parison Theorem for Lelong numbers [2], since the usual Lelong number
vanishes. So for any ǫ > 0 we can find some N ∈ N such that
A(r1, r2) ≤
for all n ≥ N . Moreover this inequality holds also for smaller r1 and r2.
δ = (
and look at the division (r1, δr1, δ
2r1, ..., δ
2pnr1 = r2) of the interval [r1, r2].
We have seen that A(δir1, δ
i+1r1) ≤
ǫ = 1
ǫ hence A(r1, r2) =
i=1A(δ
i−1r1, δ
ir1) ≤ 2
ǫ, which proves that T is a foliation current.
As in the proof of Theorem 10, T is invariant under translations in the w-
direction, hence T = pr∗1S, where S is a positive current on ∆ of dimension
zero and pr1 : ∆ × C → ∆ is the first projection. On the other hand
T = ddcu, where u is a plurisubharmonic function on ∆ × C with u ◦ f =
u− log p. The restriction of u to a general leaf {z = const.} of our foliation
is a harmonic function in w. We shall show that ∂u
= 0, the case of ∂u
being similar.
Since T is a foliation current one has that ∂
∂z̄∂w
= 0 as distributions.
In particular ∂u
= 0 is invariant by translations in the z-directions. The
invariance condition by translations in w implies that
(3) λzs
(f(z, w)) =
(z, w).
12 MATEI TOMA
Take now a small non-zero solution z′ of the equation
(4) w = λzsw +Q(z)
for fixed small w, w 6= 0. Using the invariance in the z-direction together
with (3) and (4) we obtain
(z, w) =
(z′, w) = λ(z′)s
((z′)p, w) = λ(z′)s
(z, w),
hence ∂u
(z, w) = 0 for small w. But for large w the vanishing holds as well
since we can iterate on (3).
Thus u is constant on the leaves {z = const.} and therefore descends to a
subharmonic function u on ∆ fulfilling the condition u(zp) = u(z) − log p
for all z ∈ ∆.
Take now z ∈ ∆ and a pn-th root of unity θ. The invariance condition
implies u(z) = u(zp
) + n log p = u(θz) hence each value u(z) is attained
on a dense subset of the circle {|z| = const.}. By semi-continuity it follows
that u depends only on r = |z|.
Let v = u|[0,1[. By the maximum principle and the invariance condition on
u one sees that v is strictly increasing. Using this, the semi-continuity of u
and the definition of subharmonicity (cf. [2] 1.4.13.b) one infers that v is
continuous. The Laplace operator takes the form ∂
in polar
coordinates (r, φ). We thus reduce ourselves to the search of continuous
functions v : [0, 1[→ [−∞,∞[ which satisfy
(5) v(rp) = v(r)− log p
(6) v′′(r) +
v′(r) ≥ 0.
Let h :] − ∞,∞] → [0, 1[, h(t) = exp(− exp t) and ψ = −v ◦ h − h. The
conditions (5) and (6) translate into
ψ(t+ log p) = ψ(t),
−ψ′′ + ψ′ + 1 ≥ 0
and thus ψ ∈ Klog p and the theorem is proved. �
For the case of Inoue-Hirzebruch surfaces we need some preparations. Let
f : C2 → C2, f(z, w) = (zawb, zcwd), and denote by A the matrix
We suppose that A is a product of matrices of the form
with at least one factor of the first kind. We have det(A) = ±1 and by
Lemma 2.2 from [4], trace(A) > 2 unless A =
or A =
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 13
any case one sees that the eigenvalues λ1, λ2 of A
t are real, irrational and one
of them is larger than 1. We set λ = λ1 > 1. Let (α, β) = (α1, β1), (α2, β2)
be eigenvectors associated to λ1 and λ2 respectively. An easy computation
shows that αβ = α1β1 > 0, α2β2 < 0. We choose α, β, α2 ∈ R>0 and set
φ = φ1 = α log |z|+ β log |w|, φ2 = α2 log |z|+ β2 log |w|,
U := {φ < 0} ⊂ C2.
Then φi ◦ f = λiφi for i = 1, 2.
Now we can state
Theorem 12. (a) For A, λ, α, β as above let φ = α log |z| + β log |w|,
U = φ−1([−∞, 0[), f : U → U ,
f(z, w) = (zawb, zcwd).
Then the plurisubharmonic functions u on U which satisfy u ◦ f = u− log λ
are precisely the functions of the form
u = − log(−φ)− ψ(log(−φ)),
for ψ ∈ Klog p.
(b) The cone of positive exact currents of bidimension (1, 1) on the Inoue-
Hirzebruch surface X(f) corresponds bijectively to the cone of currents of
the form cddcu on U for u as above and c ∈ R>0. In particular, the modified
Kähler rank of X(f) is one.
Proof. As in the proof of Theorem 11 we reduce ourselves to the investi-
gation of currents T on U of the form T = ddcu with u plurisubharmonic
on U and such that u ◦ f = u − log λ and ν(T, 0) = 0. Moreover we may
suppose that T is the extension to U of its restriction to U ∩ (C∗ × C∗).
There is again a holomorphic foliation on U , this time singular, which will
be shown to be compatible with T . On U∩(C∗×C∗) this foliation is induced
by the smooth positive (1, 1)-form ddcv where v := − log(−φ). As before it
is enough to show that the measure ddcv∧T is zero on any compact subset
of U ∩ (C∗ × C∗).
For a better visualisation of the present situation we introduce the map
E : C × C → C∗ × C∗, E(ζ, ω) = (exp ζ, expω). We denote the linear
automorphism induced by A on C2 again by A and get the relations
f ◦ E = E ◦ A,
φ ◦ E(ζ, ω) = αiℜeζ + βiℜeω, i = 1, 2.
In particular E−1(U) = {αℜeζ + βℜeω < 0}.
We also see that U is covered by compact subsets of the formD = D(c1, δ, c2) :=
φ−11 ([−c1δ,−c1]) ∩ φ
2 ([−c2, c2]) for c1, c2 ∈ R>0, δ ∈ [1,∞[. Since both T
and ddcv are invariant by f we have
A(c1, δ) = A(c1, δ, c2) := (dd
cv ∧ T )(D(c1, δ, c2)) =
= (ddcv ∧ T )fn(D(c1, δ, c2))) = (dd
cv ∧ T )(D(λnc1, δ, λ
−nc2)).
We need to estimate ddcv on fn(D). On D we have
14 MATEI TOMA
−c1δ ≤ α1 log |z|+ β1 log |w| ≤ −c1,
−c2 ≤ α2 log |z| + β2 log |w| ≤ c2,
hence
c1δβ2 − c2β1
α2β1 − α1β2
) ≤ |z| ≤ exp(
c1β2 + c2β1
α2β1 − α1β2
−c1δα2 − c2α1
α2β1 − α1β2
) ≤ |w| ≤ exp(
−c1α2 + c2α1
α2β1 − α1β2
Therefore we get on fn(D)
λnc1δβ2 − λ
−nc2β1
α2β1 − α1β2
) ≤ |z| ≤ exp(
λnc1β2 + λ
−nc2β1
α2β1 − α1β2
−λnc1δα2 − λ
−nc2α1
α2β1 − α1β2
) ≤ |w| ≤ exp(
−λnc1α2 + λ
−nc2α1
α2β1 − α1β2
hence fn(D) ⊂ B(0, rn), where
r2n = exp(2
λnc1β2 + λ
−nc2β1
α2β1 − α1β2
) + exp(2
−λnc1α2 + λ
−nc2α1
α2β1 − α1β2
Now on D again
i∂∂̄v =
i∂φ ∧ ∂̄φ
i(α dz
+ β dw
) ∧ (α dz̄
+ β dw̄
min{|z|2, |w|2 | (z, w) ∈ D}
i(dz ∧ dz̄ + dw ∧ dw̄)
where C1 is a constant not depending on D. This implies
A(c1, δ) ≤
min{φ2(z, w)|z|2, φ2(z, w)|w|2 | (z, w) ∈ fn(D)}
σT (B(0, r
As in the previous cases
σT (B(0,r
converges to zero, therefore for any
ǫ > 0 we get
C1σT (B(0,r
< ǫ as soon as n is large enough. Setting
2c1 max{α1,−β2}
α2β1−α1β2
+ 1 we get by our estimates r2n ≤ exp(−λ
nC2) and
min{|z|2, |w|2 | (z, w) ∈ fn(D)} ≥ exp(−λnC2δ), hence
A(c1, δ) ≤
exp(λnC2(δ − 1))
Set m = ⌊λ2n⌋ and consider the division (1, exp 1
, exp 2
, ..., exp m
= e) of
the interval [1, e]. For any subinterval our estimates hold with eC2 instead
of C2, hence
A(c1, e, c2) =
A(c1 exp
, exp
, c2) ≤
ON THE KÄHLER RANK OF COMPACT COMPLEX SURFACES 15
exp(λnC2(exp
− 1))
≤ exp(C2e
exp 1
and the last term converges to exp(C2e)
showing that A(c1, e, c2) = 0.
As in the proof of Theorem 11 we next check that u is constant on the leaves
of the foliation given by α dz
+ β dw
on U ∩ (C∗ × C∗).
Indeed, for V := u ◦ E one has V ◦ A = V − log λ on {αℜeζ + βℜeω < 0}
and ddcV is a foliation current for the foliation given by αdζ + βdω.
We consider a linear change of coordinates which diagonalizes A on C2
leading to A(ξ, τ) = (λξ, λ−1τ). Keeping the notation V for V after this
coordinate change we notice that V restricted to the leaves {ξ = const.} is
harmonic and that ∂V
are invariant by translations in the ξ-direction.
As in the case of intermediate surfaces we get
(ξ, τ).
Iterating this relation and using the continuity of ∂V
on {ξ = const.} we
obtain
τ) = λn
(ξ, τ)
which forces ∂V
to vanish. Similarly ∂V
= 0. Thus V is constant on the
leaves {ξ = const.}.
Returning now to the coordinates (ζ, ω) on C2 we shall show that V only
depends on ℜeζ and ℜeω. The function V is doubly periodic in ℑmζ ,
ℑmω since V = u ◦ E. Fix ℜeζ and ℜeω and look at a leaf {αζ + βω =
const.}. Under this restriction the imaginary parts must satisfy some rela-
tion αℑmζ + βℑmω = const. which describes a dense subset in the “torus
of the imaginary parts” R2/(2πZ)2 since α/β is irrational. On this subset V
is constant and by semi-continuity it must be constant on the whole torus.
Switching once more to coordinates (ξ, τ) we see that V depends on ℜeξ
alone. We have reduced ourselves in this way to the search of continuous
functions v : R<0 → R subject to the conditions
v(λt) = v(t)− log λ, ∀t ∈ R<0,
v′′ ≥ 0.
Remark that the continuity of v can be deduced as in Theorem 11 after first
restricting u to the line {z = w}.
Take now h : R → R<0, h(s) = − exp(s) and ψ = −v ◦ h − h. Then the
conditions on v translate into ψ ∈ Klog λ. Thus u = − log(−φ)−ψ(log(−φ))
and the proof is finished. �
Finally, the first part of Theorem 1 is a direct consequence of the description
of Pbdy(X) given in [8] and in the above Theorems. For the second part it
suffices to apply Remark 8 and to notice that the only case of a curve on a
“known” non-Kählerian surface, which is not invariant under a holomorphic
foliation, is that of the elliptic curve on a parabolic Inoue surface.
16 MATEI TOMA
References
[1] Barth W., Hulek K., Peters C., Van de Ven A.: Compact complex sur-
faces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin, Springer-Verlag,
(2004).
[2] Demailly J.P.: Complex analytic and algebraic geometry,
http://www-fourier.ujf-grenoble.fr/∼demailly/books.html.
[3] Dloussky G.: Structure des surfaces de Kato, Mémoire de la S.M.F 112.n◦14
(1984).
[4] Dloussky G.: Sur la classification des germes d’applications holomorphes con-
tractantes, Math. Ann. 280, (1988), 649-661.
[5] Dloussky G., Oeljeklaus K.: Vector fields and foliations on surfaces of class
VII0, Ann. Inst. Fourier 49, (1999), 1503-1545.
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Pures Appl. 79 (2000), 475-514.
[7] Grauert, H., Remmert, R.: Plurisubharmonische Funktionen in komplexen
Räumen, Math. Z. 65 (1956), 175-194.
[8] Harvey, R., Lawson, H. B.: An intrinsic characterization of Khler manifolds,
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[9] Kato Ma.: Compact complex manifolds containing “global spherical shells”, Pro-
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Tokyo 1978.
[10] Lamari, A.: Courants kählériens et surfaces compactes, Ann. Inst. Fourier 49
(1999), 263-285.
[11] Lamari, A.: Le cône Kählérien d’une surface, J. Math. Pures Appl. 78 (1999),
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[12] Méo, M.: Image inverse d’un courant positif fermé par une application analytique
surjective, C. R. Acad. Sci. Paris Sr. I Math. 322 (1996), 1141–1144.
[13] Nakamura I.: On surfaces of class VII0 with curves II, Tohôku Math. Jour. 42,
(1990), 475-516.
Institut de Mathématiques Elie Cartan, Nancy-Université, B.P. 239, 54506
Vandoeuvre-lès-Nancy Cedex, France and Institute of Mathematics of the
Romanian Academy.
E-mail address : toma@iecn.u-nancy.fr
URL: http://www.iecn.u-nancy.fr/∼toma/
http://www-fourier.ujf-grenoble.fr/~demailly/books.html
	1. Introduction
	2. The Kähler rank
	3. Kato surfaces
	4. The main results
	References
