arXiv:1010.2591v1 [math.CV] 13 Oct 2010
ON COMPACT COMPLEX SURFACES OF KÄHLER
RANK ONE
IONUŢ CHIOSE∗ AND MATEI TOMA
Abstract. The Kähler rank of compact complex surfaces was introduced by Harvey and Lawson in their 1983 paper on Kähler
manifolds as a measure of “kälerianity”. Here we give a partial
classification of compact complex surfaces of Kähler rank 1. These
are either elliptic surfaces, or Hopf surfaces, or they admit a holomorphic foliation of a very special type. As a consequence we give
an affirmative answer to the question raised by Harvey and Lawson
whether the Kähler rank is a birational invariant.
Introduction
In [HL], Harvey and Lawson introduced the Kähler rank of a compact
complex surface, a quantity intended to measure how far a surface is
from being Kähler. A surface has Kähler rank 2 iff it is Kähler. It
has Kähler rank 1 iff it is not Kähler but still admits a closed (semi-)
positive (1, 1)-form whose zero-locus is contained in a curve. In the
remaining cases, it has Kähler rank 0.
Harvey and Lawson computed the Kähler rank of elliptic surfaces,
Hopf surfaces and Inoue surfaces and conjectured that the Kähler rank
is a birational invariant. In this paper we prove this conjecture. One
possible approach to it is local by studying some plurisubharmonic
functions on the blow-up, and showing that they are pull-backs of
smooth functions. However, this approach leads to a rather involved
system of differential equations, and the computations are not trivial.
Instead, we use a global approach. Namely, we will study the geometry of compact complex surfaces of Kähler rank 1. The existence of a
smooth positive (1, 1) form imposes strong restrictions on the surface.
In the process we also obtain a (partial) classification of non-elliptic
surfaces of Kähler rank 1: they are either birational to a certain class
of Hopf surfaces, or else they support a very special holomorphic foliation (in which case we conjecture that they are birational to Inoue
Date: 11/10/10.
∗ Supported by a Marie Curie International Reintegration Grant within the 7th
European Community Framework Programme and the CNCSIS grant ID 1185.
1
2
CHIOSE AND TOMA
surfaces). Recall that the Kähler rank of elliptic surfaces is always 1,
[HL].
One important technical tool that we use is the compactification of
hyper-concave ends proved by Marinescu and Dinh in [MD].
1. Preliminary facts
∞
Let X be a non-Kähler compact complex surface, Pbdy
the set of
exact positive (1, 1)-forms on X and
∞
B(X) = {x ∈ X : ∃ω ∈ Pbdy
with ω(x) 6= 0}.
Then the Kähler rank of X is defined to be 1 iff the complement of
B(X) is contained in a complex curve.
Recall that a closed positive (1, 1)-form ω on a non-Kähler surface
is automatically exact, cf. [BHPV]. Such a form is also of rank 1, i.
e., ω ∧ ω = 0, hence it defines a foliation by complex curves on the set
where it does not vanish. It was remarked in [HL] that the foliations
induced by two such forms are the same on the set where they are
both defined. One thus gets a canonical foliation on B(X). In fact this
foliation is induced by some closed positive (1, 1)-form on X:
Remark 1.1. There exists an exact positive (1, 1)-form ω on X such
that B(X) = {x ∈ X : ω(x) 6= 0}.
Indeed, we may find a sequence (ωn )n of exact positive (1, 1)-forms
such that for each x ∈ X there exists some
P n with ωn (x) 6= 0. After
−n
rescaling we get kωn kC n ≤ 2 and ω := n ωn is the desired form.
Since all elliptic non-Kähler surfaces are of Kähler rank 1, we shall
restrict our attention to non-elliptic non-Kähler surfaces. By Kodaira’s
classification the only non-elliptic non-Kähler surfaces are those of class
V II, i.e. with b1 = 1 and with no meromorphic functions. The GSS
conjecture predicts that the minimal model of a class V II surface must
fall into one of the following subclasses: Hopf surfaces, Inoue surfaces
and Kato surfaces. Whereas all Inoue surfaces have Kähler rank 1 and
all Kato surfaces have Kähler rank 0, Hopf surfaces may be of Kähler
rank 0 or 1 according to their type, cf. [HL], [To].
Remark 1.2. The birational invariance of the Kähler rank holds for
the above subclasses.
It holds indeed for those surfaces whose minimal model has Kähler
rank 1. It also holds for Kato surfaces, since these contain a cycle of
rational curves and it is was shown in [To] that in this case the Kähler
rank is 0. If X is a surface whose minimal model is a Hopf surface Y of
Kähler rank 0, then Y admits an unramified finite covering Y ′ which
ON COMPACT COMPLEX SURFACES OF KÄHLER RANK ONE
3
is a primary Hopf surface of Kähler rank 0 too. It was shown in [HL]
that Y ′ admits only one exact positive current up to a multiplicative
constant and this is the current of integration along its elliptic curve.
But if X admitted a closed positive (1, 1)-form, its push-forward from
X to Y , and then the pull-back to Y ′ would give a positive, non-zero,
exact current which is not a multiple of an integration current on Y ′
(its singular locus is a finite union of points). So the Kähler rank of X
has to vanish as well.
Let now X be a non-elliptic surface of class V II and suppose the
Kähler rank of X to be 1.
Let ω be a closed, non-zero positive (1, 1) form on X. Since ω is
exact and HR1,1 (X) is naturally contained in H 2 (X, R), cf. [BHPV], we
¯
can choose a ∂-closed
(0, 1)-form γ 0,1 such that
ω = ∂γ 0,1 .
Denote by γ 1,0 the complex conjugate of γ 0,1 . Since ω is of rank 1, it
follows that
iγ 1,0 ∧ γ 0,1
¯
is a positive, i∂ ∂-closed
(1, 1) form. The following integral is thus nonnegative
Z
iγ 1,0 ∧ γ 0,1 ∧ ω.
I :=
X
We distinguish two cases, I > 0 and I = 0.
2. The case I > 0
Theorem 2.1. Let X be a class V II surface and ω = ∂γ 0,1 a closed,
positive (1, 1) form on X such that
Z
iγ 1,0 ∧ γ 0,1 ∧ ω > 0.
X
Then X is birational to a Hopf surface of Kähler rank 1.
Proof. Let Y be the minimal model of X. If X contains a cycle of
rational curves, then the Kähler rank of X is 0 (cf. [To]).
We will show that X contains an elliptic curve. If this is the case,
since it contains no cycle of rational curves, Y will have to be a Hopf
surface (cf. [Na]).
We assume to the contrary that X has no elliptic curves and no
cycles of rational curves. It is known that X can carry only a finite
number of complex curves, all rational. Let (Ci )i be the connected
components of the maximal divisor C on X. Consider p : X̃ → X a
regular covering of X with deck transformation group isomorphic to
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CHIOSE AND TOMA
Z. Since there is no cycle of rational curves on X, each connected
component of p−1 (Ci ) is isomorphic to Ci via p. Let π : X → X ′
be the normal surface obtained from X by contracting each Ci to a
point, and π ′ : X̃ → X̃ ′ the surface obtained by contracting each
connected component of p−1 (Ci ) to a point. We then have a covering
map p′ : X̃ ′ → X ′ such that p′ ◦ π ′ = π ◦ p.
R
Let ω = ∂γ 0,1 be a positive (1, 1) form on X such that I =R X iγ 1,0 ∧
¯
γ 0,1
Set η = ω + iγ 1,0R ∧ γ 0,1 . Then η is i∂ ∂-closed,
η∧η =
X
R ∧ ω1,0> 0. 0,1
2 X iγ ∧ γ ∧ ω > 0 and X η ∧ g > 0 where g is the Kähler form
of some Gauduchon metric on X. By Buchdahl’s Nakai-Moishezon
criterion (see the Remark at the end of the paper [Bu]), there exists a
real effective divisor D on X and a real C ∞ function ψ on X such that
¯ − ifD > 0
η + i∂ ∂ψ
(2.1)
where ifD is the curvature of the line bundle induced by D.
It is known (cf. [To]) that there exists ϕ a C ∞ function on X̃ and a
representation ρ : Γ → (R, +) of the deck transformation group Γ ≃ Z
¯ and g ∗ ϕ = ϕ + ρ(g), for all g ∈ Γ. Then we
of p such that p∗ ω = i∂ ∂ϕ
0,1
¯
may take γ = i∂ϕ.
Now let U ⊂ V be openSneighborhoods of C (the maximal divisor
on X) such that p−1 (V ) = n Vn is a disjoint union of copies of V and
U is relatively compact in V . We can choose the Hermitian metric on
OX (D) such that Supp(ifD ) ⊂ U. We can also assume that ψ > 0 on
X.
It is easy to check that the function (1 + a2 ψ)eaϕ is strictly plurisubharmonic on p−1 (X \ U) for 0 < a << 1.
If s is a section in OX (D) whose zero set is D, the Lelong-Poincaré
equation reads
i∂ ∂¯ ln ||s||2 = [D] − ifD
Denote by ln ||sn ||2 the function on X̃ which is ln ||s||2 ◦ p on Vn and 0
on X̃ \ Vn .
Consider Φ a C ∞ function on X̃ \ p−1 (C) of the form
X
Φ = ϕ + (1 + a2 ψ)eaϕ +
an ln ||sn ||2
n
where an > 0. Since the function (1 + a2 ψ)eaφ is strictly plurisubharmonic on p−1 (X \ U) and multiplicative automorphic on X̃, i.e.,
g ∗ [(1 + a2 ψ)eaϕ ] = eaρ(g) [(1 + a2 ψ)eaϕ ]
for g ∈ Γ, it follows that we can choose an such that Φ is a strictly
plurisubharmonic function on X̃ \ p−1 (C).
ON COMPACT COMPLEX SURFACES OF KÄHLER RANK ONE
5
Let
Φ− = ϕ + (1 + a2 ψ)eaϕ +
X
an ln ||sn ||2 ,
n<0
where we take the Vn with n < 0 in the region of X̃ where ϕ < 0. We
denote by Φ′− the function on X̃ ′ \ π ′ (p−1 (C)) induced by Φ− and with
respect to this function X̃ ′ \ π ′ (p−1 (C)) ∩ {Φ′− < 0} is a hyper-concave
end. Then the main result of [MD] implies that the hyper-concave end
of X̃ ′ \ π ′ (p−1 (C)) ∩ {Φ′− < 0} can be compactified. In particular X̃ ′
has only finitely many singularities, and in our case the singular set of
X̃ ′ has to be empty due to the transitivity of the action of Γ. Therefore
X ′ is a smooth surface with no compact curves.
Since ϕ is plurisubharmonic, it is constant on the connected components of p−1 (C), therefore it descends to a continuous function ϕ′
on X̃ ′ which satisfies g ∗ ϕ′ = ϕ′ + ρ(g), ∀g ∈ Γ. Around each point of
π ′ (p−1 (C)) consider B(1) a ball of radius 1 and Φ′ε a regularization of
Φ′ , where Φ′ is the function on X̃ ′ \ π ′ (p−1 (C)) induced by Φ. Then Φ′ε
is strictly plurisubharmonic on B(1 − ε). Take f a C ∞ function supported on B( 21 ), equal to 1 on B( 41 ). If we replace Φ′ by f Φ′ε + (1 −f )Φ′
on B(1) for 0 < ε << 1, then we obtain a C ∞ strictly plurisubharmonic
function on X̃ ′ , which becomes a hyper-concave end with the hyperconcave end given by {ϕ′ = −∞}. Let Y ′ be the compactification of
the hyper-concave end of X̃ ′ . It is a Stein complex space with finitely
many isolated singularities.
If we apply the above smoothing procedure to Φ − ϕ instead of Φ we
obtain a multiplicative automorphic strictly plurisubharmonic function
on X̃ ′ . This is a potential of a locally conformal Kähler metric on X ′ .
We are in position to apply the argument of [OV], Theorem 3.1 which
shows that Y ′ is obtained from X̃ ′ by adding only one point. Moreover,
if g0 is the generator of Γ ≃ (Z, +) such that ρ(g0 ) < 0, then g0 can be
extended to the whole Y ′ , and it becomes a contraction. From [K1],
Lemma 13., it follows that Y ′ can be embedded into some affine space
Cm such that g0 is the restriction of a contracting automorphism g̃0 of
Cm . Therefore X ′ can be embedded into the compact Hopf manifold
Cm \{0}/ < g̃0 >. From [K2], Proposition 3., we know that X ′ contains
an elliptic curve, contradiction.
Therefore the minimal model of X is a Hopf surface of Kähler rank
1.
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CHIOSE AND TOMA
3. The case I = 0
Theorem 3.1. Let X be a class V II surface of Kähler rank 1 such
that for any closed, positive (1, 1) form ω = ∂γ 0,1 on X one has
Z
iγ 1,0 ∧ γ 0,1 ∧ ω = 0.
X
Then there exists a positive, multiplicatively automorphic, non-constant,
pluriharmonic function u on a Z-covering X̃ of X.
Proof. Again we may assume that X contains no elliptic curve.
As before there exist a Z-covering p : X̃ → X, a representation of
the deck transformation group ρ : Z → (R, +) and a smooth function
¯
ϕ on X̃ such that g ∗ ϕ = ϕ + ρ(g) for g ∈ Z and p∗ γ 0,1 = i∂ϕ.
The automorphic behaviour of ϕ implies the existence of a map f :
X → R/Z ∼
= S 1 such that f ∗ dθ = iγ 1,0 − iγ 0,1 . Moreover f is surjective
and one checks easily that the map ϕ is proper. Further we may assume
that f∗ : H1 (X, Z)/Tors H1 (X, Z) → H1 (S 1 , Z) is an isomorphism.
The set K := {x ∈ X : df (x) = 0} of critical points of f is
compact and S 1 \ f (K) 6= ∅ by Sard’s theorem. Let s ∈ S 1 \ f (K)
and U ⊂ V ⊂ S 1 \ f (K) two small open connected neighbourhoods
of s such that U is relatively compact in V . Thus γ 0,1 (x) 6= 0 for all
x ∈ f −1 (V ). Let further χ be some smooth non-negative, function on
S 1 supported on U and such that χ(s) = 1. We would like to pull back
χ to one of the connected components of f −1 (V ). We explain first how
this component is chosen.
We have a commutative diagram of smooth maps
X̃
p
/
X
ϕ
f
R
/
S 1.
We denote by VR , VX , VX̃ the preimages of V in R, X, X̃. Let VR′ be
one connected component of VR and VX̃′ its preimage in X̃. It is clear
that VX̃′ disconnects X̃.
In fact any connected component VX̃′′ of VX̃′ continues to disconnect
X̃. Indeed, suppose it did not. Since ϕ is a proper submersion over VR′ ,
each connected component VX̃′′ is differentiably a product VR′ ×F ′′ , where
F ′′ is the corresponding component of a fiber of ϕ over VR′ . For p ∈ F ′′
consider now a parametrization of the segment VR′ × {p}. Its endpoints
can also be connected by some path in X̃ \ VX̃′′ , which connected to the
previous segment induces a cycle in H1 (X̃, R). This cycle is non-trivial
ON COMPACT COMPLEX SURFACES OF KÄHLER RANK ONE
7
since using dϕ and the product structure on the connected components
of VX̃ we may construct some closed 1-form on X̃ with non-vanishing
integral on it. Its image in H1 (X, R) will likewise be non-trivial. But
this would contradict the fact that f ∗ : H 1 (S 1 , R) → H 1 (X, R) is an
isomorphism.
Next we check that the image VX′′ in X of one of the components VX̃′′
does not disconnect X.
Lemma 3.2. There exists VX′′ a connected component of f −1 (V ) such
that X \ V̄X′′ is connected.
Proof. We can assume that f is differentially trivial over a small neighborhood of V̄ . Let (VX,i )i=1,N be the connected components of VX =
f −1 (V ) and suppose that X \ V̄X,i is not connected ∀i = 1, N. Since the
boundary of VX,i has two components, it follows that X \ V̄X,i has two
connected components. Set X0 = X. We will construct by induction
connected open subsets Xi of X with the following properties:
i) Xi is a connected component of Xi−1 \ V̄i
ii) the restriction H 1 (X, R) → H 1 (Xi , R) is injective.
Suppose Xi has been constructed. Then Xi \ V̄i+1 is not connected.
Indeed, suppose it is connected. Then it follows that X \ V̄X,i+1 is
connected, contradiction. Therefore Xi \ V̄i+1 is not connected and
it has two connected components, Ti1 and Ti2 . The Mayer-Vietoris
sequence for the pair (T̄i1 ∪ Vi+1 , T̄i2 ∪ Vi+1 ) implies
0 → H 0(Xi , R) → H 0 (T̄i1 ∪ Vi+1 , R) ⊕ H 0 (T̄i2 ∪ Vi+1 , R) →
→ H 0 (Vi+1 , R) → H 1 (Xi , R) → H 1 (T̄i1 ∪ Vi+1 , R) ⊕ H 1 (T̄i2 ∪ Vi+1 , R)
and since all the open subsets involved are connected, it follows that
H 1 (Xi , R) → H 1 (T̄i1 ∪ Vi+1 , R) ⊕ H 1 (T̄i2 ∪ Vi+1 , R)
is injective. Given that H 1 (X, R) → H 1 (Xi , R) is injective, it follows
that
H 1 (X, R) → H 1 (T̄i1 ∪ Vi+1 , R) ⊕ H 1 (T̄i2 ∪ Vi+1 , R)
is injective. Hence we can assume that the restriction H 1(X, R) →
H 1 (T̄i1 ∪ Vi+1 , R) is injective. Set Xi+1 = Ti1 . For i = N we obtain
XN a connected open subset of X such that XN ∩ (∪N
i=1 VX,i ) = ∅ and
H 1 (X, R) → H 1 (XN , R) injective. But this is impossible since f ∗ dθ is
exact on X \ f −1 (V̄ ).
Let VX′′ a component of VX chosen as above and χ′′ the pull-back of
χ to VX′′ extended trivially to X. We adopt the same notations for the
preimages of U. Consider the positive (1, 1)-form iχ′′ γ 1,0 ∧ γ 0,1 . Since
8
CHIOSE AND TOMA
γ 0,1 ∧ ω = 0, it follows that this form is d-closed hence d-exact. Thus
there exist a real constant c and a smooth real function ψ on X such
that
¯ = i∂ ∂v,
¯
iχ′′ γ 1,0 ∧ γ 0,1 = ∂(cγ 0,1 + i∂ψ)
where v = ϕ + ψ has additive automorphy on X̃. We shall show that
this form does not vanish identically on X \ ŪX′′ .
¯ = 0 on X \ Ū ′′ . Then the function v = cϕ+ψ
Assume that cγ 0,1 +i∂ψ
X
is plurisubharmonic on VX′′ and holomorphic on X \ ŪX′′ . But this
function is real and thus constant on X\ŪX′′ . By the maximum principle
the plurisubharmonic function cϕ + ψ reaches its maximum on the
boundary of VX′′ and since it is constant near this boundary it is constant
everywhere on VX′′ . This contradicts the fact that χ 6= 0.
Thus ∂v is a holomorphic (1, 0)-form on X \ UX′′ and the (1, 1)-form
¯ is positive and d-exact hence it induces a holomorphic foliation
i∂v ∧ ∂v
on X \ UX′′ which coincides with the canonical foliation on B(X) \ UX′′ .
We can now do this construction for two disjoint open connected
subsets U1 and U2 of V and obtain additively automorphic plurisub′′
′′
harmonic functions v1 and v2 on X̃. Set W := X \ (Ū1,X
∪ Ū2,X
) and
′′
′′
let W1 , W2 , W3 be the connected components of VX′′ \ (Ū1,X
∪ Ū2,X
)
′′
′′
counted in such a way that W2 ∪ Ū1,X ∪ Ū2,X is connected. We have
W1 ∪ W2 ∪ W3 ⊂ W . If X contains some rational curves then these are
necessarily contained in fibers of f and we have chosen V such that no
such curves are contained in f −1 (V ).
The 1-forms ∂vj are holomorphic and d-closed on W1 ∪ W2 hence
locally of the form dhj for some holomorphic functions hj . Notice that
on W1 ∪ W2 the canonical foliation is regular and induced by ∂ϕ and
at the same time by ∂(vj + ϕ) and by ∂vj . The irreducible components
of the level sets of hj are therefore contained in its leaves and further
contained in fibers of f . It follows that the zero set of ∂vj being complex
analytic and closed, it will consist of only isolated points. Since the
holomorphic 1-forms ∂v1 and ∂v2 induce the same foliation on W1 ∪ W2
they must be proportional so there exists a meromorphic function h
on W1 ∪ W2 such that ∂v1 = h∂v2 . The zeroes of ∂vj being isolated,
the function h is holomorphic and without zeroes on W1 ∪ W2 . Thus
∂h ∧ ∂v2 = 0 and the restriction of dh to the leaves of the foliation
vanishes, hence h is constant on these leaves. But the Zariski closure
of a general leaf is a connected component of W1 ∪ W2 and the function
h has to be constant on W1 and on W2 .
We thus get constants k1 , C1 such that v1 = k1 v2 + C1 on W1 . Since
v1 , v2 are real and non-constant, one sees that k1 , C1 are also real. We
ON COMPACT COMPLEX SURFACES OF KÄHLER RANK ONE
9
replace now the function v2 by k1 v2 + C1 . This new function remains
pluriharmonic on X \ f −1 (Ū2 ) and agrees with v1 on W1 . On W2 we
have now v1 = k2 v2 + C2 as before for some k2 ∈ R∗ and C2 ∈ R.
Remark that k2 6= 1, otherwise ∂v1 and ∂v2 would glue well to give
some nontrivial holomorphic 1-form on X, which is absurd.
2
Ading now the same constant C := k2C−1
to both v1 and v2 we obtain
v1 + C = k2 (v2 + C)
on W2 . We thus get by gluing a pluriharmonic function v with multiplicative automorphic factor k2 on X̃. If k2 is not positive we may pass
to a double cover of X and obtain here a positive multiplicative factor
λ = k22 .
Now we check that v has no zeros on X̃. Suppose that the zero set
Z of v is not empty. We have seen that v is constant on the leaves
of the canonical foliation, hence it is locally constant on the fibers of
ϕ. Conversely ϕ is locally constant on the fibers of v. Then Z is a
union of connected components of fibers of ϕ. Take a component of
Z contained in a domain D bordered by two regular connected fibers
of ϕ on which v does not vanish. We may assume that v has positive
values on these fibers, say a and λa. But then v|D̄ attains its minimum
in the interior of D, which is a contradiction.
We finally set u = ±v according to the sign of v and get the desired positive, multiplicatively automorphic, non-constant, pluriharmonic function on X̃.
4. Corollaries and remarks
Remark 4.1. If u is a positive, multiplicatively automorphic, nonconstant, pluriharmonic function on X̃, then ∂u gives a non-trivial
d-closed section in H 0 (Ω1X ⊗ L), where L is a flat line bundle on X.
Moreover
¯
¯ log u) = i∂u ∧ ∂u
i∂ ∂(−
u2
descends to a closed positive (1, 1)-form on X which vanishes at most
on some rational curves. In particular the Kähler rank of X is 1. The
level sets of u are compact Levi flat real hypersurfaces of X saturated
under the canonical foliation.
Since a pluriharmonic function descends by blowing down we get the
desired
Corollary 4.1. The Kähler rank is a bimeromorphic invariant.
We also get
10
CHIOSE AND TOMA
Corollary 4.2. The canonical foliation of a surface of Kähler rank 1
is always the restriction to B(X) of a (possibly singular) holomorphic
foliation on X.
Notice that we haven’t used in our proof the assuption that B(X)
contained a dense Zariski open set of X. So we get the following
Corollary 4.3. The Kähler rank of a compact complex surface X is
equal to the maximal rank that a positive closed (1, 1)-form may attain
at some point of X.
Remark 4.2. If the minimal model of X is a Hopf surface, then X
cannot admit a positive, multiplicatively automorphic, non-constant,
pluriharmonic function u on a Z-covering of X.
Indeed, if the minimal surface of X is a Hopf surface, then X will have
an elliptic curve whose preimage in X̃ is isomorphic to C∗ . But then
the restriction of any positive, pluriharmonic function to this preimage
is constant and a non-trivial multiplicatively automorphic behaviour of
such a function on X̃ is impossible.
Recall that every non-Kähler compact complex surface admits some
non-trivial d-exact positive (1, 1)-current, cf [La]. In [To] one defines
the modified Kähler rank of a compact complex surface in the following
way: it is 2 if the surface admits a Kähler metric, and if not 0 or 1 depending on whether there is one or several non-trivial d-exact positive
(1, 1)-currents up to multiplication by positive constants. One immediately checks that the modified Kähler rank is a birational invariant.
The Kähler rank and the modified Kähler rank of Kato surfaces are
computed in [To] and it shows up that they need not coincide but in
this case the modified Kähler rank is at least as large as the Kähler
rank.
From the proof of Theorem 3.1 it follows that there are infintely many
d-exact positive (1, 1)-forms up to multiplication by positive constants
on surfaces of class V II with I = 0 as in the statement. The same
is obviously true for non-Kähler elliptic surfaces and it holds for Hopf
surfaces of Kähler rank 1 by [HL].Thm. 58. Thus we get:
Corollary 4.4. The modified Kähler rank of a compact complex surface
is at least as large as its Kähler rank.
Let us mention in conclusion that the classification of compact complex surfaces of Kähler rank 1 would be complete if we had a positive
answer to the following:
ON COMPACT COMPLEX SURFACES OF KÄHLER RANK ONE
11
Conjecture 4.5. A non-elliptic surface X admitting a positive, multiplicatively automorphic, non-constant, pluriharmonic function on a
Z-covering should be bimeromorphically equivalent to an Inoue surface.
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Math. de France 136 (2008), 243-260
Addresses:
Ionuţ Chiose:
Institute of Mathematics of the Romanian Academy
Ionut.Chiose@imar.ro
Matei Toma:
Institut Elie Cartan, UMR 7502, Nancy-Université - CNRS - INRIA,
B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France
toma@iecn.u-nancy.fr
Ionut Chiose