Locally conformally Kaehler metrics on Kato surfaces

0.09 MB·English

by  Marco Brunella

LOCALLY CONFORMALLY KA¨HLER METRICS
ON KATO SURFACES
0
MARCO BRUNELLA
1
0
2 Abstract. We show that every Kato surface admits a locally confor-
n mally K¨ahler metric.
a
J

4 In this note we shall prove the following theorem, which improves and

completes our previous result [Bru].
]
V
Theorem 1. Every Kato surface admits a locally conformally Ka¨hler met-
C
ric.
.
h Wereferto[Bel]and[Bru],andreferencestherein,forsomebackgroundon
t
a locally conformally K¨ahler metrics on complex surfaces. We just recall the
m
followingfact,whichwillbeusedbelow: acomplexsurfaceS admitsalocally
e
[ conformally K¨ahler metric if and only if its universal covering S admits a

1 K¨ahler metric ω which transforms conformally under the action of the deck
v transformations (g∗(ω) = c ω, c ∈ R+, for every deck transformation g).
g g
0
In particular, we observe the following consequence of Theorem 1.
3
5 Corollary 2. The universal covering of a Kato surface is Ka¨hlerian.
0
. It would be interesting to know if this last statement holds for every
1
compact complex surface. Note, however, that there are compact complex
0
0 surfaces which do not admit locally conformally K¨ahler metrics [Bel], hence
1 ageneralizationofCorollary2toanycompactcomplexsurfaceshouldfollow
:
v a different path.
i The proof of Theorem 1 is based on a point of view already exploited in
X
[Bru, §2], but we shall not need the stability result proved in [Bru, §3].
r
a
Kato surfaces. Let us briefly recall the definition and the construction of
Kato surfaces [Kat] [Dlo]. Let B denotes the (open) ball of radius r in C2,
r
and set B = B1. Take a sequence of blow-ups
b
π : B −→ B
over the origin 0 ∈ B, subject to the following constraint: at each step, we
blow-up a point belonging to the exceptional divisor created at the previous
step. Take also a holomorphic (up to the boundary) embedding
b
σ : B −→ B
which sends the origin to a point belonging to the last exceptional divisor
created by π. Set
b
W = B\σ(B).
1
METRICS ON KATO SURFACES 2
b
Wecangluetogetherthetwoboundarycomponentsof∂W = (∂B)∪(σ(∂B))
using the real analytic CR-diffeomorphism
b
σ◦π : ∂B −→ σ(∂B).
The result is a minimal compact complex surface S = S , with first Betti
π,σ
numberequalto1andsecondBettinumberequaltothenumberofblow-ups
in π. It is called a Kato surface, or a surface with a global spherical shell.
Of course, such a surface cannot be K¨ahlerian, for its first Betti number is
odd.
The following remark is very simple, yet of capital importance for our
b
construction. For every r ∈ (0,1], the embedding σ sends B into B =
r r
π−1(B ): indeed, the composite map π ◦σ : B → B fixes the origin, and
r
hence by Schwarz Lemma it maps B inside B , for every r ≤ 1, which
r r
precisely means that σ(B ) ⊂ π−1(B ). If we set
r r
b
W = B \σ(B ),
r r r
then we can glue, as before, the two components of ∂W using σ ◦π. The
r
result is the same compact complex surface S. Indeed, the open subsets
W , r ∈ (0,1], may be understood as different fundamental domains in the
r
e
same universal covering S for the same action of the deck transformations.
This phenomenon is also at the origin of the fact that the classification of
Kato surfaces can be reduced to the classification of germs of contracting
mappings of the type π◦σ [Dlo]. In fact, we may replace the balls B with
r
any domain U ⊂ B containing 0 and such that σ(U) ⊂ π−1(U).
b b
A K¨ahler metric on B. On B, there obviously exists a K¨ahler form ω0,
smooth up to the boundary. In order to prove Theorem 1, we firstly modify
ω0 on a neighbourhood of σ(0).
Consider σ∗(ω0). It is a K¨ahler form on B, which admits a potential:
∗ c
σ (ω0) = dd ϕ
with ϕ smooth and strictly plurisubharmonic on B. Up to adding an affine
(hencepluriharmonic)function,wemayassumethatϕ(0) = 0and(dϕ)(0) =
0. The Taylor expansion of ϕ at 0 has the form
3
ϕ(z) = L(z,z)+ReM(z,z)+O(kzk )
where L is the Levi form at 0 and M is a homogeneous polynomial of degree
2. Because ReM is pluriharmonic, we may replace ϕ with ϕ−ReM, and
we obtain in this way a new potential for σ∗(ω0), still denoted by ϕ, whose
3
Taylorexpansionat0isL(z,z)+O(kzk ). Thismeansthatϕhasaminimum
point at 0, of Morse type.
We can choose λ > 0 and 0 < r < r′ < 1 such that the function
2
ρ(z) = λ(1+kzk )
is greater than ϕ on Br and smaller than ϕ on ∂Br′. The function ϕe on B
defined by ϕe= ϕ on B\Br′ and ϕe= regularized maximum between ϕ and
METRICS ON KATO SURFACES 3
ρ on Br′ is therefore smooth, strictly plurisubharmonic, and equal to ρ on
some neighbourhood of B .
r
TheK¨ahlerformσ∗(ddcϕe)onσ(B)gluestoω0 outsideσ(B),givinganew
b
K¨ahler form ω1 on B. By construction, this K¨ahler form has the property:
∗ c
σ (ω1) = dd ρ on Br.
b
Another K¨ahler metric on Br. We now modify the previous ω1 on a
b b
neighbourhood of ∂B . Choose s < r sufficiently close to r, so that B still
r s
contains σ(B ), i.e. (π◦σ)(B ) ⊂ B .
r r s
π
B B
σ
B
r B
r
B exceptional
s
divisor
Take a potential ψ for the direct image π∗(ω1):
c
π∗(ω1) = dd ψ on B.
Note that ψ is necessarily singular at 0, but it doesn’t matter. We can
choose c1 ∈ R+ and c2 ∈ R such that the function
c1ρ+c2
e
is greater than ψ on ∂B and smaller than ψ on ∂B . The function ψ
r s
e e
on B defined by ψ = ψ on B and ψ = regularized maximum between
r s
ψ and c1ρ+c2 on Br \Bs is therefore smooth, strictly plurisubharmonic,
equal to c1ρ+c2 on some neighbourhood of ∂Br and equal to ψ on some
neighbourhood of B .
s
By pulling-back ψe with π and taking ddc, we get on Bb a new K¨ahler
r
form ω2 with the following two properties:
(1) ω2 = ω1 on Bbs ⊃ σ(Br), so that σ∗(ω2) = ddcρ on Br;
(2) π∗(ω2) = ddc(c1ρ+c2) = c1ddcρ on some neighbourhood of ∂Br.
METRICS ON KATO SURFACES 4
Conformal glueing. We can now complete the proof of Theorem 1. The
b
glueing map σ◦π : B → σ(B ) satisfies the conformal property
r r
1
∗
(σ◦π) (ω2|U) = ω2|V
c
1
b
where U ⊂ σ(B ) and V ⊂ B are suitable neighbourhoods of σ(∂B ) and
r r r
b
∂B . Because the Kato surface S is obtained by identifying these two neigh-
r
bourhoods via σ◦π, we get on S a locally conformally K¨ahler metric.
Remark. Because c1 can be chosen arbitrarily large, we see also that, in
the terminology of [Bru, Rem. 9], the open subset I(S) ⊂ (0,1) contains
an interval (0,ε) (with ε, in principle, depending on S). We do not know if
I(S) is an interval.
Remark. Thepreviousconstructioncanbedoneparametrically: if{St}t∈D
is a family of Kato surfaces, then, up to restricting the base D, we can find
a family of locally conformally K¨ahler metrics {ωt}t∈D, smoothly depending
on t. This is, however, slightly weaker than the stability result of [Bru],
which states that every locally conformally K¨ahler metric ω0 on S0 extends
to a family {ωt}t∈D.
References
[Bel] F. A. Belgun, On the metric structure of non-Ka¨hler complex surfaces, Math. Ann.
317 (2000), 1–40
[Bru] M.Brunella,LocallyconformallyK¨ahlermetricsoncertainnon-Ka¨hleriansurfaces,
Math. Ann. 346 (2010), 629–639
[Dlo] G. Dloussky, Structure des surfaces de Kato, M´em. Soc. Math. France (N.S.) No.
14 (1984)
[Kat] M.Kato,Compactcomplexmanifoldscontaining“global”sphericalshells,inProc.
Int. Symp. on Algebraic Geometry (Kyoto, 1977), Kinokuniya (1978), 45–84
Marco Brunella, Institut de Mathe´matiques de Bourgogne – UMR 5584 –
9 Avenue Savary, 21078 Dijon, France