Curvature restrictions for Levi flat hypersurfaces in projective planes
Masanori Adachi
Tokyo University of Science
July 19, 2016
(partly based on joint work with Judith Brinkschulte)
1 / 15
Main Problem
Conjecture. ∄ a domain Ω⊂CP2 with C∞-boundary M such that both Ω and Ωc are Stein.
M must be Levi-flat (will be recalled soon).
Posed by Russian school and Brazilian school in 1980s.
One motivation is to pursue a complex analogue of Poincar´e–Bendixson theorem:
Conjecture. Let F be a singular holomorphic foliation of CP2 by Riemann surfaces. Then, any leaf L of F, L∩Sing(F)̸=∅.
(P–B) Let F be a foliation given by a flow on RP2.
Then, any trajectory of F has an accumulate point in Sing(F), or approaches to a periodic orbit.
Ω⊂CP2 with Cω Levi-flat boundary would give a counterexample to the second conjecture.
Levi-flatness
X: complex manifold. M ⊂X: C∞-smooth real hypersurface.
Definition. M is Levi-flat
:⇐⇒ The Levi form of M vanishes identically;
⇐⇒ M is foliated by complex hypersurfaces of X.
E.g. X =Cn, M =Cn−1×R (The local form for Cω Levi-flat).
If there exists Ω⊂CP2 with Cω Levi-flat boundary, the Levi foliation extends to a singular holomorphic foliation on CP2, giving rise to a counterexample to second conjecture.
(Fujita ’63, Takeuchi ’64) Any proper pseudoconvex domain Ω⊂CP2 is Stein.
The first conjecture just asks whether there exists a domain
⊂CP2 with C∞-smooth Levi-flat boundary.
3 / 15
Status of the non-existence conjecture
Non-existence results are established in higher dimension:
Lins Neto (’99): ∄ Cω Levi-flat in CP≥3. Siu (’00): ∄ C∞ Levi-flat in CP≥3.
Ohsawa (’07): ∄ Cω Levi-flat bounded Stein domain in compact K¨ahler manifolds of dim ≥3.
Brunella (’08), Ohsawa (’13), Biard–Iordan (’14):
∄ C∞ Levi-flat in compact K¨ahler manifolds of dim ≥3 whose holomorphic normal bundle has positive leafwise Chern
curvature.
All the (un)published proofs of ∄ Levi-flat in CP2 are incomplete.
Totally different reasoning is required in dimension two because of...
An example
Recipe
Σ: a compact Riemann surface of genus ≥2.
Uniformize Σ =D/Γ. Extend Γ ↷ D⊂CP1.
Diagonal action Γ ↷ D×CP1 gives X :=D×CP1/Γ.
The first projection gives X →Σ, aCP1-bundle.
Ω :=D×D/Γ, Ω′ :=D×D∗/Γ where D∗ :=CP1\D. Properties
∂Ω =∂Ω′ is Cω Levi-flat.
(Diederich–Ohsawa ’85) Ω is 1-convex, Ω′ is Stein.
Maximal compact analytic set in Ω is ∆D/Γ≃Σ.
Ω is diffeomorphic to TΩ, and the Levi foliation is the weak stable foliation of the geodesic flow on Σ.
(cf. Hopf ’36) Both Ω and Ω′ are Liouville.
5 / 15
Main idea and Main Theorem 1
Main idea
No Levi-flat bounded domain in Stein manifolds of dim ≥2, because of violation to maximal principle.
Study to what extent a Levi-flat bounded Stein domain is capable of holomorphic functions with slow growth.
Main Theorem 1 (A.–Brinkschulte, Ann. Inst. Fourier (’15) ) Attach to X =CP2 the Fubini–Study metric normalized so that holomorphic sectional curvature ≡+4. Suppose ∃M =∂Ω⊂CP2, C2-smooth Levi-flat. Then, ∃p ∈M such that RicM(ξp, ξp)≤ −4.
Here ξp =−Jνp, νp is the unit outward normal vector of M at p.
Bejancu–Deshmukh (’96) showed that ∃p∈M such that RicM(ξp, ξp)<0 by a differential geometric argument.
Outline of Proof for Main Theorem 1
We look at L2(Ω,KX)∩Ker∂=:A2(Ω,KX). From Takeuchi (’64) + Ohsawa–Sibony (’98), Ω has a bounded spsh exhaustion. Hence, standard argument shows that A2(Ω,KX) is infinite dimensional.
On the other hand, we showed Lemma
Let L→X2, Ω⋐X with C2 Levi-flat boundaryM =∂Ω.
If ∃ a C∞ herm. metric h of L and a C2 def. function ρ of Ω s.t.
{−log(−ρ)∈PSH(Ω)
iΘ(L)∧dρ∧dcρ <(ddcρ)2 on M =⇒ A2(Ω,L) = 0.
Apply this for X =CP2, L=O(−m), h andρ induced from the Fubini-Study. Then direct computation yields that
RicM(ξ, ξ)>2−2m on M ⇐⇒ iΘ(L)∧dρ∧dcρ <(ddcρ)2 on M. Since KX =O(−3), we complete the proof.
7 / 15
Outline of Proof for Lemma
Lemma
{−log(−ρ)∈PSH(Ω)
iΘ(L)∧dρ∧dcρ <(ddcρ)2 on M =⇒ A2(Ω,L) = 0.
Proof relies on a formula in Griffiths (’66) or Demailly (’87):
∫
{ρ=c}fdcρ∧ddcρ=
∫
{ρ<c}f(ddcρ)2+ddcf ∧dρ∧dcρ
where f :X →C is a C∞-function, andc <0 is a regular value of ρ.
Put f =|s|2h, s ∈A2(Ω,L). Formula on {−1≪a< ρ <b <0} gives
∫
{ρ=b}|s|2hdcρ∧ddcρ≥
∫
{a<ρ<b}|s|2h
((ddcρ)2−iΘh(L)∧dρ∧dcρ) .
(LHS)→0 as b →0 since∂Ω is Levi-flat &s ∈L2. Hences ≡0.
Next question
The proof consisted of, for Levi-flat bounded domain Ω⊂CP2, A2(Ω,O(−3))̸= 0;
A2(Ω,O(−m)) = 0 if RicM(ξ, ξ)>2−2m on M.
We may try to improve both results toward our main question.
For instance, Donnelly–Fefferman (’83) type vanishing result due to Berndtsson–Charpentier (’00) yields
A2(Ω,O(−3−n))̸= 0 for n< η(Ω)2 (d + 2),
where η(Ω)is the Diederich–Fornaess index of Ω andd is the degree of the Levi foliation F defined as the largest integer d such that NF >O(d+ 2) along F.
Question Is the method of B–C the best possible way to construct a non-trivial L2 holomorphic section for negatively curved line bundles over Levi-flat bounded domains?
9 / 15
Main Theorem 2
Recall
Σ =D/Γ: a compact Riemann surface of genus ≥2.
Ω :=D×D/Γ⋐X :=D×CP1/Γ, quotients by diagonal action.
Maximal compact analytic set in Ω is ∆D/Γ≃Σ.
(Diederich–Ohsawa ’85) ∃ ρ=−δ a Cω defining function of Ω such that −logδ∈PSH(Ω) which is SPSH except for ∆D/Γ.
(cf. Hopf ’36, Tsuji ’59, Garnett ’83, Feres–Zeghib ’03) The L2 Hardy space of Ω vanish, hence Ω is Liouville.
Known fact and corollary
(A. ’15) The Diederich–Fornaess index of Ω is 1/2.
For α <1/2, dimH(2)0,1(Ω, δ−α)<∞ &dimA2(Ω, δ−α) =∞. Main Theorem 2 (A., preprint, ’16)
For α <1, A2(Ω, δ−α) =∞.
Outline of Proof for Main Theorem 2
By direct construction. We shall show the well-definedness of i :
⊕∞ n=1
H0(Σ,KΣ⊗n),→ ∩
α<1
A2(Ω, δ−α)
given by a Γ-invariant holomorphic function on D×D, i(ψ)(z,w) =
∫ w
z
1 B(n,n)
((w −ζ)(ζ−z) (w −z)dζ
)⊗(n−1)
ψ(ζ)(dζ)⊗n for each ψ∈H0(Σ,KΣ⊗n), where we write ψ=ψ(ζ)(dζ)⊗n on the uniformizing coordinate ζ ∈Dand B(p,q) denotes the beta function.
The Γ-invariance follows since (w −ζ)(ζ−z)(w −z)−1(dζ)⊗(−1) is a limiting form of the cross ratio.
11 / 15
Outline of Proof for Main Theorem 2 — continued
i(ψ)∈∩
α<1A2(Ω, δ−α) follows from another description ofi: i :H0(Σ,KΣ⊗n)≃H0(∆D/Γ,I∆nD/Γ/I∆n+1D/Γ),→A2(Ω) gives the optimal L2 extension of the n-jet along ∆D/Γ.
Step 1. Deduce equations that jet coefficients formally satisfy Denote the original coordinate of D×D by (z,w). We work on a non-holomorphic coordinate (z,t) where t = (w−z)(1−zw)−1. f =f(z,w)∈ O(Ω)Γ is expanded as f =∑∞
n=0fn(z)tn and {fn} satisfy
∂fn
∂z + nz
1− |z|2fn+ n−1
1− |z|2fn−1= 0.
Put φn:=fn(z) (√
2dz 1−|z|2
)⊗n
∈C(0,0)(Σ,KΣn). Then {φn} satisfy
∂φ0 = 0, ∂φn=−n−1
√2 φn−1⊗ω (n≥1) where ω= 2dz⊗d z/(1− |z|2)2.
Outline of Proof for Main Theorem 2 — continued
2Step 2. Solve the system of ∂-equations with L2 minimal norm Let ψ∈H0(Σ,KΣ⊗N). Put φn:= 0 for n<N and φN :=ψ. We pick the L2 minimal solution to
∂φn=−n−1
√2 φn−1⊗ω
inductively and determine φn forn >N. The spectral decomposition of the complex laplacian tells us the L2 minimal solutions are
φN+m=∂∗N+mGN+m(1) (
−N+m−1
√2 φN+m−1⊗ω )
=−
√2(N+m−1)
m(2N+m−1)∂∗N+m(φN+m−1⊗ω)
where ∂∗n is the formal adjoint of ∂:C(0,0)(Σ,KΣ⊗n)→C(0,1)(Σ,KΣ⊗n) and Gn(1) is the Green operator on C(0,1)(Σ,KΣ⊗n) .
13 / 15
Outline of Proof for Main Theorem 2 — continued
3Step 3. Convergence of the formal solution The convergence of f =∑∞
n=0fn(z)tn in L2(Ω), φn=fn(z) (√
2dz 1−|z|2
)⊗n
, follows from
∥f∥2=π
∑∞ n=0
∥φn∥2 n+ 1
=π
∑∞ m=0
∥φN+m∥2 N+m+ 1
= 1
B(N,N)∥ψ∥2∑∞
m=0
{(N+m−1)!}2
m!(2N+m−1)!(N+m+ 1) <∞. Similar computation shows f ∈A2(Ω, δ−α) for any α <1.
Outline of Proof for Main Theorem 2 — continued
4Step 4. Compute the infinite series Want to show
∑∞ n=0
fn(z)tn=
∫ w
z
1 B(N,N)
((w −ζ)(ζ−z) (w−z)dζ
)⊗(N−1)
ψ(ζ)(dζ)⊗N. Enough to show the desired equality on {0} ×D.
∑∞ n=0
fn(0)tn= (2N−1)!
(N−1)!
∑∞ m=0
(N+m−1)!
(2N+m−1)!
1 m!
∂mψ
∂zm(0)tN+m
= (2N−1)!
(N−1)!tN
∫ 1
0
dtN. . .
∫ t3
0
dt2
∫ t2
0
t1N−1ψ(tt1)dt1
= (2N−1)!
(N−1)!tN
∫ 1
0
t1N−1(1−t1)N−1
(N−1)! ψ(tt1)dt1
=
∫ t
0
1 B(N,N)
((t−ζ)ζ t
)(N−1)
ψ(ζ)dζ.
15 / 15
