A mass for asymptotically complex hyperbolic manifolds

hal-00429306, version 1 - 2 Nov 2009
20 A MASS FOR ASYMPTOTICALLY COMPLEX HYPERBOLIC MANIFOLDS.
of L is c1(L) = i 1
2π [F] = π [Ω]. We also assume that (M,L) defines a spinc structure, in
that the bundle TM ⊗L admits a spin structure. We then introduce the corresponding
spinor bundle Σc , endowed with a Clifford action c and a connection ∇. The Kähler
form Ω acts on this bundle, with the eigenvalues i(m−2k), 0 ≤ k ≤ m. The eigenspaces
yield subbundles Σc k . We may therefore define a connection ˆ ∇ on Σc by requiring that
ˆ∇Xψ := ∇Xψ + ic(X 1,0 )ψl−1 + ic(X 0,1 )ψl,
where ψk is the component of ψ in Σc k . The parallel sections of Σc l−1 ⊕ Σc l for this connection
will be called twisted Kählerian Killing spinors and the corresponding subspace
will be denoted by Kc .
Every computation of section 2.3 (but one) can be carried out with twisted Kählerian
Killing spinors, leading to the same formulas. In particular, one can define uψ, αψ, ξψ
for any ψ in Kc as in section 2.3 and this yields a map Qc : Kc → N. The only
difference is that we do not get elements of N0 : (ξψ,Ω) is no longer uψ, basically
because the eigenvalues of the Kähler form are now even (compare with the proof of
Lemma 2.6).
Let us describe the model case, where (M,g,J) is the complex hyperbolic space
CHm , m = 2l, with holomorphic sectional curvature −4. With this normalization, we
have RicCHm = −2(m + 1)gCHm. Since the Ricci form ρ = Ric(.,J.) is i times the
curvature of the canonical line bundle, this implies :
c1(Λ m,0 ) = i i
+ 1
[−iρ] = [2i(m + 1)Ω] = −m [Ω] = −(m + 1)c1(L).
2π 2π π
So L ∼ = Λ m,0 − 1
m+1 in this case. It follows that
Σ c k ∼ = Σk ⊗ L ∼ = Λ 0,k ⊗ Λ m,01
1
− l
2 m+1 ∼ 0,k m,0 2l+1 = Λ ⊗ Λ .
It turns out that the sections of Kc trivialize the bundle Σc l−1 ⊕ Σc l , as noticed in [BH].
Indeed, using the same notations as in section 2.3, we can define two families of twisted
Kählerian Killing spinors in the following way. First, if a is a multi-index of length
l − 1, we set ϕ a = ϕ a l−1 + ϕa l with
ϕ a l−1
d ¯wa
= c(l)
(1 − |w| 2 l
⊗ dw 2l+1 and ϕ
) l a l
c(l)
=
2il ¯
∂
d ¯wa
(1 − |w| 2 ) l
⊗ dw l
2l+1.
l2
1−l−
The normalization we choose is c(l) = 2 2l+1. We also introduce ˘ϕ b = ˘ϕ b l−1 + ˘ϕb l
where b is a multi-index of length l and
˘ϕ b ι ¯R d ¯wb
l−1 = c(l)
(1 − |w| 2 l
⊗ dw 2l+1 and ˘ϕ
) l b c(l)
l =
2il ¯
ι ¯R d ¯wb
∂
(1 − |w| 2 ) l
⊗ dw l
2l+1
where R =
k wk∂wk as before. These families together form a basis for Kc on CHm .
One can again compute the squared norms of these spinors, like in [Kir]. If a is a
multi-index of length l − 1 and b a multi-index of length l, we have
a
ϕ
l−1
2 2
1 − |wa|
=
1 − |w| 2 , |ϕal |2 2
|wă|
=
1 − |w| 2,
˘ϕ b
l−1
2
= |wb| 2
1 − |w| 2,
˘ϕ b
l
2
= 1 −
w˘
b
2
2 ,
1 − |w|
hence
|ϕ a | 2 = 1 − |wa| 2 + |wă| 2
1 − |w| 2 ,
˘ϕ b
2
= 1 −
w˘
b
2 + |wb| 2
1 − |w| 2 .