A mass for asymptotically complex hyperbolic manifolds

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hal-00429306, version 1 - 2 Nov 2009 18 A MASS FOR ASYMPTOTICALLY COMPLEX HYPERBOLIC MANIFOLDS. This lemma leads to Re III ≈ − 1 JHX · 2 ˜ φ,φ + JHX · ˜ φ,φ = 1 JHX · 2 ˜ φ,φ . Eventually, summing I, II and III, we get : Re ζφ,φ(AX) ≈ − 1 (dTrg0 g + divg0 g) |φ|2 4 − i + i (HX · φ,φ) + Tr H (X · φ,φ) 2 2 1 JHX · 2 ˜ φ,φ . Lemmata 2.4 and 2.5 simplify this into Reζφ,φ(AX) ≈ − 1 4 (dTrg0 g + divg0 g) |φ|2 + i Tr H (X · φ,φ) 2 ≈ − 1 4 (dTrg0 g + divg0 g) |φ|2 − 1 4 Tr H d |φ|2 (X). Since g − g0 ≈ −2g0(H.,.), we have Tr(g − g0) ≈ −2Tr H, hence : Reζφ,φ(AX) ≈ − 1 4 (dTrg0 g + divg0 g) (X) |φ|2 + 1 8 Trg0 (g − g0) d |φ| 2 (X), which yields the formula of proposition 3.6. 3.2. The mass linear functional. We consider the formula (10) µg(ξ,α,u) = − 1 4 lim 1 ∗ (dTrg0 g + divg0 g) u + R→∞ 2 Trg0 (g − g0) Jα . SR If (ξ,α,u) belongs to QCHm(KCH m), the considerations of the previous paragraph imply that µg(ξ,α,u) is an element of [0,+∞]. If it is infinite at some point of QCHm(KCH m), we decide that the mass is infinite : µg = ∞. Otherwise, this defines a linear functional µg on the linear span of QCHm(KCH m) ⊂ NCHm ,0, which is NCHm ,0, since it is PU(m,1)invariant and the action of this group on Λ2 J,0R2m,2 is irreducible. The previous paragraph also indicates that µg takes non-negative values on (the convex cone generated by) QCHm(KCH m) ; since QCHm(KCH m) is PU(m,1)-invariant and contains one element of N + CHm ,0 (from the explicit computations in 2.3), it contains N + CHm ,0 . So µg is non-negative on N + CHm ,0 . Assume µg vanishes on NCHm ,0. Then Lemma 3.2, coupled to the Bochner formula, ensures that for every ψ in KCHm, there is a Kählerian Killing spinor φ on (M,g), namely φ ∈ Kg, that is asymptotic to ψ. In particular, for any element β = QCHm(ψ) of N + CHm ,0 , there is an element ˜ β = Qg(φ) of Ng,0 that is asymptotic to β. Since the linear span of N + CHm ,0 is the whole NCHm ,0, we deduce that Ng,0 has maximal dimension, which implies that Ng has maximal dimension (add the Kähler form) : (M,g) is (locally) complex hyperbolic. In view of its asymptotic, it is bound to be CHm . The holomorphic chart at infinity in Definition 3.1 is of course not unique. Any two relevant charts Ψ1 and Ψ2 differ by a biholomorphism f := Ψ2 ◦ Ψ −1 1 of CH m such that f ∗ g0 is asymptotic to g0. The map f therefore induces a CR-automorphism of the sphere at infinity, so that f is asymptotic to an element of PU(m,1). As in the remark after Definition 3.1, we conclude f is an element of PU(m,1). So the model at infinity is unique up to its natural automorphism group PU(m,1). If φ1 is a Kählerian Killing
hal-00429306, version 1 - 2 Nov 2009 A MASS FOR ASYMPTOTICALLY COMPLEX HYPERBOLIC MANIFOLDS. 19 spinor in the chart Ψ1, then φ2 := f∗φ1 is a Kählerian Killing spinor in the chart Ψ2 (by naturality). We claim that Lemma 3.8 — µ Ψ1 g (βφ1 ) = µΨ2 g (βφ2 ). Proof. Recall that in the notations introduced in the previous paragraph, µ Ψ1 g (βφ1 ) = lim R→∞ ∗ζφ1,φ1 = lim R→∞ ∗ζψ1,ψ1 . This quantity is equal to the well-defined integral M ˆ ∇ψ1 2 + 1 (Scal+4m(m + 1)) |ψ1| 4 M 2 +(m + 1) |ψ1| 2 − Ω πl−1ψ1 2 − Ω πl ψ1 2 , M S Ψ 1 R so that we can compute it with another family of spheres : µ Ψ1 g (βφ1 ) = lim = lim R→∞ R→∞ R→∞ S Ψ ∗g0 2 R f(S Ψ ∗ζψ1,ψ1 1 R ) S Ψ 1 R f(S Ψ ∗ζφ1,φ1 1 R ) . Since f ∗g0 = g0 and f(S Ψ1 R ) = SΨ2 R , we therefore obtain µ Ψ1 g (βφ1 ) = lim R→∞ f(S Ψ ∗g0 − 1 R ) 1 2 1 (dTrg0 g + divg0 g) |φ1| + g0 4 8 Trg0 (g − g0) d |φ1| 2 g0 = lim − 1 = lim R→∞ S Ψ 2 R = µ Ψ2 g (βφ2 ). ∗g0 4 (dTrg0 g + divg0 g) |f ∗ φ2| 2 g0 + 1 8 Trg0 (g − g0) d |f ∗ φ2| 2 g0 − 1 2 1 (dTrg0 g + divg0 g) |φ2| + g0 4 8 Trg0 (g − g0) d |φ2| 2 g0 In other words, changing the chart at infinity by an automorphism f ∈ PU(m,1) results in turning µg to f ∗ µg. So µg is well-defined up to the natural action of PU(m,1). We have proved the following result. Theorem 3.9 — Let (Mm ,g,J) be a spin asymptotically complex hyperbolic Kähler manifold with odd complex dimension Scalg ≥ ScalCHm. Then the linear functional µg on NCHm ,0 is well-defined up to the natural action of PU(m,1) ; it is non-negative on N + CHm ,0 and vanishes if and only if (Mm ,g,J) is the complex hyperbolic space. 4. The case of even-dimensional manifolds. 4.1. The twisted Kählerian Killing spinors. To extend the ideas above to the evendimensional case, it seems that we need Kählerian Killing spinors on even-dimensional complex hyperbolic spaces. Unfortunately, such Kählerian Killing spinors do not exist. To overcome this cruel reality, we follow [BH] and turn to the spin c realm. We refer to [LM] for basic definitions about spin c structures. We consider a Kähler manifold (M,g,J) with even complex dimension m = 2l. As in [BH], we further assume that the cohomology class of Ω is integral, i.e. in the image iπ of H2 (M, Z) → H2 (M, R). This determines a complex line bundle L endowed with a Hermitian metric and a unitary connection with curvature F = −2iΩ ; the Chern class .
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A mass for asymptotically complex hyperbolic manifolds

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