SUMMARY OF THE KÄHLER MASS PAPER

SUMMARY OF THE KÄHLER MASS PAPER HANS-OACHIM HEIN AND CLAUDE LEBRUN Let (M, g) be a complete n-dimensional Riemannian manifold. Roughly speaking, such a space is said to be ALE with l ends (l N) if there exist a compact set K M, finite subgroups Γ 1,..., Γ l of O(n) acting freely on S n 1, and a diffeomorphism Φ : l k=1 (Rn \ B1 n(0))/γ k M \ K such that, in each connected component, Φ g converges to the Euclidean metric tensor g 0 as r. Definition 1. The mass of the k-th end of an ALE Riemannian n-manifold is defined by m k (M, g, Φ) lim ( i (Φ r Σ g) ij j (Φ g) ii )ν j da (1) k (r) whenever this limit exists. Here Σ k (r) B n r (0)/Γ k denotes the Euclidean sphere of radius r in the k-th conical end of M, all sub- and superscripts refer to the standard Euclidean coordinates on R n, ν is the Euclidean exterior unit normal to Σ k (r), and da is the Euclidean area measure. Bartnik [1] and Chruściel [2] discovered sharp conditions on the decay rate of Φ g g 0 ensuring that the limit in (1) exists and is independent of Φ. We provide an alternative proof of their result under the assumption that (M, g) is Kähler. In fact, in this case, we obtain an explicit formula for the mass in terms of the total scalar curvature, the Kähler class, and the first Chern class. Using this formula, we are then able to prove the Positive Mass Theorem in the Kähler case. Remark 2. If (M, g) is ALE in any reasonable sense and Kähler of complex dimension at least 2, with parallel complex structure, then M has only one end. This follows from Theorem 6.3 of [9] applied to the Remmert reduction of (M, ), which is a normal Stein space. (In our paper, we point out several other ways of proving this, but the approach via [9] may be the most robust one.) Our two main results may now be stated as follows. Theorem 3. Let (M, g, ) be a complete Kähler manifold of real dimension n = 2m > 2. Suppose that there exist a compact set K M, a finite subgroup Γ of O(n) acting freely on the unit sphere, and a diffeomorphism Φ : (R n \ B1 n (0))/Γ M \ K such that Φ g g 0 g0 + r g0 (Φ g g 0 ) g0 = o(r 1 m ). (2) Also assume that (M, g) has scalar curvature Scal g L 1 (M, dvol g ). Then the limit in (1) exists and is independent of Φ, provided that Φ satisfies (2). More precisely, we have that [ω] m 1 m(m, g, Φ) = 4π (c 1 ), + Scal g dvol g. (3) (m 1)! Here c 1 H 2 (M) denotes the first Chern class of, [ω] H 2 (M) denotes the Kähler class of g, : H 2 (M) H 2 c (M) denotes the inverse of the natural linear map H 2 c (M) H 2 (M), which is an isomorphism, and, denotes the duality pairing of H 2 c (M) and H 2m 2 (M). Theorem 4. Let (M, g, ) be as in Theorem 3. If m = 2, strengthen (2) by assuming that either Φ g g 0 C τ 1 for some τ > 3 2, or Φ g g 0 C 2,α τ for some τ (1, 3 2 ]. If Γ = {1} and Scal g 0, then m(m, g, Φ) 0 with equality if and only if (M, g, ) is holomorphically isometric to C m. Date: March 17, 2016. M

2 Hans-oachim Hein and Claude LeBrun Proof of Theorem 3. For the sake of transparency, assume instead of (2) that Φ g g 0 g0 + r g0 (Φ g g 0 ) g0 = o(r τ ) (4) for some τ > 0 to be determined. We wish to examine the behavior as r of the quantity m(φ, r) ( i (Φ g) ij j (Φ g) ii )ν j da. (5) For convenience, we will be abusing notation by writing g Φ g from now on. Following Bartnik [1], we begin by computing that m(φ, r) = ( g x j )ν j da d log det g + o(r 2τ+n 2 ). (6) Both of the two main terms on the right-hand side of (6) are o(r τ+n 2 ). Thus, if τ n 2 2, then the error term goes to zero as r, but as long as τ < n 2, it is not clear how either one of the two main terms behaves. It is helpful to keep this numerology in mind in the following. Using (4) and = 0, we can construct a parallel complex structure 0 on R n such that Φ 0 g0 + r g0 (Φ 0 ) g0 = o(r τ ). (7) Composing Φ with a rotation, we can arrange that 0 is the usual complex structure on C m = R n and that Γ U(m) without changing the value of m(φ, r). Let Ω 0 = dz 1... dz m on C m. Then there certainly exists some smooth section Ω of Λ m,0 M (multivalued if Γ SU(m)) such that Φ Ω Ω 0 g0 + r g0 (Φ Ω Ω 0 ) g0 = o(r τ ). (8) While Ω is never unique, it is often impossible to choose Ω to be -holomorphic on M \ K. For us, the use of Ω is that it provides a link between the volume element det g featuring in (6) and the Ricci curvature of g. Let ρ(x, Y ) = Ric g (X, Y ) be the Ricci form. Then ρ = if h, where given by h(ω) = Ω Ω /ω m and F h is the curvature of the Chern connection D on Λ m,0 associated with h. Write DΩ = ϕ Ω and Ω = α Ω. Then ϕ 0,1 = α and dh = (ϕ + ϕ)h by the definition of D. Thus, h = (ϕ α + ᾱ)h. Since F h = dϕ, we get that h is the Hermitian metric on Λ m,0 ρ = dθ, θ 1 2 dc log ω m Ω Ω + 2Im α, Ω α Ω, (9) at least in the asymptotic region M \ K where Ω does not vanish. Using (6), (9), and the identity ξ = (ξ ) ω for all 1-forms ξ, we find that [ ] m(φ, r) = ( g x j )dx j Ω Ω + d log Ω 0 Ω 0 4Re α + 2 θ ω. (10) Here and below = denotes equality modulo o(1) as r, and we are now assuming that τ n 2 2. Notice that all four contributions to the right-hand side of (10) are manifestly o(r τ+n 2 ). There are now two alternative ways of continuing the proof. The first approach is more intuitive, but it requires slightly stronger assumptions than (2). Once one knows that this goes through, one is then essentially forced to discover the second, easier but less intuitive approach. First alternative. The key observation here is that, to leading order, each of the three terms in the first integral of (10) depends only on the asymptotics of. Precisely, if (7) and (8) hold with o(r σ ) for some σ τ, then these terms are o(r σ+n 2 ). We then show that there exists a new coordinate system Φ such that (4) still holds with o(r τ ) but (7), (8) indeed hold with o(r σ ) with respect to Φ, where σ n 2. (This is quite delicate in general, but is obvious in many concrete examples.) Thus, m( Φ, r) = 2 θ ω. Using that Scal g L 1, it is easy to see that 2 θ ω = RHS(3). This finishes the proof because m(φ, r) = m( Φ, r) by one of the main results of [1, 2].

Summary of the Kähler mass paper 3 Let us flesh out some of the details of this. Identify Φ. The first step is to compactify the complex manifold (R n \ B1 n(0), ) by adding a divisor biholomorphic to CPm 1 at infinity. There are several ways of doing this. Here we use the method of [5, Section 3.2], which requires us to assume 0 C 2,α τ in order to produce -holomorphic coordinates of class C2,α around infinity. (C 1,α τ should be enough. Unfortunately, as a family parametrized by CP m 1, the -holomorphic curves of [5] are only as regular as. On the upside, this method works for any τ > 0.) With more work, one can show that in each sufficiently narrow sector of R n \B1 n (0), there exist -holomorphic coordinates differing from the standard linear 0 -holomorphic ones by functions of class C 2,α 1 τ if τ N. The second step is to examine the infinitesimal neighborhoods of the compactifying divisor. The above already tells us that the divisor is τ -comfortable [8], an intermediate condition between its τ -th and τ -th neighborhoods being trivial. A careful global study of our situation reveals that, in fact, the 3rd neighborhood is trivial for m = 2 and all neighborhoods are trivial for m 3. (For more on this, see the Appendix below.) This allows us to construct global coordinates Φ that can be written as id + C 2 in terms of the above local -holomorphic ones if m = 2, and that coincide with them if m 3. In particular, Φ 0 C 3 if m = 2, and Φ 0 = 0 if m 3. Summarizing, if Φ 0 C 2,α τ for some non-integer τ > 0 (with τ < 3 if m = 2), there exist ALE coordinates Φ, differing from Φ by functions of class C 2,α 1 τ, with Φ 0 C 3 if m = 2 and Φ 0 = 0 if m 3. Moreover, if τ > m 1 and if g satisfies (2) with respect to Φ, then g clearly still satisfies (2) with respect to Φ. By the above outline, m( Φ, r) = RHS(3) if Scal g L 1. It remains to observe that [1, 2] tell us that m(φ, r) = m( Φ, r). Notice here that the hypotheses of [1] are phrased in terms of weighted Sobolev spaces, but the proof of [1, Theorem 4.2] goes through verbatim assuming (2), Scal g L 1, and Φ Φ C1 τ 2 for some τ > m 1. Second alternative. The first approach above told us that m(φ, r) = 2 θ ω if Φ g g 0 C 2,α τ for some τ > m 1 and if Scal g L 1. Moreover, 2 θ ω = RHS(3), assuming only that Scal g L 1 but no particular decay conditions on g or. Thus, under all of these assumptions, the first integral of (10) must go to zero as r. We will now show by direct computation that this holds assuming only (2), thereby proving Theorem 3 as stated. However, we would likely never have guessed this cancellation without knowing that (a slightly weaker version of) Theorem 3 is true. More precisely, we will prove that, assuming (2), the first integral of (10) goes to zero for any Φ, provided that Ω is chosen appropriately subject to (8). Of course it will then turn out after the fact that having made such a choice for Ω was just a computational convenience. The basic idea of the first proof above was to consider the asymptotics of, so we begin here by making (7) more explicit. Identifying Φ, (7) simply tells us that dz j = i(dz j + K j kd z k + L j k dzk ), K j k, L j k C1 τ, (11) where we write C τ 1 rather than C τ 1 to indicate o(r τ ) rather than O(r τ ) decay. It follows easily from 2 = id that L j k C1 2τ, which allows us to view Lj k as an inessential error term. Let us now define a complex-valued m-form Ω by Ω (dz j + 1 2 Kj kd z k ), (12) and let Ω denote the (m, 0)-part of Ω with respect to. Using (11) and the fact that L j k C1 2τ, it is not hard to see that Ω Ω C 2τ 1. Thus, in particular, Ω satisfies (8). One immediate benefit of this construction is that Ω Ω is standard modulo C 2τ 1 rather than C1 τ. In particular, the middle term in the first integral of (10) drops out to leading order. We proceed by writing the remaining contributions to the first integral of (10) in terms of K j k to leading order. All of the following computations are to be read modulo o(r 2τ 1 ).

4 Hans-oachim Hein and Claude LeBrun Claim 1. ( g x j )dx j = 2Re(K j k,k d z j ). Proof of Claim 1. It is easy to see that ( g x j )dx j = Re(( g z j )d z j ). To leading order, g z j is the contraction with ω 0 = i 2 dzk d z k of the 0 -(1, 1)-part of the -(1, 1)-form dd c z j = idk j k d z k. Claim 2. α = 1 2 Kj k,j d z k. Proof of Claim 2. Recall that α has been defined by Ω = α Ω. Also recall the formula (12) for Ω to leading order. Computing α to leading order is now straightforward from the following two facts. First, dz j = i 2 ddc z j = 1 2 dkj k d z k. Second, (K j kd z k ) = 0 to leading order; this follows from the vanishing of the Nijenhuis tensor of (the Maurer-Cartan equation in complex deformation theory) or alternatively by expanding the identity g = 0, which tells us that K j k, l = 2Γ j k l. We have now reduced our problem to proving that Re ξ 0 as r, where ξ (K j k,k K k j,k )d zj. (13) This follows from the next claim, which expresses Re ξ as a codifferential to leading order. Claim 3. Re ξ = 4d (Im ζ), where ζ denotes the (0, 2)-part of ω with respect to 0. Proof of Claim 3. Writing g = g jk dz j dz k + g j kdz j d z k + g jk d z j dz k + g j kd z j d z k, applying to the first tensor factors, and using the skew-symmetry of ω yields ζ = i 8 (Kj k K k j )d z j d z k. Abbreviate η 4Im ζ. The Kähler identities associated with the flat Kähler structure (g 0, 0 ) tell us that d η = [Λ, d c ]η = (Λdη) 0 = Im ξ 0 = Re ξ because Λdζ = 1 4ξ by the above. Proof of Theorem 4. The key technical point is to construct a -holomorphic map π : M C m that is a biholomorphism onto its image away from a compact set. 1 Given such a map, it suffices to note that π (dz 1... dz m ) is a regular m-form on M, vanishing precisely at those points where π fails to be a local diffeomorphism. In particular, (c 1 ) is Poincaré dual to a finite linear combination, with positive integer coefficients, of compact complex hypersurfaces, which implies positivity of the mass. Moreover, if the mass is zero, π is an isomorphism, so g is an AE scalar-flat Kähler metric on C m, hence Ricci-flat because its Ricci form is L 2 harmonic, hence flat by Bishop-Gromov. To construct π, we first compactify M as a complex manifold X by adding a divisor D = CP m 1 with normal bundle O(1). Since we require no additional properties of X, and are looking to prove the strongest possible version of Theorem 4, we can simply use [6] whenever possible. To apply [6], it suffices that Φ is C 0,α in inverted coordinates with α > 1 2 and that its distributional Nijenhuis tensor is L 1, which follows from Φ being L W 1,1 in inverted coordinates. Thus, X certainly exists if Φ 0 C τ 1 with τ > 3 2, which covers all relevant cases except for m = 2, τ (1, 3 2 ]. In the latter case, we can instead use [5, Section 3.2] as sketched in the first proof of Theorem 3. The desired map π will be the restriction to M of the complete linear system O X (D). For this to work, we need to know that the Kodaira deformations of D are linearly equivalent. This follows from H 1 (X, O X ) = 0, which follows from H 1,0 (X) = 0 [7, p. 555] if X satisfies Hodge symmetry. To finish the proof, we can write down a Kähler form on X by bare hands (simplifying the argument in [4] by making use of the fact that H 2 (M \ K) = 0), or use Grauert s Kodaira embedding theorem to argue that X is Moishezon as in [3, p. 527]. The latter argument actually establishes that O X (ld) (l 1) defines a degree-one holomorphic map π l from X onto some projective variety such that π l is an isomorphism onto its image near D: a weak version of the claim we set out to prove. 1 Such a map, with C m replaced by C m /Γ, still exists if Γ {1} and m 3, as well as in many (but arguably not the most interesting) examples with Γ {1} and m = 2. Here we give a simple proof for Γ = {1} that works in all dimensions. The real reason why positive mass fails for Γ {1} is the singularity of C m /Γ at the origin.

Summary of the Kähler mass paper 5 References [1] R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661 693. [2] P.T. Chruściel, Boundary conditions at spatial infinity from a Hamiltonian point of view, Topological properties and global structure of space-time (P. Bergmann, V. de Sabbata, eds.), 49 59, Plenum Press, New York, 1986. Reprint available at arxiv:1312.0254. [3] R.. Conlon, H.-. Hein, Asymptotically conical Calabi-Yau metrics on quasi-projective varieties, Geom. Funct. Anal. 25 (2015), 517 552. [4] R.. Conlon, H.-. Hein, Asymptotically conical Calabi-Yau manifolds, III, preprint, arxiv:1405.7140. [5] M. Haskins, H.-. Hein,. Nordström, Asymptotically cylindrical Calabi-Yau manifolds,. Differential Geom. 101 (2015), 213 265. [6] C.D. Hill, M. Taylor, Integrability of rough almost complex structures,. Geom. Anal. 13 (2003), 163 172. [7] C. LeBrun, B. Maskit, On optimal 4-dimensional metrics,. Geom. Anal. 18 (2008), 537 564. [8] C. Li, On sharp rates and analytic compactifications of asymptotically conical Kähler metrics, preprint, arxiv: 1405.2433. [9] H. Rossi, Vector fields on analytic spaces, Ann. of Math. (2) 78 (1963), 455 467. Appendix On any ALE Kähler manifold (M, g, ) with Φ g g 0 C τ 1, we have Φ 0 C τ 1 for some linear complex structure 0 on R n. We now take a closer look at the question of improving Φ to Φ such that Φ g g 0 C τ 1 but Φ 0 C σ 1 for some σ > τ. (For our first proof of Theorem 3 we require that σ > n 2, even if all we know a priori is that τ > n 2 2.) It is to some extent folklore that one can achieve Φ 0 = 0 for m 3. We indicate several different arguments for this in our paper, and one could alternatively also use Schlessinger s theorem (Invent. Math. 14 (1971)) or Hörmander L 2 -type techniques (as in G. Tian,. Differ. Geom. 35 (1992)). The fact that σ 3 is always attainable for m = 2 does not seem to be as widely known, but see [8]. In this appendix we sketch a more direct analytic approach to these questions. To leading order, Φ 0 is given by a (0, 1)-form K = K j kd z k z j with values in T 1,0 C m. Then K = 0 from the vanishing of the Nijenhuis tensor of, and composing Φ with exp(v ) for some smooth vector field V of sublinear growth changes K by V. Thus, using Hodge theory on cones, one can arrange that K = 0 in addition to K = 0, simply by adjusting Φ. (For this we need that Φ g g 0 C 1,α τ, but then the change of gauge will indeed preserve the decay rate of g as long as τ is not an integer.) We may now assume without loss of generality that K j k = z 2(1 m d) H j k, where each H j k is a harmonic polynomial on R n, homogeneous of some given degree d. Then obviously σ = n 2 + d, and all we need for the first proof of Theorem 3 is that K 0 implies d 0. It is actually quite easy to rule out that d = 0 by writing the equations K = 0 and K = 0 in terms of H: z 2 H j k, l + (1 m d)z l H j k is symmetric in k, l, and z 2 H j k,k + (1 m d) z k H j k = 0, so if H is constant then H = 0 as desired. However, it is also interesting to consider the next step, d = 1. Writing H j k = H j kp z p + H j k p z p, we deduce H j kp = H j δ kp and H j k p + H = 0 from the co-closed j p k condition. Then the closed condition is automatically satisfied for m = 2, while if m 3 then the closed condition forces that H j k p = 0. Now z 2m H j z k d z k z j = V with V = 1 2 2m z 2 2m H j z j in all dimensions, which is just an infinitesimal gauge. Thus, for m 3, all solutions K are gauge equivalent to zero, while for m = 2 they are gauge equivalent to z 4 ( z 2 d z 1 z 1 d z 2 ) v for some constant vector v = H j 1 2 z j C2. One can often kill this obstruction by using Γ-invariance, where Γ U(2) denotes the fundamental group of the end of M. Indeed, the obstruction is Γ-invariant only if det( γ 1 )γv = v for all γ Γ; cf. the discussion at the end of Section 3 of our paper. It seems like an interesting exercise in representation theory to push this analysis further, aiming to prove that for m 3, all solutions to K = 0, K = 0, K = z 2(1 m d) H, H a homogeneous degree d harmonic polynomial, are of the form V for some harmonic vector field V.