The eta invariant in the doubly Kählerian conformally compact Einstein case
aa r X i v : . [ m a t h . DG ] M a y THE ETA INVARIANT IN THE DOUBLY K ¨AHLERIANCONFORMALLY COMPACT EINSTEIN CASE
GIDEON MASCHLER
Abstract.
On a 3-manifold bounding a compact 4-manifold, let a conformalstructure be induced from a complete Einstein metric which conformally compact-ifies to a K¨ahler metric. Formulas are derived for the eta invariant of this conformalstructure under additional assumptions. One such assumption is that the K¨ahlermetric admits a special K¨ahler-Ricci potential in the sense defined by Derdzinskiand Maschler. Another is that the K¨ahler metric is part of an ambitoric structure,in the sense defined by Apostolov, Calderbank and Gauduchon, as well as a toricone. The formulas are derived using the Duistermaat-Heckman theorem. Thisresult is closely related to earlier work of Hitchin on the Einstein selfdual case. Introduction
In [H], Hitchin computed an obstruction, in terms of a bound on the eta invariantfor a conformal structure on the 3-sphere to be induced from a complete self-dualEinstein metric on the 4-ball. Following LeBrun [L], such conformal structures aresaid to have positive frequency , drawing on an analogy between this case and theclassical obstruction for a smooth function on the circle to be the boundary value ofa holomorphic function on the disk.In this paper we give formulas for the eta invariant of a conformal structure inducedfrom another type of asymptotically hyperbolic Einstein metric in dimension four,namely one that is conformal to a (conformally compact) K¨ahler metric, and satisfiesadditional technical assumptions. These formulas are obtained via moment maptechniques, specifically the Duistermaat-Heckman Theorem. The assumptions areeither that the K¨ahler metric admits a special K¨ahler-Ricci potential in the senseof [DM1, DM2], or that it is part of an ambitoric structure, in the sense definedrecently by Apostolov et. al. [ACG], as well as a toric one. Common to both casesis the existence of a second, oppositely oriented complex structure, with respect towhich one has a second K¨ahler metric in the conformal class, which plays a majorrole in the derivation of the formulas.Section 2 is devoted mainly to a review of the theorem of Hitchin. Section 3 givesan overview of the classification of metrics with a special K¨ahler-Ricci potential inthe case of a manifold with boundary, and also discusses the dual K¨ahler metric. InSection 4 the two formulas are derived.
Date : October 30, 2018. Preliminaries
Conformal compactifications of Einstein metrics.
In the following let(
M, g E ) be conformally compact Einstein , i.e., M is a compact manifold with anonempty boundary, g E a complete Einstein metric in the interior of M and thereexists a smooth defining function τ for the boundary ∂M (so that τ ≥ , τ | ∂M =0 , dτ | ∂M = 0) for which g = τ g E is smooth on M .The pair ( g, τ ) is called a conformal compactification for g E . The restriction γ = g | ∂M varies with τ within a fixed conformal class [ γ ] on ∂M .Later we will consider such ( M, g E ) with a K¨ahlerian (conformal) compactification ,i.e. satisfying the additional assumption that g is K¨ahler in the interior of M . TheK¨ahlerian assumption, combined with the more technical assumption (4), which willbe described in §
3, imply that the function Q := |∇ τ | is constant on each level setof τ (see [DM1]). A compactification with the latter property may be called uniform .It is easier to achieve, in a sense, than the more well-known geodesic one, where Q isconstant in a collar neighborhood of the boundary: from a given compactification,one must solve a PDE to obtain a geodesic compactification, but only an ODE toobtain a uniform one. On the other hand, the above two assumptions imply theuniformity of the compactification throughout the open and dense set of non-critical τ values, and not just in a collar neighborhood of the boundary.2.2. The theorem of Hitchin.
We recall here the result of Hitchin, emphasizingits applicability, in addition to the four-ball, also to other four-manifolds.
Theorem 1 (Hitchin) . Suppose ( g, τ ) is a conformal compactification of ( M, g E ) ,with g E Einstein and M a manifold with boundary of real dimension four. Then (1) η γ ( ∂M ) = 112 π Z M (cid:0) | W − | g − | W + | g (cid:1) vol g − σ ( M ) , where W + ( W − ) are the (anti-)self dual parts of the Weyl tensor of g , vol g its volumeform, σ ( M ) the signature of M and η γ ( ∂M ) the eta invariant of the boundary metric γ . Here the eta invariant is defined as the value at s = 0 of the series P λ =0 (sgn λ ) | λ | − s (which is holomorphic for Re s > − / ∂M , given by B ( α ) = ( − p ( ∗ d − d ∗ ) α, α ∈ Ω p ( ∂M ) . This invariant depends only on the conformal class of γ in ∂M . Proof.
The Atiyah, Patodi and Singer [APS] signature formula for a manifold withboundary reads [EGH] σ ( M ) = − π Z M tr( R ∧ R ) + 124 π Z ∂M tr(Π ∧ R ) − η γ ( ∂M ) , HE ETA INVARIANT IN THE DOUBLY K ¨AHLERIAN CONFORMALLY COMPACT. . . 3 where R is the curvature tensor of M and Π is the second fundamental form of ∂M ,considered as a one-form valued in (restrictions to ∂M of) endomorphisms of thetangent bundle of M . In the case at hand, this boundary term actually vanishes. Infact, as g is conformal to the Einstein metric g E , it satisfies a Ricci-Hessian equationof the form ∇ dτ + ( τ / σg , with r the Ricci curvature and ∇ the covariantderivative of g , for some function σ . As ∂M = { τ = 0 } , the second fundamental formof the boundary is the Hessian ∇ dτ . But the Ricci-Hessian equation above yieldsthat on the boundary this Hessian is a multiple of the metric, i.e. that ∂M is totallyumbilical. Thus, in an adapted orthonormal frame { e i } i =1 with { e i } i =1 tangent to,and e a unit normal to ∂M , Π = k P i =1 e i ⊗ e i for some constant k , which impliestr(Π ∧ R ) = k ( e ∧ R + e ∧ R + e ∧ R ) + k ( R + R + R ) e ∧ e ∧ e ) = k ( R + R + R ) e ∧ e ∧ e ) = 0 by the Bianchi identity. The result nowfollows as tr( R ∧ R ) = 2 (cid:0) | W + | g − | W − | g (cid:1) vol g . (cid:3) For the four-ball, σ ( M ) = 0, and so Hitchin concludes from (1) that the etainvariant of S is nonpositive for conformal structures induced from self-dual Einsteinmetrics on B , and zero exactly for the standard structure. More generally, thefollowing obvious general bounds hold: Corollary . (2) − π Z M | W + | g vol g − σ ( M ) ≤ η γ ( ∂M ) ≤ π Z M | W − | g vol g − σ ( M )with equality holding on the left-hand side in the self-dual case, and on the right-handside in the anti-selfdual case. Moreover, if the compactification is also Kahlerian,and g is not anti-selfdual, then the left hand inequality of (2) can be rewritten as(3) − π Z M s g vol g − σ ( M ) ≤ η γ ( ∂M ) , an inequality that holds because on a K¨ahler surface, W + is completely determinedby the scalar curvature and the K¨ahler form (see for example Derdzinski [D]).3. K¨ahler-Ricci potentials on manifolds with boundary
Metrics with a K¨ahler-Ricci potential.
We give here an outline of theanalog, for manifolds with boundary, of the main theorem in [DM2] (see Theorem 2below).Let (
M, J, g, τ ) be a quadruple consisting of a compact manifold M with a nonemptyboundary, having a defining function τ , and a complex structure J along with aK¨ahler metric g on the interior of M for which(4) τ is a special K¨ahler-Ricci potential.This requirement on τ is a technical notion which holds in the conformally Einsteincase if either the complex dimension of M is at least three, or else it is two and dτ ∧ GIDEON MASCHLER d ∆ g τ = 0, with ∆ g the g -Laplacian. Metrics satisfying (4) have been classified bothlocally and on compact manifolds in [DM1] and [DM2]. Specifically, a special K¨ahler-Ricci potential is a Killing potential on a K¨ahler manifold, i.e. J ∇ τ is a Killing vectorfield, for which the following additional condition holds: at any noncritical point of τ , the subspace H = (span( ∇ τ, J ∇ τ )) ⊥ of the tangent space is an eigenspace forboth the Hessian ∇ dτ and the Ricci curvature r.Suppose the interval of τ -values is [0 , τ ] and the τ -preimage N of the maximalvalue τ is a critical manifold. By [DM2, Lemma 7.3], N is a complex submanifoldof M which is of complex codimension one, or else consists of a single point. Fur-thermore, any critical manifold is the τ -preimage of an extreme value for τ (see theproof of Proposition 11 . τ is a defining function, the zero level set of τ does not contain any critical points (and is the boundary), so that N is the uniquecritical manifold of τ .Special K¨ahler-Ricci potentials give rise to ingredients that are used to give an ex-plicit construction of a metric g which is biholomorphically isometric to g . Namely,the function |∇ τ | is a composite consisting of τ followed by a smooth function Q ( τ )defined on [0 , τ ]. This follows from Lemma 9.1 of [DM2] along with Proposition11.5(ii) there, for the interval [0 , τ ] with the proof changed to the case of a com-pact manifold with boundary. Adopting again the same proposition, Q satisfies theboundary and positivity conditions(5) bdps Q ( τ ) = 0, Q ′ ( τ ) = 0 and Q ( τ ) > τ ∈ [0 , τ ).Another ingredient is a constant c , defined as follows: the eigenfunction φ of theHessian ∇ dτ on the subspace H can be identically zero, in which case c is undefined,or nowhere zero for τ -noncritical points, and then c = τ − Q/ (2 φ ) [DM2, Lemma3.1]. We note for future reference that(6) ct0 c / ∈ [0 , τ ] unless the critical manifold N consists of a single point and then c = τ (see [DM2, Lemma 7.5]). Furthermore, the unique nonzero eigenvalue of ∇ dτ at eachpoint of N is a constant denoted e [DM2, Proposition 7.3 (denoted a there)]. With Q, c and e one defines a metric g on a disc bundle S contained in the total space ofthe normal bundle L to N , using a Hermitian fiber metric h· , ·i whose real part is g | L and a metric h on N defined as either g | T N if φ = 0, or (2 | τ − c | ) − g | T N if φ = 0(if N consists of a single point { y } then S is a disc, L = T y M and h is undefined).The connection associated with the fiber metric is the normal connection, and itscurvature Ω satisfies(7) Ω = p ω h , where ω h is the K¨ahler form of h and the constant p equals ± e (see [DM2, Lemma12.4]). HE ETA INVARIANT IN THE DOUBLY K ¨AHLERIAN CONFORMALLY COMPACT. . . 5
Regarding N as the zero section N ⊂ L , the metric g is defined as follows:(8) met if N is a complex codimension one submanifold: i] g | H = π ∗ h if φ = 0,ii] g | H = 2 | τ − c | π ∗ h if φ = 0, if N consists of a single point: iii] g | H = (2 | τ − c | / ( | e | r )) π ∗ h, and in all cases: g | V = ( Q ( τ ) / ( er ) ) Re h· , ·i . Here, if N is a complex codimension one submanifold, H , V are the horizontal dis-tribution defined by the connection and the vertical distribution, respectively, whichare also declared g -orthogonal to each other. If N consists of a single point, then V is the distribution on the vector space T y L \ H is the distribution orthogonal to V withrespect to Re h· , ·i . Finally, r = r ( τ ) is a nonnegative function on [0 , τ ] which van-ishes only at τ = τ and satisfies the differential equation dr/dτ = e r/Q in (0 , τ ).Abusing notation by regarding r also as the norm function of h· , ·i , it is a functionon L . Using the inverse map τ ( r ) to r ( τ ), all functions of τ in (8) become functionsof the norm r on L , thus giving rise to a metric defined on L . Furthermore, thereexists a positive value r for which τ ( r ) = 0, and the disc bundle S is characterizedby the inequality r ≤ r .A version of the main theorem of [DM2], Theorem 16.3 there, now holds for M with essentially the same proof, i.e. Theorem 2.
Let M be a compact manifold with a nonempty boundary, with aK¨ahler metric g and a special K¨ahler-Ricci potential τ on its interior, both extendingsmoothly to the boundary. Suppose τ has the value zero on the boundary. Then thetriple ( M, g, τ ) is biholomorphically isometric to the disc bundle (or disc) ( S, g, τ ( r )) described above, and the orbits of the Killing vector field J ∇ τ are closed. Note that the assumption on the boundary value of τ excludes the possibilityof S being an annulus bundle, and guarantees that M has a critical manifold, thepreimage of the nonzero extremum τ of τ . Also, the normal exponential map which isa building block for the inverse of the biholomorphic isometry gives a diffeomorphism S \{ r = r } → M \ ∂M (this is [DM2, Lemma 13.2], adjusted to the case of a manifoldwith boundary).The last part of the conclusion of the theorem, which can be considered a conse-quence of the fact that the orbits of J ∇ τ coincide under the above biholomorphicisometry with the S orbits of the natural circle action on L , can be seen directlyas follows. At a critical point of a special K¨ahler-Ricci potential τ , the Hessian GIDEON MASCHLER ∇ dτ has exactly one nonzero eigenvalue (see [DM2, Proposition 7.3]). By [DM2,Lemma 10.2], the vector field J ∇ τ , being the image under the differential of theexponential map of a linear vector field with a periodic flow, has itself a periodicflow in a neighborhood of the critical point. By the unique continuation property forisometries this periodicity is in fact global, as two isometries in the flow that agreeon a nonempty open set actually agree globally. However, comparing isometries ofa flow, and, more specifically, verifying the existence of a circle action produced bya flow [DM2, Corollary 10.3], may be performed if the vector field is complete. In[DM2] completeness of the metric, which implies completeness of the vector field,was assumed. However only the latter is necessary, and in the setting at hand, ofa manifold with boundary, it occurs automatically. In fact, as g and τ are smootheverywhere, ∇ τ and the Killing field J ∇ τ are well-defined on the boundary, andsince the latter is orthogonal to to the former, and ∇ τ is normal to the boundary, J ∇ τ is in fact tangent to the boundary at boundary points, and hence necessarilycomplete. (Note as an aside that a chain of reasoning exists which, starting with thefact that J ∇ τ is Killing, concludes with smoothness of τ . Thus in the statement ofTheorem 2, it is enough to assume continuity of τ at the boundary.) Remark . If we now add to the above assumptions the requirement on (
M, g ) thatit is a (K¨ahlerian) conformal compactification of an
Einstein metric on the interiorand M is of dimension four, the metric g (and hence indirectly g ) can be specifiedby a local formula for Q [DM1, § Q ( τ ) is a member of one of thefollowing three families of rational functions, whose formulas are given below with K , α , β , A , B and C denoting constants (the precise choice of which ensures that Q ( τ ) satisfies the boundary and positivity conditions (5)):a] Q = − Kτ + ( ατ − β/ / φ = 0.b] Q = − Kτ / ατ − β/ φ = 0 and c = 0.c] Q = ( τ /c − AE ( τ /c ) + BF ( τ /c ) + C ) if φ = 0 and c = 0, where E ( x ) = x − , F ( x ) = ( x − x / ( x − . These formulas are given for completeness, and will not be used again.3.2.
The dual K¨ahler metric.
As in § M, g, τ ) be a triple consisting ofa compact manifold M with a nonempty boundary, having a defining function τ ,and a K¨ahler metric g on the interior of M for which τ is a special K¨ahler-Riccipotential. Suppose also that the isometric metric g of the last section is either oftype ii] or type iii] in (8). Such a metric was called nontrivial SKR in [M]. In thiscase the constant c is defined (see paragraph between (5) and (6)) and in fact(9) dual b g = g/ ( τ − c ) is K¨ahler for an oppositely oriented complex structure HE ETA INVARIANT IN THE DOUBLY K ¨AHLERIAN CONFORMALLY COMPACT. . . 7 while b t = 1 / ( τ − c ) is a special K¨ahler-Ricci potential for b g (see [M, Proposition 5.1]or [DM3, Remark 28.4]). Our main use of this fact is in the formula(10) hat | W − | = ( τ − c ) − | c W + | b g which follows since W − is, as a (3 ,
1) tensor, a conformal invariant (and using theconformal factor ( τ − c ) relating g with b g ).4. Moment map formulas for the η invariant The case of a special K¨ahler-Ricci potential.
In the following theorem,the meaning of the notations N , h , c , e , τ , r and p is that given in § g data used in the construction of its isometric metric g . Additionally, we continueto denote the signature of a manifold by σ ( M ), while the volume of N with respectto h will be denoted V ol h ( N ). Theorem 3.
Let M be a compact four-manifold having a nonempty boundary witha complete Einstein metric g E on its interior, admitting a K¨ahlerian conformal com-pactification ( g, τ ) satisfying (4) on its interior for a nontrivial SKR metric. Thenthe eta invariant of the boundary metric γ = g | ∂M is given by (11) η γ ( ∂M ) = 1288 π V ol h ( N ) Z τ (cid:2)(cid:0) b ( t − c ) − − a (cid:1) t (cid:3) · [ l + p t ] dt − σ ( M ) for some nonzero constants a and b , with l standing for the constant | c | if τ has acomplex codimension one critical manifold, and | c | / ( | e | r ) = 2 | τ | / ( | e | r ) (see (6))if the τ -critical manifold consists of one point.Remark . The integral in (11), is of course, elementary and equals (cid:20) − t a p − a lt − b p t − c ) − b ( l + cp )5 ( t − c ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) τ . Proof.
By Theorem 1, formula (1) gives an expression for the eta invariant in termsof the signature, the Weyl tensors W + , W − and the volume form of g . Using thedual metric b g , one can replace | W − | by expression (10) involving | c W + | b g . Since both g and b g are K¨ahler metrics, the self-dual part of their Weyl tensors can be expressedin terms of the scalar curvatures s and b s (as in (3), cf. [D]). We thus have(12) η γ ( ∂M ) = 112 π Z M h ( τ − c ) − | c W + | b g − | W + | i vol g − σ ( M )= 1288 π Z M (cid:2) ( τ − c ) − b s − s (cid:3) vol g − σ ( M )Next, in dimension four, the scalar curvature of a K¨ahler metric which is conformal toan Einstein metric is a constant nonzero multiple of the square root of the conformal GIDEON MASCHLER factor (cf. [DM1, Proposition 6.5]). That square root is τ for g , and τ / ( τ − c ) for b g = g/ ( τ − c ) . Thus, writing s = a τ , b s = b τ / ( τ − c ) for nonzero constants a , b and ω for the K¨ahler form of g we get(13) η γ ( ∂M ) = 1288 π Z M (cid:2)(cid:0) b ( τ − c ) − − a (cid:1) τ (cid:3) ω / − σ ( M ) . We now apply the Duistermaat-Heckman Theorem [DH] to the integral in (13).Namely, the Killing potential τ is a moment map for the symplectic form ω associatedwith g , with respect to the circle action guaranteed by Theorem 2. The value τ = 0is regular for this moment map, because |∇ τ | τ =0 = 0, by (5). The theorem statesthat the Duistermaat-Heckman measure, i.e. the moment map push-forward of themeasure associated with the symplectic form on M , is the function [ ω + t Ω][ N ],with ω denoting the induced symplectic form on the associated reduced space N = τ − (0) /S and Ω denoting the curvature of L (See [G, Excercise 2 . η γ ( ∂M ) = 1288 π Z τ (cid:2)(cid:0) b ( t − c ) − − a (cid:1) t (cid:3) ([ ω + t Ω] [ N ]) dt − σ ( M )= 1288 π Z τ (cid:2)(cid:0) b ( t − c ) − − a (cid:1) t (cid:3) (cid:0)(cid:2) ω + t p ω h (cid:3) [ N ] (cid:1) dt − σ ( M )= 1288 π V ol h ( N ) Z τ (cid:2)(cid:0) b ( t − c ) − − a (cid:1) t (cid:3) [ l + p t ] dt − σ ( M ) . The second line follows from (7). The third follows from cases ii] and iii] of (8), as ω is the solution of the equation π ∗ ω = ι ∗ ω , valid on ∂M with ι being the inclusion ∂M ֒ → M and π the projection of ∂M to the reduced space ∂M/S . (cid:3) Remark . The value of the signature σ ( M ) in the above formula is in fact 0 or − M is homeomorphic to a disk bundle S over a Riemann surface N .Contractibility of the disk implies that H ( S ) = H ( N ) = Z , so that the intersectionform is a number, namely the Euler number of the bundle S . The group H ( S ) isgenerated by a section, and the intersection matrix is determined by its nonpositiveself-intersection number.4.2. The ambitoric case.
Ambihermitian structures were defined in [ACG]. Theseconsist, on a 4-manifold M , of a conformal class c , with two c -orthogonal complexstructures J , b J inducing opposite orientations. A metric in such a conformal class iscalled ambihermitian. Suppose such a metric g E is Einstein, and degeneracy of theWeyl tensor components W + and W − holds, in the sense that as operators on selfdual(respectfully antiselfdual) 2-forms, at least two of their three eigenvalues coincide.Then in a neighborhood of each point the structure is ambi-K¨ahler, i.e. there existtwo metrics g , b g in the conformal class c such that ( g, J ), ( b g, b J ) are K¨ahler [ACG,Proposition 1]. The three metrics satisfy g E = s − g = b s − b g on an open and denseset (away from the zero sets of s or b s ), where s ( b s ) is the scalar curvature of g ( b g ). HE ETA INVARIANT IN THE DOUBLY K ¨AHLERIAN CONFORMALLY COMPACT. . . 9
Furthermore, J ∇ s and b J b ∇ b s are Killing vector fields with respect to both g and b g [ACG, Proposition 7], so that b J b ∇ b s = J ∇ f and J ∇ s = b J b ∇ b f for two functions f , b f . Generically, these vector fields span a 2-dimensional space of Poisson-commutingKilling vector fields, and the whole structure is called ambitoric. We will also assumethat ( s, f ) and ( b f , b s ) are each moment maps for a toric group action (in the case of amanifold without boundary, this follows from the assumption of having an ambitoricstructure). We now show that in this case, a formula also exists for the eta invariantof the boundary. Theorem 4.
Let M be a compact four-manifold having a nonempty boundary witha complete Einstein metric g E whose conformal class admits an ambitoric structureconsisting of the metrics g , b g , with all metrics defined on the interior. If the vectorfields J ∇ s and b J b ∇ b s span an ambitoric structure and each gives rise to a toric struc-ture as described above, then the eta invariant of the boundary metric γ = g | ∂M isgiven by η γ ( ∂M ) = 1288 π Z b P b y v euc − π Z P x v euc − σ ( M ) , where P , b P are the images of the moment maps ( s, f ) , ( b f , b s ) respectively. Here x , b y are the functions on these images of the moment maps whose pullbacks are s and b s ,respectively, and v euc represents integration with respect to Lebesgue measure.Proof. Suppose f is the function such that g = f b g . We begin by noticing that therelation generalizing (10) is | W − | = f − | c W + | b g , so that by (1) η γ ( ∂M ) = 112 π Z M h f − | c W + | b g − | W + | i vol g − σ ( M ) . Next, vol g = f vol b g , so that in the first line below, f cancels to give: η γ ( ∂M ) = 112 π Z M | c W + | b g vol b g − π Z M | W + | vol g − σ ( M )= 1288 π Z M b s vol b g − π Z M s vol g − σ ( M )= 1288 π Z b P b y v euc − π Z P x v euc − σ ( M ) . The last line follows from the fact that the Duistermaat-Heckman measure in thetoric case equals Lebesgue measure on the image of the moment map. (cid:3)
Acknowledgements
The author thanks the anonymous referee for a comment that led to a correctedstatement and improved proof outline of Theorem 2.
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