Uniqueness of Kähler-Einstein cone metrics

Publicacions Matem`atiques, Vol. 44 (2000), 437–448 UNIQUENESS OF K¨ AHLER-EINSTEIN CONE METRICS Thalia D. Jeffres Abstract The purpose of this paper is to describe a method to construct a K¨ahler metric with cone singularity along a divisor and to illustrate a type of maximum principle for these incomplete metrics by showing that K¨ahler-Einstein metrics are unique in geometric H¨older spaces. 1. Introduction Outline of results. We show that if Mis a compact complex manifold of complex dimension two or greater, and Da divisor with one irreducible component and if the cohomology class C1(KM)+αC1(O(D)), for α∈(0,1), contains a positive representative, then we can construct an initial K¨ahler cone metric ωwith cone angle α. Here KMdenotes the canonical bundle of the manifold and O(D) the line bundle associated to the divisor. This metric is incomplete along the divisor. Functions describing geometric quantities will often be continuous on all of M, but may achieve nonsmooth extrema over the divisor. We develope a generalized maximum principle for such functions. The technique is illustrated by proving uniqueness of K¨ahler-Einstein cone metrics. For the existence of such metrics, [JM], more special function spaces are needed, which simultaneously yield refined regularity properties at the divisor, but for uniqueness it suffices to work within the larger geometric H¨older spaces. It is useful to provide an example of the technique in this more general and more geometrically intuitive setting because the method has other applications. In [J], for example, it is used to prove a Schwarz Lemma for K¨ahler metrics with cone singularities. Background. Singular spaces are of interest in differential geometry, algebraic geometry, and in analysis. They occur naturally in many settings. For example, many algebraic varieties are not smooth. Differential geometers interested in special metrics on smooth manifolds will naturally study the moduli space of such, and singularities often develope at 438 T. D. Jeffres the boundary of the moduli space. Interesting analytic features arise in the resolution of geometric problems, and this paper is an example of such. Uniqueness of partial differential equations, and also the estimates needed to prove existence of K¨ahler-Einstein metrics, often rely upon the maximum principle. The importance of the maximum principle is that it enables one to deduce, from elliptic inequalities, a priori estimates on solutions of differential equations. In the case of a singular or noncompact space, direct application of the maximum principle may be impossible. There is a close relationship between singular and noncompact spaces, because removal of the singular set leaves a noncompact space. One obvious difficulty in applying the maximum principle is that a maximum may simply fail to exist. An instance of this may be found in the paper of Cheng and Yau [CY] wherein they prove existence of K¨ahler-Einstein metrics on pseudoconvex domains. Indeed, a nontrivial step along the way is a generalized maximum principle which asserts, roughly speaking, the existence of a sequence of points approaching the boundary for which the first derivatives of the solution go to zero and the Hessian becomes negative semidefinite. In some cases, the problem can be circumvented. Incomplete metrics on the complement of a divisor were studied by Tian and Yau, [TY], but restrictions on the cone angle make it possible to pass to a finite branched cover and apply techniques similar to the smooth, compact case. The construction described below yields a metric with singularity along a divisor. Because the metric is incomplete, it is not possible to regard the singular set as being out at infinity; it is reached in finite time. If a function achieves an extremum over the divisor, the singular background metric allows the extremum to be achieved nonsmoothly or with a cusp shape. In some sense, this phenomenon is the opposite of that encountered by Cheng and Yau; their function did not achieve a maximum but had the correct shape, while ours has a maximum but with the wrong shape. The maximum principle is really an analysis of the shape of a function, and the technique described below consists in using a barrier function to push the maximum off the divisor and into the interior where it will be achieved smoothly and with the correct shape. Naturally, this barrier function must be chosen so that the resulting estimates are uniform. Uniqueness of K¨ ahler-Einstein Cone Metrics 439 2. Cone metrics Cone singularities are very natural singularities and have been studied from different points of view by many people. In the Riemannian context one might consult, for example the papers of Cheeger, among them [C]. The point of view there is to construct a singular space by forming a cone over a given compact Riemannian manifold, and the author describes the analysis of such a space. The special case of a sphere has also been considered by Troyanov in [Tr1], who has also studied cone singularities from the point of view of Riemann surfaces and conformal maps; see also [Tr2]. In this section we describe a construction of a K¨ahler cone metric with negative curvature on a compact complex manifold. An important application of this construction is to use it as an initial approximation toaK¨ahler-Einstein cone metric and then prove existence by perturbing away from it. In joint work with Rafe Mazzeo, [JM], we used this approach to prove the existence of K¨ahler-Einstein metrics with cone singularities at least for certain cone angles. To construct this cone metric, we need to assume some global conditions, namely, we suppose that Mcontains a smooth divisor Dwith one irreducible component, and fix a constant αwith 0 <α<1. This constant will be referred to as the cone angle. Now let KMdenote the canonical bundle of M, and O(D) the line bundle associated to the divisor D. In order to make sure that what we construct is really a metric we need to assume that C1(KM)+αC1(O(D)) ∈H2 DR(M) contains a positive definite real, closed (1,1) form. Some choices of M and Dthat satisfy this are smooth algebraic varieties Vin CPndescribed as the zero locus of a homogeneous polynomial of degree k>n+ 1, and D=V∩H, where His a hyperplane section of CPnthat intersects V in one smooth irreducible component. We can now proceed to construct the K¨ahler cone metric. Let sbe a defining section of O(D). Make provisional choices of a smooth volume function Von Mand a Hermitian metric ·in O(D) and write down ˆ V=V s2α(1 −s2(1−α))2. In the denominator, we only want the first term to vanish, so multiply sby a constant if necessary so that s≤δ<1; this is no problem because sis a smooth section defined on all of the compact manifold M. We would like to make sense of this as a singular K¨ahler potential so that 440 T. D. Jeffres our cone metric will be given by ω=i∂∂log ˆ V, so we compute directly ωdef =i∂∂ log ˆ V =i∂∂log V−iα∂∂log s2−2i∂∂ log(1 −s2(1−α)). Noting that a volume function on Mis the same thing as a metric hin the anticanonical line bundle K−1 M, and writing Θ(K−1 M) for the curvature of this metric, the first term here is i∂∂log V=i∂∂ log h(K−1 M)=−i∂∂ log h(KM)=iΘ(KM). How can we interpret the second term? Locally, suppose that e0is a nonvanishing holomorphic section of O(D), so that s=s0e0for a holomorphic function s0. Then −iα∂∂log s2=−iα∂∂log |s0|2−iα∂∂log e02. This is actually independent of the local choices, because any other choice e1of nonvanishing section would give s=s1e1=(s1g10)e0=s0e0 and ∂∂log |s0|2=∂∂ log |s1|2+∂∂ log |g10|2=∂∂ log |s1|2+0. This term gives a singular current supported over the divisor, since ∂∂log |s0|2=πδds0∧ds0. We denote this current by TD. So we have ω=i∂∂log ˆ V=iΘ(KM)+iαΘ(O(D)) −2παTD−2i∂∂ log(1 −s2(1−α)). Direct computation shows that the last term also produces a singularity; it looks like s−2αtimes a bounded form. Now the assumption that 2π(C1(KM)) + αC1(O(D))) >0 comes into play, because the sum of the first two terms, iΘ(KM)+iαΘ(O(D)), is a representative of this cohomology class. Therefore, the initial choices of volume function on Mand metric in O(D) can be made in such a way that the sum of the first three terms is positive definite. In fact, in local holomorphic coordinates (z,w2,... ,w n) in which D={z=0}, the sum of the first three terms is equivalent to √−11 |z|2αdz ∧dz + n  2 dwi∧dwi. Uniqueness of K¨ ahler-Einstein Cone Metrics 441 So ωconsists of a positive definite metric on Ω def =M\Dtogether with a singular term supported by D.ωis a current on all of M, and a genuine metric on Ω but we just refer to it as a singular metric on M. More descriptively, we say it has a cone singularity and that αis the cone angle, because in the zdirection, the surface with metric (1/|z|2α)dz∧dz is a cone. Let us now explain what it means to say that such a metric is K¨ahlerEinstein. Since ωdefines a metric on Ω, one may compute the Ricci curvature ρof this metric, and then extend as a current to all of M.A local expression for ρis ρ=−√−1∂∂ log det gi, and again choosing local coordinates (z,w2,... ,w n) for which Dappears as the zero set of z, this is ρ=−i∂∂log |z|−2αb, where bis a smooth nonzero bounded function which makes sense on all of M,or ρ=iα∂∂log |z|2−∂∂ log b=2παTD−i∂∂log b. Since −ωand ρboth contain the singular term 2παTDsupported by the divisor, the correct K¨ahler -Einstein condition is: ρ=−ω as currents on all of M, and pointwise on Ω. As in the smooth case, the existence of a K¨ahler-Einstein cone metric may be reformulated analytically as the existence of a solution uto the Monge-Amp`ere equation. This is the same equation as in the smooth case, except in this context a K¨ahler-Einstein metric is determined by a solution uover the noncompact set Ω. The derivation too is the same as in appearance as in the smooth case; one has only to remember that everything must be interpreted in the sense of currents and distributions. Excellent references for the smooth case are the expos´e of Bourguignon, [B], and also the lecture notes of Siu, [S]. With ωthe original K¨ahler cone metric, we consider new metrics of the form ωdef =ω+i∂∂u with u∈C2,δ g(Ω), the so-called geometric H¨older space, the definition of which will be given in the next section. For the moment, suffice it to 442 T. D. Jeffres say that uis a function that has 2 +δcovariant or geometric derivatives bounded on Ω. Then umust satisfy det(gi +∂i∂u) det(gi)=ef+u; we also require ω+i∂∂u > 0 so that the solution is a metric. 3. Geometric H¨older spaces A natural setting in which we solve nonlinear problems on Ω is the geometric H¨older spaces Ck,δ g(Ω). These spaces consist of the functions which are continuous on all of Mand whose covariant derivatives up to order k+δare bounded on Ω with respect to the singular cone metric ω constructed above. Namely, Ck g(Ω) consists in functions uwhich are continuous on all of Mand for which sup Ω|u|+···+ sup Ω∇kug is finite. Note that the singular metric appears in this expression twice —both in the covariant derivative ∇and in the norm ·. For the H¨older part we first define C0,δ g(Ω) to consist in those functions ufor which sup Ω|u|+ sup p=q∈Ω |u(z,w2,... ,w n)−u(z0,w 2,0,... ,w n,0)| |z|αδ|z−z0|δ+|w2−w2,0|δ+···+|wn−wn,0|δ is bounded. Here p=(z,w2,... ,w n), and q=(z0,w 2,0,... ,w n,0). Successive Ck,δ g(Ω) are then obtained by replacing the numerator by |X1...X ku(p)−X1...X ku(q)|where the Xiare vector fields for which Xiis bounded on Ω. The definitions of Ck g(Ω) and Ck,δ g(Ω) are consistent with each other because ∇ugis bounded if and only if for every vector field Xthere is a constant Cso that |Xu|≤CXg. 4. Maximum principle technique To obtain the uniqueness result, in the next section we will use the maximum principle to show that the difference between two solutions, u1−u2, must be zero. We demonstrate the idea here by explaining how to obtain a C0estimate. This is also an important step in the proof of existence, [JM]. Then in the following section a refinement gives uniqueness. Uniqueness of K¨ ahler-Einstein Cone Metrics 443 We begin by recalling how C0estimates on solutions of the MongeAmp`ere equation for negative first Chern class were obtained in the smooth case by Aubin, [A1] and [A2] and by Yau, [Y]. If uis a solution of the Monge-Amp`ere equation, then locally det(gi +∂i∂u) det(gi)=ef+u. Remember that fis determined by the original geometry. At a point P where uachieves a maximum, (∂i∂u) is a negative semidefinite Hermitian matrix, and so at this point, ef+u(P)=det(gi +∂i∂u) det(gi)(P)≤1, and so f(P)+u(P)≤0. Therefore, for all x,wehaveu(x)≤max{−f(x)}. In our singular case, if a maximum of uoccurs over D, it could have a cusp shape, but because it is an element of the function space C2,δ g(Ω), we know exactly how fast the derivatives can blow up. So our modification is to add a function Fwhich just fails to be in this space, and show that uniform control is maintained. Put v=u+Ffor an unknown function Fto be determined. Then u=v−Fso the Monge Amp`ere equation becomes det(gi +∂i∂v−∂i∂F) det(gi)=ef+u. Suppose vachieves a maximum on Ω. Then at that point, (∂i∂v)is negative semidefinite, so that det(gi +∂i∂v−∂i∂F) det(gi)≤det(gi −∂i∂F) det(gi) or ef+u≤det(gi −∂i∂F) det(gi), that is, ev≤e−f+F·det(gi −∂i∂F) det(gi). If the right hand side can be bounded, then we will have obtained a bound for vand hence of u. So we can write down the conditions that the choice of Fmust satisfy. They are: 444 T. D. Jeffres 1. Max voccurs on Ω. 2. max u≤max v. 3. Fis uniformly bounded. 4. For some C,Cgi ≤∂i∂F, uniformly. Let sbe a section with s≤1 and put F=s2βfor a positive power βto be determined in a moment. Then v=u+Fwill agree with ualong D, and s2βwill be a function which is initially increasing in directions perpendicular to the divisor. If Fincreases more rapidly than any element of C2,δ g(Ω), then vwill achieve a maximum on Ω. We compare the gradient of Fto that of functions in C2,δ g(Ω); if the gradient is unbounded with respect to the flat cone metric ω0then it will also be unbounded with respect to ωg. Write s2β=|z|2βe2βlocally, with ea basis section for O(D), or s2β=|z|2βb. Because eis bounded away from zero, bis smooth. In diagonal coordinates at one point grad f,grad fg= ∂f ∂zi 2 giı. We compute the first term i= 1, since it corresponds to the singular (z,z) direction ∂ ∂z(|z|2βb)∂ ∂z (|z|2βb)g11 0=β2b2|z|2+4(β−1)+2α+···. This is unbounded if 2 + 4(β−1) + 2α<0or2β<1−α. Because F rises more steeply than ucan fall, max u+Foccurs over Ω. We now show that i∂∂F ≥Cωgfor some C, noting first the formula ∂∂ef= ef(∂∂f +∂f ∧∂f). Then i∂∂s2β=i∂∂eβlog s =is2β(β∂∂log s2+β2∂log s2∧∂log s2) ≥is2ββ∂∂log s2, since for a real-valued form h,∂h ∧∂h ≥0. But i∂∂log s2=−R(·), so we need to show that there exists C such that −βs2βR(·)≥Cωg or βs2βR(·)≤−Cωg. Since s≤1 and R(·) is bounded, there is such a constant C. Uniqueness of K¨ ahler-Einstein Cone Metrics 445 It is interesting to note that Cdepends on the original metric and on the curvature of the line bundle, while in the smooth case of course it only depends on the metric of M. That makes sense, because we had some freedom in how to choose the metric in the line bundle O(D). In the above instance, all that was needed was some bound for the maximum and the minimum of u. That is not good enough to obtain a uniqueness result, for that would only show that the difference between two solutions was at most one. So we need a sharpening of this technique. 5. Uniqueness of K¨ahler-Einstein cone metrics Now turning to the uniqueness question, the main result is: Theorem. Suppose that u∈C2,δ g(Ω) is a solution to the Monge-Amp`ere equation (ω+i∂∂u)n=ef+uωn, with ω+i∂∂u positive definite, where ωis equivalent to √−11 |z|2αdz ∧dz + n  2 dwi∧dwi at the divisor. Then uis unique. To prove this, we suppose that u1and u2are two solutions to the Monge-Amp`ere equation, lying in C2,δ g(Ω). That is, on the interior Ω, (ω+i∂∂u1)n=ef+u1ωnand (ω+i∂∂u2)n=ef+u2ωn, with both solution metrics positive definite. If a maximum or minimum of the difference u2−u1occurs over the interior, then these can be handled as in the smooth case. So we imagine that these extrema occur over the divisor. Since this equation is nonlinear, u2−u1is not a solution, so the first step is to find a nonlinear equation for which u2−u1is a solution. Similarly to the expos´e of Bourguignon, [B], equating the two expressions for efωngives (ω+i∂∂u1)n=eu1−u2(ω+i∂∂u1+i∂∂(u2−u1))n. Setting u=u2−u1and ω1=ω+i∂∂u1, this reads euωn 1=(ω1+i∂∂u)n. This is the nonlinear equation solved by the difference of the two solutions.