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Henri Guenancia Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie
Paris & Département de Mathématiques et Applications
École Normale Supérieure
Paris
guenancia@math.jussieu.fr
25 août 2019
Résumé

Let be a Kähler manifold and be a -divisor with simple normal crossing support and coefficients between and . Assuming that is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on having mixed Poincaré and cone singularities according to the coefficients of . As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair .

Kähler-Einstein metrics with mixed Poincaré and cone singularities]Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

\urladdr

www.math.jussieu.fr/ guenancia

Introduction

Let be a compact Kähler manifold of dimension , and an effective -divisor with simple normal crossing support such that the ’s satisfy the following inequality: . We write .

Our local model is given by the product where (resp. ) is the disc (resp. punctured disc) of radius in , the divisor being , with . We will say that a metric on has mixed Poincaré and cone growth (or singularities) along the divisor if there exists such that

where

is simply the product metric of the standard cone metric on , the Poincaré metric on , and the euclidian metric on .

This notion makes sense for global (Kähler) metrics on the manifold ; indeed, we can require that on each trivializing chart of where the pair becomes (those charts cover ), is equivalent to just like above, and this does not depend on the chosen chart.

Our goal will then be to find, whenever this is possible, Kähler metrics on having constant Ricci curvature and mixed Poincaré and cone growth along . Those metrics will naturally be called Kähler-Einstein metrics. For reasons which will appear in section 1.2 and more precisely in Remark 1.2, we will restrict ourselves to looking for Kähler-Einstein metrics with negative curvature.

The existence of Kähler-Einstein metrics (in the previously specified sense) has already been studied in various contexts and for multiple motivations. The logarithmic case (all coefficients of are equal to ) has been solved when is assumed to be ample by R. Kobayashi [Kob84] and G.Tian-S.T.Yau [TY87], the latter considering also orbifold coefficients for the fractional part of , that is of the form for some integers . Our main result extends this when the coefficients of are no longer orbifold coefficients, but are any real numbers (condition which is realized if is of orbifold type):

Theorem A.

Let be a compact Kähler manifold and a -divisor with simple normal crossing support such that is ample. We assume furthermore that the coefficients of satisfy the following inequalities:

Then carries a unique Kähler-Einstein metric with curvature having mixed Poincaré and cone singularities along .

The conic case, ie when the coefficients of are stricly less than ), under the assumption that is positive or zero, has been studied by R. Mazzeo [Maz99], T. Jeffres [Jef00] and recently resolved independently by S. Brendle [Bre11] and R. Mazzeo, T. Jeffres, Y. Rubinstein [JMR11] in the case of an (irreducible) smooth divisor, and by [CGP11] in the general case of a simple normal crossing divisor (having though all its coefficients greater than ). In the conic case where , some interesting existence results were obtained by R. Berman in [Ber11] and T. Jeffres, R. Mazzeo and Y. Rubinstein in [JMR11]. Let us finally mention that in [JMR11], it is proved that the potential of the Kähler-Einstein metric has polyhomogeneous expansion, which is much stronger than the assertion on the cone singularities of this metric.

Let us now give a sketch of the proof by detailing the organization of the paper.

The first step is, as usual, to relate the existence of Kähler-Einstein metrics to some particular Monge-Ampère equations. We explain this link in Proposition 2.2. The idea is that any negatively curved normalized Kähler-Einstein metric on with appropriate boundary conditions extends to a Kähler current of finite energy in satisfying on a Monge-Ampère equation of the type where is a Kähler form on , and . One may observe that as soon as some equals , the measure has infinite mass.

The uniqueness of the solution metric will then follow from the so-called comparison principle established by V.Guedj and A.Zeriahi for this special class of finite energy currents.

We are then reduced to solving some singular Monge-Ampère equation. The strategy consists in working on the open manifold , and we are led to the following equation: where this time is a Kähler form on with Poincaré singularities along , and . If were smooth, one could simply apply the results of Kobayashi and Tian-Yau. As it is not the case, we adapt the strategy of Campana-Guenancia-Păun to this setting:

We start in section 4.1 by regularizing into a smooth function (on ) and introducing smooth approximations of the cone metric on having Poincaré singularities along . Then we consider the regularized equation which we can solve for every (we are in the logarithmic case). The point is to construct our desired solution as the limit of ; this is made possible by controlling (among other things) the curvature of , and applying appropriate a priori laplacian estimates which we briefly explain in section 1.4. The final step is standard: it consists in invoking Evans-Krylov interior estimates, and concluding that is smooth on using Schauder estimates.

In the last part of the paper, and as in [CGP11], we try to use the Kähler-Einstein metric constructed in the previous sections to obtain the vanishing of some particular holomorphic tensors attached to a pair , being still a -divisor with simple normal crossing support and having coefficients in . This specific class consists in the holomorphic tensors which are the global sections of the locally free sheaf introduced by Campana in [Cam10]: they are holomorphic tensors with prescribed zeros or poles along . Thanks to their realization as bounded tensors with respect to some (or equivalently, any) twisted metric with mixed cone and Poincaré singularities along , given in Proposition 5 we can use Theorem A to prove the following:

Theorem B.

Let be a pair satisfying the assumptions of Theorem A. Then, there is no non-zero holomorphic tensor of type whenever :

The proof of this results follows closely the one of its analogue in [CGP11]: we use a Bochner formula applied to the truncated holomorphic tensors, and the key point is to control the error term. However, a new difficulty pops up here, namely we have to deal with an additional term coming from the curvature of the line bundle ; fortunately, it has the right sign.

Acknowledgments

I am very grateful to Sébastien Boucksom for his patient and careful reading of the preliminary versions, and his several highly valuable comments and suggestions to improve both the organization and the content of this paper.

I would like to also thank warmly Mihai Păun for the precious help he gave me to elaborate the last section of this article.

1 Preliminaries

In this first section devoted to the preliminaries, we intend to fix the notations and the scope of this paper. We also recall some useful objects introduced in [Kob84] and [TY87] within the framework of the logarithmic case; finally, we explain briefly some a priori estimates which are going to be some of our main tools in the proof of the main theorem.

1.1 Notations and definitions

All along this work, will be a compact Kähler manifold of complex dimension . We will consider effective -divisors with simple normal crossing support, and such that their coefficients belong to .

It will be practical to separate the hypersurfaces appearing with coefficient in from the other ones. For this, we write:

These notations come from the framework of the pairs in birational geometry; klt stands for Kawamata log-terminal whereas lc means log-canonical. In this language, is called a log-smooth lc pair, and is a log-smooth klt pair. Apart from these practical notations, we will not use this terminology.

We will denote by a section of whose zero locus is the (smooth) hypersurface , and, omitting the dependance in the metric, we write the curvature form of for some hermitian metric on . Up to scaling the ’s, one can assume that , and we will make this assumption all along the paper. Finally, we set and .

In the introduction, we introduced a natural class of growth of Kähler metrics near the divisor which we called metrics with mixed Poincaré and cone singularities along . They are the Kähler metrics locally equivalent to the model metric whenever the pair is locally isomorphic to with and , with .

The following elementary lemma ensures that given a pair as above, Kähler metrics with mixed Poincaré and cone singularities along always exist:

{lemm}

The following -form

defines a Kähler form on as soon as is a sufficiently positive Kähler metric on . Moreover, it has mixed Poincaré and cone singularities along .

Démonstration.

This can be seen by a simple computation: combine e.g. [Cla08, Proposition 2.1] with [CG72, Proposition 2.1] or [Gri76, Proposition 2.17]. ∎

Before we end this paragraph, we would like to emphasize the different role played by the ’s whether they appear in with coefficient or stricly less than . Here is some explanation: let be a real number, and ; its curvature is constant equal to on the punctured disc , and it has a cone singularity along the divisor . Then, when goes to , converges pointwise to the Poincaré metric .

In the following, any pair will be implicitely assumed to be composed of a compact Kähler manifold and a -divisor on having simple normal crossing support and coefficients belonging to .

1.2 Kähler-Einstein metrics for pairs

As explained in the introduction, the goal of this paper is to find a Kähler metric on with constant Ricci curvature, and having mixed Poincaré and cone singularities along the given divisor . The second condition is essential and as important as the first one; the proof of the vanishing theorem for holomorphic tensors in the last section will render an account of this and shall surely convince the reader. Let us state properly the definition:

{defi}

A Kähler-Einstein metric for a pair is defined to be a Kähler metric on satisfying the following properties:

  1. for some real number ;

  2. has mixed Poincaré and cone singularities along .

{rema}

Unlike cone singularities, Poincaré singularities are intrinsically related to negative curvature geometry:

  1. The Bonnet-Myers Theorem tells us that in the case where (so that we work with complete metrics), there cannot exist Kähler-Einstein metrics in the previous sense with . However, if , there may exist Kähler-Einstein metrics with positive curvature, and the question of their existence is often a difficult question (see e.g. [BBE] or [Ber11]).

  2. As for the Ricci-flat case (), it also has to be excluded. Indeed, there cannot be any Ricci-flat metric on the punctured disc with Poincaré singularity at ; to see this, we write such a metric, and then has to satisfy the following properties: is harmonic on and behaves like near , up to constants. But it is well-known that any harmonic function on can be written for some holomorphic function on and some constant . Clearly, cannot have an essential singularity at ; moreover, because of the logarithmic term in the Poincaré metric, can neither be bounded, nor have a pole at . This ends to show that in general (and for local reasons), there does not exist Ricci-flat Kähler-Einstein metric in the sense of the previous definition (whenever ).

For these reasons, we will focus in the following on the case of negative curvature, which we will normalize in .

1.3 The logarithmic case

For the sake of completeness, we will briefly recall in this section the proof of the main result (Theorem 3) in the logarithmic case, namely when , ie when . As we already explained, this was achieved by Kobayashi [Kob84] and Tian-Yau [TY87] in a very similar way. In this section, we will assume that is logarithmic, so that .

We will use the following terminology which is convenient for the following:

{defi}

We say that a Kähler metric on is of Carlson-Griffiths type if there exists a Kähler form on such that .

As observed in Lemma 1.1, such a metric always exists, and it has Poincaré singularities along . In [CG72], Carlson and Griffiths introduced such a metric for some . The reason why we exhibit this particular class of Kähler metric on having Poincaré singularities along is that we have an exact knowledge on its behaviour along , much more precise that its membership of the previously cited class. For example, Lemma 1.3 mirrors this fact.

We start from a compact Kähler manifold with a simple normal crossing divisor such that is ample. We want to find a Kähler metric on with , and having Poincaré singularities along . If we temporarily forget the boundary condition, the problem amounts to solve the following Monge-Ampère equation on :

where is a Kähler metric on of Carlson-Griffiths type (cf. Definition 1.3), and ) for some Kähler metric on .

The key point is that has bounded geometry at any order. Let us get a bit more into the details. To simplify the notations, we will assume that is irreducible, so that locally near a point of , is biholomorphic to , where (resp. ) is the unit disc (resp. punctured disc) of . We want to show that, roughly speaking, the components of in some appropriate coordinates have bounded derivatives at any order. The right way to formalize it consists in introducing quasi-coordinates: they are maps from an open subset to having maximal rank everywhere. So they are just locally invertible, but these maps are not injective in general.
To construct such quasi-coordinates on , we start from the univeral covering map , given by . Formally, it sends to . The idea is to restrict to some fixed ball with , and compose it (at the source) with a biholomorphism of sending to , where is a real parameter which we will take close to . If want to write a formula, we set , so that the quasi-coordinate maps are given by , with .
Once we have said this, it is easy to see that is covered by the images when goes to , and for all the trivializing charts for , which are in finite number. Now, an easy computation shows that the derivatives of the components of with respect to the ’s are bounded uniformly in . This can be thought as a consequence of the fact that the Poincaré metric is invariant by any biholomorphism of the disc.

At this point, it is natural to introduce the Hölder space of -functions on using the previously introduced quasi-coordinates: {defi} For a non-negative integer , a real number , we define:

where the supremum is taken over all our quasi-coordinate maps (which cover ). Here denotes the standard -norm for functions defined on a open subset of .

The following fact, though easy, is very important for our matter: {lemm} Let be a Carlson-Griffiths type metric on , and some Kähler metric on . Then

belongs to the space for every and .

Démonstration.

The first remark is that is bounded (cf. [Kob84, Lemma 1.(ii)] or the beginning of section 4.2.3), and if and only if . So in the following, we will deal with .
Then, as the (elementary) computations of Lemma 4.2.3 show, it is enough to check that the functions on (say with radius 1/2) defined by and are in . But in the quasi-coordinates given by , and , for with , and where . Now there is no difficulty in seeing that these two functions of are bounded when goes to (actually this property does not depend on the chosen coordinates), and so are their derivatives (still with respect to ); this is obvious for the first function, and for the second one, it relies on the fact that goes to as , for all . ∎

The end of the proof consists in showing that the Monge-Ampère equation has a unique solution for all functions with . This can be done using the continuity method in the quasi-coordinates. In particular, applying this to (cf beginning of the section), which the previous lemma allows to do, this will prove the existence of a negatively curved Kähler-Einstein metric, which is equivalent to (in the strong sense: for all ).

To summarize, the theorem of Kobayashi and Tian-Yau is the following:

{theo}

Let be a logarithmic pair, a Kähler form of Carlson-Griffiths type on , and for some . Then there exists solution to the following equation:

In particular if is ample, then there exists a (unique) Kähler-Einstein metric of curvature equivalent to .

1.4 A priori estimates

In this section, we recall the classical estimates valid for a large class of complete Kähler manifolds; they are derived from the classical estimates over compact manifolds using the generalized maximum principle of Yau [Yau78]. We will use them in an essential manner in the course of the proof of our main theorem. Indeed, our proof is based upon a regularization process, and in order to guarantee the existence of the limiting object, we need to have a control on the norms.

{theo}

Let be a complete Riemannian manifold with Ricci curvature bounded from below. Let be a function which is bounded from below on . Then for every , there exists such that at ,

From this, we easily deduce the following result, stated in [CY80, Proposition 4.1]. {prop} Let be a -dimensional complete Kähler manifold, and a bounded function. We assume that we are given satisfying and

Suppose that the bisectional curvature of is bounded below by some constant, and that is a bounded function. Then

We emphasize the fact that the previous estimate does not depend on the lower bound for the bisectional curvature of .

As for the Laplacian estimate, we have the following (we could also have used [CY80, Proposition 4.2]):

{prop}

Suppose that the bisectional curvature of is bounded below by some constant , and that as well as its Laplacian are bounded functions on . If defines a complete Kähler metric on with Ricci curvature bounded from below, then

where only depends on , , and .

Sketch of the proof.

We set , and is defined to be the Laplacian with respect to .

Using [CGP11, Lemma 2.2], we obtain , and from this we may deduce that

where are constant depending only , and . The assumptions allow us to use the generalized maximum principle stated as Theorem 1.4 to show that . As , and as we have at our disposal uniform estimates on thanks to 1.4, the usual arguments work here to give a uniform bound . We refer e.g. to [CGP11, section 2] for more details. ∎

2 Uniqueness of the Kähler-Einstein metric

In this section, we begin to investigate the questions raised in the introduction concerning the existence of Kähler-Einstein metrics for pairs . The first thing to do is, as usual, to relate the existence of theses metrics to the existence of solutions for some Monge-Ampère equations. We will be in a singular case, so we have to specify the class of -psh functions to which we are going to apply the Monge-Ampère operators. This is the aim of the few following lines, where we will recall some recent (but relatively basic) results of pluripotential theory. We refer to [GZ07] or [BEGZ] for a detailed treatment.

2.1 Energy classes for quasi-psh functions

Let be a Kähler metric on ; the class is defined to be composed of -psh functions such that their non-pluripolar Monge-Ampère has full mass (cf. [GZ07], [BEGZ]). An alternate way to apprehend those functions is to see them as the largest class where one can define as a measure which does not charge pluripolar sets. Those functions satisfy the so-called comparison principle, which we are going to use in an essential manner for the uniqueness of our Kähler-Einstein metric:

{prop}

[Comparison Principle, [GZ07]] Let . Then we have:

An important subset of is the class of functions in the class having finite -energy, namely . Every smooth (or even bounded) -psh function belongs to this class.

In order to state an useful result for us, we recall the notion of capacity attached to a compact Kähler manifold , as introduced in [GZ05], generalizing the usual capacity of Bedford-Taylor ([BT82]): for every Borel subset of , we set:

There is an useful criteria to show that some -psh function belongs to the class without checking that it has full Monge-Ampère mass, but only using the capacity decay of the sublevel sets. It appears in different papers, among which [GZ07, Lemma 5.1], [BGZ08, Proposition 2.2], [BBGZ09, Lemma 2.9]: {lemm} Let . If

then .

Now we have enough background about these objects to state and prove the result we will use in the next section. Let us first fix the notations.

Let be a Kähler manifold, and a simple normal crossing divisor. We choose sections of whose divisor is precisely , and we fix some smooth hermitian metrics on those line bundles. We can assume that , and we know that, up to scaling the metrics, one may assume that is positive on , and defines a Kähler current on . {prop} The function

belongs to the class .

Démonstration.

We want to apply Lemma 2.1. To compute the global capacity as defined above, or at least know the capacity decay of the sublevel sets, it is convenient to use the Bedford-Taylor capacity. But a result due to Kołodziej [Koł01] (see also [GZ05, Proposition 2.10]), states that up to universal multiplicative constants, the capacity can be computed by the local Bedford-Taylor capacities on the trivializing charts of .

Therefore, we are led to bound from above in the unit polydisc of , where for some . As

one can now assume that . But (see e.g [Dem, Example 13.10]), whence . The result follows. ∎

{rema}

An alternate way to proceed is to show that the smooth approximations of have (uniformly) bounded -energy, which also allows to conclude that thanks to [BEGZ, Proposition 2.10 & 2.11].

2.2 From Kähler-Einstein metrics to Monge-Ampère equations

The following proposition explains how to relate Kähler-Einstein metrics for a pair and some Monge-Ampère equations, the difficulty being here that we have to deal with singular weights/potentials for which the definitions and properties of the Monge-Ampère operators are more complicated than in the smooth case. Note that this result generalizes [Ber11, Proposition 5.1]:

{prop}

Let be a compact Kähler manifold, and an effective -divisor with simple normal crossing support, such that for all . We assume that is ample, and we choose a Kähler metric . Then any Kähler metric on satisfying:

  1. on ;

  2. There exists such that:

extends to a Kähler current on where is a solution of

and for some . Furthermore there exists at most one such metric on .

{rema}

One can observe that although has finite mass, does not (as soon as .

Démonstration.

We recall that denotes the curvature of , and we write , and . All those forms are smooth on .

Let us define a smooth function on by:

By assumption, is bounded on , so that is bounded above on . On this set, we have

so that is -psh for some big enough. As it is bounded above, it extends to a (unique) -psh function on the whole , which we will also denote by . Let now be a smooth potential on of . It is easily shown that satisfies on .

From the definition of , we see that , where . Therefore, Proposition 2.1 ensures that , so that its non-pluripolar Monge-Ampère satisfies the equation

on the whole , with the notations of the statement. By the comparison principle (Proposition 2.1), if the previous equation had two solutions , then on the set , we would have

but on , so that has zero measure with repect to the measure , so it has zero measure with respect to . We can do the same for , so that has full measure with respect to . As are -psh, they are determined by their data almost everywhere, so they are equal on . This finishes to conclude that our is unique, so that the proposition is proved. ∎

{rema}

In the logarithmic case (), the metrics at stake are complete, so that their uniqueness follow from the generalized maximum principle of Yau (cf. [Kob84], [TY87] e.g). In the conic case, Kołodziej’s theorem [Koł98] ensures that the potentials we are dealing with are continuous, and the unicity follows from the classical comparison principle established in [BT82, Theorem 4.1].

As Kähler metrics with mixed Poincaré and cone singularities clearly satisfy the second condition of the proposition, we deduce that any negatively curved normalized Kähler-Einstein metric must be obtained by solving the global equation on , for , and for some . We will now show how to solve the previous equation, and derive from this the existence of negatively curved Kähler-Einstein metrics and their zero-th order asymptotic along .

3 Statement of the main result

Here is a result which encompasses the previous results of [CGP11], Kobayashi ([Kob84]) and Tian-Yau ([TY87]). This provides a (positive) partial answer to a question raised in [CGP11, section 10].

{theo}

Let be a compact Kähler manifold, and an effective -divisor with simple normal crossing support such that its coefficients satisfy the inequalities:

Then for any Kähler form on of Carlson-Griffiths type and any function with , there exists a Kähler metric on solution to the following equation:

such that has mixed Poincaré and cone singularities along .

We refer to section 1.3 and more precisely to Definition 1.3 for the definition of the space ; one important class of functions belonging to is pointed out in Lemma 1.3, and we will use it for proving the following result.

{coro}

Let be a pair such that is a divisor with simple normal crossing support whose coefficients satisfy the inequalities

If is ample, then carries a unique Kähler-Einstein metric of curvature having mixed Poincaré and cone singularities along .

Here, by ample, we mean that contains a Kähler metric, or equivalently that is a positive combination of ample -divisors.

Démonstration.

We choose and some smooth hermitian metrics on the line bundles and respectively such that the product metric on has positive curvature , and up to renormalizing the metrics , one can assume that defines a Kähler metric on with Poincaré singularities along ; more precisely it is of Carlson-Griffiths type.
Lemma 1.3 shows that one can write

with the smooth volume form on attached to (in particular , the curvature of ), and for all and .
Now we use Theorem 3 with , and as reference metric. We then get a Kähler metric on with mixed Poincaré and cone singularities along satisfying

Therefore,

Moreover, has mixed Poincaré and cone singularities along , so it is a Kähler-Einstein metric for the pair .
As for the uniqueness of , it follows directly from Proposition 2.2. ∎

4 Proof of the main result

As we explained in the introduction, the natural strategy is to combine the approaches of [CGP11] and Kobayashi ([Kob84]). More precisely we will produce a sequence of Kähler metrics on having Poincaré singularities along and acquiring cone singularities along at the end of the process when .

4.1 The approximation process

We keep the notation of Theorem 3, so that is a Kähler form on of Carlson-Griffiths type; in particular it has Poincaré singularities along .
We define, for any sufficiently small , a Kähler form on by

where for functions defined by:

for any . The important facts to remember about this construction are the following ones, extracted from [CGP11, section 3]:

  1. For big enough, dominates (as a current) a Kähler form on because already does;

  2. is uniformly bounded (on ) in ;

  3. When goes to , converges on to having mixed Poincaré and cone singularities along .

As is a Kähler metric on with Poincaré singularities along , the case treated by Kobayashi ([Kob84]) and Tian-Yau ([TY87]), cf section 1.3, Theorem 1.3, enables us to find a smooth -psh function on satisfying:

(1)

where