Analytic test configurations and geodesic rays
aa r X i v : . [ m a t h . AG ] J u l Analytic test configurations and geodesic rays
Julius Ross and David Witt Nystr¨om5 June 2013
Abstract
Starting with the data of a curve of singularity types, we use the Legendretransform to construct weak geodesic rays in the space of locally bounded metricson an ample line bundle L over a compact manifold. Using this we associate weakgeodesics to suitable filtrations of the algebra of sections of L . In particular thisworks for the natural filtration coming from an algebraic test configuration, and weshow how this recovers the weak geodesic ray of Phong-Sturm. Let H ( L ) be the space of smooth strictly positive hermitian metrics on an ample holo-morphic line bundle L over a compact manifold X . Then, by the work of Mabuchi[31], Semmes [43] and Donaldson [23], formally H ( L ) has the structure of an infinitedimensional symmetric space with a canonical Riemannian metric. Thus a natural wayto study this space is through its geodesics, an approach that has been taken up bya number of authors (e.g. Berndtsson [11], Chen-Tian [18], Donaldson, Phong-Sturm[34, 35], Mabuchi, and Semmes among others).In this paper we give a general method for constructing weak geodesic rays inthe space of locally bounded positive metrics on L . In the following we shall, in thestandard way, identify a (positive) metric h on L with its (plurisubharmonic) potential φ = log h . Our initial data consists of a fixed smooth plurisubharmonic potential φ and a curve of singular plurisubharmonic potentials ψ λ on L for λ ∈ R . We are reallyonly interested in the singularity type of ψ λ , so we consider the equivalence class of ψ λ under the relation ψ λ ∼ ψ ′ λ if ψ λ − ψ ′ λ is bounded globally on X . We define the maximal envelope of this data to be φ λ := sup ∗ { ψ : ψ ≤ φ and ψ ∼ ψ λ } (1)where the supremum is over positive metrics ψ with the same singularity type as ψ λ ,and the star denotes the operation of taking the upper-semicontinuous regularization.This equivalence relation on potentials was considered by Demailly-Peternell-Schn-eider [22] and is relevant to metrics with minimal singularities. The envelope in (1) wasstudied in a special case by Berman [6], and is the global analogue of a constructionof Rashkovskii-Sigurdsson [32]. Observe that if ψ λ is itself locally bounded then φ is a candidate for this envelope implying φ λ = φ , and thus we are most interested in1 INTRODUCTION 2the case that ψ λ is singular along some non-trivial subset of X . When ψ λ has analyticsingularities, it can be shown that φ λ is the largest plurisubharmonic potential boundedabove by φ with the same singularity type as ψ λ (see Remark 4.6).We shall call a curve ψ λ of plurisubharmonic potentials a test curve if it is concavein λ , locally bounded for λ sufficiently negative and identically −∞ for λ sufficientlylarge (we also make one further technical condition concerning the kind of singularityallowed, see Definition 5.1). A justification of this terminology will be given below,and as a simple example to have in mind, suppose that s is a holomorphic section of L scaled so | s | φ ≤ and let ψ λ = φ λ < − λ ) φ + λ ln | s | ≤ λ < −∞ λ ≥ , (2)so ψ λ is decreasing and concave in λ . Geometrically then φ λ = φ for λ ≤ and for λ ∈ (0 , it is the largest plurisubharmonic potential bounded above by φ with Lelongnumber at least λ along the divisor Y = { s = 0 } . Theorem 1.1.
Let ψ λ be a test curve and φ be a smooth positive potential, and considerthe Legendre transform of its maximal envelope φ λ given by b φ t := sup ∗ λ { φ λ + λt } for t ∈ [0 , ∞ ) . Then b φ t is a weak geodesic ray in the space of locally bounded plurisubharmonicpotentials on L that emanates from φ . Looking back at (2) it is clear that specific examples of test curves are easy toproduce, and thus this theorem gives a new construction of many weak geodesic raysemanating from φ . In fact, as we shall observe in Remark 6.4, essentially every weakgeodesic ray can be produced in this way.We recall precisely what is meant by a weak geodesic. Let I ⊂ R be an intervaland consider the annulus A := { w ∈ C : − ln | w | ∈ I } and π : X × A → X be theprojection. Then a curve of plurisubharmonic potentials φ t for t ∈ I can be identifiedwith a rotation invariant potential Φ( x, w ) := φ − log | w | on π ∗ L . A simple calculation[43, 2.3,2.4] reveals that if φ t is a smooth curve of smooth potentials then the geodesicequation for φ t is equivalent to the homogeneous complex Monge-Amp`ere equation M A (Φ) := ( dd c Φ) n +1 = 0 on X × A ◦ . (3)A curve ( φ t ) t ∈ I of locally bounded (not necessarily smooth) plurisubharmonic poten-tials given by Φ = Φ( x, w ) is said to be a weak geodesic if dd c Φ is positive and solves(3) in the sense of currents. When I = [0 , ∞ ) (so A is the punctured unit disc) we saythat φ t is a weak geodesic ray .The first step in our approach to Theorem 1.1 is showing that the Monge-Amp`eremeasure of the maximal envelope φ λ satisfies M A ( φ λ ) = { φ λ = φ } M A ( φ λ ) for all λ, (4) INTRODUCTION 3where S denotes the characteristic function of a set S . We say that a positive metric φ λ bounded by φ and having property (4) is maximal with respect to φ (see Definition4.5), and a test curve φ λ where φ λ is maximal with respect to φ for all λ is referred toas a maximal test curve . We shall show that the Legendre transform b φ t of a maximaltest curve φ λ is a subgeodesic, and also that Aubin-Mabuchi energy is linear in t , whichtogether imply that it is a weak geodesic (in the smooth case this is well known andgoes back to Mabuchi [30] and more generally can be deduced from a Theorem ofBerman-Boucksom-Guedj-Zeriahi, see Lemma 3.11).The well known Yau-Tian-Donaldson conjecture states that for a smooth projectivemanifold it should be possible to detect the existence of a constant scalar curvatureK¨ahler metric algebraically. Through ideas developed by many authors (e.g. Chen,Donaldson, Mabuchi, Tian) a general picture has emerged in which such metrics appearas critical points of certain energy functionals that are convex along smooth geodesics.The input from algebraic geometry arises through notion of an (algebraic) test con-figuration, originally due to Tian [46] and then extended by Donaldson [24], which,roughly speaking, is a one-parameter algebraic degeneration of the original projectivemanifold X .In a series of papers, Phong-Sturm show how one can naturally associate a weakgeodesic ray to a test configuration [34, 35, 37] (see also [2, 15, 16, 17, 44, 45] forother constructions of geodesic rays related to test configurations). We show how thegeodesic of Phong-Sturm can be viewed as a particular case of the Legendre transformconstruction. For example, applying the Legendre transform construction to the ex-ample of the test curve (2) given above recovers the geodesic of Phong-Sturm that isassociated to the test configuration given by the degeneration to the normal cone of thedivisor Y .Generalizing slightly, suppose that F k,λ , for k ∈ N , λ ∈ R is a multiplicative filtra-tion of the graded algebra ⊕ k H ( X, kL ) . Using the underlying smooth positive metric φ and some auxiliary volume form we have an L -inner product on each H ( X, kL ) ,and thus can consider the associated Bergman metric φ k,λ = 1 k ln X α | s α | where { s α } is an orthonormal basis for F k,λk ⊂ H ( X, kL ) . So, for example, if Y isa divisor on X we can let F k,λ = { s ∈ H ( kL ) : ord Y ( s ) ≥ λ } be the multiplicativefiltration given by order of vanishing along Y , and then φ k,λ is the potential associatedto the partial Bergman kernel coming from sections of kL that vanish to at least order kλ along Y . Theorem 1.2.
Suppose that F k,λ is left continuous and decreasing in λ and bounded(see (7.2) ). Then there is a well-defined limit φ F λ = lim ∗ k →∞ φ k,λ where the star denotes taking the upper semicontinuous regularization after the limit.Furthermore this limit is maximal except possibly for one critical value of λ , and itsLegendre transform is a weak geodesic ray. INTRODUCTION 4In particular this applies to a natural filtration associated to a test configuration,and thus we have associated a weak geodesic to any such test configuration. We provethat, in this case, we recover precisely Phong-Sturm geodesic, thus reproving the mainresult of [35]. Hence one interpretation of Theorem 1.1 is that in the problem of findingweak geodesic rays, the algebraic data of a test configuration can be replaced with asuitable concave curve of singularity types which we thus refer to as an analytic testconfiguration .The relationship between a algebraic and analytic test configurations is analogousto the relationship between holomorphic and plurisubharmonic functions. Being givenby a concave curve of singularity types, analytic test configurations are easier to pro-duce and manipulate. For example it is possible to interpolate between two analytictest configurations merely by taking the line between them, whereas it is not so clearwhat the analogous construction is for algebraic test configurations. With regard to theYau-Tian-Donaldson conjecture it is now expected that the notion of algebraic test con-figuration needs to be extended in some way for it to detect a constant scalar curvaturemetric (see the examples of Apostolov-Calderbank-Gauduchon-Friedman [1] and alsowork of Sz´ekelyhidi [38] which considers general filtrations of the ring of sections).Thus analytic test configurations provide a context for such extensions.Interesting examples of analytic test configurations that do not correspond to alge-braic ones appear for example in the setting of Arakelov geometry. In [12] Boucksom-Chen construct multiplicative filtrations of the section ring which encodes arithmeticproperties of the sections such as their adelic norm. This filtration then gives rise to ananalytic test configuration whose geometrical significance remains unclear.Another advantage of the notion of analytical test configurations is that they can bedefined independent of the polarization, and even in the non-projective case (althoughwe will not consider that further here). If we pick an arbitrary K¨ahler form ω we canconsider a concave curve of singularity types [ ψ λ ] of ω -plurisubharmonic functions.Then choosing a polarization L we get an analytic test configuration [ φ λ ] by letting φ λ be the envelope of all positive singular metrics of L bounded from above by φ + ψ λ , where φ is some locally bounded metric of L. In fact, rather than choosing apolarization we could have picked an arbitrary big (1 , -coholomogy class and workedwith that instead.It should be stressed that in the problem of finding constant scalar curvature met-rics it is important to have control of the regularity of geodesics under consideration.By using approximations to known regularity results of solutions of Monge-Amp`ereequations, Phong-Sturm prove that their weak geodesic is in fact C ,α for < α < (see [37]). It is interesting to ask whether such regularity holds more generally, whichis a topic we take up in [40]. Organization:
We start in Section 2 with some motivation from convex analysisand toric geometry, and Section 3 contains preliminary material on the space of sin-gular metrics, the Monge-Amp`ere measure and the Aubin-Mabuchi functional. Thereal work starts in Section 4 where we consider the maximal envelopes associated toa given singularity type. Along the way we prove a generalization of a theorem of CONVEXMOTIVATION 5Bedford-Taylor which says that such envelopes are maximal (Theorem 4.10). This isthen extended to the case of a test curve of singularities, and in Section 6 we discussthe Legendre transform and prove Theorem 1.1.Following these analytic results, we move on to the algebraic picture. In Section7 we associate a test curve to a suitable filtration of the coordinate ring of ( X, L ) , andprove Theorem 1.2. We then recall how such filtrations arise from test configurations,and in Section 9 show how this agrees with the construction of Phong and Sturm. Acknowledgments : We would like to thank Robert Berman, Bo Berndtsson, Sebas-tian Boucksom, Yanir Rubinstein and Richard Thomas for helpful discussions and thereferees for their constructive suggestions. We also thank Dano Kim for pointing out amistake in a previous version of this paper. The first author also acknowledges supportfrom a Marie Curie International Reintegration Grant within the 7 th European Commu-nity Framework Programme (PIRG-GA-2008-230920).
This section contains some motivation from convex analysis in the study of the ho-mogeneous Monge-Amp`ere equation. Much of this material is standard; our mainreferences are the two papers [41] and [42] by Rubinstein-Zelditch. We shall presentlysee how solutions to this equation can be found using the Legendre transform in twodifferent, but ultimately equivalent, ways. Although this is logically independent of therest of the paper, the techniques used are similar and give an illustration in the toricsetting.Let
Conv( R n ) denote the space of convex functions on R n . We take the conventionthat the function identically equal to −∞ is in Conv( R n ) . Definition 2.1.
Let φ be a C convex function on an open subset of R n . The (real)Monge-Amp`ere measure of φ, denoted by M A ( φ ) , is the Borel measure defined as M A ( φ ) := d ∂φ∂x ∧ ... ∧ d ∂φ∂x n +1 . The operator
M A has an unique extension to a continuous operator on the cone of(finite-valued) convex functions (see [42] for references). If φ is C then M A ( φ ) = det ( ∇ φ ) dx = ( ∇ φ ) ∗ dx, (5)i.e. the Monge-Amp`ere measure is the pullback of the Lebesgue measure under thegradient map.If φ ∈ Conv( R n ) , we say y is a subgradient of φ at x if the function x φ ( x ) − φ ( x ) − y · ( x − x ) is bounded from below, and we let ∆ φ = { y : y is a subgradient for some x } be the set of all subgradients of φ . So, if φ is differentiable then ∆ φ is simply the imageof ∇ φ . One can easily check that ∆ φ is convex, that if r > then ∆ rφ = ∆ φ and ∆ φ + ψ ⊆ ∆ φ + ∆ ψ . CONVEXMOTIVATION 6When φ is C it follows from equation (5) that the total mass of the Monge-Amp`eremeasure M A ( φ ) equals the Lebesgue volume of ∆ φ . An important fact [42] is that thisis true for all convex functions on R n with linear growth, i.e. Z R n M A ( φ ) = vol (∆ φ ) . (6)We say two convex functions φ and ψ are equivalent if | φ − ψ | is bounded, anddenote this by φ ∼ ψ . Since for two equivalent convex functions φ and ψ with lineargrowth we clearly have that ∆ φ = ∆ ψ , it follows from (6) that Z R n M A ( φ ) = Z R n M A ( ψ ) whenever φ ∼ ψ. Definition 2.2.
Let φ ∈ Conv( R n ) and let ˙ φ be a bounded continuous function on R n . A curve φ t in Conv( R n ) , t ∈ [ a, b ] , is said to solve the Cauchy problem for thehomogeneous real Monge-Amp`ere equation, abbreviated as HRMA, with initial data ( φ, ˙ φ ) , if the function Φ( x, t ) := φ t ( x ) is convex on R n × [ a, b ] , and satisfies theequation M A (Φ) = 0 on the strip R n × ( a, b ) , with initial data φ = φ, ∂∂t | t =0 + φ t = ˙ φ. Remark 2.3.
This convex geometry has particular geometric significance when ∆ isthe moment polytope of a polarised toric manifold ( X, L ) . Then, through the notion ofsymplectic potentials, there is a correspondence between hermitian metrics on L andconvex functions on ∆ (where due care is to be taken to ensure that a given convexfunction determines a smooth or positive metric on L ) and the Cauchy problem for theHRMA translates to the Cauchy problem for finding geodesics in the space of hermitianmetrics on L . We refer the reader to [41, 42] for a detailed discussion of this idea.Now let φ and φ be two equivalent convex functions with linear growth, and φ t bethe affine curve between them. The energy of φ relative to φ , denoted by E ( φ , φ ) is defined as E ( φ , φ ) := Z t =0 (cid:18)Z R n ( φ − φ ) M A ( φ t ) (cid:19) dt. We observe that by the linear growth assumption it follows that the relative energy E ( φ , φ ) is finite. This energy has a cocycle property, namely if φ , φ and φ areequivalent with finite energy then E ( φ , φ ) = E ( φ , φ ) + E ( φ , φ ) , which is easily seen to be equivalent to the fact that ∂∂t E ( φ t , φ ) = Z R n ∂∂t φ t M A ( φ t ) . CONVEXMOTIVATION 7The energy along a smooth curve φ t of convex functions with linear growth is relatedto the Monge-Amp`ere measure of Φ( x, t ) := φ t ( x ) by the identity Z R n × [ a,b ] M A (Φ) = ∂∂t | t = b E ( φ t , φ a ) − ∂∂t | t = a E ( φ t , φ a ) . (7)Thus a smooth curve φ t of equivalent convex functions of linear growth solves theHRMA equation if and only if Φ is convex and the energy E ( φ t , φ a ) is linear in t .As is noted in [42] this Cauchy problem is not always solvable. Nevertheless thereis a standard way to produce solutions φ t with t ∈ [0 , ∞ ) to the homogeneous Monge-Amp`ere equation with given starting point φ = φ using the Legendre transform. Wegive a brief account of this.For simplicity assume from now on that φ is differentiable and strictly convex.Recall that the Legendre transform of φ is the function on ∆ φ defined as φ ∗ ( y ) := sup x { x · y − φ ( x ) } (which we can also think of as being defined on the whole of R n , by being −∞ outsideof ∆ φ ). Since φ ∗ is defined as the supremum of the linear functions x · y − φ ( x ) , it isconvex. In fact, one can show that φ being differentiable and strictly convex essentiallyimplies that φ ∗ is also differentiable and strictly convex (see [33, Theorem 1] for aprecise statement that requires a further boundary condition).For a given y ∈ ∆ φ , the function x · y − φ ( x ) is strictly concave, and is maximizedat the point where the gradient is zero. Thus φ ∗ ( y ) = x · y − φ ( x ) where ∇ φ ( x ) = y, (8)and hence ∇ φ ∗ ( y ) = x where ∇ φ ( x ) = y. The Legendre transform is an involution. For using the above formula ∇ φ ∗∗ ( x ) = y for x such that ∇ φ ∗ ( y ) = x which holds when ∇ φ ( x ) = y, i.e. ∇ φ ∗∗ ( x ) = ∇ φ ( x ) . If ∇ φ ( x ) = y , then φ ∗ ( y ) = x · y − φ ( x ) , therefore φ ∗∗ ( x ) = x · y − φ ∗ ( y ) = x · y − ( x · y − φ ( x )) = φ ( x ) , as claimed. Lemma 2.4. If φ t is a curve of convex functions, then for any point y ∈ ∆ φ t ∂∂t φ ∗ t ( y ) = − ∂∂t φ t ( x ) , where x is the point such that ∇ φ ( x ) = y . CONVEXMOTIVATION 8
Proof.
Let x t be the solution to the equation ∇ φ t ( x t ) = y . By the implicit functiontheorem x t varies smoothly with t . By equation (8) we know ∂∂t φ ∗ t ( y ) = ∂∂t ( x t · y − φ t ( x )) = ∂∂t ( x t · y − φ ( x )) − ∂∂t φ t ( x ) . Since x t · y − φ ( x ) is maximized at x = x the derivative of that part vanishes at t = 0 , so we get the lemma for t = 0 , and similarly for all t .This leads us to the following formula relating the energy with the Legendre trans-form, Lemma 2.5.
We have E ( φ t , φ ) = Z ∆ φ ( φ ∗ − φ ∗ t ) dy. (9) Proof.
We noted above that the derivative with respect to t of the left-hand side of (9)is equal to Z R n ∂∂t φ t M A ( φ t ) . On the other hand, differentiating the right-hand side yields ∂∂t Z ∆ φ ( φ ∗ − φ ∗ t ) dy = − Z ∆ φ ∂∂t φ ∗ t dy = Z ∆ φ ∂∂t φ t ( ∇ φ − t ( y )) dy = Z R n ∂∂t φ t ( ∇ φ t ) ∗ dy = Z R n ∂∂t φ t M A ( φ t ) , where we used Lemma 2.4 and the fact that the pullback ( ∇ φ t ) ∗ dy of the Lebesguemeasure is M A ( φ t ) . Since both sides of the equation (9) is zero when φ t = φ and thederivatives coincide, we get that they must be equal for all t .Now fix a smooth bounded strictly concave function u on ∆ φ and let ˜ φ t := ( φ ∗ − tu ) ∗ . (10) Proposition 2.6.
The curve ˜ φ t , t ∈ [0 , ∞ ) solves the HRMA equation with initial data ˜ φ = φ and ∂∂t | t =0 + ˜ φ t = u (( ∇ φ ) − ) . To see this note that from (9) it follows that E ( ˜ φ t , φ ) = Z ∆ φ ( φ ∗ − ˜ φ ∗ t ) dy = Z ∆ φ ( φ ∗ − φ ∗ + tu ) dy = t Z ∆ φ udy, which is linear in t , and the initial conditions follow from (10) and Lemma 2.4. Theconvexity of ˜Φ( t, x ) = ˜ φ t ( x ) can of course be shown directly, but it also followsfrom another characterization of ˜ φ t that also involves a Legendre transform, but in the t -coordinate instead of in the x -coordinates which we now discuss.Let A λ be the subset of ∆ φ where u is greater than or equal to λ and let φ λ bedefined as φ λ := sup { ψ ≤ φ : ψ ∈ Conv( R n ) , ∆ ψ ⊆ A λ } . (11) CONVEXMOTIVATION 9 Lemma 2.7.
The curve of functions φ λ is concave in λ and { φ λ = φ } = { x : ∇ φ ( x ) ∈ A λ } . Proof.
Let ψ i ≤ φ be such that ∆ ψ i ⊆ A λ i with i = 1 , . Let < t < . From ourdiscussion above it follows that tψ + (1 − t ) ψ ≤ φ and ∆ tψ +(1 − t ) ψ ⊆ t ∆ ψ + (1 − t )∆ ψ ⊆ tA λ + (1 − t ) A λ ⊆ A tλ +(1 − t ) λ , where the last inclusion follows from the fact that u was assumed to be concave. Forthe second statement, it is easy to see that in fact φ λ is equal to the supremum of affinefunctions x · y + C bounded by φ and y lying in A λ . Definition 2.8.
For t ≥ let b φ t be defined as b φ t := sup λ { φ λ + tλ } . Since for each λ the function ( x, t ) φ λ ( x ) + tλ is convex in all its variables, andthe supremum of convex functions is convex, b Φ( x, t ) := b φ t ( x ) is convex. Proposition 2.9.
Recalling that ˜ φ t = ( φ ∗ − tu ) ∗ we have ˜ φ t = b φ t . In particular thisproves that ˜Φ is convex, thereby proving ˜ φ t solves the HRMA equation (Proposition2.6).Proof. We claim ∂∂t b φ t ( x ) = u ( ∇ b φ t ( x )) . (12)To see this first consider the right-derivative at t = 0 . As we noted above, the gradientof a Legendre transform is the point where the maximum is attained, thus in this case ∂∂t | t =0 + b φ t ( x ) = sup { λ : φ λ ( x ) = φ ( x ) } . By the second statement in Lemma 2.7 it follows that this supremum is equal to u ( ∇ φ ( x )) , and we are done for t = 0 . On the other hand it is easy to see that b φ t + t = b ψ t , with ψ := b φ t . Using this, the equation (12) holds for all t . Thus by Lemma 2.4the Legendre transform of b φ t is equal to φ − tu, so by the involution property of theLegendre transform b φ t coincides with ˜ φ t .Now the above discussion can be reformulated as follows. Let ψ λ be a curve in Conv( R n ) . We say that ψ λ is a test curve if there is a C such that1. ψ λ is concave in λ ψ λ − φ bounded for λ < − C and3. ψ λ ≡ −∞ for λ > C . CONVEXMOTIVATION 10Given such a test curve let u be the function on ∆ φ defined by u ( y ) := sup { λ : y ∈ ∆ ψ λ } . and observe that since ψ λ is assumed to be concave the function u is also concave.Thus our definition of φ λ above (11) is φ λ = sup { ψ ≤ φ : ψ ∈ Conv( R n ) , ∆ ψ ⊆ A λ } = sup { ψ ≤ φ : ψ ∈ Conv( R n ) , ∆ ψ ⊆ { u ≥ λ }} . (13)Hence we in fact have φ λ := sup { ψ : ψ ≤ φ, ψ ≤ ψ λ + o (1) , ψ ∈ Conv( R n ) } . (14)From Proposition 2.9, b φ t solves the homogeneous real Monge-Amp`ere equation.Thus in order to get solutions to the HRMA, instead of starting with a concave function u on ∆ φ we can just as well start with a test curve ψ λ . It is this second reformulation(14) that extends more naturally to the context of positive metrics on line bundles. Remark 2.10.
This convex picture can be given some geometric context by consid-ering toric manifolds (compare [5, Example 5.2]). Consider a complex toric manifold ( X, L ) of dimension n , so L carries a compatible torus invariant hermitian metric φ .Pick complex coordinates z , . . . , z n on a dense complex torus T = ( C ∗ ) n ⊂ X andlet t i = ln | z i | so we consider t , . . . , t n as coordinates on R n . Then φ ( z ) descendsto a convex function on R n which, by abuse of notation, we denote by φ ( t ) . Con-versely any function φ on R n induces a metric on L | T , which will be (strictly) positiveas long as φ is (strictly) convex (one must make some additional hypothesis as to thebehaviour of φ ( t ) at infinity if one wishes to ensure that this induced metric extends toa smooth or locally bounded metric over all of X , but we will not consider that furtherhere).As an illustration, consider the simplest case of dimension 1 so X = P . Fix asmooth strictly positive metric on L and assume the strictly convex potential φ ( t ) on R satisfies φ ′ (0) = 0 and lim t →∞ φ ′ ( t ) = 1 . Then, for λ ∈ (0 , define ψ λ ( t ) = (cid:26) λt t ≥ t ≤ (this can be made into a test curve by extending it to be φ + C ′ for negative λ and to be −∞ for λ ≥ similar to the example (2) in the introduction). Now for λ ∈ (0 , let t λ be the point where φ ′ ( t λ ) = λ . Then one sees from (13) that φ λ = φ for t ≤ t λ and φ λ = λ ( t − t λ ) + φ ( t λ ) for t ≥ t λ . One can check that φ λ defines a singular metric on L with the same singularity type as ψ λ . Observe that φ λ is linear for t > t λ making thismetric pluriharmonic (and thus maximal) over this set, and it is this maximal propertythat will be crucial in the generalisation that follows. PRELIMINARYMATERIAL 11 We collect here some preliminary material on the space of positive metrics, the (nonpluripolar) Monge-Amp`ere measure and the Aubin-Mabuchi energy functional. Mostof this material is standard, and we give proofs only for those results for which we didnot find a convenient reference.
Let X be a complex projective manifold of complex dimension n, and let L be anample line bundle on X . We start with some preliminaries on singular metrics, forwhich a convenient reference is [19]. A continuous (or smooth) hermitian metric on L is a continuous (or smooth) choice of scalar product on the complex line L p at eachpoint p on the manifold. If f is a local holomorphic frame for L on U f , then one writes | f | h = h f = e − φ f , where φ f is a continuous (or smooth) function on U f . We will use the convention to let φ denote the metric h = e − φ , thus if φ is a metric on L, kφ is a metric on kL := L ⊗ k .The curvature of a smooth metric is given by dd c φ which is the (1 , -form locallydefined as dd c φ f , where f is any local holomorphic frame. Here d c is short-hand forthe differential operator i π ( ¯ ∂ − ∂ ) , so dd c = i/π∂ ¯ ∂ . A classic fact is that the curvature form of a smooth metric φ isa representative for the first Chern class of c ( L ) . The metric φ is said to be strictlypositive if the curvature dd c φ is strictly positive as a (1 , -form, i.e. if for any localholomorphic frame f, the function φ f is strictly plurisubharmonic. We let H ( L ) denotethe space of smooth strictly positive (i.e. locally strictly plurisubharmonic) metrics on L, which is non-empty since we assumed that L was ample.A positive singular metric ψ is a metric that can be written as ψ := φ + u, where φ is a smooth metric and u is a globally defined dd c φ -psh function, i.e. u is uppersemicontinuous and dd c ψ := dd c φ + dd c u is a positive (1 , -current. For conveniencewe also allow u ≡ −∞ . We let P SH ( L ) denote the space of positive singular metricson L .As an important example, if { s i } is a finite collection of holomorphic sections of kL, we get a positive metric ψ := k ln( P | s i | ) which is defined by letting for anylocal frame f , e − ψ f := | f | ( P | s i | ) /k . We note that
P SH ( L ) is a convex set, since any convex combination of positivemetrics yields a positive metric. Another important fact is that if ψ i ∈ P SH ( L ) forall i ∈ I are uniformly bounded above by some fixed positive metric, then the uppersemicontinuous regularization of the supremum denoted by sup ∗ { ψ i : i ∈ I } lies in P SH ( L ) as well. Similarly the upper semicontinuous regularization of the pointwiselimit lim i ψ i (when defined) will be denote by lim ∗ i ψ i . PRELIMINARYMATERIAL 12If ψ is in P SH ( L ) , then the translate ψ + c where c is a real constant is also in P SH ( L ) . For any ψ ∈ P SH ( L ) , dd c ψ is a closed positive (1 , -current, and fromthe dd c lemma it follows that any closed positive current cohomologous with dd c ψ can be written as dd c φ for some φ in P SH ( L ) . By the maximum principle this φ isuniquely determined up to translation.If there exists a constant C such that ψ ≤ φ + C, we say that ψ is more singularthan φ, and we will write this as ψ (cid:22) φ. If both ψ (cid:23) φ and φ (cid:23) ψ we say that ψ and φ are equivalent , which we write as ψ ∼ φ . Following [13] an equivalence class [ ψ ] is called a singularity type , and we usethe notation Sing ( L ) for the set of singularity types. If ψ is equivalent to an elementin H ( L ) we say that ψ is locally bounded .The singularity locus of a positive metric ψ is the set where ψ is minus infinity,i.e. the set where ψ f = −∞ when f is a local frame and the unbounded locus of ψ is the set where ψ is not locally bounded. Recall that a set is said to be completepluripolar if it is locally the singularity locus of a plurisubharmonic function, while it iscalled pluripolar if you only have a local inclusion in the singularity set. Pluripolar setshave zero measure with respect to any smooth volume form (since this is true locallywith respect to the Lebesgue measure [29, Corollary 2.9.10]). In [13] Boucksom-Eyssidieux-Guedj-Zeriahi give the following definition. Definition 3.1.
A positive metric ψ is said to have small unbounded locus if its un-bounded locus is contained in a closed complete pluripolar subset of X .We note that metrics of the form k ln( P | s i | ) have small unbounded locus, sincethey are locally bounded away from the algebraic set ∪ i { s i = 0 } which is a closedpluripolar set. If f is a plurisubharmonic function on an open subset U of C n then using a convolu-tion we can write f as the limit of a decreasing sequence of smooth plurisubharmonicfunctions on any relatively compact subset of U [29, Theorem 2.9.2].If ψ is a positive singular metric, we can use a partition of unity with respect tosome open cover U f i to patch together the smooth decreasing approximations of ψ f i .Thus any positive singular metric can be written as the pointwise limit of a decreas-ing sequence of smooth metrics, but of course because of the patching these smoothapproximations will in general not be positive.A fundamental result due to Demailly [20] is that any positive singular metric canbe approximated by metrics of the form k − ln( P i | s i | ) , where s i are sections of kL .Let I ( ψ ) denote the multiplier ideal sheaf of germs of holomorphic functions locallyintegrable against e − ψ f dV, where f is a local frame for L and dV is an arbitrary volumeform. We get a scalar product ( ., . ) kψ on the space H ( kL ⊗ I ( kψ )) by letting k s k kψ := Z X | s | e − kψ dV. PRELIMINARYMATERIAL 13Let { s i } be an orthonormal basis for H ( kL ⊗ I ( kψ )) and set ψ k := 1 k ln( X | s i | ) . Theorem 3.2.
The sequence of metrics ψ k converge pointwise to ψ as k tends to infin-ity, and there exists a constant C such that for large k,ψ ≤ ψ k + Ck .
As a reference see [21, (A4)] (note in fact the results of Demailly are in fact muchstronger than that stated here, and hold in greater generality [20]). When ψ is assumedto be smooth and strictly positive, a celebrated result by Bouche-Catlin-Tian-Zelditch[10, 14, 45, 50] on Bergman kernel asymptotics implies that the ψ k in fact converge to ψ in any C m norm.Using a variation of this construction, Guedj-Zeriahi prove [27, Theorem 7.1] thatany positive singular metric on an ample line bundle over a compact X is the pointwiselimit of a decreasing sequence of smooth positive metrics. Let ψ i , ≤ i ≤ n, be an n-tuple of positive metrics, so for each i, dd c ψ i is a positive (1 , -current. If all ψ i are smooth one can consider the wedge product dd c ψ ∧ ... ∧ dd c ψ n , (15)which is a positive measure on X . The fundamental work of Bedford-Taylor shows thatone can still take the wedge product of positive currents dd c ψ i to get a positive measureas long as the metrics ψ i are all locally bounded. The Monge-Amp`ere measure of alocally bounded positive metric ψ, is then defined as the positive measure M A ( ψ ) := ( dd c ψ ) n . This measure does not put any mass on pluripolar sets. We recall the following impor-tant continuity property, proved in [3].
Theorem 3.3 (Bedford-Taylor) . If ψ i,k , ≤ i ≤ n + 2 , k ∈ N , are sequences oflocally bounded positive metrics such that each ψ i,k decreases to a locally boundedpositive metric ψ i , then the signed measures ( ψ ,k − ψ ,k ) dd c ψ ,k ∧ ... ∧ dd c ψ n +2 ,k converge weakly to ( ψ − ψ ) dd c ψ ∧ ... ∧ dd c ψ n +2 . If each sequence of locallybounded positive metrics ψ i,k instead increase pointwise a.e. to a positive metric ψ i , then again the measures ( ψ ,k − ψ ,k ) dd c ψ ,k ∧ ... ∧ dd c ψ n +2 ,k converge weakly to ( ψ − ψ ) dd c ψ ∧ ... ∧ dd c ψ n +2 . Since the curvature form dd c φ of any smooth metric φ is a representative of c ( L ) ,we see that if φ i is any n -tuple of smooth metrics then Z X dd c φ ∧ ... ∧ dd c φ n = Z X c ( L ) n (16) PRELIMINARYMATERIAL 14which is just a topological invariant of L . Since any positive metric can be approxi-mated from above in the manner of Theorem 3.3 by positive metrics that are smooth,we see that (16) still holds if the φ i are merely assumed to be locally bounded insteadof smooth.Our proof that maximal envelopes are maximal (Theorem 4.10) is based on anapproximation argument that requires some technical results concerning convergenceof plurisubharmonic functions. Recall that a plurisubharmonic function is, by defi-nition, upper semicontinuous, so if ψ is a positive metric then for each local frame f the function ψ f is upper semicontinuous. The plurifine topology is defined as thecoarsest topology in which all local plurisubharmonic functions are continuous; a ba-sis for this topology is given by sets of the form A ∩ { u > } , where A is open inthe standard topology and u is a local plurisubharmonic function. This topology hasthe quasi-Lindel¨of property [4, Thm 2.7], meaning that an arbitrary union of plurifineopen sets differs from a countable subunion by at most a pluripolar set. Any basis set A ∩ { u > } is Borel, so it follows from the quasi-Lindel¨of property that the plurifineopen (and closed) sets lie in the completion of the Borel σ -algebra with respect to anyMonge-Amp`ere measure [4, Prop 3.1]. Definition 3.4.
A function f is said to be quasi-continuous on a set Ω if for every ǫ > there exists an open set U with capacity less than ǫ so that f is continuous on Ω \ U .We refer to [3, (1.3)] for the definition of capacity, and in [3, Thm 3.5] it is shownthat plurisubharmonic functions are quasi-continuous.If f k is a sequence of non-negative continuous functions increasing to the character-istic function of an open set A then the characteristic function of a basis set A ∩{ u > } is the increasing limit of the non-negative quasi-continuous functions kf k (max { u, } − max { u − /k, } ) . From this fact and the quasi-Lindel¨of property it follows that the characteristic func-tion of any plurifine open set differs from an increasing limit of non-negative quasi-continuous functions at most on a pluripolar set.A fundamental property of the Bedford-Taylor product is that it is local in the plu-rifine topology, so if ψ i = ψ ′ i for all i on some plurifine open set O then O dd c ψ ∧ ... ∧ dd c ψ n = O dd c ψ ′ ∧ ... ∧ dd c ψ ′ n , where O denotes the characteristic function of O . We also have that the convergencein Theorem 3.3 is local in this topology [4, Thm 3.2], i.e. we get convergence whentesting against bounded quasi-continuous functions. Lemma 3.5.
Let ψ k be a sequence of locally bounded positive metrics that decreasespointwise (or increases a.e.) to a locally bounded positive metric ψ, and let O be aplurifine open set. Then O M A ( ψ ) ≤ lim inf k →∞ O M A ( ψ k ) , where the lim inf is to be understood in the weak sense, i.e. when testing against non-negative continuous functions. PRELIMINARYMATERIAL 15
Proof.
Let u i be a sequence of quasi-continuous functions increasing to O excepton a pluripolar set. Let f be a non-negative continuous function. Since u i M A ( ψ k ) converges weakly to u i M A ( ψ ) , and M A ( ψ k ) does not put any mass on a pluripolarset, Z X f u i M A ( ψ ) = lim k →∞ Z X f u i M A ( ψ k ) ≤ lim inf k →∞ Z O f M A ( ψ k ) . (17)Now u i increases to the characteristic function of O except possibly on a pluripolar set,so letting i tend to infinity in (17) yields Z O f M A ( ψ ) ≤ lim inf k →∞ Z O f M A ( ψ k ) . For singular ψ i there is a (non pluripolar) product constructed by Boucksom-Eyssi-dieux-Guedj-Zeriahi [13], building on a local construction due to Bedford-Taylor [4].Fix a locally bounded metric φ , and consider the auxiliary metrics ψ i,k := max { ψ i , φ − k } for k ∈ N , and the sets O k := T i { ψ i > φ − k } . The non-pluripolar product of thecurrents dd c ψ i , here denoted by dd c ψ ∧ ... ∧ dd c ψ n is defined as the limit dd c ψ ∧ ... ∧ dd c ψ n := lim k →∞ O k dd c ψ ,k ∧ ... ∧ dd c ψ n,k . Since we are assuming that X is compact this limit is well defined [13, Prop. 1.6].The (non-pluripolar) Monge-Amp`ere measure of a positive metric is ψ is defined as M A ( ψ ) := ( dd c ψ ) n . Essentially by construction, the non-pluripolar product is localin the plurifine topology [13, Prop. 1.4], and is multilinear [13, Prop 4.4].Clearly from the definition and (16), for any n -tuple of positive metrics ψ i on L , Z X dd c ψ ∧ ... ∧ dd c ψ n ≤ Z X c ( L ) n , however the inequality may well be strict.Combining Lemma 3.5 with the fact that the Monge-Amp`ere measure is local inthe plurifine topology yields the following continuity result. Lemma 3.6.
Let ψ k be a sequence of positive metrics decreasing to a positive metric ψ, and let φ be some locally bounded positive metric. If O is a plurifine open setcontained in { ψ > φ − C } for some constant C then O M A ( ψ ) ≤ lim inf k →∞ O M A ( ψ k ) , (18) where again the lim inf is to be understood in the weak sense. If ψ k instead is increas-ing a.e. to ψ, and O is a plurifine open set contained in { ψ j > φ − C } for some naturalnumber j and some constant C then once again O M A ( ψ ) ≤ lim inf k →∞ O M A ( ψ k ) . PRELIMINARYMATERIAL 16
Proof.
First assume that ψ k is decreasing to ψ . Let ψ ′ k := max { ψ k , φ − C } and ψ ′ := max { ψ, φ − C } . From Lemma 3.5 it follows that O M A ( ψ ′ ) ≤ lim inf k →∞ O M A ( ψ ′ k ) , and since by assumption ψ ′ = ψ and ψ ′ k = ψ k on O the lemma follows from thelocality of the non-pluripolar product. The case where ψ k is increasing a.e. follows bythe same reasoning.In [13, Thm 1.16] it is shown that the non-pluripolar product, when restricted tometrics with small unbounded locus, has the following monotonicity property. Theorem 3.7.
Let ψ i , ψ ′ i be two n -tuples of positive metrics with small unboundedlocus, and suppose that for all i, ψ i is more singular than ψ ′ i . Then Z X dd c ψ ∧ ... ∧ dd c ψ n ≤ Z X dd c ψ ′ ∧ ... ∧ dd c ψ ′ n . There is also a comparison principle for metrics with small unbounded locus [13,Cor 2.3] and a domination principle [13, Cor 2.5]. When combined with the compar-ison principle, the proof of the domination principle in [13] in fact yields a slightlystronger version:
Theorem 3.8.
Let φ be a positive metric with small unbounded locus and suppose thatthere exists a positive metric ρ, more singular than φ , with small unbounded locus andsuch that M A ( ρ ) dominates a volume form. If ψ is a positive metric more singularthan φ and such that ψ ≤ φ a.e. with respect to M A ( φ ) , then it follows that ψ ≤ φ onthe whole of X . The
Aubin-Mabuchi energy bifunctional maps any pair of equivalent positive metrics ψ and ψ to the number E ( ψ , ψ ) := 1 n + 1 n X i =0 Z X ( ψ − ψ )( dd c ψ ) i ∧ ( dd c ψ ) n − i . Observe E ( ψ + t, ψ ) = t Z X M A ( ψ ) . The Aubin-Mabuchi energy restricted to the class of locally bounded metrics has acocycle property (see, for example, [7, Cor 4.2]), namely if φ , φ and φ are locallybounded equivalent metrics then E ( φ , φ ) = E ( φ , φ ) + E ( φ , φ ) . In fact the proof in [7] of the cocycle property extends to the case where the equiv-alent metrics are only assumed to have small unbounded locus, since the integration-by-parts formula of [13] used in the proof holds in that case. ENVELOPESANDMAXIMALMETRICS 17This leads to an important monotonicity property. If ψ , ψ and ψ are equivalentwith small unbounded locus, and ψ ≥ ψ , then E ( ψ , ψ ) ≥ E ( ψ , ψ ) since E ( ψ , ψ ) = E ( ψ , ψ ) + E ( ψ , ψ ) , and E ( ψ , ψ ) ≥ as it is the integral ofthe positive function ψ − ψ against a positive measure.We also record the following lemma, which comes from the locality of the non-pluripolar product in the plurifine topology. Lemma 3.9.
Let ψ ∼ ψ be such that ψ ≥ ψ . Let ψ ′ and ψ ′ be two other metricssuch that ψ ′ ∼ ψ ′ and assume that { ψ ′ = ψ ′ } = { ψ = ψ } and that ψ ′ = ψ and ψ ′ = ψ on the set where ψ > ψ . Then E ( ψ ′ , ψ ′ ) = E ( ψ , ψ ) . Following Phong-Sturm in [34] we can relate weak geodesics to the energy func-tional. Let I ⊂ R be an interval and consider the annulus A := { w ∈ C : − ln | w | ∈ I } and π : X × A → X be the projection. Definition 3.10.
A curve of positive metrics φ t , for t ∈ I is said to be a weak sub-geodesic if there exists a locally bounded positive metric Φ on π ∗ L that is rotationinvariant and whose restriction to X × { w } equals φ − ln | w | . The curve φ t is said tobe a weak geodesic if it is a weak subgeodesic and furthermore Φ solves the HCMAequation, i.e. M A (Φ) = 0 on X × A ◦ . A weak geodesic φ t defined for ≤ t < ∞ will be called a weak geodesicray .As in the convex setting (7) there is a formula [8, Proposition 6.2] relating theAubin-Mabuchi energy of a locally bounded subgeodesic φ t with the Monge-Amp`eremeasure of Φ , namely dd ct E ( φ t , φ a ) = π ∗ ( M A (Φ)) , (19)where π ∗ ( M A (Φ)) denotes the push-forward of the measure
M A (Φ) with respect tothe projection π . From this we immediately get the following lemma. Lemma 3.11.
A curve φ t of locally bounded positive metrics defined for t ∈ [0 , ∞ ) is a weak geodesic ray if and only if it is a subgeodesic and for any a ∈ (0 , ∞ ) theAubin-Mabuchi energy E ( φ t , φ a ) is linear in t . In studying the Dirichlet problem for the HCMA equation it is often possible to givea solution as an envelope in some space of plurisubharmonic functions (or positivemetrics). Such envelopes will be crucial in our setting as well. ENVELOPESANDMAXIMALMETRICS 18
Definition 4.1. If φ is a continuous metric, not necessarily positive, let P φ denote theenvelope
P φ := sup { ψ ≤ φ, ψ ∈ P SH ( L ) } . Since φ is assumed to be continuous it follows that the upper semicontinuous regular-ization of P φ is bounded from above by φ , and hence also by P φ . Thus
P φ is itselfupper semicontinuous and so
P φ ∈ P SH ( L ) .The next theorem is essentially just a reformulation of a local result of Bedford-Taylor [3, Corollary 9.2] in our global setting. It follows as a special case of [7, Propo-sition 1.10] (letting K = X ). Theorem 4.2. If φ is a continuous metric then P φ = φ a.e. with respect to M A ( P φ ) . Recall that if A is a closed set and µ is a Borel measure we say that µ is said tobe concentrated on A if A µ = µ, or equivalently µ ( A c ) = 0 . Thus another way offormulating Theorem 4.2 is to say that M A ( P φ ) is concentrated on { P φ = φ } . Wenow extend this result to more general envelopes that arise from the additional data ofsingularity type. Definition 4.3.
Given a positive metric ψ ∈ P SH ( L ) let P ψ denote the projectionoperator on P SH ( L ) defined by P ψ φ := sup { ψ ′ ≤ min { φ, ψ } , ψ ′ ∈ P SH ( L ) } . We also define P [ ψ ] by P [ ψ ] φ := lim C →∞ P ψ + C φ = sup { ψ ′ ≤ φ, ψ ′ ∼ ψ, ψ ′ ∈ P SH ( L ) } . Clearly P ψ φ is monotone with respect to both ψ and φ . Since min { φ, ψ } is uppersemicontinuous, it follows that the upper semicontinuous regularization of P ψ φ is stillless than min { φ, ψ } , and thus P ψ φ ∈ P SH ( L ) . By this it follows that P ψ ( P ψ φ ) = P ψ φ, i.e. that P ψ is indeed a projection operator on P SH ( L ) . One also notes that theupper semicontinuous regularization of P [ ψ ] φ, lies in P SH ( L ) and is bounded by φ . Definition 4.4.
The maximal envelope of φ with respect to the singularity type [ ψ ] isdefined to be φ [ ψ ] := usc( P [ ψ ] φ ) where usc denotes the process of taking the upper-semicontinuous regularization. Definition 4.5. If ψ ∈ P SH ( L ) , then ψ is said to be maximal with respect to a metric φ if ψ ≤ φ and furthermore ψ = φ a.e. with respect to M A ( ψ ) . Similarly, if A is ameasurable set, we say that ψ is maximal with respect to φ on A if ψ ≤ φ and ψ = φ a.e. on A with respect to M A ( ψ ) . Remark 4.6.
The terminology is justified by a proof below that the maximal envelopeof a continuous metric φ is maximal with respect to φ . As it is defined as a limit, itis not clear from the definition if φ [ ψ ] is equivalent to ψ (this can be shown if ψ hasalgebraic singularities by passing to a suitable resolution, and we refer the reader to[40] for a further study of maximal envelopes). For this reason, the method in the proofof Theorem 4.2 in [7] does not directly apply, so instead we will use an approximationargument. ENVELOPESANDMAXIMALMETRICS 19Our use of the word maximal is motivated by the following property: Proposition 4.7.
Let ψ be maximal with respect to a metric φ . Suppose also thatthere exists a positive metric ρ (cid:23) ψ with small unbounded locus and such that M A ( ρ ) dominates a volume form. Then for any ψ ′ ∼ ψ with ψ ≤ φ we have ψ ′ ≤ ψ .Proof. Since ψ ′ ≤ φ , the maximality assumption yields ψ ′ ≤ ψ a.e. with respectto M A ( ψ ) , so the proposition thus follows from the domination principle (Theorem3.8).The next two lemmas are the main steps in showing that maximal envelopes aremaximal. Lemma 4.8.
Let ψ k be a sequence of positive metrics increasing a.e. to a positivemetric ψ, and assume that all ψ k are maximal with respect to a fixed continuous metric φ on some plurifine open set O . Then ψ is maximal with respect to φ on O .Proof. Since φ was assumed to be continuous, ψ ≤ φ . Now, for all k { ψ k = φ } ⊆ { ψ = φ } and thus by the the maximality of ψ k , we know O M A ( ψ k ) is concentrated on { ψ = φ } . Since ψ ≤ φ we have that { ψ = φ } = { ψ ≥ φ } , and since φ is continuous this isa closed set. Let C be a constant. The set O ∩ { ψ > φ − C } is plurifinely open, so byLemma 3.6 it follows that O { ψ >φ − C } M A ( ψ ) ≤ lim inf k →∞ O { ψ >φ − C } M A ( ψ k ) . (20)It is easy to see that if µ k is a sequence of measures all concentrated on a closed set A ,and µ ≤ lim inf k →∞ µ k in the weak sense, then µ is also concentrated on A . It thus follows from (20) that O { ψ >φ − C } M A ( ψ ) is concentrated on { ψ = φ } . Since M A ( ψ ) puts no mass onthe pluripolar set { ψ = −∞} the lemma follows by letting C tend to infinity. Lemma 4.9.
Let ψ ∈ P SH ( L ) and let φ be a continuous metric. Then the envelope P ψ φ is maximal with respect to φ on the plurifine open set { ψ > φ } .Proof. Clearly P ψ φ ≤ φ , so we have to show P ψ φ is equal to φ almost everywhereon { ψ > φ } with respect to M A ( P ψ φ ) . Let φ k be a sequence of continuous met-rics decreasing pointwise to min { φ, ψ } , so that φ k ≤ φ for all k and φ k = φ on theset { ψ > φ } . For example let φ k := min { φ, ψ k } where ψ k is a sequence of smoothmetrics decreasing pointwise to ψ . From Theorem 4.2 it follows that M A ( P φ k ) is con-centrated on { P φ k = φ k } , and since φ k = φ when ψ > φ we see { ψ>φ } M A ( P φ k ) is concentrated on { P φ k = φ } . Now P φ k is decreasing in k and lim k →∞ P φ k ≤ min { φ, ψ } . At the same time, for any k ∈ N we clearly have that P ψ φ ≤ P φ k , whichtaken together means that lim k →∞ P φ k = P ψ φ. ENVELOPESANDMAXIMALMETRICS 20Since
P φ k ≤ φ this implies that { P φ k = φ } is decreasing in k and { P ψ φ = φ } = \ k ∈ Z { P φ k = φ } . (21)Let O denote the plurifine open set { ψ > φ } ∩ { P ψ φ > φ − C } . By Lemma 3.6 weknow O M A ( P ψ φ ) ≤ lim inf k →∞ O M A ( P φ k ) , and thus we conclude that O M A ( P ψ φ ) is concentrated on { P φ k = φ } for any k, soby (21) we get that O M A ( P ψ φ ) is concentrated on { P ψ φ = φ } . Since M A ( P ψ φ ) puts no mass on the pluripolar set { P ψ φ = −∞} , letting C tend to infinity yields thelemma. Theorem 4.10.
Let ψ ∈ P SH ( L ) and let φ be a continuous metric. Then φ [ ψ ] ismaximal with respect to φ, i.e. φ [ ψ ] = φ a.e. with respect to M A ( φ [ ψ ] ) .Proof. P [ ψ ] φ = φ [ ψ ] a.e., and since P ψ + C φ increases to P [ ψ ] φ, it thus increases to φ [ ψ ] a.e.. By Lemma 4.9 we get that P ψ + C φ is maximal with respect to φ on the plurifineopen set { ψ > φ − C } and thus also on any set { ψ > φ − C ′ } whenever C ′ ≤ C . FromLemma 4.8 it thus follows that φ [ ψ ] is maximal with respect to φ on the set { ψ > φ − C } for any C . Since M A ( φ [ ψ ] ) puts no mass on { ψ = −∞} the theorem follows. Example 4.11.
Consider the case that s is a section of rL that vanishes along a divisor D , and set ψ = r ln | s | . Then the maximal envelope φ [ ψ ] is considered by Berman [6,Sec. 4], and equals sup ∗ { ψ ′ ≤ φ : ψ ′ ∈ P SH ( L ) , ν D ( ψ ′ ) ≥ } where ν D denotes the Lelong number along D . This metric governs the Bergmankernel asymptotics of sections of kL for k ≫ that vanish along the divisor D . Themore general case when ψ has analytic singularities is also considered in [6].The maximal property gives the following bounds on the energy functional whichwill be crucial for our construction of weak geodesics (Theorem 6.8). Proposition 4.12.
Suppose that ψ is maximal with respect to a positive metric φ withsmall unbounded locus, and let t > . Then t Z X M A ( ψ ) ≤ E (max { ψ + t, φ } , φ ) ≤ t Z X M A ( φ ) . (22) Proof.
Since by assumption ψ ≤ φ we know max { ψ + t, φ } ≤ φ + t, so from themonotonicity of the Aubin-Mabuchi energy it follows that E (max { ψ + t, φ } , φ ) ≤ E ( φ + t, φ ) = t Z X M A ( φ ) which gives the upper bound. For the lower bound, first choose an ǫ with < ǫ < t .Again by monotonicity, E (max { ψ + t, φ } , φ ) ≥ E (max { ψ + t, φ } , max { ψ + ǫ, φ } ) . (23) TESTCURVESANDANALYTICTESTCONFIGURATIONS 21Now clearly E (max { ψ + t, φ } , max { ψ + ǫ, φ } ) ≥ ( t − ǫ ) Z { ψ + ǫ>φ } M A ( ψ ) . (24)By the assumption that ψ is maximal with respect to φ Z { ψ = φ } M A ( ψ ) = Z X M A ( ψ ) and since { ψ = φ } ⊆ { ψ + ǫ > φ } , the combination of (23) and (24) yields E (max { ψ + t, φ } , φ ) ≥ ( t − ǫ ) Z X M A ( ψ ) . Since ǫ > was chosen arbitrarily the lower bound in (22) follows. Definition 5.1.
A map λ ψ λ from R to P SH ( L ) is called a test curve if there is aconstant C such that(i) ψ λ is equal to some locally bounded positive metric ψ −∞ for λ < − C ,(ii) ψ λ ≡ −∞ for λ > C ,(iii) ψ λ has small unbounded locus whenever ψ λ
6≡ −∞ , and(iiii) ψ λ is concave in λ .Observe that since ψ λ is concave and constant for λ sufficiently negative it is de-creasing in λ . The set of test curves forms a convex set, by letting ( X r i γ i )( λ ) := X r i γ i ( λ ) . It is also clear that any translate γ a ( λ ) := γ ( λ − a ) of a test curve γ is a new test curve.We introduce the notation λ c for the critical value of a test curve defined as λ c := inf { λ : ψ λ ≡ −∞} . For later use we record here two continuity properties of test curves.
Lemma 5.2.
1. A test curve ψ λ is left-continuous in λ as long as λ < λ c .2. Suppose that λ < λ c and λ k is a decreasing sequence that tends to λ . Then lim ∗ k →∞ ψ λ k = ψ λ . (25) (So a a test curve is right continuous modulo taking an upper semicontinuousregularization.) TESTCURVESANDANALYTICTESTCONFIGURATIONS 22
Proof.
For (1), let λ k increase to some λ < λ c , so we need to show that lim k →∞ ψ λ k = ψ λ . By our hypothesis there exists a λ ′ such that λ < λ ′ < λ c , and thus ψ λ ′
6≡ −∞ . Since ψ λ ( x ) is concave in λ it is continuous for all x such that ψ λ ′ ( x ) = −∞ . Thus ψ λ k converges to ψ λ at least away from a pluripolar set, i.e. a.e. with respect to a volumeform. On the other hand we have that ψ λ k is decreasing in k, so the limit is a positivemetric. Now if two positive metrics coincide a.e. with respect to a volume form it fol-lows that they are equal everywhere, because this is true locally for plurisubharmonicfunction [29, Corollary 2.9.8].The proof of (2) is essentially the same. This time λ k is a decreasing sequence, soas λ < λ c we may as well assume that each λ k < λ ′ and so in particular ψ λ k
6≡ −∞ .Then the ψ λ k form an increasing sequence so the left hand side of (25) is a positivemetric. But for the same reason as above, the limit lim k →∞ ψ λ k equals ψ λ away froma pluripolar set, and thus the left and right hand side of (25) agree a.e. with respect toa volume form, and thus are equal everywhere. Definition 5.3.
A map γ from R to Sing ( L ) is called an analytic test configuration if itis the composition of a test curve with the natural projection of P SH ( L ) to Sing ( L ) .As with the set of test curves, the set of analytic test configurations is convex. Wenow extend the definition of the maximal envelope (Definition 4.4) to test curves. Definition 5.4.
Let ψ λ be a test curve and φ an element in H ( L ) . The maximal enve-lope of φ with respect to ψ λ is the map λ φ λ := φ [ ψ λ ] = usc( P [ ψ λ ] φ ) . It is easy to see that φ λ only depends on φ and the analytic test configuration [ ψ λ ] , since if ψ ′ λ ∼ ψ λ we trivially have φ [ ψ λ ] = φ [ ψ ′ λ ] . Observe also that since ψ −∞ islocally bounded, we have φ λ = φ for λ < − C . Definition 5.5.
We say that a test curve ψ λ is maximal if for all λ the metric ψ λ ismaximal with respect to ψ −∞ .Since ψ λ is decreasing in λ, { ψ λ ′ = ψ λ } ⊇ { ψ λ ′ = ψ −∞ } if λ ≤ λ ′ . It follows that if ψ λ is a maximal test curve, ψ λ ′ is maximal with respect to ψ λ when-ever λ ≤ λ ′ . We shall show in the next section how the Legendre transform of amaximal test curve gives a weak geodesic ray, and end this section by showing howmaximal envelopes give rise to maximal test curves. Proposition 5.6.
The maximal envelope φ λ is a maximal test curve. THELEGENDRETRANSFORMANDGEODESICRAYS 23
Proof.
We first show it is a test curve. Pick a real number C . Let λ and λ ′ be two realnumbers, and let ≤ t ≤ . By the concavity of ψ λ , tP ψ λ + C φ + (1 − t ) P ψ λ ′ + C φ ≤ tψ λ + (1 − t ) ψ λ ′ + C ≤ ψ tλ +(1 − t ) λ ′ + C. Thus from the definition of the projection operator, tP ψ λ + C φ + (1 − t ) P ψ λ ′ + C φ ≤ P ψ tλ +(1 − t ) λ ′ + C φ, which means that P ψ λ + C φ is concave in λ for all C . Since P ψ λ + C φ increases to P [ ψ λ ] φ, and an increasing sequence of concave functions is concave, we get that P [ ψ λ ] φ is concave, and because of the monotonicity of the upper semicontinuous regularizationit follows that usc( P [ ψ λ ] φ ) = φ λ also is concave. The other properties of a test curveare immediate.Clearly φ −∞ = φ, so that φ λ is maximal follows from Theorem 4.10. If f is a convex function in the real variable λ, the set of subgradients of f, denoted by ∆ f , is the set of t ∈ R such that f ( λ ) − tλ is bounded from below. If f happens to bedifferentiable, then the set subgradients coincides with the image of the derivative of f . By convexity of f , the set of subgradients is convex, i.e. an interval. Recall that theLegendre transform of f, here denoted by b f , is the function on ∆ f defined as b f ( t ) := sup λ { tλ − f ( λ ) } . Since b f is defined as the supremum of the linear functions tλ − f ( λ ) , it follows that b f is convex.If f is concave then of course − f is convex, and one can define the Legendretransform of f, also denoted by b f , as the Legendre transform of − f, i.e. b f ( t ) := sup λ { f ( λ ) + tλ } , which is thus convex. Definition 6.1.
The
Legendre transform of a test curve ψ λ is b ψ t := sup ∗ λ ∈ R { ψ λ + tλ } , where t ∈ [0 , ∞ ) .Recall that the star denotes the operation of taking the upper semicontinuous regu-larization. Lemma 6.2.
Let ψ λ be any test curve (not necessarily maximal). Then the Legendretransform b ψ t is locally bounded for all t , and the map t b ψ t is a subgeodesic rayemanating from ψ −∞ . THELEGENDRETRANSFORMANDGEODESICRAYS 24
Proof.
By assumption, for some λ, ψ λ is locally bounded, and trivially b ψ t ≥ ψ λ + tλ, thus b ψ t is locally bounded. It is clear that for a fixed λ, the curve ψ λ + tλ is asubgeodesic. Clearly sup λ ∈ R { ψ λ + tλ } is convex and Lipschitz in t, and the sameis easily seen to hold for b ψ t . Thus b ψ t is upper semicontinuous in the directions in X and also Lipschitz in t, which implies that it is upper semicontinuous on the product X × R ≥ . Therefore b ψ t (thought of as a function on the product) coincides with theupper semicontinuous regularization of of sup λ ∈ R { ψ λ + tλ } .Now, taking the upper semicontinuous regularization of the supremum of sub-geodesics yields a subgeodesic, as long as it is bounded from above. We observedabove that ψ λ ≤ ψ −∞ . Now for some constant C , ψ C ≡ −∞ . It follows that b ψ t ≤ ψ −∞ + tC, so it is bounded from above and thus it is a subgeodesic.Finally by definition b ψ = sup ∗ λ { ψ λ } , which clearly is equal to ψ −∞ since ψ λ ≤ ψ −∞ ( ψ λ being decreasing in λ ) and ψ −∞ itself being upper-semicontinuous.One can also consider the inverse Legendre transform, going from subgeodesic raysto concave curves of positive metrics. Definition 6.3.
The Legendre transform of a subgeodesic ray φ t , t ∈ [0 , ∞ ) , denotedby b φ λ , λ ∈ R , is defined as b φ λ := inf t ∈ [0 , ∞ ) { φ t − tλ } . Remark 6.4.
It follows from Kiselman’s minimum principle (see [28]) that for any λ ∈ R , b φ λ is a positive metric (we would like to thank Bo Berndtsson for this observation).Furthermore it is clear that b φ λ is concave and decreasing in λ . From the involutionproperty of the (real) Legendre transform it follows that the Legendre transform of b φ λ is φ t , thus any subgeodesic ray is the Legendre transform of a concave curve of positivemetrics.The goal of this section is to prove that if ψ λ is an maximal test curve then theLegendre transform b ψ t of ψ λ is a weak geodesic ray emanating from ψ −∞ . By Lemma6.2 we know b ψ t is a subgeodesic ray emanating from ψ −∞ . What remains then is toshow that if ψ λ is maximal then the Aubin-Mabuchi energy E ( b ψ t , b ψ ) is linear in t ,which we now do with an approximation argument.For N ∈ N consider the approximation b ψ Nt to b ψ t , given by b ψ Nt := sup k ∈ Z { ψ k − N + tk − N } . Since ψ λ is concave it is continuous in λ at all points such that ψ λ ( x ) > −∞ . From thecontinuity it follows that b ψ Nt will increase pointwise to b ψ t a.e. as N tends to infinity.Also let b ψ N,Mt denote the curve b ψ N,Mt := sup k ∈ Z ,k ≤ M { ψ k − N + tk − N } , and observe that b ψ Nt and b ψ N,Mt are all locally bounded. THELEGENDRETRANSFORMANDGEODESICRAYS 25
Lemma 6.5.
Let
M < M ′ be two integers. Then b ψ N,M ′ t = ψ M ′ − N + tM ′ − N implies that b ψ N,Mt = ψ M − N + tM − N . Proof.
Certainly f ( λ ) := ψ λ ( x ) + tλ is concave in λ . If b ψ N,Mt > ψ M − N + tM − N at x, then f would be strictly decreasing at λ = M − N , so by concavity we would getthat f ( M ′ − N ) < f ( M − N ) < b ψ N,Mt ( x ) , which would be a contradiction. Lemma 6.6. If ψ λ is a maximal test curve then t − N Z X M A ( ψ ( M +1)2 − N ) ≤ E ( b ψ N,M +1 t , b ψ N,Mt ) ≤ t − N Z X M A ( ψ M − N ) . Proof.
By Lemma 6.5 it follows that b ψ N,Mt = ψ M − N + tM − N on the support of b ψ N,M +1 t − b ψ N,Mt and thus Lemma 3.9 yields E ( b ψ N,M +1 t , b ψ N,Mt ) = E (max { ψ M − N , ψ ( M +1)2 − N + t − N } , ψ M − N ) . (26)Since we assumed that ψ λ is maximal, ψ ( M +1)2 − N is maximal with respect to ψ M − N , and thus the lemma follows immediately from Lemma 4.12.Let ψ λ be a maximal test curve, and let F ( λ ) denote the function F ( λ ) := Z X M A ( ψ λ ) . Whenever λ < λ ′ , ψ λ ′ ≤ ψ λ and therefore it follows from Theorem 3.7 that F ( λ ) isdecreasing in λ, hence F ( λ ) is Riemann integrable. Proposition 6.7. If ψ λ is a maximal test curve then E ( b ψ t , b ψ ) = − t Z ∞ λ = −∞ λdF ( λ ) . (27) Proof.
Suppose first m ∈ Z is such that ψ m = ψ −∞ . For a given N ∈ N set M = m N . Then b ψ N,Mt = ψ −∞ + tm = b ψ + tm. By repeatedly using the cocycle property of the Aubin-Mabuchi energy in combi-nation with Lemma 6.6 we get that t X k>M − N F (( k + 1)2 − N ) ≤ E ( b ψ Nt , b ψ N,Mt ) ≤ t X k>M − N F ( k − N ) . (28) THELEGENDRETRANSFORMANDGEODESICRAYS 26We noted above that b ψ Nt increases pointwise to b ψ t a.e. as N tends to infinity. Bythe continuity of the Aubin-Mabuchi energy under a.e. pointwise increasing sequences(3.6), E ( b ψ t , b ψ + tm ) = t Z ∞ λ = m λF ( λ ) dλ, since both the left- and the right-hand side of (28) converges to this. Again using thecocycle property we get that E ( b ψ t , b ψ ) = E ( b ψ t , b ψ + tm ) + E ( b ψ + tm, b ψ ) == t Z ∞ λ = m λF ( λ ) dλ + tm Z X M A ( ψ −∞ ) = t Z ∞ λ = m F ( λ ) dλ + tmF ( m ) . (29)Since by our assumption the measure dF is zero on ( −∞ , m ) , integration by partsyields − t Z ∞ λ = −∞ λdF ( λ ) = − tλF ( λ ) | ∞ m + t Z ∞ λ = m F ( λ ) dλ == tmF ( m ) + t Z ∞ λ = m F ( λ ) dλ. (30)The proposition follows from combining equation (29) and equation (30). Theorem 6.8.
The Legendre transform b ψ t of a maximal test curve ψ λ is a weakgeodesic ray emanating from ψ −∞ .Proof. That b ψ t is a subgeodesic emanating from ψ −∞ was proved in Lemma 6.2. Ac-cording to Proposition 6.7 the energy E ( b ψ t , b ψ ) is linear in t, and therefore by Lemma3.11 we get that b ψ t is a weak geodesic ray.These weak geodesics are continuous in φ in the following sense: Proposition 6.9.
Let ψ λ be a test curve and φ, φ ′ ∈ H ( L ) . Suppose φ λ is the maximalcurve of φ (with respect to ψ λ ) and similarly for φ ′ λ . If k φ − φ ′ k ∞ < C then k b φ t − b φ ′ t k ∞ < C for all t. Proof.
We claim that k φ λ − φ ′ λ k ∞ < C for all λ . But this is clear since φ ≤ φ ′ impliesthat φ λ ≤ φ ′ λ for all λ . It is also clear that ( φ + C ) λ = φ λ + C when C is a constant.Thus b φ t ≤ b φ ′ t for all t and ( φ + C ) λ = φ λ + C , so consequently \ φ + C t = b φ t + C which proves the lemma.Let [ ψ λ ] be an analytic test configuration, and let φ λ be the associated maximal testcurve. Then [ φ λ ] defines a new analytic test configuration. This could possibly differfrom [ ψ λ ] , but the following proposition tells us that the associated geodesic rays arethe same. Proposition 6.10.
Let φ ′ ∈ H ( L ) . Then the Legendre transform of φ ′ [ φ λ ] coincideswith the Legendre transform of φ ′ λ := φ ′ [ ψ λ ] . FILTRATIONSOFTHERINGOFSECTIONS 27
Proof.
Since φ ′ λ ∼ φ λ we get that φ ′ [ φ λ ] = φ ′ [ φ ′ λ ] , thus without loss of generality wecan assume that φ ′ = φ . Recall that the critical value λ c was defined as λ c := inf { λ : φ λ ≡ −∞} . If λ < λ c there exists a λ ′ such that λ < λ ′ < λ c , and thus by the assumption φ λ ′ has small unbounded locus. Let C be a constant less than λ such that φ C = φ . Byconcavity it follows that φ λ ≥ rφ + (1 − r ) φ λ ′ , where < r < , is chosen such that λ = rC + (1 − r ) λ ′ . If we let ρ := rφ + (1 − r ) φ λ ′ , by the multilinearity of the Monge-Amp´ere operator it follows that M A ( ρ ) dominatesthe volume form r n M A ( φ ) . Furthermore ρ has small unbounded locus and is moresingular than φ λ . Thus by Proposition 4.7 we have P φ λ + C φ ≤ φ λ for any constant C and therefore φ [ φ λ ] = φ λ , (31)whenever λ < λ c . If λ > λ c then clearly equation (31) holds as well since both sidesare identically equal to minus infinity. It follows that for any ǫ > ,φ λ ≤ φ [ φ λ ] ≤ φ λ − ǫ , which implies that d ( φ λ ) t ≤ \ ( φ [ φ λ ] ) t ≤ \ ( φ λ − ǫ ) t = d ( φ λ ) t + ǫt. Since ǫ > was arbitrary the proposition follows. First we recall what is meant by a filtration of a graded algebra.
Definition 7.1. A filtration F of a graded algebra ⊕ k V k is a vector space-valued mapfrom R × N , F : ( t, k ) t V k , such that for any k, F t V k is a family of subspaces of V k that is decreasing and left-continuous in t . FILTRATIONSOFTHERINGOFSECTIONS 28In [12] Boucksom-Chen consider certain filtrations which behaves well with re-spect to the multiplicative structure of the algebra. They give the following definition. Definition 7.2.
Let F be a filtration of a graded algebra ⊕ k V k . We shall say that(i) F is multiplicative if ( F t V k )( F s V m ) ⊆ F t + s V k + m for all k, m ∈ N and s, t ∈ R .(ii) F is (linearly) bounded if there exists a constant C such that F − kC V k = V k and F kC V k = { } for all k .The goal in this section is to associate an “analytic test configuration” φ F λ to anybounded multiplicative filtration of the section ring R ( L ) = ⊕ k H ( kL ) , which willbe used in the next section to construct an associated geodesic. Example 7.3.
An important example for our purpose will be the filtrations constructedfrom an algebraic test configuration (Section 8). Another kind of example to havein mind come when ( X, L ) is toric with moment polytope ∆ . Let f : ∆ → R be abounded positive concave function. Recalling that ∆ ∩ k − Z n parametrizes a toricbasis { s ( k ) α } for H ( kL ) define F t H ( kL ) = linspan { s ( k ) α : f ( α ) ≥ k − t } ⊂ H ( kL ) . It is easy to check that the concavity of f implies that F is multiplicative and that f is bounded implies that F is linearly bounded. When f is rational piecewise linear itturns out that this is precisely the filtration associated to the algebraic test configurationdefined by f as considered by Donaldson [24] (see also [38]). In this way one sees howthe analytic test configurations and associated geodesics considered in this section andthe next generalize the algebraic picture.Now fix φ ∈ H ( L ) , and let dV be a smooth volume form on X with unit mass.This gives the L -scalar product on H ( kL ) by letting ( s, t ) kφ := Z X s ( z ) t ( z ) e − kφ ( z ) dV ( z ) . For any λ ∈ R let { s i,λ } be an orthonormal basis for F kλ H ( kL ) and define φ k,λ := 1 k ln( X | s i,λ | ) , which is a positive metric on L . Lemma 7.4.
For any λ , the sequence of metrics φ k,λ converges to a limit as k tends toinfinity, and the upper semicontinuous regularization of the limit φ F λ := lim ∗ k →∞ φ k,λ is a positive metric. FILTRATIONSOFTHERINGOFSECTIONS 29
Proof.
Since K λ ( z, w ) := X i s i,λ ( z ) s i,λ ( w ) is a reproducing kernel of F kλ H ( kL ) with respect to ( · , · ) kφ , as for the full Bergmankernel we have the following useful characterization (see, for example, [5, (4.3)]) X | s i,λ | = sup {| s | : s ∈ F kλ H ( kL ) , k s k kφ ≤ } . (32)Let k s k ∞ := sup z ∈ X {| s ( z ) | e − kφ } and define F k,λ ( z ) := sup {| s ( z ) | : s ∈ F kλ H ( kL ) , k s k ∞ ≤ } . We trivially have the upper bound F k,λ ( z ) ≤ e − kφ ( z ) . It follows that usc( 1 k ln F k,λ ) = sup ∗ { k ln | s | : s ∈ F kλ H ( kL ) , k s k ∞ ≤ } ) is a positive metric. Let λ be fixed, pick a point z ∈ X, and let for all k, s k ∈F kλ H ( kL ) be such that k s k k ∞ = 1 and F k,λ ( z ) = | s k ( z ) | . Since the product s k s m lies in F ( k + m ) λ H (( k + m ) L ) by the multiplicativity of F , and k s k s m k ∞ ≤ k s k k ∞ k s m k ∞ , we get that F k + m,λ ( z ) ≥ F k,λ ( z ) F m,λ ( z ) , (33)so the map k F k,λ ( z ) is multiplicative. The existence of a limit lim k →∞ k ln F k,λ ( z ) thus follows from Fekete’s lemma (see e.g. [9, p37]). Since we assumed that dV hadunit mass, for any section s k s k kφ ≤ k s k ∞ , and thus by equation (32) X | s i,λ ( z ) | ≥ F k,λ ( z ) . On the other hand, by the Bernstein-Markov property of any volume form dV we havethat for any ǫ > there exists a constant C ǫ so that k s k ∞ ≤ C ǫ e ǫk k s k kφ , and thus X | s i,λ ( z ) | ≤ C ǫ e ǫk F k,λ ( z ) . (34) FILTRATIONSOFTHERINGOFSECTIONS 30It follows that the difference φ k,λ ( z ) − k ln F k,λ ( z ) tends to zero as k tends to infinity,thus the convergence of φ k,λ follows.By the multiplicativity, for any k ∈ N k ln F k,λ ≤ lim l →∞ l ln F l,λ = lim l →∞ φ l,λ , and thus usc( 1 k ln F k,λ ) ≤ lim ∗ l →∞ φ l,λ =: φ F λ . (35)On the other hand, clearly lim l →∞ φ l,λ ≤ sup k { usc( 1 k ln F k,λ ) } , and it follows that φ F λ = sup ∗ k { usc( 1 k ln F k,λ ) } so φ F λ is indeed a positive metric. Remark 7.5.
Since all volume forms dV on X are equivalent, the limit φ λ does notdepend on the choice of volume form dV . Lemma 7.6.
We have that φ k,λ ≤ φ F λ + ǫ ( k ) , where ǫ ( k ) is a constant independent of λ that tends to zero as k tends to infinity.Proof. By combining the inequalities (34) and (35) from the proof of the the previouslemma we see that for any ǫ > there exists a constant C ǫ independent of λ such that φ k,λ ≤ φ F λ + ǫ + (1 /k ) ln C ǫ . This yields the lemma.
Proposition 7.7.
The map λ φ F λ is a test curve.Proof. Let λ be such that F kλ H ( kL ) = H ( kL ) for all k . Then φ k,λ is the usualBergman metric, and by the result on Bergman kernel asymptotics due to Bouche-Catlin-Tian-Zelditch (see Section 3) we get that φ k,λ converges to φ . Trivially we seethat if F kλ H ( kL ) = { } for all k then φ F λ ≡ −∞ . By the boundedness of thefiltration we thus have φ F λ = φ for λ < − C and φ F λ ≡ −∞ for λ > C .By the multiplicativity of the filtration φ λ ≡ −∞ if and only if for all k, F kλ H ( kL ) = { } . Pick a λ such that φ F λ
6≡ −∞ , then for some k, F kλ H ( kL ) is non-trivial. FromLemma 7.6 it follows that φ F λ has small unbounded locus since φ k,λ has small un-bounded locus.It remains to prove concavity. Let λ , λ ∈ R and let t be a rational point in the unitinterval. Let m be a natural number such that mt is an integer. Given a point z ∈ X, let FILTRATIONSOFTHERINGOFSECTIONS 31 s ∈ F kλ H ( kL ) and s ∈ F kλ H ( kL ) be two sections with k s k ∞ = k s k ∞ = 1 such that F k,λ = | s ( z ) | and F k,λ = | s ( z ) | . By the multiplicativity of the filtration, s mt s m (1 − t )2 ∈ F mk ( tλ +(1 − t ) λ ) H ( mkL ) , and trivially k s mt s m (1 − t )2 k ∞ ≤ . It follows that F mk,tλ +(1 − t ) λ ( z ) ≥ F k,λ ( z ) mt F k,λ ( z ) m (1 − t ) . Taking the logarithm on both sides, dividing by mk, and taking the limit yields φ F tλ +(1 − t ) λ ≥ tφ F λ + (1 − t ) φ F λ (36)except possibly on the pluripolar set where the limits are not equal to their upper semi-continuous regularization.Now if two plurisubharmonic functions ζ and ζ are equal almost everywhere thenthey are equal everywhere [29, 2.9.8]. Applying this to ζ and max { ζ , ζ } we see thatif ζ ≥ ζ almost everywhere then this is true everywhere. Thus in fact (36) holds onthe whole of X .Recall that t was assumed to be rational. If λ ≤ λ , the left-hand side of (36)is decreasing in t since clearly φ F λ is decreasing in λ . The right-hand side of (36) iscontinuous in t, so it follows that the equation (36) holds for all t ∈ (0 , , i.e. φ F λ isconcave in λ . Lemma 7.8.
For any two φ, ψ ∈ H ( L ) and any λ ∈ R we have φ F λ ∼ ψ F λ .Proof. If φ ≤ ψ then for all k and λ we have that φ k,λ ≤ ψ k,λ , and so φ F λ ≤ ψ F λ . Alsoit is clear that ( φ + C ) k,λ = φ k,λ + C, which proves the lemma. Definition 7.9.
We call the map λ [ φ F λ ] the analytic test configuration associatedto the filtration F .So by the previous lemma this analytic test configuration depends only on F andnot on the choice of φ ∈ H ( L ) . Our next goal is to show the curve φ F λ is maximal for λ < λ c , for which we will need a Skoda-type division theorem. Theorem 7.10.
Let L be an ample line bundle. Assume that L has a smooth positivemetric φ with the property that dd c φ ≥ dd c φ K X for some smooth metric φ K X on thecanonical bundle K X . Let { s i } be a finite collection of holomorphic sections of L and m > n + 2 where n = dim X .Suppose s is a section of mL such that Z X | s | ( P | s i | ) m dV < ∞ . FILTRATIONSOFTHERINGOFSECTIONS 32
Then there exists sections h α ∈ H (( n + 1) L ) such that s = X α h α s α , where α is a multiindex α = ( α i ) with P i α i = m − n − , and s α are the monomials s α := Π i s α i i .Proof. Let k be an integer such that n + 2 ≤ k ≤ m . Then given a section t ∈ H ( kL ) with Z X | t | ( P i | s i | ) k dV < ∞ an application of the Skoda division theorem [47, Thm. 2.1] yields sections { t i } of ( k − L such that t = P i t i s i and Z X | t i | ( P i | s i | ) k − dV < ∞ . (To apply the cited theorem replace F, E, ψ, η with kL − K X , L, kφ − φ K X , φ respec-tively and replace αq with k − > n + 1 .)Now we first apply the above with k = m to the section s , and then apply againwith k = m − to each of the sections t i . Repeating this process with k = m, m − , . . . , n + 2 we see that s can be written as a linear sum of monomials in the s i asrequired. Proposition 7.11.
For λ less than the critical value λ c , φ F λ = lim k →∞ φ [ φ k,λ ] . Proof.
Let φ k := φ k, −∞ , i.e. the Bergman metric /k ln( P | s i | ) , where { s i } is an or-thonormal basis for the whole space H ( kL ) with respect to ( · , · ) kφ . By the Bernstein-Markov property of any volume form dV (see e.g. [48]), or simply the maximum prin-ciple, φ k ≤ φ + ǫ k , (37)where ǫ k tends to zero as k tends to infinity. Since φ k,λ is decreasing in λ, the inequality(37) still holds when φ k is replaced by φ k,λ , i.e. φ k,λ − ǫ k ≤ φ . Therefore φ k,λ − ǫ k belongs to the class of metrics the supremum of which yields P [ φ k,λ ] φ, so φ k,λ ≤ P [ φ k,λ ] φ + ǫ k ≤ φ [ φ k,λ ] + ǫ k , so letting k tend to infinity φ F λ ≤ φ [ φ k,λ ] . For the other inequality it is enough to show that for any constant
C,P φ k,λ + C φ ≤ φ F λ . (38) FILTRATIONSOFTHERINGOFSECTIONS 33By the assumption that λ < λ c it follows that φ F λ
6≡ −∞ . Let ψ be a positive metricdominated by both φ k,λ + C and φ, where k is large enough so that kL fulfills the re-quirements of Theorem 7.10. We denote by J ( kψ ) the multiplier ideal sheaf of germsof holomorphic functions locally integrable against e − kψ . Let { s i } be an orthonormalbasis of H ( kL ⊗ J ( kψ )) , and denote by ψ k the Bergman metric ψ k := 1 k ln( X | s i | ) . By Theorem 3.2, ψ ≤ ψ k + δ k where δ k tends to zero as k tends to infinity, and ψ k converges pointwise to ψ . If s liesin H ( kL ⊗ J ( kψ )) , specifically we must have that Z X | s | P | s i,λ | dV < ∞ , since we assumed that ψ was dominated by φ k,λ + C = 1 /k ln( P | s i,λ | ) + C . Simi-larly if s lies in H ( kmL ⊗ J ( kmψ )) we have Z X | s | ( P | s i,λ | ) m dV < ∞ . From Theorem 7.10 applied to the sections { s i,λ } it thus follows that s = X h α s α , where h α ∈ H ( k ( n + 1) L ) , and the s α are monomials in the { s i,λ } of degree m − n − . Because of the multiplicativity of the filtration each s α lies in F k ( m − n − λ H ( k ( m − n − L ) , and by the boundedness of the filtration we also have that each h α lies in F − k ( n +1) C H ( k ( n + 1) L ) for some fixed constant C . We thus get that H ( kmL ⊗ J ( kmψ )) is contained in ( F − k ( n +1) C H ( k ( n + 1) L ))( F k ( m − n − λ H ( k ( m − n − L )) ⊆ F k ( m − n − λ − k ( n +1) C H ( kmL ) . (39)Since we assumed that ψ ≤ φ we have that ψ km is less than or equal to the Bergmanmetric using an orthonormal basis for H ( kmL ⊗J ( kmψ )) with respect to φ . Becauseof (39) this Bergman metric is certainly less than or equal to φ km,λ ′ , where λ ′ := 1 km ( k ( m − n − λ − k ( n + 1) C ) . Hence ψ km ≤ φ km,λ ′ . FILTRATIONSASSOCIATEDTOALGEBRAICTESTCONFIGURATIONS 34On the other hand, by Lemma 7.6 we get φ km,λ ′ ≤ φ F λ ′ + ǫ ( km ) , where ǫ ( km ) is a constant independent of λ ′ that tends to zero as km tends to infinity.Since λ ′ tends to λ as m tends to infinity this implies ψ ≤ lim λ ′ → λ φ F λ ′ , and thus byLemma 5.2 ψ ≤ φ F λ . Taking the supremum over all such ψ completes the proof. Corollary 7.12.
Suppose F is a multiplicative linearly bounded filtration of ⊕ k H ( kL ) .Then the associated test curve φ F λ is maximal for λ < λ c and its Legendre transformis a geodesic ray.Proof. Theorem 4.10 tells us that φ [ φ k,λ ] is maximal with respect to φ = φ −∞ . ByLemma 4.8 it follows that this is true for the limit φ F λ = lim k →∞ φ [ φ k,λ ] as well. Let φ λ be the test curve defined by φ λ := φ F λ for λ < λ c and φ λ ≡ −∞ for λ ≥ λ c . Thus φ λ is a maximal test curve, thus its Legendre transform is a geodesic ray. On the otherhand, for every ǫ > φ λ ≤ φ F λ ≤ φ λ − ǫ , and therefore b φ t ≤ [ ( φ F ) t ≤ b φ t + ǫt. Since ǫ was arbitrary we get that the Legendre transform of φ F λ coincides with that of φ λ , and thus it is a geodesic ray. Remark 7.13.
Given an analytic test configuration [ ψ λ ] there is a naturally associatedfiltration F of the section ring, defined as F kλ H ( kL ) := H ( kL ⊗ J ( kψ λ )) . This filtration is bounded, but in general not multiplicative.
We recall briefly Donaldson’s definition of a test configuration [24, 25]. In order to notconfuse them with the our analytic test configurations, we will in this article refer tothem as algebraic test configurations.
Definition 8.1. An algebraic test configuration T for an ample line bundle L over X consists of:(i) a scheme X with a C × -action ρ, (ii) a C × -equivariant line bundle L over X , (iii) and a flat C × -equivariant projection π : X → C where C × acts on C bymultiplication, such that L is relatively ample, and such that if we denote by X := π − (1) , then L | X → X is isomorphic to rL → X for some r > . FILTRATIONSASSOCIATEDTOALGEBRAICTESTCONFIGURATIONS 35By rescaling we can without loss of generality assume that r = 1 in the definition.An algebraic test configuration is called a product test configuration if there is a C × -action ρ ′ on L → X such that L = L × C with ρ acting on L by ρ ′ and on C bymultiplication. An algebraic test configuration is called trivial if it is a product testconfiguration with the action ρ ′ being the trivial C × -action.Since the zero-fiber X := π − (0) is invariant under the action ρ , we get an in-duced action on the space H ( kL ) , also denoted by ρ, where we have denoted therestriction of L to X by L . Specifically, we let ρ ( τ ) act on a section s ∈ H ( kL ) by ( ρ ( τ )( s ))( x ) := ρ ( τ )( s ( ρ − ( τ )( x ))) . (40)By standard theory any vector space V with a C × -action can be split into weight spaces V λ i on which ρ ( τ ) acts as multiplication by τ λ i , (see e.g. [24]). The numbers λ i withnon-trivial weight spaces are called the weights of the action. Thus we may write H ( kL ) as H ( kL ) = ⊕ λ V λ with respect to the induced action ρ .In [35, Lem. 4] Phong-Sturm give the following linear bound on the absolute valueof the weights. Lemma 8.2.
Given a test configuration there is a constant C such that | λ i | < Ck whenever dim V λ i > . In [49] the second author showed how to get an associated filtration F of the sectionring ⊕ k H ( kL ) given a test configuration T of L which we now recall.First note that the C × -action ρ on L via the equation (40) gives rise to an inducedaction on H ( X , k L ) as well as H ( X \ X , k L ) , since X \ X is invariant. Let s ∈ H ( kL ) be a holomorphic section. Then using the C × -action ρ we get a canonicalextension ¯ s ∈ H ( X \ X , k L ) which is invariant under the action ρ , simply by letting ¯ s ( ρ ( τ ) x ) := ρ ( τ ) s ( x ) (41)for any τ ∈ C × and x ∈ X .We identify the coordinate z with the projection function π ( x ) , and we also con-sider it as a section of the trivial bundle over X . Exactly as for H ( X , k L ) , ρ givesrise to an induced action on sections of the trivial bundle, using the same formula (40).From this one sees ( ρ ( τ ) z )( x ) = ρ ( τ )( z ( ρ − ( τ ) x ) = ρ ( τ )( τ − z ( x )) = τ − z ( x ) , (42)where we used that ρ acts on the trivial bundle by multiplication on the z -coordinate.Thus ρ ( τ ) z = τ − z, which shows that the section z has weight − .By this it follows that for any section s ∈ H ( kL ) and any integer λ, we get asection z − λ ¯ s ∈ H ( X \ X , k L ) , which has weight λ . FILTRATIONSASSOCIATEDTOALGEBRAICTESTCONFIGURATIONS 36 Lemma 8.3.
For any section s ∈ H ( kL ) and any integer λ the section z − λ ¯ s extendsto a meromorphic section of k L over the whole of X , which we also will denote by z − λ ¯ s .Proof. It is equivalent to saying that for any section s there exists an integer λ suchthat z λ ¯ s extends to a holomorphic section S ∈ H ( X , k L ) . By flatness, which wasassumed in the definition of a test configuration, the direct image bundle π ∗ L is in facta vector bundle over C . Thus it is trivial, since any vector bundle over C is trivial.Therefore there exists a global section S ′ ∈ H ( X , k L ) such that s = S ′| X . Onthe other hand, as for H ( kL ) , H ( X , k L ) may be decomposed as a direct sum ofinvariant subspaces W λ ′ such that ρ ( τ ) restricted to W λ ′ acts as multiplication by τ λ ′ .Let us write S ′ = X S ′ λ ′ , (43)where S λ ′ ∈ W λ ′ . Restricting the equation (43) to X gives a decomposition of s,s = X s λ ′ , where s λ ′ := S ′ λ ′ | X . From (41) and the fact that S ′ λ ′ lies in W λ ′ we get that for x ∈ X and τ ∈ C × ¯ s λ ′ ( ρ ( τ )( x )) = ρ ( τ )( s λ ′ ( x )) = ρ ( τ )( S ′ λ ′ ( x )) = ( ρ ( τ ) S ′ λ ′ )( ρ ( τ )( x ))) == τ λ ′ S ′ λ ′ ( ρ ( τ )( x )) , and therefore ¯ s λ ′ = τ λ ′ S ′ λ ′ . Since trivially ¯ s = X ¯ s λ ′ it follows that t λ ¯ s extends holomorphically as long as λ ≥ max − λ ′ . Definition 8.4.
Given a test configuration T we define a vector space-valued map F from Z × N by letting ( λ, k ) s ∈ H ( kL ) : z − λ ¯ s ∈ H ( X , k L ) } =: F λ H ( kL ) . It is immediate that F λ is decreasing since H ( X , k L ) is a C [ z ] -module. We canextend F to a filtration by letting F λ H ( kL ) := F ⌈ λ ⌉ H ( kL ) for non-integers λ, thus making F left-continuous. Since z − ( λ + λ ′ ) ss ′ = ( z − λ ¯ s )( z − λ ′ ¯ s ′ ) ∈ H ( X , k L ) H ( X , m L ) ⊆ H ( X , ( k + m ) L ) whenever s ∈ F λ H ( kL ) and s ′ ∈ F λ ′ H ( kL ) , we see that ( F λ H ( kL ))( F λ ′ H ( mL )) ⊆ F λ + λ ′ H (( k + m ) L ) , i.e. F is multiplicative.Recall that we had the decomposition of H ( kL ) into weight spaces V λ . THEGEODESICRAYSOFPHONGANDSTURM 37 Lemma 8.5.
For each λ, we have that dim F λ H ( kL ) = X λ ′ ≥ λ dim V λ ′ . Proof.
We have the following isomorphism: ( π ∗ k L ) |{ } ∼ = H ( X , k L ) /zH ( X , k L ) , the right-to-left arrow being given by the restriction map (see e.g. [39, p12]). Also, for k ≫ , ( π ∗ k L ) |{ } = H ( kL ) , therefore for large kH ( kL ) ∼ = H ( X , k L ) /zH ( X , k L ) , (44)We also had a decomposition of H ( X , k L ) into the sum of its invariant weight spaces W λ . By Lemma 8.3 it is clear that a section S ∈ H ( X , k L ) lies in W λ if and only ifit can be written as z − λ ¯ s for some s ∈ H ( kL ) , in fact we have that s = S | X . Thus W λ ∼ = F λ H ( kL ) , and by the isomorphism (44) then V λ ∼ = F λ H ( kL ) / F λ +1 H ( kL ) . Therefore dim F λ H ( kL ) = X λ ′ ≥ λ dim V λ ′ . (45)Using Lemma 8.5 together with Lemma 8.2 shows that the filtration F is bounded. In [35] Phong-Sturm show how to construct a weak geodesic ray, starting with a φ ∈H ( L ) and an algebraic test configuration T (see also [44] for how this works in thetoric setting). In the previous section we showed how to associate an analytic testconfiguration [ φ F λ ] to an algebraic test configuration, and thus get a weak geodesicusing the Legendre transform of its maximal envelope. Recall by Proposition 6.10 thisgeodesic is the same as the Legendre transform of the original test curve φ F λ . The goalin this section is to prove that this ray coincides with the one constructed by Phong-Sturm.To describe what we aim to show, recall that if V is a vector space with a scalarproduct, and F is a filtration of V, there is a unique decomposition of V into a directsum of mutually orthogonal subspaces V λ i such that F λ V = ⊕ λ i ≥ λ V λ i . THEGEODESICRAYSOFPHONGANDSTURM 38Furthermore we allow for λ i to be equal to λ j even when i = j, so we can assume thatall the subspaces V λ i are one dimensional. This additional decomposition is of coursenot unique, but it will not matter in what follows.Let φ ∈ H ( L ) and H ( kL ) = ⊕ V λ i be the decomposition of H ( kL ) with respectto the scalar product ( · , · ) kφ coming from the volume form ( dd c φ ) n . Consider next thefiltration coming from an algebraic test configuration (note that then the collection of λ i will depend also on k but we omit that from our notation) and define the normalizedweights to be ¯ λ i := λ i k , which form a bounded family by Lemma 8.2.Now if s i is a vector of unit length in V λ i , then { s i } will be an orthonormal basisfor H ( kL ) . Since the filtration F encodes the C ∗ -action on H ( k L ) it is easy to seethat the basis { s i } is the same one as in [35, Lem 7]. In terms of the notation in theprevious sections φ k,λ = 1 k ln( X λ i ≥ kλ | s i | ) and φ F λ = lim ∗ k →∞ φ k,λ . Definition 9.1.
Let Φ k ( t ) := 1 k ln( X i e tλ i | s i | ) The
Phong-Sturm ray is the limit Φ( t ) := lim ∗ k →∞ (sup l ≥ k Φ l ( t )) . (46)Our goal is the following: Theorem 9.2.
Let φ F be the analytic test configuration associated to the filtration F from a test configuration. Then Φ( t ) = [ ( φ F ) t . In particular, the results from the previous section yield another proof of [35, Thm1] which says that Φ( t ) is a weak geodesic ray emanating from φ . Lemma 9.3. Φ( t ) = lim ∗ k →∞ (sup l ≥ k Φ l ( t )) = lim ∗ k →∞ (sup l ≥ k max i { φ l, ¯ λ i + t ¯ λ i } ) . (47) Proof.
Our proof will be based on the elementary fact that if { a l,i : i ∈ I l } is a set ofreal numbers then max i ∈ I l a l,i ≤ l ln X i ∈ I l e la l.i ≤ max i ∈ I l a l,i + 1 l ln | I l | . (48) THEGEODESICRAYSOFPHONGANDSTURM 39Now pick x ∈ X and t > . Let a l,i := 1 l ln | s i ( x ) | + t ¯ λ i and I l be the indexing set for the λ i . Then | I l | = O ( l n ) and Φ l ( t ) = 1 l ln X i e la l,i ! . Thus by (48) max i { a l,i } ≤ Φ l ( t ) ≤ max i { a l,i } + | I l | l . (49)Now set b l,i := φ l, ¯ λ i + t ¯ λ i = 1 l ln X λ j ≥ λ i | s j ( x ) | + t ¯ λ i . For fixed i , pick any j such that max λ j ≥ λ i | s j ( x ) | = | s j | and λ j ≥ λ i . Then b l,i ≤ l ln( | I l || s j | + t ¯ λ i ≤ l ln | s j | + t ¯ λ j + ln | I l | l = a j ,l + ln | I l | l . Clearly a l,i ≤ b l,i for all i , so we in fact have max i { a l,i } ≤ max { b l,i } ≤ max i { a l,i } + ln | I l | l , which combined with (49) yields max i { b l,i } − ln | I l | l ≤ Φ l ( t ) ≤ max i { b l,i } + ln | I l | l . Now taking the supremum over all l ≥ k followed by the upper semicontinuous regu-larization and then the limit as k tends to infinity gives the result since k − ln | I k | tendsto zero. Proof of Theorem 9.2.
From Lemma 7.6 there is a constant ǫ ( l ) such that φ l, ¯ λ i + t ¯ λ i ≤ φ F ¯ λ i + t ¯ λ i + ǫ ( l ) , where ǫ ( l ) is independent of λ i and tends to zero as l tends to infinity. Thus max i { φ l, ¯ λ i + t ¯ λ i } ≤ sup λ { φ F λ + tλ } + ǫ ( l ) , and so lim ∗ k →∞ sup l ≥ k max i { φ l, ¯ λ i + t ¯ λ i } ≤ ( c φ F ) t + lim k →∞ sup l ≥ k ǫ ( l ) , EFERENCES 40so using Lemma 9.3 gives Φ( t ) ≤ ( c φ F ) t . For the opposite inequality, let λ ∈ R be arbitrary. Trivially Φ k ( t ) = 1 k ln( X i e tλ i | s i | ) ≥ k ln( X λ i ≥ kλ e tkλ | s i | ) = φ k,λ + tλ. Hence Φ( t ) ≥ φ F λ + tλ for any λ, and thus Φ( t ) ≥ ( c φ F ) t . Remark 9.4.
Phong-Sturm prove in [35] that the geodesic ray one gets from an alge-braic test configuration F is non-trivial if the norm of F is non-zero. From the abovewe see that the weak geodesic ray is trivial if and only if the associated analytic testconfiguration is trivial, i.e. if there exists a number λ c such that φ λ = φ when λ < λ c and φ λ ≡ −∞ when λ > λ c . References [1] V Apostolov, D Calderbank, P Gauduchon and C Tønnesen-Friedman
Hamilto-nian 2-forms in K¨ahler geometry III, extremal metrics and stability
Invent. Math173(3)(2008) 547–601.[2] C Arezzo and G Tian
Infinite geodesic rays in the space of K¨ahler potentials
Ann.Sc. Norm. Sup. Pisa (5) 2 (2003), no. 4, 617-630.[3] E Bedford and B A Taylor
A new capacity for plurisubharmonic functions
ActaMath. 149 (1982), no. 1-2, 1-40.[4] E Bedford and B A Taylor
Fine topology, Silov boundary, and ( dd c ) n J. Funct.Anal. 72 (1987), no. 2, 225-251.[5] R Berman
Bergman kernels and equilibrium measures for line bundles over pro-jective manifolds
Amer. J. Math. 131 (2009), no. 5, 1485–1524.[6] R Berman
Bergman kernels and equilibrium measures for ample line bundles (2007) Preprint arXiv:0704.1640.[7] R Berman and S Boucksom
Growth of balls of holomorphic sections and energyat equilibrium
Invent. Math. 181 (2010), no. 2, 337-394.[8] R Berman, S Boucksom, V Guedj and A Zeriahi
A variational approachto complex Monge-Amp`ere equations
To appear in Publ. Math. de l’IHES,arXiv:0907.4490.[9] B Bollob´as and O Riordan
Percolation
Cambridge University Press, New York,2006. x+323 pp. ISBN: 978-0-521-87232-4EFERENCES 41[10] T Bouche
Convergence de la m´etrique de Fubini-Study d’un fibr´e lin´eaire positif
Ann. Inst. Fourier (1) 40 (1990), 117-130.[11] B Berndtsson
Probability measures associated to geodesics in the space of K¨ahlermetrics (2009) Preprint arXiv:math/0907.1806.[12] S Boucksom and H Chen
Okounkov bodies of filtered linear series
Compos. Math.147 (2011), no. 4, 1205–1229[13] S Boucksom, P Eyssidieux, V Guedj and A Zeriahi
Monge-Amp`ere equations inbig cohomology classes
Acta Math. 205 (2010), no. 2, 199–262.[14] D Catlin
The Bergman kernel and a theorem of Tian
In Analysis and geometry inseveral complex variables (Katata, 1997), Trends Math., pages 1-23. Birkh¨auserBoston, Boston, MA, 1999.[15] X X Chen
The space of K¨ahler metrics
J. Differential Geom. 56 (2000), 189–234.[16] X X Chen
Space of K¨ahler metrics III: on the lower bound of the Calabi energyand geodesic distance
Invent. Math. 175 (2009), no. 3, 453-503.[17] X X Chen and S Sun
Space of K¨ahler metrics V-K¨ahler quantization
In Met-ric and Differential Geometry, the Jeff Cheeger anniversary volume, Progress inMathematics 297 (2012), 19-42.[18] X X Chen and G Tian
Geometry of K¨ahler metrics and foliations by holomorphicdiscs . Publ. Math. Inst. Hautes ´Etudes Sci. No. 107 (2008).[19] J-P Demailly
Complex analytic and differential geometry
Regularization of closed positive currents and intersection theory
J.Algebraic Geom. 1 (1992), no. 3, 361-409.[21] J-P Demailly
Appendix to I. Cheltsov and C. Shramov’s article “Log canonicalthresholds of smooth Fano Threefolds”
Russian Mathematical surveys 63 (2008),859-958.[22] J-P Demailly, Thomas Peternell and Michael Schneider
Pseudo-effective line bun-dles on compact K¨ahler manifolds
Internat. J. Math. 12 (2001), no. 6, 689–741.[23] S K Donaldson
Scalar curvature and projective embeddings. I.
J. DifferentialGeom. 59 (2001), no. 3, 479-522.[24] S K Donaldson
Scalar curvature and stability of toric varieties
J. DifferentialGeom. 62 (2002), no. 2, 289-349.[25] S K Donaldson
Lower bounds on the Calabi functional
J. Differential Geom. 70(2005), no. 3, 453-472.[26] S K Donaldson
Symmetric spaces, K¨ahler geometry and Hamiltonian dynam-ics . Northern California Symplectic Geometry Seminar, 13-33. Amer. Math. Soc.Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999.[27] V Guedj and A Zeriahi
Intrinsic capacities on compact K¨ahler manifolds
J.Geom. Anal. 15 (2005), no. 4, 607-639.[28] C Kiselman
The partial Legendre transformation for plurisubharmonic functions
Invent. Math. 49 (1978), no. 2, 137-148.EFERENCES 42[29] M Klimek
Pluripotential Theory
London Mathematical Society Monographs.New Series, 6. Oxford Science Publications. The Clarendon Press, Oxford Uni-versity Press, New York, 1991. ISBN: 0-19-853568-6[30] T Mabuchi
Some symplectic geometry on compact K¨ahler manifolds. I.
Osaka J.Math. 24 (1987), no. 2, 227.[31] T Mabuchi
K-energy maps integrationg Futaki invariants
Tohoku Math. J. (2) 38(1986), no. 4, 575-593.[32] A Rashkovskii and R. Sigurdsson
Green functions with singularities along com-plex spaces
Internat. J. Math. 16 (2005), no. 4, 333–355[33] R. Rockafellar
Conjugates and Legendre transforms of convex functions
Canad.J. Math. 19 1967 200–205.[34] D H Phong and J Sturm
The Monge-Ampe`ere operator and geodesics in the spaceof K¨ahler potentials
Invent. Math. 166 (2006), no. 1, 125-149.[35] D H Phong and J Sturm
Test configurations for K-stability and geodesic rays
Jour.Symplectic geom. 5 (2007), no. 2, 221-247.[36] D H Phong and J Sturm
Lectures on stability and constant scalar curvature
Cur-rent developments in mathematics, 2007, 101-176, Int. Press, Somerville, MA,2009[37] D H Phong and J Sturm
Regularity of geodesic rays and Monge-Ampere equations
Proc. Amer. Math. Soc. 138 (2010), no. 10, 3637-3650[38] G Sz´ekelyhidi
Filtrations and test-configurations (2011) PreprintarXiv:1111.4986[39] J Ross and R P Thomas
A study of the Hilbert-Mumford criterion for the stabilityof projective varieties
J. Alg. Geom. 16 (2007), 201-255.[40] J Ross and D Witt Nystr¨om
Envelopes of plurisubharmonic metrics with pre-scribed singularities (2012) Preprint arXiv:1210.2220[41] Y A Rubinstein and S Zelditch
The Cauchy problem for the homogeneous Monge-Amp`ere equation, I. Toeplitz quantization
J. Differential Geom. 90 (2012), no. 2,303–327[42] Y A Rubinstein and S Zelditch
The Cauchy problem for the homogeneous Monge-Amp`ere equation, II. Legendre transform
Adv. Math. 228 (2011), no. 6, 2989–3025.[43] S Semmes
Complex Monge-Amp`ere and symplectic manifolds
Amer. J. Math. 114(1992), no. 3, 495-550.[44] J Song and S Zelditch
Test configurations, large deviations and geodesic rays ontoric varieites
Adv. Math. 229 (2012), no. 4, 2338-2378.[45] G Tian
On a set of polarized K¨ahler metrics on algebraic manifolds
J. DifferentialGeom. 32 (1) (1990), 99-130.[46] G Tian
K¨ahler-Einstein metrics with positive scalar curvature
Invent. Math. 130(1997), no. 1, 1–37.[47] D Varolin
Division theorems and twisted complexes
Math. Z. 259 (2008), no. 1,1-20.EFERENCES 43[48] D Witt Nystr¨om
Transforming metrics on a line bundle to the Okounkov body (2009) Preprint arXiv:math/0903.5167.[49] D Witt Nystr¨om
Test configurations and Okounkov bodies
Comp. Math. 148(2012), no. 6, 1736-1756.[50] S Zelditch
Szeg¨o kernels and a theorem of Tian
Int. Math. Res. Notices 6 (1998),317-331.J
ULIUS R OSS , U
NIVERSITY OF C AMBRIDGE , UK. J . ROSS @ DPMMS . CAM . AC . UK D AVID W ITT N YSTR ¨ OM , U NIVERSITY OF G OTHENBURG , S
WEDEN . WITTNYST @ CHALMERS ..