ENERGY FUNCTIONALS AND CANONICAL KÄHLER METRICS

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1 ENERGY FUNCTIONALS AND CANONICAL KÄHLER ETRICS Jan Song Ben Wenkove Johns Hopkns Unversty Harvard Unversty Department of athematcs Department of athematcs Baltmore D Cambrdge A 2138 jsong@math.jhu.edu wenkove@math.harvard.edu 1. Introducton The problem of fndng necessary and suffcent condtons for the exstence of extremal metrcs, whch nclude Kähler-Ensten metrcs, on a compact Kähler manfold has been the subject of ntense study over the last few decades and s stll largely open. If has zero or negatve frst Chern class then t s known by the work of Yau [Ya1] and Yau, Aubn [Ya1], [Au] that has a Kähler-Ensten metrc. When c 1 ( >, so that s Fano, there s a well-known conjecture of Yau [Ya2] that the manfold admts a Kähler-Ensten metrc f and only f t s stable n the sense of geometrc nvarant theory. There are now several dfferent notons of stablty for manfolds [T2], [PhSt1], [Do2], [RoTh]. Donaldson showed that the exstence of a constant scalar curvature metrc s suffcent for the manfold to be asymptotcally Chow stable [Do1] (under an assumpton on the automorphsm group. It s conjectured by Tan [T2] that the exstence of a Kähler-Ensten metrc should be equvalent to hs K-stablty. Ths stablty s defned n terms of the Futak nvarant [Fu], [DT] of the central fber of degeneratons of the manfold. Donaldson [Do2] ntroduced a varant of K-stablty extendng Tan s defnton. The behavor of abuch s [a] energy functonal s central to ths problem. It was shown by Bando and abuch [Baa], [Ba] that f a Fano manfold admts a Kähler-Ensten metrc then the abuch energy s bounded below. Recently, t has been shown by Chen and Tan [ChT3] that f admts an extremal metrc n a gven class then the abuch energy s bounded below n that class. Donaldson has gven an alternatve proof for constant scalar curvature metrcs wth a condton on the space of automorphsms [Do3]. oreover, f a lower bound on the abuch energy s gven then the class s K-semstable [T2], [PaT]. Conversely, Donaldson [Do2] showed that, for torc surfaces, K-stablty mples the lower boundedness of the abuch energy. In addton, the exstence of a Kähler-Ensten metrc on a Fano manfold has been shown to be equvalent to the properness of the abuch energy [T2]. Tan conjectured [T3] that the exstence of a constant scalar curvature Kähler metrc be equvalent to ths condton on the abuch energy. Ths holds when the frst Chern class s a multple of the Kähler class. (If c 1 ( <, t has been shown [Ch1], [We], [SoWe] that the abuch energy s proper on certan classes whch are not multples of the canoncal 1

2 class. It s not yet known whether there exsts a constant scalar curvature metrc n such classes. In ths paper, we dscuss a famly of functonals E k, for k =,..., n, whch were ntroduced by Chen and Tan [ChT1]. They are generalzatons of the abuch energy, wth E beng precsely abuch s functonal. The functonals E k can be descrbed n terms of the Delgne parng [De]. Ths constructon of Delgne has provded a very useful way to understand questons of stablty [Zh], [PhSt1], [PhSt2]. Phong and Sturm [PhSt3] show that, up to a normalzaton term, the abuch energy corresponds to the Delgne parng L,..., L, K 1. Generalzng ths, the functonals E k can be descrbed n terms of the parng: n k k+1 {}}{{}}{ L,..., L, K 1,..., K 1. The fact that the functonals E k can be formulated n ths way seems now to be known by some experts n the feld, and was ponted out to us by Jacob Sturm n 22. However, snce t does not appear n the lterature, we have ncluded a short explanaton on ths correspondence (see secton 2. A crtcal metrc ω of E k s a soluton of the equaton σ k+1 (ω (σ k (ω = constant, where σ k (ω s the kth elementary symmetrc polynomal n the egenvalues of the Rcc tensor of the metrc ω. Notce that the crtcal metrcs for E are precsely the constant scalar curvature metrcs. Kähler-Ensten metrcs are solutons to the above equaton for all k. The crtcal metrcs are dscussed more n secton 2. The functonals E k were used by Chen and Tan [ChT1, ChT2] to obtan convergence of the normalzed Kähler-Rcc flow on Kähler-Ensten manfolds wth postve bsectonal curvature (see [PhSt4] for a related result. The abuch energy s decreasng along the flow. The functonal E 1 s also decreasng, as long as the sum of the Rcc curvature and the metrc s nonnegatve. A major part of the argument n [ChT1] s to show that the E k can be bounded from below along the Kähler-Rcc flow assumng nonnegatve Rcc curvature and the exstence of a Kähler-Ensten metrc. In a recent preprnt, Chen [Ch2] has proved a stablty result for E 1 for Fano manfolds n the sense of the Kähler-Rcc flow. In the same paper, Chen asked whether E 1 s bounded below or proper on the full space of potentals (not just along the flow f there exsts a Kähler-Ensten metrc. In ths paper we answer Chen s queston: E 1 s bounded below f there exsts a Kähler-Ensten metrc, and the lower bound s attaned by ths metrc. oreover, modulo holomorphc vector felds, E 1 s proper f and only f there exsts a Kähler-Ensten metrc. We also show that, agan assumng the exstence of a Kähler-Ensten metrc, the functonals E k are bounded below on the space of metrcs wth nonnegatve Rcc curvature. We now state these results more precsely. Let ω be a Kähler form on the compact manfold of complex dmenson n. Wrte P (, ω for the space of all smooth functons on such that ω = ω + 1 >. 2

3 For n P (, ω, let t be a path n P (, ω wth = and 1 =. The functonal E k,ω for k =,..., n s defned by E k,ω ( = k ( t t Rc(ω t k ω n k t n k 1 t (Rc(ω t k+1 µ k ω k+1 t ω n k 1 t, (1.1 where s the volume ωn, and µ k s the constant, dependng only on the classes [ω] and c 1 ( gven by µ k = Rc(ωk+1 ω n k 1 = (2π k+1 [K 1 ]k+1 [ω]n k 1 ωn [ω] n. The functonal s ndependent of the choce of path. We wll often wrte E k (ω, ω nstead of E k,ω (. In fact, Chen and Tan [ChT1] frst defne E k by a dfferent (and explct formula, makng use of a generalzaton of the Louvlle energy, whch they call Ek. The Delgne parng provdes another explct formula (Proposton 2.1. Suppose now that has postve frst Chern class and denote by K the space of Kähler metrcs n 2πc 1 (. Notce that for ω n K, the correspondng constant µ k s equal to 1. We have the followng theorem on the lower boundedness of the functonals E k. Theorem 1.1 Let (, ω KE be a Kähler-Ensten manfold wth c 1 ( >. Then, for k =,..., n, and for all ω K wth Rc( ω, E k (ω KE, ω, and equalty s attaned f and only f ω s a Kähler-Ensten metrc. In the case of E 1 we obtan lower boundedness on the whole space K. In addton, t s an easy result that for a Calab-Yau manfold, E 1 s bounded below on every class admttng a Rcc-flat metrc (and hence, by Yau s theorem, on every class. Puttng these two cases together we obtan: Theorem 1.2 Let (, ω KE be a Kähler-Ensten manfold wth c 1 ( > or c 1 ( =. Then for all Kähler metrcs ω n the class [ω KE ], E 1 (ω KE, ω, and equalty s attaned f and only f ω s a Kähler-Ensten metrc. We show that f (, ω KE s Kähler-Ensten wth c 1 ( > and f there are no holomorphc vector felds, E 1 s bounded below by the Aubn-Yau energy functonal J rased to a small power. Ths mples that E 1 s proper on P (, ω KE (for the defntons, see secton 2.4. If there exst holomorphc vector felds, then the statement changes slghtly (c.f. [T2]. 3

4 Theorem 1.3 Let (, ω KE be a compact Kähler-Ensten manfold wth c 1 ( >. Then there exsts δ dependng only on n such that the followng hold: ( If admts no nontrval holomorphc vector felds then there exst postve constants C and C dependng only on ω KE such that for all θ n P (, ω KE, E 1,ωKE (θ CJ ωke (θ δ C. ( In general, let Λ 1 be the space of egenfunctons of the Laplacan for ω KE wth egenvalue 1. Then for all θ n P (, ω KE satsfyng θ ψ ωke n =, for all ψ Λ 1, there exst constants C and C dependng only on ω KE such that E 1,ωKE (θ CJ ωke (θ δ C. Remark 1.1 The analagous result above s proved n [T2], [TZh] for the F functonal [D], gvng a generalzed oser-trudnger nequalty. We use a smlar argument. The correspondng nequalty for the abuch energy also holds [T3] snce, up to a constant, t can be bounded below by F. It would be nterestng to fnd the best constant δ = δ(n for whch these nequaltes hold. Wth some work, modfyng the argument n [T2], one can show that δ can be taken to be arbtrarly close to 1/(4n + 1, but we doubt that ths s optmal. Remark 1.2 We expect that the above results on the E 1 functonal have applcatons to the study of the stablty of. Indeed, let π 1 : X Z be an SL(N + 1, C- equvarant holomorphc fbraton between smooth varetes such that X Z CP N s a famly of subvaretes of dmenson n wth an acton of SL(N + 1, C on CP N. Tan defnes C-stablty [T2] for X z = π1 1 (z n terms of the vrtual bundle: E = (K 1 K (L L 1 n nµ n + 1 (L L 1 n+1, where K = K X K 1 Z s the relatve canoncal bundle and L s the pullback of the hyperplane bundle on CP n va the second projecton π 2 (alternatvely, one can use the language of the Delgne parng. When X z s Fano, Tan proved that X z s weakly C-stable f t s Kähler-Ensten, usng the properness of the abuch energy E. One can defne a smlar noton of stablty for X z wth respect to the vrtual bundle E k = (K 1 K k+1 (L L 1 n k (n kµ k (L L 1 n+1. n + 1 For k = 1, we would expect X z to be stable f t s Kähler-Ensten, snce E 1 s proper. It would be nterestng to try to relate ths noton of stablty to an analogue of K- stablty expressed n terms of the holomorphc nvarants F k [ChT1] whch generalze the Futak nvarant. 4

5 We also have a converse to Theorem 1.3. Theorem 1.4 Let (, ω be a compact Kähler manfold wth c 1 ( >. Suppose that ω 2πc 1 (. Then the followng hold: ( Suppose that (, ω admts no nontrval holomorphc vector felds. Then admts a Kähler-Ensten metrc f and only f E 1 s proper on P (, ω. ( In general, let G be a maxmal compact subgroup n Aut(. Suppose that ω s a G-nvarant Kähler metrc. Then admts a G-nvarant Kähler-Ensten metrc f and only f E 1 s proper on P G (, ω, the space of G-nvarant potentals for ω. Ths gves a new analytc condton for a Fano manfold to admt a Kähler-Ensten metrc. Indeed, together wth the result of Tan [T2], at least modulo holomorphc vector felds, we have: admts a E 1 s proper Kähler-Ensten abuch energy s proper metrc and one would expect some versons of stablty to be equvalent to these as well. Remark 1.3 It s natural to ask whether there exst crtcal metrcs for E k whch are not Kähler-Ensten. Chen and Tan [ChT1] observe that for k = n the only crtcal metrcs wth postve Rcc curvature are Kähler-Ensten. We see from Theorem 1.2 that on a Kähler-Ensten manfold wth c 1 ( >, a crtcal metrc for E 1 whch s not Kähler-Ensten could not gve an absolute mnmum of E 1. In secton 2, we descrbe some of the propertes of the E k functonals and the Aubn- Yau functonals I and J. In sectons 3 and 4 we prove Theorems 1.1 and 1.2 respectvely. Our general method follows that of Bando and abuch [Baa] and Bando [Ba]. They consder the onge-ampère equatons (3.1 and (4.1 respectvely. The abuch energy s decreasng n t for both of these equatons, and one can solve them backwards to show that the abuch energy s bounded below on the space of metrcs wth postve Rcc curvature [Baa], and on all metrcs n the gven class [Ba]. The functonals E k may not be decreasng n t. However, we are able to show, wth some extra calculaton, that the value at t = s greater than the value at t = 1. Ths works for all the E k for the frst equaton, but only seems to work for E 1 for the second. Fnally, n secton 5 we adapt the results of Tan [T2] and Tan-Zhu [TZh], makng use of our formulas from sectons 3 and 4, to prove Theorem 1.3 and Theorem

6 2. Propertes of the E k and I, J functonals 2.1 Basc propertes of E k It s mmedate from the formula (1.1 that the functonals E k satsfy the cocycle condton E k (ω 1, ω 2 + E k (ω 2, ω 3 + E k (ω 3, ω 1 =, for any metrcs ω 1, ω 2 and ω 3 n the same Kähler class. We wll make use of ths relaton n sectons 3 and 4. Observe also that E k,ω s nvarant under addton of constants: for all n P (, ω and any constant c. E k,ω ( + c = E k,ω ( 2.2 Crtcal metrcs We can wrte the E k functonals as: E k,ω ( = 1 1 where σ k (ω s gven by ( n 1 t [(k + 1 ( t k (σ k(ω σ t k+1(ω + (n kµ t k] ω n t, ( n (ω + t Rc(ω n = σ k (ω t k ω n. If at a pont p on we pck a normal coordnate system n whch the Rcc tensor s dagonal wth entres λ 1,..., λ n then σ k (ω at p s the k-th elementary symmetrc polynomal n the λ, σ k (ω = λ 1 λ 2 λ k. 1 < < k For example, σ = 1, σ 1 = R and σ 2 = 1 2 (R2 Rc 2. The crtcal ponts of E k are metrcs ω satsfyng ( n σ k+1 (ω (σ k (ω = µ k, (2.1 k + 1 The crtcal ponts for k = are of course constant scalar curvature metrcs. For k = 1, the crtcal equaton can be wrtten k= R 2 Rc 2 2 R = n(n 1µ 1. Note that we always mean the Laplacan gven by n 1 f ω n 1 = f ω n. 6

7 For any k, Kähler-Ensten metrcs are solutons to (2.1, but n general we do not know f they are the only solutons. It would be nterestng to undertake a more detaled study of these canoncal metrcs. 2.3 Delgne Parng We wll now descrbe the functonals E k n terms of the Delgne parng. Let π : X S be a flat projectve morphsm of ntegral schemes of relatve dmenson n. For each s, X s = π 1 (s s a projectve varety of dmenson n. Let L,..., L n be Hermtan lne bundles on X. We denote the correspondng Delgne parng - a Hermtan lne bundle on S - by L,..., L n (X /S. For ts defnton, we refer the reader to the references [De], [Zh] and [PhSt3]. For a smooth functon on X (or S denote by O( the trval lne bundle on X (or S wth metrc e. The only property of the Delgne parng we wll use s the change of metrc formula: L O(,..., L n O( n (X /S = L,..., L n (X /S O(E, where E s the functon on S gven by n E(s = j c 1(L O( c 1(L, X s j= <j >j and where c 1 (L = 1 2π 1 log L. In our case, X wll be the varety and S wll be a sngle pont. In the followng we wll omt any reference to X or S. We fx a metrc ω 2πc 1 (L for some lne bundle L on, and an element of P (, ω. Let L be the Hermtan lne bundle (L, h wth 1 log h = ω. Defne L by L = L O(. Let K -1 be the antcanoncal bundle K 1 equpped wth the metrc ω n, and let K -1 be the same bundle wth the metrc ω n. In other words, K -1 = K -1 O(log ωn ω n. Then we have the followng formula for the E k functonal. Proposton 2.1 n k k+1 {}}{{}}{ L,..., L, K -1,..., K -1 ( L,..., L (n kµ k n+1 n k k+1 {}}{{}}{ = L,..., L, K -1,..., K -1 ( L,..., L (n kµ k n+1 O( (2π n E k,ω( 7

8 Proof Defne a functonal a k,ω by n k k+1 k+1 {}}{{}}{ n k L,..., L, K -1,..., K {}}{{}}{ -1 = L,..., L, K -1,..., K -1 O From the change n metrc formula we see that Now calculate d a k,ω( = a k,ω ( = n k 1 j= n k j= j= j= n k 1 j= + j= ω j Rc(ωk+1 ω n j k 1 ( 1 (2π n a k,ω(. (2.2 ( ω n log ω n Rc(ω j ω n k Rc(ω k j. ω j Rc(ωk+1 ω n j k 1 j ω j 1 1 Rc(ω k+1 ω n j k 1 ( Rc(ω j ω n k j log (n k j= Integratng by parts, we have j= ( ω n ω n Rc(ω k j Rc(ω j 1 1 ( ω n k ( ω n log ω n n k 1 d a k,ω( = (j + 1 n k 1 + j= j (j + 1 j= j= Rc(ω k j Rc(ω j ω n k 1 1 Rc(ω k j. ω j Rc(ωk+1 ω n j k 1 ω j 1 Rc(ω k+1 ω n j k ( Rc(ω j ω n k Rc(ω k j j ( Rc(ω j 1 ω n k Rc(ω k j+1 8

9 + (n k Rearrangng ths expresson gves d a k,ω( = (n k ( Rc(ω k+1 Rc(ω k+1 ω n k 1. (k (n k (n k = (k (n k ω n k 1 Rc(ω k+1 ( Rc(ω k ω n k Rc(ω k+1 ω n k 1 Rc(ω k+1 ω n k 1 ( Rc(ω k ω n k Rc(ω k+1 ω n k 1. (2.3 We recognze ths (after dvdng by as the frst two terms of the dervatve of E k,ω. We are left wth the term nvolvng the constant µ k. As n [Zh], defne a functonal b ω by ( L,..., L 1 = L,..., L O (2π n b ω(. (2.4 From the change n metrc formula, we see that b ω s the well-known functonal: b ω ( = n = ω ωn. Its dervatve along a path n P (, ω s gven by d b ω( = (n + 1 ω n. (2.5 Comparng wth (1.1, we see from (2.3 and (2.5 that E k,ω ( = 1 a k,ω( + (n kµ k (n + 1 b ω(. The proposton then follows from ths along wth the defnng equatons (2.2 and (2.4. Q.E.D. 9

10 2.4 The I and J functonals The Aubn-Yau energy functonals I and J are defned as follows. For P (, ω, set I ω ( = 1 J ω ( = 1 Ther dfference s gven by Observe that n 1 = n 1 = (I ω J ω ( = n + 1 n 1 = 1 ω ω n 1, 1 ω ω n 1. n 1 ω ω n 1 n n J ω( (I ω J ω ( nj ω (. A well-known calculaton shows that for any famly of functons = t P (, ω, d (I ω J ω ( t = 1 t ( t t ω n t. (2.6 We now defne the noton of properness for a functonal. Followng Tan [T3], we say that a functonal T on a P (, ω s proper f there there exsts an ncreasng functon f : [, R, satsfyng f(x as x, such that for any P (, ω, T ( f (J ω (. The condton of properness s ndependent of the choce of metrc n the class. 3. The lower boundedness of the E k functonals In the followng, we wll denote by K + the set of metrcs ω n K wth strctly postve Rcc curvature. We wll wrte E for the space of metrcs ω K wth Rc(ω = ω. We wll show that the functonals E k are bounded below on K +. One can then easly extend to metrcs wth nonnegatve Rcc curvature by a peturbaton argument, usng (4.1 for small t. We need the followng result from [Baa] (see also [S] for a good exposton of ther work: 1

11 Lemma 3.1 Let ω K + and set ω = Rc( ω. If E s non-empty, there exsts a Kähler- Ensten metrc ω KE E and, for δ > suffcently small, a famly of metrcs ωɛ K+ for ɛ (, δ such that: ( ω ɛ ω n C as ɛ. ( Let f ɛ be the unque smooth functon on satsfyng Rc(ω ɛ ω ɛ = 1 f ɛ and e f ɛ (ω ɛ n = (ω ɛ n. Then for each ɛ (, δ, there exsts a smooth soluton ψ ɛ t (for t 1 of the equaton (ω ɛ + 1 ψ ɛ t n = e tψɛ t +f ɛ (ω ɛ n, wth ( Set ω ɛ = ω ɛ + 1 ψ ɛ. Then ω ɛ + 1 ψ ɛ 1 = ω KE, ω ɛ ω n C as ɛ. In order to make use of ths for the proof of Theorem 1.1, we need the followng lemma: Lemma 3.2 Let ω be a metrc n K, and let t be a soluton of ω n t = e tt+f ω n, (3.1 where f s gven by Then E k (ω, ω E k (ω, ω 1. Rc(ω ω = 1 f and e f ω n =. Before provng ths lemma, we wll complete the frst part of the proof of Theorem 1.1. From the above lemmas, gven ω K +, we have that By the cocycle condton, we have E k (ω ɛ, ω ɛ E k (ω ɛ, ω KE. E k (ω KE, ω ɛ E k (ω KE, ω KE. Lettng ɛ, we obtan from the contnuty of E k, E k (ω KE, ω E k (ω KE, ω KE. We need a lemma to show that the rght hand sde of ths equaton s zero. 11

12 Lemma 3.3 Let ω KE and ω KE be n E. Then E k (ω KE, ω KE =. Proof Chen and Tan [ChT1] defne an nvarant of a Fano manfold whch s generalzes the Futak nvarant. For each holomorphc vector feld X, they defne for any ω K, F k (X = (n k h X ω n + ( (k + 1 ω h X Rc(ω k ω n k (n kh X Rc(ω k+1 ω n k 1, where h X s a functon (unque up to constants satsfyng L X (ω = 1 h X. They prove that F k (X does not depend on the choce of ω n K, and that f E s nonempty, the nvarant vanshes. oreover, they show that f X s a holomorphc vector feld and {Φ(t} the one-parameter subgroup of automorphsms nduced by Re(X, then for any ω, d E k(ω, Φ t ω = 1 Re(F k(x. Gven ths, the lemma s an mmedate consequence of the theorem of Bando and abuch [Baa] that the space E s a sngle orbt of the acton of the group of holomorphc automorphsms of. Q.E.D. We wll now prove Lemma 3.2. Proof of Lemma 3.2 Let t be a soluton of (3.1. Dfferentatng (3.1 wth respect to t, we have Applyng the operator 1 log to (3.1 we obtan t t = t t t. (3.2 Rc(ω t = Rc(ω + t 1 t 1 f, whch, from the defnton of f, can be wrtten Rc(ω t = ω t + (t 1 1 t. (3.3 We wll need to make use of the followng fact whch s well known: for t a soluton of (3.1, the frst egenvalue λ 1 of t satsfes λ 1 t f t 1 (and the nequalty s strct f t < 1. An mmedate consequence of ths and (3.2 s the followng nequalty: t ( t t ω n t, (3.4 12

13 whch by (2.6 s equvalent to the fact that (I ω J ω ( t s ncreasng n t. We wll fnd a formula for the expresson (E k,ω ( 1 E k,ω ( by calculatng 1 d E k,ω( t. Lemma 3.2 and the frst part of Theorem 1.1 wll follow from (3.4 and the next lemma, whch s the key result of ths secton. In the statement and proof of the lemma, we wll smplfy the notaton by omttng the subscrpt t. Lemma 3.4 Let = t be a soluton of (3.1. Then E k,ω ( 1 E k,ω ( = k n (1 t( ω 1 k 1 (k 1 ω ω n 1. t= = Proof Usng (3.2 and (3.3, calculate where Now, 1 d E k,ω( = k n k 1 = 1 ( k (k k+1 ( k + 1 (n k A( = B( = C( = ( t (ω (1 t 1 k ω n k ((ω (1 t 1 k+1 ω k+1 ( 1 +1 (A( + B( C( + 1 = 1 n 1 ω n k 1 ( 1 +1 C(, (3.5 (1 t ( 1 ω n t(1 t ( 1 ω n (1 t ( 1 ω n. (1 t +1 ( 1 d (ωn 13

14 where = + 1 n n 1 n 1 (1 t ( 1 ω n (1 t +1 ( 1 ω n ( 1 ω n t= = (A( + B( C( D(, (3.6 n n D( = Usng the relaton (3.6 n (3.5, we obtan 1 d E k,ω( = 1 ( k + 1 (n k+1 ( k + 1 (n k = 1 k+1 [ ( k + 1 (n + 1 = + n(k + 1 C(1 + 1 n(k + 1 C(1 + 1 ( 1 ω n. t= ( 1 +1 C( ( 1 +1 C( + 1 (k + 1 usng the fact that ( k + 1 (n + 1 (k + 1 Now observe that and C(1 = 1 n ( k ( k (k + 1 ( 1 +1 C( ( k + 1 ( 1 +1 D( + 1 (n k ( k + 1 ] ( 1 C( ( k + 1 ( 1 +1 D( + 1 ( k + 1 ( 1 +1 D(, ( ( k (n k ( k + 1 =. (3.8 (1 t( ω n, (3.9 ( k + 1 ( 1 +1 D( + 1 ( k + 1 = ( 1 1 (ω ω 1 ω n t=

15 1 ( ( k = ( 1 +j 1 ω j ω n j j j= t= k 1 ( ( = k ( 1 +j 1 ω j ω n j j j= =j+1 t= k 1 = (j k 1 ω j ω n j 1, (3.1 t= j= where we are usng the dentty =j+1 ( k ( 1 j ( 1 +j = j k. Combnng (3.7, (3.9 and (3.1 completes the proof of lemma. Q.E.D. Ths proves the frst part of Theorem 1.1. We also have to show that f E k (ω KE, ω = for some ω wth Rc( ω then ω s Kähler-Ensten. We may assume that k 1. Let us also assume that Rc( ω > and we wll leave the sem-defnte case to the reader. Usng the notaton above we see that E k ( ω ɛ, ω KE as ɛ. But by Lemma 3.4 we see that f ω KE = ωɛ + 1 ɛ then ɛ must tend to a constant as ɛ tends to zero. Hence ω = ω KE. Q.E.D. 4. The lower bound of E 1 In ths secton we gve a proof of Theorem 1.2. Let us frst deal wth the easy case when c 1 ( =. From the formula obtaned n secton 2, we see that E 1,ωKE ( = 1 log ( ω n ω n Rc(ω ω n 1. Usng the defnton of Rc(ω and ntegraton by parts, we obtan: E 1,ωKE ( = 1 ( ω n ( ω n 1 log ω n log ω n ω n 1, as requred. We turn now to the case when c 1 ( >. Consder the equaton ω n ψ t = e tf+ct ω n, (4.1 15

16 where f satsfes Rc(ω ω = 1 f, and c t s the constant chosen so that e tf+ct ω n =. e f ω n = By the theorem of Yau [Ya1], we know that there s a unque ψ t wth ψ tω n = solvng (4.1 for t n [, 1]. Notce that ψ =. Dfferentatng the equaton we obtan and ψt ψ t = f + constant, (4.2 Rc(ω ψt = ω ψt + (1 t 1 f 1 ψ t. (4.3 The followng lemma s the key result of ths secton. Lemma 4.1 Let ψ = ψ t be a soluton of (4.1. Then In partcular, E k,ω (ψ 1 = 1 k 1 (n k( ψ ψ ω ω n 1 n + 1 ψ = 1 n (k + 1 n 1 ψ ψ ω ω n 1 n + 1 ψ =k k (1 t( ψ ψ 2 ωψ n + 1 ( k + 1 f ( 1 f ω n. + 1 E 1,ω (ψ 1. Gven ths lemma, we wll prove Theorem 1.2. Let ω K be gven. Then by Yau s theorem there exsts ω K + such that Rc( ω = ω. By the cocycle condton and Theorem 1.1, E 1 (ω KE, ω = E 1 (ω KE, ω + E 1 ( ω, ω E 1 ( ω, ω. We apply Lemma 4.1 wth ω = ω. Then we see that ω ψ1 = ω, and E 1,ω (ψ 1 = E 1 (ω, ω, thus completng the frst part of the proof of Theorem 1.2. The second part, that equalty mples Kähler-Ensten, wll follow by a smlar argument to that gven n secton 3. It remans to prove Lemma t=1 t=1

17 Proof of Lemma 4.1 Calculate usng (4.2 and (4.3: E k,ω (ψ 1 = k n k = 1 ( k (k + 1 ( ψ ψ(ω ψ + (1 t 1 f 1 ψ k ω n k ψ ( ψ (ω ψ + (1 t 1 f 1 ψ k+1 ω k+1 ψ Â( 1 ω n k ψ k+1 ( k + 1 (n k ˆB(, (4.4 where and Now calculate Â( = ˆB( = 1 1 f ( (1 t 1 f 1 ψ ω n ψ, ψ ( (1 t 1 f 1 ψ ω n ψ. ˆB( + 1 = 1 n = + 1 n 1 1 ((1 tf ψ ( (1 t 1 f 1 ψ ( d ωn ψ (f + ψ ( (1 t 1 f 1 ψ ω n ψ 1 ψ ( 1 ψ ω n n ψ 1 f ( 1 f ω n t=1 n = + 1 n Â( n ˆB( 1 n (( 1 Ĉ( + ˆD(, (4.5 where Ĉ( = ψ ( 1 ψ ω n ψ and ˆD( = t=1 akng use of the relaton (4.5 n (4.4, we see that f ( 1 f ω n. E k,ω (ψ 1 = 1 ( ( k n (k ˆB( + 1 ˆB( (( 1 Ĉ( + ˆD( 1 k+1 ( k + 1 (n k ˆB( 17

18 = 1 k+1 ( k n + 1 (k + 1 ˆB( 1 1 =2 1 k+1 ( k + 1 (n k ˆB( + 1 n(k + 1 = ˆB(1 + 1 ( k (k + 1 ˆB( k where we have used (3.8 agan. Now ˆB(1 = 1 1 t( ψ ψ n (1 2 ωψ n 1 n Usng (2.6 we have 1 n 1 ( k (( 1 Ĉ( + ˆD( ( k + 1 (( 1 Ĉ( ˆD(, (4.6 = n 1 ψ( ψ ψω n ψ 1 d (I ω J ω (ψ ψ( ψ ψωψ n. (4.7 = n (I ω J ω (ψ 1. (4.8 Let us put ths all together. By the same calculaton as n (3.1, we have 1 ( k + 1 ( 1 Ĉ( = k 1 (k = Combnng ths wth equatons (4.6, (4.7 and (4.8, 1 ψ ψ ω ω n 1 ψ E k,ω (ψ 1 = (k + 1(I ω J ω (ψ 1 (k (1 t( ψ ψ 2 ωψ n + 1 k 1 (k 1 ψ ψ ω ω n 1 ψ + 1 = ( k t=1. t=1 f ( 1 f ω n. (4.9 Combnng the frst and thrd terms completes the frst part of the lemma. Fnally, to show that E 1 (ψ 1, just observe that, by ntegraton by parts, the last term s nonpostve for k = 1. Notce that ths s the only step that fals for k > 1. Ths completes the proof of the lemma. Q.E.D. 18

19 5. Properness of E 1 In ths secton we wll prove Theorem 1.3 and Theorem 1.4. In fact we wll assume that has no nontrval holomorphc vector felds, whch wll prove only the frst part of these theorems. The proof of the second part n each case s dentcal except for an addtonal dffculty n applyng the mplct functon theorem n the method of contnuty at t = 1. For ths step, we refer the reader to [Baa] and [T2]. Proof of Theorem 1.3 We wll now prove part ( of Theorem 1.3. Let θ be n P (, ω KE and set ω = ω KE + 1 θ. We consder the famly of onge-ampère equatons (3.1, correspondng to ths partcular choce of ω. That s: ω n t = e tt+f ω n, where f s gven by Rc(ω ω = 1 f and e f ω n =. Snce there are no holomorphc vector felds, we can obtan a soluton t for t 1 [Baa]. oreover, 1 = θ + constant. We have the followng lemma. Lemma 5.1 E 1,ωKE (θ 2 1 (I ω J ω ( t. Proof We wll work n more generalty and derve a formula for E k,ωke (θ. Usng the cocycle condton for E k and Lemma 3.4 we have E k,ωke (θ = E k (ω KE, ω = E k (ω + 1 1, ω = E k,ω ( 1 = E k,ω ( k k 1 (k = 1 (1 t t ( t t ω n t 1 ω ω n 1. 19

20 Now observe that, usng (2.6, we have Hence k = (k + 1 = (k + 1 (1 t t ( t t ω n t 1 1 E k,ωke (θ = E k,ω ( + (k k 1 (k = (1 t d (I ω J ω ( t (I ω J ω ( t + (k + 1(I ω J ω (. 1 (I ω J ω ( (k + 1(I ω J ω ( 1 ω ω n 1. (5.1 Now, as n secton 4, let ψ t be a soluton of (4.1, wth ω as above: ω n ψ t = e tf+ct ω n, wth c t the approprate constant. Then observe that ψ 1 = + constant. Then makng use of the calculaton (4.9 from secton 4, we have E k,ω ( = E k,ω (ψ 1 = (k + 1(I ω J ω ( k 1 (k = ( k Combnng (5.1 and (5.2 we obtan E k,ωke (θ = (k (k ω ω n 1 (I ω J ω ( t + ( k (1 t( ψ ψ 2 ω n ψ f ( 1 f ω n. (5.2 (k f ( 1 f ω n. (1 t( ψt ψ t 2 ω n ψ t We see that for k = 1, the last two terms are nonnegatve, and ths proves the lemma. Q.E.D. 2

21 Now observe that we can now apply the argument of Tan [T2]. Recall that Tan proves the properness of a functonal F ω usng the formula: F ω (θ = 1 (I ω J ω ( t, wth t as above. The nequalty of Lemma 5.1 s even stronger than ths. We can drectly apply Tan s argument to obtan the followng: Theorem 5.1 There exsts δ = δ(n such that for every constant K > there exst postve constants C 1 and C 2 dependng on K such that for all θ n P (, ω KE satfsyng we have Here, the oscllaton osc s defned by osc (θ K(1 + J ωke (θ, (5.3 E 1,ωKE (θ C 1 J ωke (θ δ C 2. osc (θ = sup(θ nf (θ. We wll now apply the argument of Tan and Zhu [TZh] to show that E 1,ωKE s proper on the full space of potentals. The argument for E 1 dffers n only one step, but we wll outlne the whole argument here. We need some lemmas. Lemma 5.2 For t as above, there exsts a constant C 3 dependng only on ω KE such that for t 1 2, osc ( t 1 C 3 (1 + J ωke ( t 1. Proof We omt the proof, snce t can be found n [TZh]. Q.E.D. We need the followng lemma, whch we state n more generalty than s actually needed here. Lemma 5.3 Let t be a soluton of Then for t 1 t 2 1, we have E 1,ω ( t2 E 1,ω ( t1 ω n t = e tt+f ω n. = 2(1 t 2 (I ω J ω ( t2 + 2(1 t 1 (I ω J ω ( t1 t2 ( (1 t 2 t=t 2 2 (I ω J ω ( t 1 t ω n 1 t 1 t. t=t 1 21

22 Proof Calculate, as n secton 3, droppng the subscrpt t, E 1,ω ( t2 E 1,ω ( t1 = 2 t2 (1 t( t 1 ω n 1 t 1 n 1 t2 ((ω (1 t 1 2 ω 2 ωn 2 t 1 = 2 t2 (1 t 1 ω n t2 1 ω n 1 t 1 t 1 + n 1 t2 (1 t 1 ω n 1 t 1 1 t2 (1 t 2 1 d (ωn 1. t 1 Integratng by parts n t, and makng use of (2.6, we obtan E 1,ω ( t2 E 1,ω ( t1 = 2n t2 (1 t 1 ω n 1 = 2 t 1 (1 t2 t2 t 1 t 1 t ω n 1 t t=t 2 (1 t d (I ω J ω ( ( (1 t 2 t=t 1 t=t 2 1 ω n 1, t=t 1 and the lemma follows after ntegratng by parts n the frst term. Q.E.D. We wll use ths lemma n the proof of the followng: Lemma 5.4 E 1,ωKE ( t 1 2n(1 tj ωke (θ. Proof We use the cocycle condton for E 1 together wth Lemma 5.3 for t 1 = t and t 2 = 1 to obtan E 1,ωKE ( t 1 = E 1,ω ( t E 1,ω ( 1 1 = 2(1 t(i ω J ω ( t + 2 (I ω J ω ( s ds t (1 t2 + 1 t t ω n 1 t 22

23 2 1 t (I ω J ω ( s ds 2(1 t(i ω J ω ( 1 2n(1 tj ω ( 1 = 2n(1 tj ωke (θ where we have used the fact (3.4 that (I ω J ω ( t s ncreasng n t. Q.E.D. Now note from Lemma 5.2 that t 1 satsfes the condton (5.3 for K = C 3 and so we can apply Theorem 5.1 to obtan E 1,ωKE ( t 1 C 4 (J ωke ( t 1 δ C 5, for t 1 2 and for some constants C 4 and C 5 dependng only on ω KE. Applyng Lemma 5.2 and Lemma 5.4, we see that 2n(1 tj ωke (θ C 6 (osc ( t 1 δ C 7, for constants C 6 and C 7 dependng on ω KE. We need another lemma from [T2]. Lemma 5.5 For any t [, 1], 1 (I ω J ω ( s ds (1 t(i ω J ω ( 1 2n(1 tosc ( t 1. Proof Ths s essentally contaned n [T2], but we wll provde a proof for the reader s convenence. Frst, snce (I ω J ω ( t s ncreasng n t, From (2.6, 1 (I ω J ω ( s ds (1 t(i ω J ω ( t. Let us deal wth the frst term. Recall that (I ω J ω ( t (I ω J ω ( 1 = 1 1 s ( s s (ω s n ds t = 1 1 ( d s t ds (ω s n ds = 1 1 ( d ( s 1 t ds (ω s n ds ( d 1 ds (ω s n ds. (5.4 t ω s = ω + 1 s = ω KE + 1 ( s 1. 23

24 Then, regardng s 1 as a path of potentals for ω KE, 1 1 ( d ( s 1 t ds (ω s n ds = 1 1 ( d ( s 1 ds (ω KE + 1 ( s 1 n ds = t 1 t d ds ((I ω KE J ωke ( s 1 ds = (I ωke J ωke ( t 1 (5.5 Let us turn now to the second term of (5.4. Integratng by parts n t we see that 1 1 t = 1 = 1 = 1 1 ( d ds (ω s n ds 1 ((ω 1 n (ω t n n (1 t (ω 1 (ω t n 1 = n 1 ( 1 t (ω 1 ω (ω 1 (ω t n 1 = 2n osc( t 1. (5.6 Insertng (5.5 and (5.6 n (5.4, we have Q.E.D. (I ω J ω ( 1 (I ω J ω ( t 2n osc( t 1. We can now complete the proof of Theorem 1.3. We have for t 1 2, Settng E 1,ωKE (θ 2 1 (I ω J ω ( t 2(1 t(i ω J ω ( 1 4n(1 tosc ( t 1 2(1 t J ωke (θ 4n(1 tc 1/δ 6 (2n(1 tj ωke (θ + C 7 1/δ. n 1 t = 1 C 8 (1 + J ωke (θ 1+δ, for C 8 >> max{c 6, C 7 } completes the proof. Q.E.D. 24

25 Proof of Theorem 1.4 We have to show that f E 1 s proper then there exsts a Kähler-Ensten metrc on. Fx a metrc ω and consder agan the famly of onge-ampère equatons (3.1: ω n t = e tt+f ω n, where f s gven by Rc(ω ω = 1 f and e f ω n =. If we have a soluton for t = 1 then ω s a Kähler-Ensten metrc. We use the contnuty method. We have a soluton at t = by Yau s theorem [Ya1]. It s well known that we can fnd a soluton t for t 1 as long as we can bound the C norm of t unformly n t. oreover, t C C(1 + J ω ( t, for a constant C dependng only on ω. So t suffces to show that J ω ( t s unformly bounded from above. Snce E 1,ω s proper, t suffces to show that E 1,ω ( t s bounded from above unformly n t. We wll use the followng lemma. We apply Lemma 5.3 n the case when t 2 = t and t 1 =. Then E 1,ω ( t E 1,ω ( = 2(1 t(i ω J ω ( t + 2(I ω J ω ( t 2 (I ω J ω ( s ds (1 t t t ω n 1 1 ω n 1 t 2(I ω J ω ( 2 (I ω J ω ( s ds 1 1 ω n 1, where we are usng the defnton of I ω J ω. It then follows mmedately that E 1,ω ( t s bounded from above by a constant ndependent of t. Q.E.D. References [Au] Aubn, T. Equatons du type onge-ampère sur les varétés Kählerennes compacts, Bull. Sc. ath. 12 (1976, [Ba] Bando, S. The K-energy map, almost Ensten Kähler metrcs and an nequalty of the yaoka-yau type, Tohuku ath. Journ. 39 (1987,

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