A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS

Transcription

1 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS HAOZHAO LI, YALONG SHI 2, AND YI YAO ariv:3.032v2 [math.dg] 28 Jun 204 Abstract. In this paper, we give a criterion for the properness of the K-energy in a general Kähler class of a compact Kähler manifold by using Song-Weinkove s result in [24]. As applications, we give some Kähler classes on CP 2 #3CP 2 and CP 2 #8CP 2 in which the K-energy is proper. Finally, we prove Song-Weinkove s result on the existence of critical points of Ĵ functional by the continuity method. Contents. Introduction 2. Preliminaries 5 3. Proofs of Theorem. and Corollary Toric varieties 9 5. An example of Dervan 3 6. Existence of critical points by the continuity method 4 References 9. Introduction The behavior of the K-energy plays an important role in Kähler geometry. It is conjectured by Tian [27] that there exists a constant scalar curvature Kähler (csck) metric in a Kähler class Ω if and only if the K-energy is proper on Ω. For the Kähler-Einstein case, this was proved by Tian when M has no nontrivial holomorphic vector fields. For the general case, Chen-Tian [4] showed that the K-energy is bounded from below if M has a csck metric. On toric manifolds, using Donaldson s idea in [8] Zhou-Zhu [33] gave a sufficient condition on the properness of the (modified) K-energy on the space of invariant potentials. In a series of papers [7][8][9], S. Paul gave a sufficient and necessary condition on the lower boundedness and properness of the K-energy on the finite dimensional spaces of Bergman metrics. However, it is still difficult to analyze the behavior of the K-energy on Research partially supported by NSFC grant No and No Research partially supported by NSFC grants No. 0206, No. 743 and by a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

2 2 HAOZHAO LI, YALONG SHI 2, AND YI YAO general Kähler manifolds. In this paper, we give a sufficient condition for the properness ofthek-energyinageneralkählerclassonacompactkählermanifoldbyusingthej-flow. The J-flow was introduced by Donaldson[7] and Chen [2] independently, and it was used to obtain the properness or the lower bound of the K-energy on a compact Kähler manifold with negative first Chern class. As pointed out by Chen [2], the J-flow is a gradient flow of the functional Ĵ ω,χ0, which is strictly convex along any C, geodesics. Thus, if there is a critical point of Ĵ ω,χ0 in a Kähler class, then Ĵω,χ 0 is bounded from below and the K- energy is proper when the first Chern class is negative by the formula of K-energy relating Ĵ. Therefore, to obtain the properness of the K-energy it suffices to study the existence of the critical point of Ĵω,χ 0. In [24] Song-Weinkove gave a sufficient and necessary condition to this problem, and their result directly implies that the K-energy is proper on a Kähler class [χ 0 ] on a n-dimensional Kähler manifold of c () < 0 with the property that there is a Kähler metric χ [χ 0 ] such that (.) ( n c () [χ 0 ] n ) χ +(n )c [χ 0 ] n () χ n 2 > 0. Moreover, Song-Weinkove asked whether the conclusion still holds if the inequality (.) is not strict. In [0] Fang-Lai-Song-Weinkove studied the J-flow on the boundary of the Kähler cone and gave an affirmative answer in complex dimension 2. Later, Song-Weinkove [25] gave a result on the properness of the K-energy on a minimal surface with general type. Besides, Lejmi-Székelyhidi [6] studied the relation of the convergence of J-flow to a notion of stability. To state our main results, we recall Tian s α-invariant for a Kähler class [χ 0 ]: { } α ([χ 0 ]) = sup α > 0 C > 0, e α(ϕ supϕ) χ n 0 C, ϕ H(,χ 0 ), where H(,χ 0 ) denotes the space of Kähler potentials with respect to the metric χ 0. For any compact subgroup G of Aut(), and a G-invariant Kähler class [χ 0 ], we can similarly define the α,g invariant by using G-invariant potentials in the definition. Theorem.. Let be a n-dimensional compact Kähler manifold. If the Kähler class [χ 0 ] satisfies the following conditions for some constant ǫ : () 0 ǫ < n+α n ([χ 0 ]), (2) πc () < ǫ[χ 0 ], (3) ( n πc () [χ 0 ] n ) +ǫ [χ [χ 0 ] n 0 ]+(n )πc () > 0,

3 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 3 then the K-energy is proper on H(,χ 0 ). If instead of (), we assume [χ 0 ] is G-invariant for a compact subgroup G of Aut(), and 0 ǫ < n+ α n,g([χ 0 ]), then the K-energy is proper on the space of G-invariant potentials. In the polarized algebraic case, our theorem translates into the following Corollary.2. Let be a n-dimensional compact Kähler manifold and L an ample holomorphic line bundle on. If there is a positive number ǫ > 0 such that the following conditions hold: () α (ǫπc (L)) > n n+, (or equivalently, α (πc (L)) > nǫ n+,) (2) K +ǫl > 0, (3) nk L n +ǫl n L n L (n )K > 0, then the K-energy is proper on the Kähler class πc (L). A direct corollary of Theorem. is the following result, which gives a partial answer to the question of Song-Weinkove in [24] and generalize a result of Fang-Lai-Song-Weinkove [0] to higher dimensions. Corollary.3. Let be a compact Kähler manifold with c () < 0. If the Kähler class [χ 0 ] satisfies (.2) n c () [χ 0 ] n then the K-energy is proper on H(,χ 0 ). [χ 0 ] n [χ 0 ]+(n )c () 0 Note that when the strict inequality holds, our condition (.2) is stronger than (.). For some technical reasons, we cannot weaken (.2) to Song-Weinkove s original condition. Moreover, in Theorem. we don t require the condition that c() < 0. In the case of c () > 0, we have the following result. Although this result is very simple and its original proof is very direct (see, for example, Tian s book [27], page 95), we feel it is still interesting to write it as a corollary of Theorem.. Corollary.4. Let be a compact Kähler manifold with c () > 0. If the α-invariant α (πc ()) > n n+, then then the K-energy is proper on the Kähler class πc (). An application of Corollary.3 is the following result, which says that the properness of the K-energy is not a sufficient condition of the smooth convergence of the J-flow. This answers a question of J. Ross [2]. This result is also implied by Corollary.2 of [0]. Corollary.5. There exists a compact Kähler surface with a Kähler class Ω such that the K-energy is proper on Ω but the J-flow doesn t converge smoothly.

4 4 HAOZHAO LI, YALONG SHI 2, AND YI YAO One might ask whether the conditions of Theorem. are optimal. We calculate two concrete examples here and apply Theorem. to determine the Kähler classes on which the K energy is proper. Let be the blowup of CP 2 at three general points and E,...,E 3 the exceptional divisors of the blowing up map. Denote by F i, i =,2,3 the strict transforms of lines through two of the three blowing up centers. Consider the class L λ = (E +E 2 +E 3 )+λ(f +F 2 +F 3 ) for a positive rational number λ. Then applying Theorem., if λ satisfies 5 (.3) 6 < λ < 6 5, then the K-energy is proper on πc (L λ ). The details are contained in section 4. This example was also studied by Zhou-Zhu in [33]. They analyzed the expression of the K- energy carefully on toric manifolds and showed that the K-energy is is proper on G- invariant metrics if 0 (.4) < λ < We see that the conditions (.4) is less restrictive than (.3). Therefore, the conditions in Theorem. is not sharp. Theorem. shows that the K-energy and α-invariants are closely related. In [6] Dervan proved that the α invariant also closely relates the K-stability. In [6], Dervan studied the α-invariant and K-stability for general polarizations on Fano manifolds. An example of [6] is the the blowup of CP 2 at eight points in general positions. Let E i be the exceptional divisors, and L λ = 3H 7 i= E i λe 8, where λ is a positive rational number. Dervan proved that when (.5) (0 0) < λ < 0 2.6, (,L λ ) is K-stable. Applying Theorem. to this example, we know that the K-energy is proper on πc (L λ ) if 4 (.6) 5 < λ < 0 9. Note that the interval (.6) is strictly contained in (.5). According to Tian s conjecture and general Yau-Tian-Donaldson conjecture, this example hints that the conditions in Theorem. is not optimal. There are some overlap regions between our results and Dervan s for the properness of the K-energy and K-stability. All these results provide some support to general Yau-Tian-Donaldson conjecture for general csck metrics. Finally, we reprove the existence of the critical point of Ĵ functional under the condition (2.4) by the continuity method. In [9], Fang-Lai-Ma discussed a class of fully nonlinear

5 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 5 flows in Kähler geometry, which includes the J-flow as a special case. In Remark.3 of [9], they asked whether the critical points of those fully nonlinear flows can be solved by using the elliptic method instead of the geometric flow method. Later, in a series of papers [3][4][5] Guan and his collaborators gave some C 2 estimates for these critical points on Hermitian manifolds. Then Sun proved Song-Weinkove s result by the elliptic method on general Hermitian manifolds [26]. Here we give a different proof only using the estimates in [24] [30] and [3]. Theorem.6. (cf. [24]) If there is a metric χ [χ 0 ] satisfying (ncχ (n )ω) χ n 2 > 0, then there is a smooth Kähler metric χ [χ 0 ] satisfying the equation ω χ n = cχ n. In a forthcoming paper, we will generalize the results in this paper to the properness of the log K-energy and discuss the existence of critical points of the J flow with conical singularities. Our method can also be applied to study the modified K-energy for extremal Kähler metrics and we will discuss this elsewhere. Acknowledgements: The authors would like to thank Professor Julius Ross for bringing our attention to this problem and many helpful discussions. We are also grateful to the anonymous referee for valuable comments and suggestions. 2. Preliminaries In this section, we recall some basic facts on J-flow. We follow the notations in [24]. Let (,ω)bean-dimensionalcompactkählermanifoldwithakählerformω = g 2 i jdz i d z j, and χ 0 another Kähler form on. We denote by H(,χ 0 ) the space of Kähler potentials H(,χ 0 ) = {ϕ C (,R) χ ϕ = χ ϕ > 0}. The J-flow is defined by (2.) ϕ t where c is a constant defined by ω χn ϕ = c, ϕ χ n t=0 = ϕ 0 H, ϕ (2.2) c = [ω] [χ 0] n [χ 0 ] n. A critical point of J-flow is a Kähler form χ satisfying (2.3) ω χ n = cχ n.

6 6 HAOZHAO LI, YALONG SHI 2, AND YI YAO Donaldson [7] showed that a necessary condition for the existence of the critical metrics (2.3) is [cχ 0 ω] > 0, and Chen [2] showed that it is also sufficient in complex dimension 2. In a series of papers [30][24] Weinkove and Song-Weinkove obtain a sufficient and necessary condition for any dimension: Theorem 2.. (cf. [24])The following conditions are equivalent: (). There is a metric χ [χ 0 ] satisfying (2.4) (ncχ (n )ω) χ n 2 > 0. (2). For any initial data ϕ 0 H, the J-flow (2.) converges smoothly to ϕ H with the limit metric χ satisfying (2.3). (3). There is a smooth Kähler metric χ [χ 0 ] satisfying the equation (2.3). The convergence of the J-flow can be used to determine in which Kähler class the K- energy is proper or bounded from below. Note that the J-flow is the gradient flow of the functional (2.5) Ĵ ω,χ0 (ϕ) = 0 ϕ (ω χn ϕ cχ n ϕ t ) dt (n )!. When ω is a positive (,) form, the functional Ĵ ω,χ0 is strictly convex along any C, geodesics(cf. [2]). Therefore, under the assumption () of Theorem 2., the critical point of Ĵ ω,χ0 exists and Ĵω,χ 0 is bounded from below. When c () < 0, we can choose ω = Ric(χ 0 ) > 0, then the K-energy can be written as (2.6) µ χ0 (ϕ) = log χn ϕ χ n 0 χ n ϕ n! +Ĵω,χ 0 (ϕ). where ω = Ric(χ 0 ) > 0. Since the first term of this formula is always proper (see Lemma 4. of [24]), one can conclude that the K-energy is proper on [χ 0 ], provided (2.4) holds for [ω] = πc (). 3. Proofs of Theorem. and Corollary.3-.5 In this section, we prove Theorem. and Corollary Proof of Theorem.. We focus on the G = {} case, the proof in the general case is identical. Recall the Aubin-Yau functionals I χ0 (ϕ) = ϕ( χn 0 n! χn ϕ n! ), ϕ t J χ0 (ϕ) = dt t (χn 0 n! χn ϕ n! ). 0

7 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 7 Direct calculation shows that I χ0 (ϕ) J χ0 (ϕ) = = 0 0 dt ϕ t χ ϕ ϕ χn ϕ n! ϕ t (χn ϕ χ 0 χ n ϕ ) dt (n )!. By the assumption (2), there exists χ [χ 0 ] such that ω := Ric(χ ) satisfies (3.) ω +ǫχ 0 > 0. By the assumption (3), there exists χ, χ [χ 0 ] such that where ω := Ric( χ) and Note that c+ǫ > 0 by assumption (2). Set χ := n(c+ǫ) Then χ [χ 0 ] and χ satisfies (nc+ǫ) χ (n ) ω > 0, c = πc () [χ 0 ] n [χ 0 ] n. ( (nc+ǫ) χ +(n )ǫχ 0 +(n ) 2 log χn χ n ). (3.2) n(c+ǫ)χ (n )(ω +ǫχ 0 ) = (nc+ǫ) χ (n ) ω > 0. In particular, χ > 0. We define the modified Ĵ functional by ( ) Ĵω ǫ,χ 0 (ϕ) = Ĵω,χ 0 (ϕ)+ǫ I χ0 (ϕ) J χ0 (ϕ) = 0 ϕ ( ) (ω +ǫχ 0 ) χϕ n (c+ǫ)χ n ϕ t dt (n )!, which is exactly the functional Ĵ ω +ǫχ 0,χ 0 defined by (2.5). Thus, by Chen s result in [2] if there is a Kähler metric χ satisfying (3.3) (ω +ǫχ 0 ) χ n = (c+ǫ)χ n, then Ĵǫ ω,χ 0 is bounded from below on H(,χ 0 ). By Theorem 2. the critical metric (3.3) exists if there exists a Kähler metric χ [χ 0 ] such that ( ) (3.4) n(c+ǫ)χ (n )(ω +ǫχ 0 ) χ n 2 > 0. Clearly, (3.4) can be implied by (3.2). Therefore, if (3.) and (3.2) hold, then Ĵǫ ω,χ 0 is bounded from below on H(,χ 0 ) and we have ( ) (3.5) Ĵ ω,χ 0 (ϕ) ǫ I χ0 (ϕ) J χ0 (ϕ) C, ϕ H(,χ 0 ).

8 8 HAOZHAO LI, YALONG SHI 2, AND YI YAO Next, we claim that for ω := Ric(χ 0 ), there is a constant C(χ,χ 0 ) such that for any ϕ H(,χ 0 ) (3.6) Ĵω,χ 0 (ϕ) Ĵω,χ 0 (ϕ) C(χ,χ 0 ). In fact, by the explicit expression of the Ĵ functional from [2] we have n Ĵ ω,χ0 (ϕ) = ϕω χ n p 0 ( ϕ) p nc (p+)!(n p )! = = p=0 n p=0 n p=0 c p c p ϕω χ n p 0 (χ ϕ χ 0 ) p nc dt 0 ϕω χ n p 0 χ p ϕ nc 0 dt ϕ t t χ n ϕ t n!, ϕ t t χ n ϕ t n! 0 dt where c p,c p are universal constants. Since ω ω = f where f = log χn, we have 2 χ n 0 n c p Ĵω,χ 0 (ϕ) Ĵω,χ 0 (ϕ) ϕ f χ n p 0 χ p ϕ 2 Therefore, (3.6) is proved. = p=0 n p=0 c p f (χ ϕ χ 0 ) χ n p 0 χ p ϕ Now using Tian s α-invariant we have (see Lemma 4. of [24] ) log χn ϕ χ n ϕ αi χ n χ0 (ϕ) C 0 n! (3.7) n+ n α (I χ 0 (ϕ) J χ0 (ϕ)) C, ϕ H for any α (0,α [χ0 ]()). Combining the inequalities (3.5)-(3.7) we have Therefore, if α ([χ 0 ]) > n n+ µ χ0 (ϕ) = log χn ϕ χ n ϕ χ n 0 n! +Ĵω,χ 0 (ϕ) log χn ϕ χ n ϕ χ n 0 n! +Ĵω,χ 0 (ϕ) C(χ 0,χ ) ( n+ )( ) n α ǫ I χ0 (ϕ) J χ0 (ϕ) C. ǫ then the K energy is proper. C f C 0. ϕ t t χ n ϕ t n!

9 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 9 Remark 3.. We see from the above proof that the K-energy is proper if (3.) and (3.4) hold. However, we are unable to show that (3.4) holds if there exists a Kähler metric χ [χ 0 ] such that ( n πc () [χ 0 ] n ) χ (n )ω +ǫχ χ n 2 > 0. [χ 0 ] n If it were true, then Song-Weinkove s question would be answered as a corollary. Proof of Corollary.3. Let ǫ (0, n+ n α ([χ 0 ])). Since the manifold has negative first Chern class, the condition (2) in Theorem. automatically holds. The third condition in Theorem. follows directly from (.2). Thus, the corollary is proved. Proof of Corollary.4. Let [χ 0 ] = πc (). By the assumption α ([χ 0 ]) > n, we can n+ choose ǫ (, n+α n ([χ 0 ])). Therefore, the three conditions of Theorem. hold and the corollary is proved. Proof of Corollary.5. In complex dimension 2, the J flow converges smoothly if and only if the inequality (3.8) 2 c () [χ 0 ] n [χ 0 ] n χ +c () > 0 holds for some χ [χ 0 ]. By Corollary.4, the K-energy is still proper on H(,χ 0 ) if the inequality (3.8) is not strictly. Therefore, on any Kähler class lying on the boundary of the cone defined by (3.8), the K-energy is proper but the J-flow doesn t converge smoothly. 4. Toric varieties In this section, we apply Theorem. to projective toric manifolds. First we recall some basic facts on toric varieties from Fulton s book []. Let N be a lattice of rank n, M = Hom(N,Z) is its dual. The complex torus group T is defined to be T = N Z C = Hom(M,C ), and we have the real torus group T R = N Z S = Hom(M,S ). A complete toric variety is defined by a fan in N R = N Z R, consists of strongly convex rational polyhedral cones, such that the union of these cones is the whole of N R. We assume each cone is generated by a subset of a Z-basis of N, then is smooth. Each -dimensional cone ρ i of corresponds to an irreducible divisor D i and it is well known that the Picard group of is generated by these D i s.

10 0 HAOZHAO LI, YALONG SHI 2, AND YI YAO Let u i N be the primitive generator of the -dimensional cone ρ i. For any T-invariant divisor D = i a id i, a i Z, we associate to it a piecewise linear function φ D on N R, defined by φ D (u i ) = a i. The divisor D is ample if and only if φ D is strictly concave in the sense that for any n-dimensional cone σ of, the graph of φ D on the compliment of σ is strictly under the graph of the linear function u σ whose restriction on σ equals φ D. For such a D, we can also associate to it a polytope P D in M R, defined by We have P D := {m M R m,v φd (v), v N R }. H 0 (,O(D)) = m P D M Cχ m, where χ m is the rational function on defined by m M. If we restrict χ m to (C ) n, the χ m has the form z m zn mn. Besides the natural T-action, the toric manifold also has some discrete symmetries from the fan. Let W = {g GL(n,Z) g preserves }. Since each g W is decided by a permutation of {u i }, so W is finite. Every g induces a ḡ Aut(). For a given ample divisor D, we consider the subgroup of W preserving the class of D: K D := {g W ḡ D D}, where means linearly equivalent. Note that ḡ D corresponds to the function φ D g, so we have a combinatorial characterization of K D : K D = {g W m M, s.t. φd g = φ D + m, }. Let G be the compact subgroup of Aut() generated by T R and K D, we compute the α-invariant α G (π[d]), extending previous works of Song [23] and Cheltsov-Shramov [] in the toric Fano case. First, we make a normalization: assume the barycenter of P D is the origin of M R. This is equivalent to taking power of the line bundle O(D) and change the divisor in its linear equivalent class. Since we are computing the α-invariant, this does note lose any generality. Under this assumption, for any g K D, the induced linear action g on M R preserves P D. Actually, by definition of K D, g P D is a translation of P D. However, since g is linear, the barycenter of g P D is also the origin. This implies g P D = P D. Now the result is: In Fulton s terminology, it is called convex. We choose this name according to the usual notation.

11 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS Theorem 4.. The α G invariant equals α G (π[d]) = min min, i y PD K y,u i +a i where P K D is the set of K D (induced action on M R ) fixed points in P D. Proof. To prove this, we use the result of J.-P. D ly [5], saying that α G (π[d]) = inf k Z + inf lct( Σ kd k Σ ) where the second infimum is taken for all G invariant linear systems. If Σ Σ 2 kd, obviously lct( Σ k ) lct( Σ k 2 ), so we only need take all G invariant and irreducible linear systems Σ. Note that H 0 (,kd) = span{s m m kp D M} is just the decomposition into one dimensional invariant subspaces of the T-action, and the torus acts on these lines with different characters. Take a G invariant linear system Σ H 0 (,kd), Σ must be spanned by {s m,m Γ} for some Γ kp D M, and Γ is K D invariant. We can assume Σ is irreducible, so Γ is an orbit of the K D action. Denoted #Γ = N, then s = s m m Γ H 0 (,knd), and it spans a one dimensional G-invariant linear system. Moreover, we have lct( (s)) kn lct( Σ ) by Hölder inequality. So without loss of generality, we can k take Σ to be one dimensional in the following. Let s H 0 (,kd). Assume s corresponds to the lattice point m kp D M, and m is fixed by K D. Now the divisor of s is given by (s) = i ( m,u i +ka i )D i. Since is smooth, D i s have simple normal crossing intersections with each other. So we have lct( k (s)) = min. k i m,u i +ka i From this and the above discussion, we have α G (π[d]) = inf k Z + m inf min k P D k M m,u = min min k i +a i i y PD K i y,u i +a i. Now we consider a concrete example: the blowup of CP 2 at three general points. The fan of M is generated by the following primitive vectors: u = (,0), u 2 = (,), u 3 = (0,), u 4 = (0, ), u 5 = (, ), u 6 = (0, ).

12 2 HAOZHAO LI, YALONG SHI 2, AND YI YAO They correspond to all the ( )-curves D,...,D 6 on M. The intersection numbers satisfy : D i D i =, D i D i+ =,i =,...,6, where D 7 = D is understood. Note also that the anti-canonical divisor is given by K = D + +D 6. The bundle we choose is L λ = (D +D 3 +D 5 )+λ(d 2 +D 4 +D 6 ), where λ Q. This is the same class as we mentioned in the introduction. Actually if we blow down D,D 3 and D 5, we will get CP 2, and the image of D 2,D 4,D 6 are lines. Proposition 4.2. Under the above assumptions, if 5 < λ < on the space of G-invariant potentials for the class πc (L λ ). then the K-energy is proper Proof. For this class to be ample, we need the function φ Lλ to be strictly concave, and this is easily seen to be 2 < λ < 2. Nowweapplyourtheoremtothiscase. WriteD = al λ forsomepositivea. Theα-invariant can be computed using our Theorem 4.: Since the elements of K D can be enumerated, one can check easily that P K D = {0 M R}. So α G (π[d]) = min{ a, λa }. So the condition α G > 2 translates to 0 < a < 3,0 < a < 3. Similarly, by considering the 3 2 2λ concexity of φ K+D, we can translate the condition K +D > 0 into The last condition says that (2 λ)a >, (2λ )a >. ( R)D+K > 0, where R is the mean value of the scalar curvature of the class π[d]. Direct computation 2 shows that 2(+λ) R = aλ(4λ λ 2 ). Again, use the piecewise linear function φ ( R)D K, we get the condition In conclusion, we have 2(λ+) 4λ λ 2 + λ 2 < a, 2 < λ < 2, 2(λ+) 4λ λ + 2 2λ < a. 2 Just compute the intersection number, or use the fact that R = V ol( PL) Vol(P L), where Vol( P L) is computed using Donaldson s special boundary measure, see [8] and[33].

13 and A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 3 { max 2 λ, 2λ, 2(λ+) 4λ λ + 2 λ 2, 2(λ+) 4λ λ + } < a 2 2λ < min{ 3 2, 3 2λ }. Then for any 5 < λ < 6, we can find a suitable a satisfying these conditions. So we can 6 5 apply Theorem. (or Corollary.2 ) to conclude that the K-energy is proper on the space of G-invariant potentials for the class πc (al λ ) and hence on πc (L λ ). 5. An example of Dervan In this section, we would like to compare our result with that of Dervan [6] on the α-invariant and K-stability for general polarizations on Fano manifolds. His sufficient condition involves the quantity µ(,l) := K L n L n. And he proved that if α (L) > n µ(,l) and K n+ n µ(,l)l, then (,L) is n+ K-stable. This condition is quite similar to ours, but at present we don t know whose condition is stronger. We know compare our result with that of Dervan on a concrete example, the Del Pezzo surface of index one. It is the blowup of CP 2 at eight points in general positions. Let E i be the exceptional divisors, and L λ = 3H 7 i= E i λe 8, then L = K and we set L = al λ, where a and λ are both positive and λ Q. We know the exceptional curves on are E i s and the strictly transforms of following curves in CP 2 : lines through 2 points, conics through 5 points, cubics through 7 points, vanishing doubly at of them, quartics through 8 points, vanishing doubly at 3 points, quintics through 8 points, vanishing doubly at 6 points, sextics through 8 points, vanishing doubly at 7 points and triply at another point. Proposition 5.. Under the above assumptions, if 4 < λ < the Kähler class πc (L λ ). the K-energy is proper on Proof. ByKleiman sampleness criterion, weknowthatl λ isamplewhenλ < 4. According 3 to Dervan[6], we have α(π[l λ ]) min{, 2 λ }.

14 4 HAOZHAO LI, YALONG SHI 2, AND YI YAO Now we look for the λ for which the class L = al λ satisfies our conditions for some a > 0. The condition on the α-invariant gives a min{, 2 λ } > 2 3. For the ampleness of L+K, by computing the intersection number with the above exceptional curves we have a >, when λ, 4 3λ and a >, when λ <. λ For the last condition, denote b = 4 2λ a, we require the divisor 3( b)h 7 2 λ 2 i= ( b)e i ( λb)e 8 to be ample. This is equivalent to the conditions b <, when λ, λ and b <, when λ <. 4 3λ From these inequalities, now we can conclude that when 4 < λ < 0, the K-energy is 5 9 proper on the Kähler class 3H 7 i= E i λe Existence of critical points by the continuity method In this section, we prove Theorem.6 by the continuity method by using the estimates of [24][30] and [3]. Proof of Theorem.6. By the assumption, we can choose the reference metric χ 0 as χ in (2.4). Consider the continuity equation (6.) ω t χ n t = c t χ n t, where χ t = χ ϕ t, ω t = ( t)χ 0 +tω. Clearly, ω 0 = χ 0 and ω = ω. The constant c t is given by (6.2) c t = [ω t] [χ 0 ] n [χ 0 ] n = ( t)+c t, where the constant c is given by (2.2). We denote by g t the corresponding metric tensor of the Kähler form ω t. Clearly, ϕ = 0 is a solution to the equation (6.) when t = 0. Define the set S = {s [0,] the equation (6.) has a solution when t = s}.

15 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 5 We first prove the openness of S. Consider the operator L t (ϕ) = χ i j ϕ g t,i j : U C0 α, where U := {ϕ C 2,α (;R) χ ϕ > 0}/R, and C0 α := Cα (;R)/R. Here we take the quotient space norm, and the Hölder semi-norms are taken with respect to a fixed metric, say, χ 0. For any S, the linearization of the operator L t (ϕ) at (,ϕ t0 ), DL (t0,ϕ t0 ) : C 2,α 0 C0 α is given by DL (t0,ϕ t0 )(f) = h p q f p q, h p q t = χ i q t χ p j t g t,i j. Here C 2,α 0 := C 2,α (;R)/R. It is obvious to see that DL (t0,ϕ t0 ) (We write DL for short.) is elliptic. By strong maximum principle, its kernel in C 2,α 0 is trivial. We claim that DL is a self-adjoint operator in L 2. Actually, for any real-valued smooth function η, we have 3 (DL(f),η) L 2 = ηdl(f) χn = ηχ i q t n! 0 χ p j χ n t g 0 t0,i jf p q n! = η p χ i q χ p j χ n t g 0 t0,i jf q n! + ηχ i q χ p j χ n t g 0 t0,i j,pf q n! Here all the covariant derivatives are taken with respect to χ t0. Observe that χ i j g t0,i j = c t0, so χ i j g t0,i j,p = 0. Since both ω t0 and χ t0 are Kähler, we have g t0,i j,p = g t0,p j,i by comparing the expressions in local coordinates. So χ p j g t0,i j,p = χ p j g t0,p j,i = 0. We have (DL(f),η) L 2 = η p χ i q χ p j χ n t g 0 t0,i jf q n!. Then another integration by parts gives the result: (DL(f),η) L 2 = (f,dl(η)) L 2. Now we conclude that the kernel of the adjoint of DL is also trivial (modulo constant functions), and hence DL (t0,ϕ t0 ) is invertible. By standard inverse function theorem, the set S is open. Since 0 S, by the openness there exists t > 0 such that (6.) has a smooth solution for t [0,t ). By Lemma 6. and Lemma 6.2 below, for any t S the metric χ t is uniformly equivalent to χ 0. By the elliptic estimates in [2] we can get C 2,α (,ω) estimates of ϕ t. 4 This together with the classical Schauder estimates can improve the regularity to C. Thus, S is closed and the theorem is proved. 3 We can also prove this by observing that (DL(f),η) L 2 = η < 2 f,ω χ t > n χt0 n! = η 2 f ω t. Here is the Hodge star operator associated with χ t0. Since and commute with It is obvious that this equals f 2 η ω t = (f,dl(η)) L 2. A good reference for Hodge star operator on Kähler manifold is [32]. 4 The adaptation of the classical Evans-Krylov theorem to the complex case has been carried out by Siu Y.-T. in [22] P00-07.

16 6 HAOZHAO LI, YALONG SHI 2, AND YI YAO Lemma 6.. Fix some t 2 (0,t ). Let ϕ t be a solution of (6.) with t t 2 > 0. If the inequality (2.4) holds, then there exist constants A,λ,C > 0 independent of t such that Λ ω χ t λλ ωt χ t λce A(ϕt inf ϕ t), t t 2. Proof. Here we follow the estimates in [24] and [30]. Sometimes we omit the subscript t for simplicity. We choose χ 0 to be the χ in (2.4). Write We calculate f = n χk j χ i lg t,i j k lf = n hk l k lf. (6.3) (Λωt χ) = i j hk lr (g n k l t )χ i j + i j hk lg t k lχ n i j, where R i j k l (g t ) denotes the curvature tensor of g t. Note that by equation (6.), 0 = g i j t i j(χ k lg t,k l) Therefore, we have = g i j t h p q i jχ p q g i j t h r q χ p s i χ r s jχ p q g i j t h p s χ r q i χ r s jχ p q +χ k lr k l(g t ). log(λ ωt χ) = (Λ ωt χ) Λ ωt χ (Λ ωt χ) 2 (Λ ωt χ) 2 = nλ ωt χ ( h k lr i j (g k l t )χ i j +g i j t h r q χ p s i χ r s jχ p q +g i j t h p s χ r q i χ r s jχ p q χ k lr k l(g t ) n (Λ ωt χ) 2 ) Λ ωt χ ( h k lr i j (g nλ ωt χ k l t )χ i j χ k lr k l(g t ) where we used the inequality by Lemma 3.2 in [30] (6.4) n (Λ ωt χ) 2 (Λ ωt χ)g i j t h r q χ p s i χ r s jχ p q. For any A, we have ( ) log(λ ωt χ) Aϕ ), ( h k lr i j (g nλ ωt χ k l t )χ i j χ k lr ) k l(g t ) n (nc ta Ah k lχ 0,k l). Fix [0,]. We choose A large enough such that (6.5) ( h k lr i j (g AΛ ωt χ k l t )χ i j χ k lr ) k l(g t ) where we used the fact that Λ ωt χ C > 0 ǫ,

17 A CRITERION FOR THE PROPERNESS OF THE K-ENERGY IN A GENERAL KÄHLER CLASS 7 for some constant C independent of t. We assume that log(λ ωt χ) Aϕ achieves its maximum at the point (x t,t). Then at the point (x t,t) we have 0 ( ) log(λ ωt χ) Aϕ ( h k lr i j (g nλ ωt χ k l t )χ i j χ k lr ) k l(g t ) n (nc ta Ah k lχ 0,k l) A ( ǫ nc t +h k lχ ) n 0,k l. Therefore, at the point (x t,t) we have h k lχ 0,k l nc t ǫ. We choose normal coordinates for the metric χ 0 so that the metric χ t is diagonal with entries λ,,λ n. We denote the diagonal entries of g t by µ,,µ n. Thus, we have n µ i nc t ǫ, which implies the inequality n ǫ (6.6) i=,i k nc t i= λ 2 i ( ) 2 n µ i λ i n i=,i k µ i 2 µ k λ k. i=,i k µ i + µ k λ 2 k 2 µ k λ k +nc t On the other hand, by the assumption and the choice of the metric χ 0 we have (6.7) (ncχ 0 (n )ω) χ n 2 0 β k > Bǫχ n 0 β k for sufficiently small ǫ > 0. Here β k denotes the (,) form dz k d z k and we choose the constant B such that Bt 2 >. Combining (6.7) with (6.2), we have (6.8) (nc t χ 0 (n )ω t ) χ0 n 2 β k ( ) = ( t)χ0 n β k +t ncχ 0 (n )ω Bǫtχ n 0 β k. By the argument of [24], the inequality (6.8) implies that n (6.9) µ i < nc t Btǫ. Combining (6.9) with (6.6), we have λ k µ k < i=,i k 2 (Bt )ǫ 2 (Bt 2 )ǫ, t t 2 χ n 2 0 β k

18 8 HAOZHAO LI, YALONG SHI 2, AND YI YAO for k =,,n. Hence, at the point (x t,t) we have the estimate Thus, for any x we have Λ ωt χ Λ ωt χ 2n (Bt 2 )ǫ, t t 2. 2n (Bt 2 )ǫ ea(ϕ inf ϕ). Since ω t χ 0 +ω λω and χ is a positive (,) form, we have The lemma is proved. Λ ω χ 2nλ (Bt 2 )ǫ ea(ϕ inf ϕ). Lemma 6.2. Under the assumptions of Lemma 6., there exists a uniform constant C such that osc ϕ t C, t t 2. Proof. The argument follows directly from [25] and [24] and we sketch the details here. We normalize ϕ t by (6.0) ϕ t ωt n = 0. Therefore, we have sup ϕ t 0 and inf ϕ t 0. Define u = e N ˆϕ, ˆϕ = ϕ supϕ, where N = A. Here A is the constant in Lemma 6. and δ is a small positive constant δ to be determined later. Using the argument in [30], there is a constant C independent of t such that for all p and t t 2 we have u p 2 2 ω n t ωt n! Cp u δ C 0 u p ( δ)ωn t n!. Note that the Sobolev constant of ω t is uniformly bounded for all t [0,]. Thus, following the argument of [3] we can prove that u is bounded and hence inf ˆϕ t C for some constant C. In other words, we have 0 inf ϕ t sup ϕ t C C, which implies that osc ϕ t C. The lemma is proved.

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20 20 HAOZHAO LI, YALONG SHI 2, AND YI YAO [26] Sun, W., On a class of fully nonlinear elliptic equations on closed Hermitian manifolds, ariv: [27] Gang Tian. Canonical metrics in Kähler geometry. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, Notes taken by Meike Akveld. [28] Wang, L., On the regularity theory of fully nonlinear parabolic equations. I. Comm. Pure Appl. Math. 45 (992), no., [29] Wang, L., On the regularity theory of fully nonlinear parabolic equations. II. Comm. Pure Appl. Math. 45 (992), no. 2, [30] Weinkove, B. Convergence of the J-flow on Kähler surfaces. Comm. Anal. Geom. 2 (2004), no. 4, [3] Weinkove, B. On the J-flow in higher dimensions and the lower boundedness of the Mabuchi energy. J. Differential Geom. 73 (2006), no. 2, [32] Wells, R.O. Differential analysis on complex manifolds, GTM 65, Springer,980. [33] Zhou, B. and Zhu,. Relative K-stability and modified K-energy on toric manifolds. Advances in Math. 29 (2008), Department of Mathematics, University of Science and Technology of China, Hefei, , Anhui province, China and Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei , Anhui, China address: hzli@ustc.edu.cn Department of Mathematics and Institute of Mathematical Science, Nanjing University, Nanjing, 20093, Jiangsu province, China address: shiyl@nju.edu.cn Department of Mathematics and Institute of Mathematical Science, Nanjing University, Nanjing, 20093, Jiangsu province, China address: yeeyoe@63.com