Existence of Kahler-Ricci solitons on smoothable Q-Fano varities
aa r X i v : . [ m a t h . DG ] A ug EXISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLEQ-FANO VARIETIES
YAN LI
Abstract.
In this article we prove the existence of K¨ahler-Ricci solitons on smoothable,K-stable Q -Fano varieties. We also investigate the behavior of twisted K¨ahler-Riccisolitons in the Gromov-Hausdorff topology under this smoothing family. Contents
1. Introduction 12. Preliminaries 33. The variational approach for twisted K¨ahler-Ricci solitons 84. L ∞ -bound on the potentials 144.1. Existence of the twisted K¨ahler-Ricci solitons with small r ( λ ) 154.2. Uniform lower bounds on Ding functional and Mabuchi functional 154.3. L ∞ -estimates and locally higher order estimates for the potentials 195. Gromov-Hausdorff convergence under L ∞ -bound on K¨ahler potentials 236. Existence of K¨ahler-Ricci solitons 25References 27 Introduction
A basic problem in K¨ahler geometry is whether the Fano manifold M admits a K¨ahler-Einstein metric. This problem is confirmed recently by Chen-Donaldson-Sun [11] [12][13] and Tian [46] [47], which claims that the existence of K¨ahler-Einstein metric on M is equivalent to the algebro-geometric notion of K-stability. These techniques solvingthis problem play an important role in many other aspects. For instance, on one hand,this problem is reproved through the Aubin’s continuity method developed by Datar-Sz´ekelyhidi [42] [15]. Moreover, this continuity method is also adapted to deal withthe problem that whether the Fano manifold M admits a K¨ahler-Ricci soliton [15]. Onthe other hand, motivated by the study of the compactification of the moduli spaces ofsmooth K¨ahler-Einstein Fano manifold, Spotti-Sun-Yao [41] investigate the existence ofK¨ahler-Einstein metrics on smoothable Q -Fano varieties by using the conic continuitymethod on a flat family. Combining these arguments, a natural problem is whether theexistence of K¨ahler-Ricci soliton on a smoothable Q -Fano variety M is equivalent to thealgebro-geometric notion of K-stable which is defined in [15]. It is notable that Berman-Nystr¨om [10] show that the existence of K¨ahler-Ricci soliton implies K-stable withoutany assumptions. Therefore, in this article we mainly consider the other side by applyingthe Aubin’s continuity method on a flat family. Before stating main results, we recall some basic definitions. A Q -Fano variety M isa normal projective variety with at worst log-terminal singularities and with ample Q -Cartier anticanonical divisor K − M . A Q -Fano variety M is called Q -Gorestein smoothableif there is a flat projective family π : M → ∆ over a disk ∆ in C such that M ∼ = M := π − (0), M t := π − ( t ) are smooth for t = 0 and M has a relatively ample Q -Cartieranticanonical divisor K − M / ∆ . Proposition 1.41 [20] says that, by possibly shrinking ∆, M t is a Fano manifold for t = 0 and there exists an integer m > K − mM t are very ample line bundles for all t ∈ ∆. Let V be a holomorphic vector field on M which is only tangent to the fibers and belongs to a reductive algebra of reductiveautomorphism subgroup (c.f.[51]) and T be the compact group induced by Im V . Embed M into ∆ × CP N by using T -invariant sections of K − m M / ∆ and denote α t by the suitablescaling of the Fubuni-Study metric m ω F S on M t for t ∈ ∆.Next we recall the definition of K-stable (c.f.[15]). Suppose that there exists a C ∗ action ρ generated by a holomorphic vector field W on M which commutes with V := V| M .Assume that X := lim t → ρ ( t ) · M is a Q -Fano variety. We take the limit α ∗ := lim t → ρ ( t ) · α , V ∗ := lim t → ρ ( t ) · V . The C ∗ action ρ defines a T -equivariant special degeneration ( T := T | M ) and its twistedFutaki invariant is defined to beFut (1 − λ ) α ,V ( M , W ) := Fut (1 − λ ) α ∗ ,V ∗ ( X, W ) = Fut V ∗ ( X, W ) − − λ R X ω nφ h Z X θ W ( e θ V ∗ − ω nφ + n Z X θ W ( α ∗ − ω φ ) ∧ ω n − φ i , where W is the induced vector field on X by W , λ ∈ (0 , ω φ is the restriction of asuitable scaling of the Fubini-Study form on X , θ W and θ V ∗ are Hamiltonian functionsand Fut V ∗ ( X, W ) := R X θ W e θ V ∗ ω nφ R X ω nφ . Definition 1.1.
The triple ( M , (1 − λ ) α , V ) is K-semistable if Fut (1 − λ ) α ,V ( M , W ) ≥ for all W as above. The triple is K-stable if in addition equality holds only when ( X, (1 − λ ) α ∗ , V ∗ ) is biholomorphic to ( M , (1 − λ ) α , V ) . The main theorem of this article is the following result, which extends the consequencesof [15].
Theorem 1.1.
Let π : M → ∆ be a Q -Gorestein smoothing of a Q -Fano variety M and V be a reductive holomorphic vector field on M , which preserves the fibers. If ( M , V ) isK-stable, then M admits a K¨ahler-Ricci soliton. We now briefly describe the structure of this article and sketch the main argumentsneeded to prove our main Theorem 1.1. The strategy of the proof is based on Aubin’scontinuity method.The first result, which is the subject of section 2 and 3, shows that there exists a uniquetwisted K¨ahler-Ricci soliton on a Q -Fano variety when the parameter λ close to 1 − m − .We mainly apply the pluripotential theory developed by Berman-Boucksom-Eyssidieux-Guedj-Zeriahi, see [17] [8] [3] [4] [10] and [19], to show that the properness of Mabuchi XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 3 functional implies the existence and uniqueness of twisted K¨ahler-Ricci soliton. When λ close to 1 − m − , the Mabuchi functional is proper due to the α -invariant.In section 4, we obtain the uniform L ∞ -estimate for the K¨ahler potentials. First, weshow that if there exist twisted K¨ahler-Ricci solitons on M t for t ∈ ∆ ∗ = ∆ \{ } when λ = λ , then the Mabuchi functional has a uniformly lower bound for λ < λ . Note thatwhen λ < − m − , the Mabuchi functional is uniformly proper. Thus, the boundednessof the functional I is obtained due to the fact that the Mabuchi functional is linear in λ .This is the reason why we take the parameter λ ∈ (1 − m − ,
1] and m > λ t := sup { λ ∈ (1 − m − , |∃ twisted K¨ahler-Ricci soliton on M t for all κ ≤ λ } is lower semi-continuous, which we present in section 6. It implies the openness andclosedness in the Aubin’s continuity method.Next we give some remarks on the main Theorem 1.1. First, the main technical pointwhere the smoothability is used here is given by the application of smooth Riemannianconvergence theory with Bakry- ´Emery Ricci curvature bounded below which is developedby Wang-Zhu [50] and Datar-Sz´ekelyhidi [15]. Second, we expect that Theorem 1.1 holdsfor general, not necessarily smoothable, Q -Fano varieties. But it is difficult. From now on,Li-Tian-Wang [27] show that the existence of weak K¨ahler-Einstein metric is equivalentto the algebraic notion K-stability on a Q -Fano variety with admissible singularities. Theadmissible singularities imply that the metrics they deal with always have at worst conicsingularities. Thus, this problem is still open for general case. It is notable that recentlyLi [24] claims that the uniform K-stability is equivalent to the existence of weak K¨ahler-Einstein metric on a Q -Fano variety without any assumptions by applying the argumentof [5].There are also fundamental results about the moduli spaces of smooth K¨ahler-Einsteinmanifolds, see [35] [38] [26] [39] [40] [32] and [31]. This is another motivation for thisarticle. 2. Preliminaries
In this section we will establish some elementary estimates which will be used in thelater. Let M be a Q -Fano variety and V be a reductive holomorphic vector field definedon the regular part of M . If π : M → M is a log resolution, by normality, the vectorfield V admits a unique extension V to M (c.f. section 2.3 [10]). Denote T and T by thecompacts groups induced by Im V and Im V . There exists an integer m > M can be embedded into CP N by using T -invariant sections of K − mM . α denotes the scalingof the Fubini-Study form m ω F S . Set ω = π ∗ α , then ω is T -invariant. We introducePSH( M, ω ) := { ϕ | ω + √− ∂ ¯ ∂ϕ ≥ } and PSH( M, ω ) T := { ϕ | ω + √− ∂ ¯ ∂ϕ ≥ ϕ is T -invariant } . YAN LI
Lemma 2.1. If ϕ ∈ PSH(
M, ω ) T , then V ( ϕ ) is well-defined and | V ( ϕ ) | ≤ C a.e. [ χ n ] ,where C is a constant independent of ϕ and χ is a T -invariant K¨ahler metric on M .Proof. There is a strictly decreasing sequence ϕ j of smooth functions with limit ϕ suchthat ω + ǫ j χ + √− ∂ ¯ ∂ϕ j > ǫ j decreases to 0 due to [9]. By averaging we can assumethat ϕ j are T -invariant. Lemma 5.1 [51] and Corollary 5.3 [51] imply that | V ( ϕ j ) | ≤ C ,where C is a constant independent of ϕ j . By Theorem 1.48 [19] and locally argument, ∇ ϕ j converge to ∇ ϕ in L q for all 1 ≤ q <
2, where ∇ denotes the gradient of functions.Furthermore, there exists a subsequence j k such that ∇ ϕ j k converges to ∇ ϕ a.e. [ χ n ]. So V ( ϕ ) is well-defined and | V ( ϕ ) | ≤ C . (cid:3) In [8], the finite energy classPSH full ( M, ω ) := n ϕ ∈ PSH(
M, ω ) (cid:12)(cid:12)(cid:12) Z M ( ω + √− ∂ ¯ ∂ϕ ) n = Z M ω n = a o has been investigated. Similarly, we need the following definition. Definition 2.1.
The T -invariant finite energy class is PSH full ( M, ω ) T := n ϕ ∈ PSH(
M, ω ) T (cid:12)(cid:12)(cid:12) Z M ( ω + √− ∂ ¯ ∂ϕ ) n = Z M ω n = a o Lemma 2.2. If ϕ ∈ PSH(
M, ω ) T and ψ ∈ PSH full ( M, ω ) T , then | V ( ϕ ) | ≤ C a.e. [( ω + √− ∂ ¯ ∂ψ ) n ] .Proof. By the Lemma 2.1, there exists a constant C independent of ϕ such that | V ( ϕ ) | ≤ C a.e. [ χ n ]. We introduce the set S := { x ∈ M || V ( ϕ )( x ) | > C } , then there is a Borelset B ⊃ S which is G δ such that R B χ n = 0. We take the canonical approximation ψ j := max( ψ, − j ). Proposition 10.15 [19] claims that lim j →∞ R B ( ω + √− ∂ ¯ ∂ψ j ) n = R B ( ω + √− ∂ ¯ ∂ψ ) n . For each j , if R B ( ω + √− ∂ ¯ ∂ψ j ) n = 0, then this lemma is true.Next we assume that ψ ∈ PSH full ( M, ω ) T ∩ L ∞ ( M ). Choosing a decreasing sequence ψ k of smooth functions with limit ψ such that ω + ǫ k χ + √− ∂ ¯ ∂ψ k > ǫ k decreases to 0.Theorem 3.18 [19] implies that lim k →∞ R B ( ω + ǫ k χ + √− ∂ ¯ ∂ψ k ) n = R B ( ω + √− ∂ ¯ ∂ψ ) n .Note that R B ( ω + ǫ k χ + √− ∂ ¯ ∂ψ k ) n = 0 for each k since R B χ n = 0. Therefore the proofis completed. (cid:3) PSH full ( M, ω ) T is convex according to the same argument of Proposition 10.7 [19].Next we introduce the following functionals. Definition 2.2.
For φ ∈ PSH(
M, ω ) T ∩ L ∞ ( M ) , E V ( φ ) := Z Z M φe θ M + s · V ( φ ) ω nsφ ∧ ds and E ( φ ) := 1 n + 1 n X j =0 Z M φω jφ ∧ ω n − j where ω sφ := ω + √− ∂ ¯ ∂ ( sφ ) and θ M is defined by L V ω = √− ∂ ¯ ∂θ M . XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 5
Definition 2.3.
For ϕ ∈ PSH(
M, ω ) T , E ( ϕ ) := inf { E ( φ ) | φ ≥ ϕ, φ ∈ PSH(
M, ω ) ∩ L ∞ ( M ) } ,E V ( ϕ ) := inf { E V ( φ ) | φ ≥ ϕ, φ ∈ PSH(
M, ω ) T ∩ L ∞ ( M ) } , E ( M, ω ) := { ϕ ∈ PSH full ( M, ω ) | E ( ϕ ) > −∞} , E V ( M, ω ) := { ϕ ∈ PSH full ( M, ω ) T | E V ( ϕ ) > −∞} . The following lemma is standard due to Proposition 2.15 [10].
Lemma 2.3.
The map ϕ E V ( ϕ ) is upper semi-continuous for the L -topology on PSH(
M, ω ) T . It is continuous along decreasing sequences in PSH(
M, ω ) T . Lemma 2.4. If ϕ ∈ E V ( M, ω ) , then ϕ ∈ E ( M, ω ) .Proof. Define ϕ j := max( ϕ, − j ), then Lemma 2.3 and Proposition 10.19 [19] imply thatlim j →∞ E V ( ϕ j ) = E V ( ϕ ) and lim j →∞ E ( ϕ j ) = E ( ϕ ). Assume that ϕ ≤ C where C is apositive constant, we have the following calculations E V ( ϕ j ) = Z Z M ( ϕ j − C ) e θ M + s · V ( ϕ j ) ω nsϕ j ∧ ds + Z Z M C e θ M + s · V ( ϕ j ) ω nsϕ j ∧ ds ≥ e C Z Z M ( ϕ j − C ) ω nsϕ j ∧ ds + C e − C Z Z M ω nsϕ j ∧ ds = e C E ( ϕ j ) + C ( e − C − e C ) a where the second inequality holds due to Lemma 2.2. By the same argument, we have E V ( ϕ j ) ≤ e − C Z Z M ( ϕ j − C ) ω nsϕ j ∧ ds + C e C Z Z M ω nsϕ j ∧ ds = e − C E ( ϕ j ) + C ( e C − e − C ) a. Taking limit on both sides e C E ( ϕ ) + C ( e − C − e C ) a ≤ E V ( ϕ ) ≤ e − C E ( ϕ ) + C ( e C − e − C ) a. Therefore, E V ( ϕ ) > −∞ gives E ( ϕ ) > −∞ . (cid:3) Proposition 2.1.
Let ϕ ∈ PSH(
M, ω ) T and ϕ j := max( ϕ, − j ) . Assume that V ( ϕ j ) pointwise converges to V ( ϕ ) , then e θ M + V ( ϕ j ) ω nϕ j → e θ M + V ( ϕ ) ω nϕ weakly as j → ∞ .If ϕ ∈ E V ( M, ω ) and V ( ϕ j ) pointwise converges to V ( ϕ ) , then ϕ j e θ M + V ( ϕ j ) ω nϕ j → ϕe θ M + V ( ϕ ) ω nϕ weakly as j → ∞ .Proof. The first argument is obtained according to Theorem 2.7 [10]. Next we prove thesecond argument which is similar as Theorem 2.17 [8]. Let h be a continuous function on M , then it is enough to establish thatlim j →∞ Z M hϕ j e θ M + V ( ϕ j ) ω nϕ j = Z M hϕe θ M + V ( ϕ ) ω nϕ . YAN LI
We have (cid:12)(cid:12)(cid:12) Z M h ( ϕ j e θ M + V ( ϕ j ) ω nϕ j − ϕe θ M + V ( ϕ ) ω nϕ ) (cid:12)(cid:12)(cid:12) ≤ Z { ϕ> − j } | h || ϕ | (cid:12)(cid:12) e θ M + V ( ϕ j ) − e θ M + V ( ϕ ) (cid:12)(cid:12) ω nϕ + Z { ϕ ≤− j } | h || ϕ j | e θ M + V ( ϕ j ) ω nϕ j + Z { ϕ ≤− j } | h || ϕ | e θ M + V ( ϕ ) ω nϕ . The condition that V ( ϕ j ) pointwise converges to V ( ϕ ) implieslim j →∞ Z { ϕ> − j } | h || ϕ | (cid:12)(cid:12) e θ M + V ( ϕ j ) − e θ M + V ( ϕ ) (cid:12)(cid:12) ω nϕ ≤ lim j →∞ Z M | h || ϕ | (cid:12)(cid:12) e θ M + V ( ϕ j ) − e θ M + V ( ϕ ) (cid:12)(cid:12) ω nϕ = 0 . Lemma 2.4 and Exercise 10.5 [19] show that ϕ ∈ E ( M, ω ) and there exists a convexweight γ such that lim k →∞ − kγ ( − k ) = 0 and R M γ ( ϕ ) ω nϕ > −∞ , where a weight denotes asmooth increasing function γ : R → R such that γ ( −∞ ) = −∞ . According to Lemma2.2, we have Z { ϕ ≤− j } | h || ϕ j | e θ M + V ( ϕ j ) ω nϕ j ≤ sup M | h | e C Z { ϕ ≤− j } | ϕ j | ω nϕ j = sup M | h | e C Z { ϕ ≤− j } | γ ( ϕ j ) | · | ϕ j || γ ( ϕ j ) | ω nϕ j ≤ sup M | h | e C · − jγ ( − j ) · Z M | γ ( ϕ j ) | ω nϕ j , which yields lim j →∞ Z { ϕ ≤− j } | h || ϕ j | e θ M + V ( ϕ j ) ω nϕ j = 0 . Also Z { ϕ ≤− j } | h || ϕ | e θ M + V ( ϕ ) ω nϕ ≤ sup M | h | e C · lim k →∞ Z {− k<ϕ ≤ j } | ϕ k | ω nϕ k ≤ sup M | h | e C · − jγ ( − j ) · lim sup k →∞ Z M | γ ( ϕ k ) | ω nϕ k , which gives lim j →∞ Z { ϕ ≤− j } | h || ϕ | e θ M + V ( ϕ ) ω nϕ = 0 . This proposition is proved. (cid:3)
Corollary 2.1. If ϕ ∈ E V ( M, ω ) , then E V ( ϕ ) = Z Z M ϕe θ M + s · V ( ϕ ) ω nsϕ ∧ ds. XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 7
Proof.
Set ϕ j := max( ϕ, − j ), f j ( s ) := R M ϕ j e θ M + s · V ( ϕ j ) ω nsϕ j and f ( s ) := R M ϕe θ M + s · V ( ϕ ) ω nsϕ ,then Proposition 2.1 shows that lim j →∞ f j ( s ) = f ( s ) for each s ∈ [0 , | f j ( s ) | ≤ e C Z M | ϕ j | ((1 − s ) ω + sω ϕ j ) n = e C Z M | ϕ j | · n X m =0 C nm s n − m (1 − s ) m ω m ∧ ω n − mϕ j ≤ C ′ Z M | ϕ j | ω nϕ j ≤ C ′′ , where C ′ and C ′′ are positive constants and the third inequality bases on ϕ ∈ E ( M, ω ).By the Lebesgue dominated convergence theorem, lim j →∞ R f j ( s ) ds = R f ( s ) ds . (cid:3) Lemma 2.5.
Let E V,C ( M, ω ) := { ϕ ∈ E V ( M, ω ) | E V ( ϕ ) ≥ − C and sup M ϕ ≤ } , thenit is a compact subset for the L -topology.Proof. For ϕ ∈ E V,C ( M, ω ), we see − C ≤ E V ( ϕ ) = Z Z M ϕe θ M + s · V ( ϕ ) ω nsϕ ∧ ds ≤ e − C · a · (sup M ϕ ) . So there exists a constant
C > ϕ such that − C ≤ sup M ϕ ≤
0, whichimplies E V,C ( M, ω ) ⊂ { ϕ ∈ PSH(
M, ω ) T | − C ≤ sup M ϕ ≤ } . The latter set is a compact subset of PSH(
M, ω ) T by Hartog’s Lemma, see Theorem 1.46[19]. Since ϕ E V ( ϕ ) is upper semi-continuous by Lemma 2.3, the set E V,C ( M, ω ) isclosed, hence compact for L -topology. (cid:3) To deal with K¨ahler-Ricci soliton, the following functionals are introduced (c.f.[48]).For φ ∈ E V ( M, ω ), we define I V ( φ ) = Z M φ ( e θ M ω n − e θ M + V ( φ ) ω nφ )and J V ( φ ) = Z Z M φ ( e θ M ω n − e θ M + s · V ( φ ) ω nsφ ) ∧ ds. Proposition 2.2.
Define α M = inf M θ M and β M = sup M θ M which are independent ofthe choice of ω , the we have I V ( φ ) ≤ ( n + 1 + β M − α M )( I V ( φ ) − J V ( φ )) ≤ ( n + β M − α M ) I V ( φ ) . Proof.
Taking φ j := max( φ, − j ), then lim j →∞ I V ( φ j ) = I V ( φ ) and lim j →∞ J V ( φ j ) = J V ( φ ) according to Proposition 2.1 when V ( φ j ) pointwise converges to V ( φ ). Withoutloss of generality, we can assume that φ ∈ E V ( M, ω ) ∩ L ∞ ( M ). By the approximationtheorem [9], there is a strictly decreasing sequence φ k of smooth functions with limit φ such that ω + ǫ k χ + √− ∂ ¯ ∂φ k >
0. We further assume that φ k are T -invariant and V ( φ k ) pointwise converges to V ( φ ). Define θ M,k by L V ( ω + ǫ k χ ) = √− ∂ ¯ ∂θ M,k and α M,k := inf M θ M,k , β M,k := sup M θ M,k . We denote ω k by ω + ǫ k χ and define I V ( φ k ) = Z M φ k ( e θ M,k ω nk − e θ M,k + V ( φ k ) ω nφ k ) YAN LI and J V ( φ k ) = Z Z M φ k ( e θ M,k ω nk − e θ M,k + s · V ( φ k ) ω nsφ k ) ∧ ds where ω sφ k := ω k + √− ∂ ¯ ∂ ( sφ k ). Proposition A.1 [30] implies that(2.1) I V ( φ k ) ≤ ( n + 1 + β M,k − α M,k )( I V ( φ k ) − J V ( φ k )) ≤ ( n + β M,k − α M,k ) I V ( φ k ) . By Theorem 2.7 [10] and the fact that lim k →∞ θ M,k = θ M , we know that lim k →∞ I V ( φ k ) = I V ( φ ) and lim k →∞ J V ( φ k ) = J V ( φ ). Therefore, by taking the limit on inequality (2.1) wededuce this proposition. (cid:3) The variational approach for twisted K¨ahler-Ricci solitons
This section is devoted to explain a variational approach developed in [3] to solve thetwisted K¨ahler-Ricci soliton equation.Recall that if π : M → M is a log resolution, then there exist rational numbers a i ≥ < b j < K M = π ∗ K M + X i a i E i − X j b j F j where E i and F j are exceptional prime divisors. We embed M into CP N by using T -invariant sections of K − mM . α denotes m ω F S . Let ν be an adapted measure with √− ∂ ¯ ∂ log ν = − α on ( M ) reg , where ( M ) reg denotes the regular part of M . θ M is aHamiltonian function defined by L V α = √− ∂ ¯ ∂θ M . Definition 3.1.
For λ ∈ (1 − m − , , a twisted K¨ahler-Ricci soliton for the triple ( M , V , (1 − λ ) ω F S ) is a current ω φ := α + √− ∂ ¯ ∂φ with full Monge-Amp`ere mass,i.e. φ ∈ PSH full ( M , α ) T such that e θ M + V ( φ ) ( α + √− ∂ ¯ ∂φ ) n = e − r ( λ ) φ ν R M e − r ( λ ) φ ν where we assume that e θ M + V ( φ ) ( α + √− ∂ ¯ ∂φ ) n is a probability measure on M and r ( λ ) = 1 − (1 − λ ) m . Remark 3.1.
The existence of the twisted K¨ahler-Ricci soliton is also equivalent to solvethe following degenerated complex Monge-Amp`ere equation on M (3.2) e θ M + V ( φ ) ( ω + √− ∂ ¯ ∂φ ) n = e − r ( λ ) φ µ R M e − r ( λ ) φ µ , where ω = π ∗ α , µ = π ∗ ν and θ M is a Hamiltonian function defined by L V ω = √− ∂ ¯ ∂θ M . Next, some consequences about α -invariant defined by Tian [45] will be recalled (c.f.[4]). Definition 3.2.
The α -invariant of a measure µ is defined as α µ ( ω ) := sup n α > (cid:12)(cid:12)(cid:12) sup ϕ ∈ PSH(
M,ω ) Z M e − αϕ dµ < + ∞ o . Remark 3.2.
The α -invariant α µ ( ω ) > due to Proposition 1.4 [4] . XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 9
Mabuchi functional and Ding functional play important roles in the research of theexistence of K¨ahler-Einstein metrics on Fano manifolds. Similarly, we need the following
Definition 3.3.
For ϕ ∈ E V ( M, ω ) , we define the twisted Ding functional to be F V,λ ( ϕ ) = − r ( λ ) · E V ( ϕ ) − log Z M e − r ( λ ) ϕ dµ = − r ( λ ) · Z Z M ϕe θ M + s · V ( ϕ ) ω nsϕ ∧ ds − log Z M e − r ( λ ) ϕ dµ Proposition 3.1. F V,λ is lower semi-continuous on each E V,C ( M, ω ) defined in Lemma2.5.Proof. Proposition 11.3 (iii) [19] implies that ϕ log R M e − r ( λ ) ϕ dµ is continuous on E V,C ( M, ω ). The conclusion follows due to Lemma 2.3. (cid:3) Definition 3.4.
For ϕ ∈ E V ( M, ω ) , we define the twisted Mabuchi functional to be M V,λ ( ϕ ) = − r ( λ ) · (cid:16) E V ( ϕ ) − Z M ϕe θ M + V ( ϕ ) ω nϕ (cid:17) + Z M log e θ M + V ( ϕ ) ω nϕ µ e θ M + V ( ϕ ) ω nϕ . Set µ ϕ = e − r ( λ ) ϕ µ R M e − r ( λ ) ϕ dµ , we have Lemma 3.1.
For ϕ ∈ E V ( M, ω ) , F V,λ ( ϕ ) = M V,λ ( ϕ ) − Z M log e θ M + V ( ϕ ) ω nϕ µ ϕ e θ M + V ( ϕ ) ω nϕ ≤ M V,λ ( ϕ ) . Proof.
Observing that Z M log e θ M + V ( ϕ ) ω nϕ µ ϕ e θ M + V ( ϕ ) ω nϕ = Z M log e θ M + V ( ϕ ) ω nϕ µ e θ M + V ( ϕ ) ω nϕ + r ( λ ) · Z M ϕe θ M + V ( ϕ ) ω nϕ + log Z M e − r ( λ ) ϕ dµ. By definition, we have M V,λ ( ϕ ) − Z M log e θ M + V ( ϕ ) ω nϕ µ ϕ e θ M + V ( ϕ ) ω nϕ = F V,λ ( ϕ ) . Jensen’s inequality implies that Z M log e θ M + V ( ϕ ) ω nϕ µ ϕ e θ M + V ( ϕ ) ω nϕ ≥ . Thus the proof is completed. (cid:3)
Definition 3.5.
We say that the functional M V,λ ( F V,λ ) is proper if whenever ϕ j ∈E V ( M, ω ) is a sequence of functions such that J V ( ϕ j ) → + ∞ , then M V,λ ( ϕ j ) → + ∞ ( F V,λ ( ϕ j ) → + ∞ ) . Lemma 3.2.
Fix < σ < α µ ( ω ) . There exists a constant C σ such that M V,λ ( ϕ ) ≥ (cid:16) σ − r ( λ ) · n + β M − α M n + 1 + β M − α M (cid:17) I V ( ϕ ) − C σ for all ϕ ∈ E V ( M, ω ) and sup M ϕ = 0 , where α M and β M are defined in Proposition 2.2.In particular, if α µ ( ω ) > r ( λ ) · n + β M − α M n +1+ β M − α M , then M V,λ is proper.Proof.
By assumption Z M e − σϕ − log eθM + V ( ϕ ) ωnϕµ · e θ M + V ( ϕ ) ω nϕ = Z M e − σϕ dµ ≤ e C σ . Jensen’s inequality implies that − σ · Z M ϕe θ M + V ( ϕ ) ω nϕ − C σ ≤ Z M log e θ M + V ( ϕ ) ω nϕ µ · e θ M + V ( ϕ ) ω nϕ . By a direct calculation and Proposition 2.2 we have M V,λ ( ϕ ) ≥ − r ( λ ) · (cid:16) E V ( ϕ ) − Z M ϕe θ M + V ( ϕ ) ω nϕ (cid:17) − σ · Z M ϕe θ M + V ( ϕ ) ω nϕ − C σ ≥ − r ( λ ) · ( I V ( ϕ ) − J V ( ϕ )) + σ · I V ( ϕ ) − C σ ≥ − r ( λ ) · n + β M − α M n + 1 + β M − α M I V ( ϕ ) + σ · I V ( ϕ ) − C σ = (cid:16) σ − r ( λ ) · n + β M − α M n + 1 + β M − α M (cid:17) I V ( ϕ ) − C σ . For the second argument, we choose σ > r ( λ ) · n + β M − α M n +1+ β M − α M . According to Proposition 2.2, I V ( ϕ ) ≥ J V ( ϕ ) · n +1+ β M − α M n + β M − α M . Thus M V,λ ( ϕ ) ≥ (cid:16) σ − r ( λ ) · n + β M − α M n + 1 + β M − α M (cid:17) · n + 1 + β M − α M n + β M − α M J V ( ϕ ) − C σ = (cid:16) σ · n + 1 + β M − α M n + β M − α M − r ( λ ) (cid:17) J V ( ϕ ) − C σ . So the second argument holds. (cid:3)
For each ϕ ∈ E V ( M, ω ), Theorem 2.18 [10] says that there exists a unique ψ ∈E V ( M, ω ) modulo constants such that e θ M + V ( ψ ) ω nψ = e − r ( λ ) ϕ µ R M e − r ( λ ) ϕ dµ . This argument has a connection with the so-called Ricci iteration, which is introduced in[36].
Lemma 3.3.
For ϕ, ψ ∈ E V ( M, ω ) as above, we have F V,λ ( ψ ) ≤ F V,λ ( ϕ ) and M V,λ ( ψ ) ≤ F V,λ ( ϕ ) . XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 11
Proof.
By the Proposition 2.15 [10], we see that F V,λ ( ϕ + C ) = F V,λ ( ϕ ) and M V,λ ( ϕ + C ) = M V,λ ( ϕ ). To getting the first inequality we only show that E V ( ψ ) ≥ E V ( ϕ ) by assumingthat R M e − r ( λ ) ϕ dµ = R M e − r ( λ ) ψ dµ = 1. Let φ s := sϕ + (1 − s ) ψ, s ∈ [0 , E V ( φ s ) is concave about s . This implies E V ( ϕ ) − E V ( ψ ) ≤ Z M ( ϕ − ψ ) e θ M + V ( ψ ) ω nψ = Z M ( ϕ − ψ ) e − r ( λ ) ϕ dµ = 1 r ( λ ) · Z M log µ ψ µ ϕ dµ ϕ ≤ r ( λ ) · log Z M dµ ψ = 0Therefore the first inequality holds. Observing that M V,λ ( ψ ) = − r ( λ ) · E V ( ψ ) + r ( λ ) · Z M ψe − r ( λ ) ϕ dµ + Z M log e − r ( λ ) ϕ µµ e − r ( λ ) ϕ dµ = − r ( λ ) · E V ( ψ ) − r ( λ ) · Z M ( ϕ − ψ ) e − r ( λ ) ϕ dµ = − r ( λ ) · E V ( ψ ) − r ( λ ) · Z M ( ϕ − ψ ) e θ M + V ( ψ ) ω nψ ≤ − r ( λ ) · E V ( ϕ ) = F V,λ ( ϕ ) . The proof is completed. (cid:3)
Lemma 3.4.
Fix C > , there exists a constant C ′ such that the sublevel set { ϕ ∈E V ( M, ω ) | J V ( ϕ ) ≤ C and sup M ϕ = 0 } is contained in E V,C ′ ( M, ω ) .Proof. For ϕ ∈ E V ( M, ω ) and sup M ϕ = 0, Lemma 3.45 [14] says that there exists aconstant A such that Z M ϕω n ≤ sup M ϕ ≤ Z M ϕω n + A. Furthermore we have Z M ϕe θ M ω n ≥ e C · Z M ϕω n ≥ − Ae C . Let C ′ = C + Ae C , by the definition of J V ( ϕ ) we conclude that E V ( ϕ ) ≥ − C ′ . (cid:3) Given an upper semi-continuous T -invariant function h , we define P ( h )( x ) := sup { ψ ( x ) ∈ R | ψ ∈ PSH(
M, ω ) T and ψ ≤ h } . Remark 3.3.
If we define P ( h ) ′ ( x ) := sup { ψ ( x ) ∈ R | ψ ∈ PSH(
M, ω ) and ψ ≤ h } ,then P ( h ) = P ( h ) ′ . In fact, on one hand P ( h ) ≤ P ( h ) ′ by the definitions. On theother hand, we denote P ( h ) ′ by the average of P ( h ) ′ along the compact group T , then P ( h ) ′ ≤ h and max( P ( h ) ′ , P ( h ) ′ ) ∈ PSH(
M, ω ) . By the definition of P ( h ) ′ , we have max( P ( h ) ′ , P ( h ) ′ ) = P ( h ) ′ = P ( h ) ′ . Proposition 2.16 [10] gives
Lemma 3.5.
Let w be a non-negative T -invariant continuous function and ϕ ∈ E V ( M, ω ) .Then we have ddt E V ( p ( ϕ + tw )) (cid:12)(cid:12)(cid:12) t =0 = Z M we θ M + V ( ϕ ) ω nϕ . Next we give the main theorems of this section.
Theorem 3.1.
If the twisted Mabuchi functional M V,λ is proper, then there exists ϕ ∈E V ( M, ω ) solving e θ M + V ( ϕ ) ω nϕ = e − r ( λ ) ϕ µ R M e − r ( λ ) ϕ dµ . Proof.
By the assumption that M V,λ is proper and Lemma 3.4, we haveinf E V ( M,ω ) M V,λ = inf E V,C ( M,ω ) M V,λ where C is a constant as Lemma 3.4. It follows from Lemma 3.1 and Lemma 3.3 thatinf E V,C ( M,ω ) M V,λ = inf E V,C ( M,ω ) F V,λ = inf E V ( M,ω ) F V,λ . Since F V,λ is lower semi-continuous on the compact set E V,C ( M, ω ), we can find ϕ ∈E V,C ( M, ω ) which minimizes the functional F V,λ on E V ( M, ω ). Fix an arbitrary non-negative T -invariant continuous function w and consider g ( t ) := − r ( λ ) · E V ( P ( ϕ + tw )) − log Z M e − r ( λ )( ϕ + tw ) dµ. Lemma 3.5 implies that ddt g ( t ) (cid:12)(cid:12) t =0 = − r ( λ ) · Z M we θ M + V ( ϕ ) ω nϕ + r ( λ ) · R M we − r ( λ ) ϕ dµ R M e − r ( λ ) ϕ dµ Now P ( ϕ + tw ) ≤ ϕ + tw gives g (0) ≤ F V,λ ( P ( ϕ + tw )) ≤ g ( t ) , since ϕ is a minimizer. Therefore, Z M we θ M + V ( ϕ ) ω nϕ = R M we − r ( λ ) ϕ dµ R M e − r ( λ ) ϕ dµ . Finally we note that e θ M + V ( ϕ ) ω nϕ and e − r ( λ ) ϕ dµ are T -invariant measures, so given a con-tinuous function f , the integral of f with respect to these measures equal to that of theaverage of f along the compact group T . (cid:3) Theorem 3.2.
Assume that the twisted Mabuchi functional M V,λ is proper, then we have (1)
Aut ( M, V, ω ) = 1 , where Aut ( M, V, ω ) denotes the identity component of au-tomorphism group which preserves the form ω and the holomorphic vector field V . (2) M admits a unique twisted K¨ahler-Ricci soliton.Proof. (2) is the direct corollary of (1), Proposition 23 [15] and Theorem 3.1. Let usprove (1), we follow the argument of [4]. There exists a twisted K¨ahler-Ricci soliton ω by Theorem 3.1. Let γ be a 1-parameter subgroup of Aut ( M, V, ω ) and observing that γ ( s ) ∗ ω is also a twisted K¨ahler-Ricci soliton for each s ∈ C . We assume that φ is a metricon π ∗ K − M with curvature ω and set ϕ s := γ ( s ) ∗ φ − φ where φ is a metric with curvature ω , then ϕ ( x, s ) := ϕ s ( x ) is a Φ ∗ ω -psh function on M × C such that(Φ ∗ ω + √− ∂ ¯ ∂ϕ ) n +1 = 0 , XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 13 where Φ is a projection from M × C to M . By Proposition 2.17 [10], E V ( ϕ s ) is harmonicon C , while R M ( ϕ s − ϕ ) e θ M + V ( ϕ ) ω nϕ is subharmonic since s ϕ s ( x ) is subharmonic foreach x ∈ M . It follows that E V ( ϕ ) − E V ( ϕ s ) + Z M ( ϕ s − ϕ ) e θ M + V ( ϕ ) ω nϕ is subharmonic and bounded on C , hence it is vanishing.Set µ = e − r ( λ ) ϕ µ R M e − r ( λ ) ϕ dµ and F V ( ϕ ) = − E V ( ϕ ) + Z M ϕdµ for ϕ ∈ E V ( M, ω ) . Claim 3.1.
Given ϕ ∈ E V ( M, ω ) , we have F V ( ϕ ) = inf E V ( M,ω ) F V if and only if µ = e θ M + V ( ϕ ) ω nϕ . Proof. If µ = e θ M + V ( ϕ ) ω nϕ , then by the concavity of E V , we have E V ( ϕ ) − Z M ϕdµ ≥ E V ( ψ ) − Z M ψdµ for any ψ ∈ E V ( M, ω ). It follows that F V ( ϕ ) = inf E V ( M,ω ) F V . Conversely we assume that ϕ is the minimizer of F V and consider g ( t ) := − E V ( P ( ϕ + tw )) + Z M ( ϕ + tw ) dµ where w is a non-negative T -invariant continuous function. The argument of Theorem3.1 implies that ddt g ( t ) (cid:12)(cid:12) t =0 = − Z M we θ M + V ( ϕ ) ω nϕ + Z M wdµ Since P ( ϕ + tw ) ≤ ϕ + tw , we see g (0) ≤ − E V ( P ( ϕ + tw )) + Z M P ( ϕ + tw ) dµ ≤ g ( t )which gives Z M we θ M + V ( ϕ ) ω nϕ = Z M wdµ (cid:3) By this claim we know that e θ M + V ( ϕ s ) ω nϕ s = e − r ( λ ) ϕ µ R M e − r ( λ ) ϕ dµ . According to Theorem 2.18 [10], ϕ s = ϕ + C s where C s is a constant dependent of s . Hence γ ( s ) ∗ ω = ω . The automorphism subgroup Aut ( M, V, ω ) is contained in thecompact group of isometries of ω and hence it is trivial. (cid:3) L ∞ -bound on the potentials Let π : M → ∆ be the flat family as section 1. In this section we concern twoarguments. One is the existence of twisted K¨ahler-Ricci solitons on M t when | t | and r ( λ ) = 1 − (1 − λ ) m are sufficiently small. The other is L ∞ -estimate (relatively to theambient Fubini-Study metric) for the potentials of twisted K¨ahler-Ricci solitons (if exist)when | t | is small enough. To begin with, we recall some concepts in K¨ahler geometrywhich will be used in this section.Let M be a smooth Fano manifold, V be a holomorphic vector field belonging to areductive Lie subalgebra and T be the compact group induced by Im V . M is embeddedinto CP N by using the T -invariant sections of K − mM and ω F S is the Fubini-Study metric.We denote ω by m ω F S and choose a smooth volume form Ω such that Ric(Ω) = ω (i.e.Ric( ω ) = ω + √− ∂ ¯ ∂h by the relation Ω = e h ω m ). θ M is a Hamiltonian function on M defined by L V ω = √− ∂ ¯ ∂θ M and R M e θ M ω n = 1.The twisted K¨ahler-Ricci soliton ω φ = ω + √− ∂ ¯ ∂φ on M is defined as the followingequation Ric( ω φ ) − L V ω φ = (1 − λ ) ω F S + r ( λ ) ω φ where r ( λ ) = 1 − (1 − λ ) m , which is equivalent to the complex Monge-Amp`ere equation e θ M + V ( φ ) ω nφ = e − r ( λ ) φ Ω R M e − r ( λ ) φ Ω . Denote PSH(
M, ω ) by the space of ω -psh functions on M . For the convenience, we givethe various functionals on smooth Fano manifold M as section 2 and 3 (see [48] for acollection of them). Definition 4.1.
For φ ∈ C ∞ ( M ) ∩ PSH(
M, ω ) T , we define I V ( φ ) = Z M φ ( e θ M ω n − e θ M + V ( φ ) ω nφ ) ,E V ( φ ) = Z Z M φe θ M + s · V ( φ ) ω nsφ ∧ ds,J V ( φ ) = Z M φe θ M ω n − E V ( φ ) ,F V,λ ( φ ) = − r ( λ ) · E V ( φ ) − log Z M e − r ( λ ) φ Ω ,M V,λ ( φ ) = − r ( λ ) · ( I V ( φ ) − J V ( φ )) + Z M log e θ M + V ( φ ) ω nφ Ω e θ M + V ( φ ) ω nφ . Remark 4.1.
Lemma 3.1 implies M V,λ ( φ ) ≥ F V,λ ( φ ) . From now on, we return to the setting of Theorem 1.1. Namely, we consider a Q -Gorestein smoothing M of a Q -Fano variety M . The notations V t , T t , ω F S,t , ω t , Ω t , h t , θ M t , I V t ( φ t ), E V t ( φ t ), J V t ( φ t ), F V t ,λ ( φ t ) and M V t ,λ ( φ t ) on M t for t = 0 represent the samemeaning as above. XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 15
Existence of the twisted K¨ahler-Ricci solitons with small r ( λ ) . Denote S bythe singular set of the central fiber M of the flat family M . Let G : ( M \ S ) × ∆ → M bea smooth embedding such that G t ( M \ S ) := G (( M \ S ) × { t } ) ⊂ M t and G : M \ S → M \ S is the identity map. We have the following lemma. Lemma 4.1. θ M t ◦ G t smoothly converge to θ M . In particular, lim t → β M t = β M and lim t → α M t = α M , where β M t and α M t denote the maximum and minimum of θ M t .Proof. The smooth embedding map G satisfies that G ∗ t ω F S,t C ∞ -converges to ω F S, ( G ∗ t ω t C ∞ -converges to ω ), ( G − t ) ∗ V t C ∞ -converges to V and G ∗ t J t C ∞ -converges to J , where J t denotes the complex structure of M t . We also have L ( G − t ) ∗ V t G ∗ t ω t = − d ( G ∗ t J t ) d ( θ M t ◦ G t )due to L V t ω t = √− ∂ ¯ ∂θ M t . Furthermore, θ M t ◦ G t smoothly converges to θ M on M \ S .The second argument is deduced from the first. (cid:3) Next we concern the uniform lower bound of the α -invariant α Ω t ( ω t ) on M t for t ∈ ∆ ∗ := ∆ \{ } . Lemma 4.2.
There exists a positive constant l only dependent of the upper bound of Vol( M t ) such that α Ω t ( ω t ) > l , where Vol( M t ) denotes the volume of M t .Proof. Proposition 2.8 [41] implies this argument. (cid:3)
According to Lemma 4.1, Lemma 4.2, Lemma 3.2 and Theorem 3.2, we have
Proposition 4.1.
There exists a number λ such that for any λ ∈ (1 − m − , λ ] and t ∈ ∆ , M t has a unique twisted K¨ahler-Ricci soliton. Note that on each M t ( t = 0), we obtain the existence and uniqueness of the twistedK¨ahler-Ricci soliton in the sense of definition 3.1 when r ( λ ) is small enough, but we donot know the regularity about this solution. So we need the following proposition. Proposition 4.2.
Assume that ω φ = ω + √− ∂ ¯ ∂φ is the twisted K¨ahler-Ricci soliton inthe sense of definition 3.1 on a smooth Fano manifold M , then φ is smooth.Proof. Proposition 1.4 [4] says that e − r ( λ ) φ ∈ L p ( M ) for all p ≥
1. Lemma 5.1 [51] andCorollary 5.3 [51] imply that | V ( φ ) | is bounded. So by [17], φ is continuous on M . We canobtain the Laplacian estimate for φ according to Proposition 6.1 [51]. By the standardelliptic regularity theory, φ is smooth. (cid:3) Uniform lower bounds on Ding functional and Mabuchi functional.
Wedefine F ′ V t ,λ and M ′ V t ,λ to be the infimum of the twisted Ding functional and Mabuchifunctional on M t with base metric ω t . If the twisted K¨ahler-Ricci soliton ω φ t,λ exists,then F ′ V t ,λ can be achieved at φ t,λ (c.f. P1006 [15]). The goal of this subsection is to provethe following theorem. Theorem 4.1.
Suppose that for a fixed λ ∈ (1 − m − , , there are twisted K¨ahler-Riccisolitons ω φ t,λ for all t ∈ ∆ . Then we have lim sup t → F ′ V t ,λ > −∞ and lim sup t → M ′ V t ,λ > −∞ . We first prove the statement about the twisted Ding functional. We follow the argumentof [41] and [25]. For r ∈ (0 , M r = M| ∆ r ⊂ CP N × ∆ r , where ∆ r is a discwith radius r in C . So M r can be viewed as a complex analytic variety with smoothboundary, endowed with a natural K¨ahler metric W = ω t + √− dt ∧ d ¯ t . Let ω φ t,λ be theunique twisted K¨ahler-Ricci soliton on each M t for λ ∈ (1 − m − , t, · ) := φ t,λ ( · ) on M r . We consider the Dirichlet problem for the following homogeneouscomplex Monge-Amp`ere equation(4.3) ( W + √− ∂ ¯ ∂ Φ) n +1 = 0 , W + √− ∂ ¯ ∂ Φ ≥ , Φ | ∂ M r = Ψ . Proposition 2.7 [2] claims that Φ := sup { Φ ′ ∈ PSH( M r , W ) | Φ ′ ≤ Ψ on ∂ M r } is theunique solution of the equation (4.3). Note that Φ is T -invariant by the same argumentof Remark 3.3, where T is the compact group induced by Im V .We need the following auxiliary lemma (c.f. Proposition 2.17 [41] and [33]). Lemma 4.3.
The Dirichlet problem (4.3) has a unique solution which is bounded on M r (i.e. || Φ || L ∞ ≤ C ) and locally C ,α away from the singular set of M r Denote Φ by the solution of (4.3). For t ∈ ∆ r , set f ( t ) = − r ( λ ) · E V t (Φ t ) = − r ( λ ) · Z Z M t Φ t e θ Mt + s · V t (Φ t ) ω ns Φ t ∧ ds and g ( t ) = − log Z M t e − r ( λ )Φ t Ω t where Φ t = Φ | M t . Then the twisted Ding functional is the sum of these two functions. Proposition 4.3.
The function g ( t ) is continuous and subharmonic on ∆ r .Proof. From the C ,α regularity of Φ, g ( t ) is continuous on ∆ ∗ r . Next we prove that g is subharmonic on ∆ ∗ r . It suffices to prove this for t in a small disk ∆ ′ ⊂ ∆ ∗ . e − r ( λ )Φ t Ω t can be viewed as a smooth Hermitian metric on K − M / ∆ with positive cur-vature r ( λ )( W + √− ∂ ¯ ∂ Φ) + (1 − r ( λ )) W . Consider the direct image bundle D withfibers D t = Γ( M t , K − M t ⊗ K M t ), the trivial section e has L -norm given by || e || t = Z M t e − r ( λ )Φ t Ω t . Berndtsson’s positivity of the direct image bundle (see Lemma 2.1 [6] and Theorem 3.1[7]) implies that − log || e || t is a smooth subharmonic function over ∆ ′ . Note that g iscontinuous at t = 0 by the same calculation of Lemma 2 [23]. Therefore g is subharmonicon ∆ r . (cid:3) Proposition 4.4.
The function f ( t ) is continuous on ∆ r .Proof. We only prove the continuity at t = 0. For any δ >
0, we choose U δ to be thecomplement of a small neighborhood S (the singular set of M ) such that Z Z M \ U δ e θ M + s · V (Φ ) ω ns Φ ∧ ds < δ. XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 17
We then extend U δ to a smooth family of open subsets U δt in M t . Using the same notationsas the beginning of subsection 4.1, we let G : U δ × ∆ r → M r be the smooth embedding.By the C ,α regularity of Φ, we see that Φ t ◦ G t C ,α -converges to Φ and ( G − t ) ∗ V t (Φ t ◦ G t ) C α -converges to V (Φ ) on U δ . By [1], G ∗ t ω ns Φ t converges to ω ns Φ as currents on U δ . Sowe have thatlim t → Z Z U δt Φ t e θ Mt + s · V t (Φ t ) ω ns Φ t ∧ ds = Z Z U δ Φ e θ M + s · V (Φ ) ω ns Φ ∧ ds, lim t → Z Z U δt e θ Mt + s · V t (Φ t ) ω ns Φ t ∧ ds = Z Z U δ e θ M + s · V (Φ ) ω ns Φ ∧ ds, lim t → Z Z U δt ω ns Φ t ∧ ds = Z Z U δ ω ns Φ ∧ ds. The following calculation (cid:12)(cid:12)(cid:12) Z Z M t \ U δt Φ t e θ Mt + s · V t (Φ t ) ω ns Φ t ∧ ds (cid:12)(cid:12)(cid:12) ≤ e C · || Φ || L ∞ · (cid:16) Z Z M t ω ns Φ t ∧ ds − Z Z U δt ω ns Φ t ∧ ds (cid:17) implies lim t → (cid:12)(cid:12)(cid:12) Z Z M t \ U δt Φ t e θ Mt + s · V t (Φ t ) ω ns Φ t ∧ ds (cid:12)(cid:12)(cid:12) ≤ δe C · || Φ || L ∞ . Let δ →
0, we conclude this proposition. (cid:3)
Proposition 4.5.
The function f ( t ) is subharmonic on ∆ r .Proof. Choose a small dick ∆ ′ ⊂ ∆ ∗ r , we want to show that f is subharmonic on ∆ ′ .Let h be an arbitrary non-negative function supported on ∆ ′ , then we have the followingcalculation Z ∆ ′ f √− ∂ ¯ ∂h = − r ( λ ) Z Z π − (∆ ′ ) π ∗ h √− ∂ ¯ ∂ (Φ e θ M + s ·V (Φ) ) W ns Φ ∧ ds = − r ( λ ) Z Z π − (∆ ′ ) π ∗ h √− ∂ ¯ ∂ Φ · e θ M + s ·V (Φ) W ns Φ ∧ ds + r ( λ ) Z Z π − (∆ ′ ) π ∗ h √− ∂ Φ ∧ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds − r ( λ ) Z Z π − (∆ ′ ) π ∗ h √− ∂ Φ ∧ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds − r ( λ ) Z Z π − (∆ ′ ) π ∗ h · Φ √− ∂ ( θ M + s · V (Φ)) ∧ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds − r ( λ ) Z Z π − (∆ ′ ) π ∗ h · Φ √− ∂ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds. Note that i V W s Φ = √− ∂ ( θ M + s · V (Φ)) and i ¯ V W s Φ = −√− ∂ ( θ M + s · V (Φ)) . So we have Z Z π − (∆ ′ ) π ∗ h √− ∂ Φ ∧ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds = − n + 1 Z Z π − (∆ ′ ) π ∗ h · V (Φ) · e θ M + s ·V (Φ) W n +1 s Φ ∧ ds and Z Z π − (∆ ′ ) π ∗ h √− ∂ Φ ∧ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds = 1 n + 1 Z Z π − (∆ ′ ) π ∗ h · V (Φ) · e θ M + s ·V (Φ) W n +1 s Φ ∧ ds. The integral by parts implies that Z Z π − (∆ ′ ) π ∗ h · Φ √− ∂ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds = − Z Z π − (∆ ′ ) ( π ∗ h∂ Φ + Φ ∂ ( π ∗ h )) ∧ √− ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds − Z Z π − (∆ ′ ) π ∗ h · Φ √− ∂ ( θ M + s · V (Φ)) ∧ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds = − n + 1 Z Z π − (∆ ′ ) π ∗ h · V (Φ) · e θ M + s ·V (Φ) W n +1 s Φ ∧ ds − Z Z π − (∆ ′ ) π ∗ h · Φ √− ∂ ( θ M + s · V (Φ)) ∧ ¯ ∂ ( θ M + s · V (Φ)) e θ M + s ·V (Φ) W ns Φ ∧ ds where the second equality holds due to V ( π ∗ h ) = 0. Therefore, we obtain Z ∆ ′ f √− ∂ ¯ ∂h = − r ( λ ) Z Z π − (∆ ′ ) π ∗ h √− ∂ ¯ ∂ Φ · e θ M + s ·V (Φ) W ns Φ ∧ ds − r ( λ ) n + 1 Z Z π − (∆ ′ ) π ∗ h · V (Φ) · e θ M + s ·V (Φ) W n +1 s Φ ∧ ds = − r ( λ ) n + 1 Z π − (∆ ′ ) π ∗ h Z dds ( e θ M + s ·V (Φ) W n +1 s Φ ) ds = − r ( λ ) n + 1 Z π − (∆ ′ ) π ∗ h · ( e θ M + V (Φ) W n +1Φ − e θ M W n +1 )= r ( λ ) n + 1 Z π − (∆ ′ ) π ∗ h · e θ M W n +1 ≥ f is subharmonic on ∆ r since it is continuous at t = 0. (cid:3) Now we give the proof of Theorem 4.1.
XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 19
Proof of Theorem 4.1.
From Proposition 4.3, 4.4 and 4.5, we know that F V t ,λ (Φ t ) is acontinuous subharmonic function on ∆ r , so by the maximum principlesup ∆ r F V t ,λ (Φ t ) = sup ∂ ∆ r F ′ V t ,λ ≥ F V ,λ (Φ ) ≥ F ′ V ,λ , where the last inequality bases on the fact that the twisted K¨ahler-Ricci soliton minimizesthe twisted Ding functional. Finally, letting r →
0, we obtainlim sup t → F ′ V t ,λ ≥ F ′ V ,λ . This proves the statement about the twisted Ding functional. Remark 4.1 claims thatlim sup t → M ′ V t ,λ ≥ F ′ V ,λ , so we complete the proof. (cid:3) L ∞ -estimates and locally higher order estimates for the potentials. Thegoal of this subsection is to obtain L ∞ -estimates and Laplacian estimates for the poten-tials. First we establish some auxiliary lemmas. Let M be a smooth Fano manifold, theAubin’s functional are given by I ( φ ) = Z M φ ( ω n − ω nφ ) and J ( φ ) = Z Z M φ ( ω n − ω nsφ ) ∧ ds where φ ∈ C ∞ ( M ) ∩ PSH(
M, ω ) T . Lemma 4.4.
There are positive constants C ( α M ) and C ( β M ) such that C ( I ( φ ) − J ( φ )) ≤ I V ( φ ) − J V ( φ ) ≤ C ( I ( φ ) − J ( φ )) . Proof.
Take a path φ s = sφ , then by Lemma 3.3 [48], we have dds ( I V ( φ s ) − J V ( φ s )) = s Z M | ∂φ | ω sφ e θ M + s · V ( φ ) ω nsφ . We also know that dds ( I ( φ s ) − J ( φ s )) = s Z M | ∂φ | ω sφ ω nsφ . Thus e α M dds ( I ( φ s ) − J ( φ s )) ≤ dds ( I V ( φ s ) − J V ( φ s )) ≤ e β M dds ( I ( φ s ) − J ( φ s )) . So we obtain this lemma. (cid:3)
Lemma 4.5.
For λ ∈ [ λ , λ ] and − m − < λ < λ < , if φ t,λ are twisted K¨ahler-Riccisolitons for t ∈ ∆ ∗ and I V t ( φ t,λ ) is uniformly bounded for t and λ , then there is a uniformconstant C such that || φ t,λ || L ∞ ≤ C for t ∈ ∆ ∗ and λ ∈ [ λ , λ ] .Proof. Let G ( · , · ) be the Green function of m ω F S for Laplacian operator △ m ω F S on CP N , then there exists a constant C > G ( · , · ) ≥ − C . Denote G t ( · , · )by G ( · , · ) | M t × M t . By the inequality △ ω t φ t,λ > − n , we have r ( λ ) (cid:16) sup M t φ t,λ − a t Z M t φ t,λ ω nt (cid:17) ≤ r ( λ ) a t · n · Z M t ( G t ( · , · ) + C ) ω nt ≤ C where a t = R M t ω nt . Next, set ω t,λ := ω φ t,λ . Since Ric( ω t,λ ) − √− ∂ ¯ ∂ ( θ M t + V t ( φ t,λ )) > r ( λ ) ω t,λ , TheoremB [29] implies that G ω t,λ ( x, y ) ≥ − C where G ω t,λ ( · , · ) is the Green function of ω t,λ for operator Re D t,λ ( D t,λ := △ ω t,λ + V t ). Bythe inequality D t,λ ( φ t,λ ) ≤ n + C , we have r ( λ ) (cid:16) inf M t φ t,λ − a t Z M t φ t,λ ω nt,λ (cid:17) ≥ − C . Therefore, osc φ t,λ := sup M t φ t,λ − inf M t φ t,λ ≤ C + 1 a t Z M t φ t,λ ( ω nt − ω nt,λ ) . This lemma is followed by the boundedness of I V t ( φ t,λ ) and Lemma 4.4. (cid:3) Remark 4.2. If I ( φ t,λ ) is uniformly bounded for t and λ , then the above lemma can alsobe deduced. Let K be a compact subset of M \ S , then we construct a compact subset K := S t ∈ ∆ ∗ G t ( K ) on M| ∆ ∗ , where G t is explained at the beginning of subsection 4.1. Thenwe have Lemma 4.6.
For λ ∈ [ λ , λ ] and − m − < λ < λ < , suppose that φ t,λ are twistedK¨ahler-Ricci solitons for t ∈ ∆ ∗ and φ t,λ is uniformly bounded for t and λ . If K is acompact subset of M| ∆ ∗ as above, then for each k ≥ , we have || φ t,λ || C k ( K ∩ M t ) ≤ C ,where C is a positive constant only depending on λ , λ , K , k and || φ t,λ || L ∞ .Proof. By a direct calculation, we have △ ω t,λ log tr ω t ω t,λ ≥ − tr ω t (Ric( ω t,λ )) tr ω t ω t,λ − C · tr ω t,λ ω t , where C is a constant of the lower bound for the holomorphic bisectional curvature of ω t . Note that, by the definition of φ t,λ , − Ric( ω t,λ ) + √− ∂ ¯ ∂θ ( φ t,λ ) = − r ( λ ) ω t,λ − (1 − λ ) ω F S where θ ( φ t,λ ) := θ M t + V t ( φ t,λ ). Applying the inequality n ≤ ( tr ω t ω t,λ ) · ( tr ω t,λ ω t ), we get △ ω t,λ log tr ω t ω t,λ ≥ − C · tr ω t,λ ω t − C − △ ω t θ ( φ t,λ ) tr ω t ω t,λ . Set H = log tr ω t ω t,λ − ( C + 1) φ t,λ , so we have △ ω t,λ H ≥ tr ω t,λ ω t − C − △ ω t θ ( φ t,λ ) tr ω t ω t,λ . Assume that the function H achieves its maximum at some point x , then at this point(4.4) ∇ (cid:0) e − ( C +1) φ t,λ ( n + △ ω t φ t,λ ) (cid:1) = 0 . At x , we choose the normal coordinate so that g t,i ¯ j = δ ij and ( φ t,λ ) i ¯ j = δ ij · ( φ t,λ ) i ¯ i .Therefore, (4.4) gives (cid:0) n + ( φ t,λ ) i ¯ i (cid:1) l − (cid:2) ( C + 1)( φ t,λ ) l (cid:3) ( n + △ ω t φ t,λ ) = 0 , XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 21 which yields V lt · ( φ t,λ ) i ¯ il = V lt · (cid:2) ( C + 1)( φ t,λ ) l (cid:3) ( n + △ ω t φ t,λ ) = ( C + 1) · V t ( φ t,λ )( n + △ ω t φ t,λ ) . Lemma 5.1 [51] and Corollary 5.3 [51] imply that | V t ( φ t,λ ) | ≤ C , so V lt · ( φ t,λ ) i ¯ il ≤ C · ( n + △ ω t φ t,λ ) . At point x , we also have △ ω t θ ( φ t,λ ) = θ ( φ t,λ ) i ¯ i = (cid:2) V lt · ( g t,l ¯ i + ( φ t,λ ) l ¯ i ) (cid:3) i = V lt · g t,l ¯ ii + V lt · ( φ t,λ ) l ¯ ii + V lt,i · ( g t,l ¯ i + ( φ t,λ ) l ¯ i ) ≤ C · ( n + △ ω t φ t,λ ) + sup M t | V lt,i | · ( n + △ ω t φ t,λ ) ≤ C · ( n + △ ω t φ t,λ )Thus we obtain tr ω t,λ ω t ( x ) ≤ C . The inequality(4.5) tr ω t ω t,λ ≤ n · ω nt,λ ω nt · ( tr ω t,λ ω t ) n − gives log tr ω t ω t,λ ≤ log n − θ ( φ t,λ ) − r ( λ ) φ t,λ + log Ω t ω nt + ( n −
1) log tr ω t,λ ω t . Therefore, the boundedness of θ ( φ t,λ ) and φ t,λ imply H ≤ H ( x ) ≤ C + log Ω t ω nt . Note that there exists a constant C ′ K such that Ω t ω nt ≤ e C ′ K on K ∩ M t , so we have tr ω t ω t,λ ≤ C K . Using (4.5) again, we get tr ω t,λ ω t ≤ C K . This lemma holds due to the standardEvans-Krylov theory [18] [21] for the complex Monge-Amp`ere equation. (cid:3) The next lemma illustrates that the functional I is continuous under the above conti-nuity of K¨ahler potentials (c.f. Lemma 2.14 [41]). Lemma 4.7.
Suppose that φ t,λ are twisted K¨ahler-Ricci solitons and φ t,λ is uniformlybounded. If φ t,λ ◦ G t converges to φ ,λ in the C sense on any compact subset away from S on M , then we have lim t → I ( φ t,λ ) = I ( φ ,λ ) . Fix ˆ λ ∈ (0 , − m − ), by the definition of the twisted Mabuchi functional and theelementary inequality x log x ≥ − e − , we see that for any φ ∈ PSH( M t , ω t ) T t ∩ C ∞ ( M t ), M V t , ˆ λ ( φ ) = − r (ˆ λ ) (cid:0) I V t ( φ ) − J V t ( φ ) (cid:1) + Z M log e θ Mt + V t ( φ ) ω nφ t Ω t e θ Mt + V t ( φ ) ω nφ t ≥ − r (ˆ λ ) (cid:0) I V t ( φ ) − J V t ( φ ) (cid:1) − e − · Z M t Ω t ≥ − r (ˆ λ ) (cid:0) I V t ( φ ) − J V t ( φ ) (cid:1) − C where C is a constant independent of t since the volume of M t can be bounded. On theother hand, assume that there exists a twisted K¨ahler-Ricci soliton on each M t for t ∈ ∆ ∗ when λ = ¯ λ , then by Theorem 4.1, we can find C > t i → M V i , ¯ λ ( φ ) ≥ − C ( V t i = V i ) for any φ ∈ PSH( M t i , ω t i ) T ti ∩ C ∞ ( M t i ).Note that the twisted Mabuchi functional is linear in λ , i.e. sM V i , ˆ λ ( φ ) + (1 − s ) M V i , ¯ λ ( φ ) = M V i ,s ˆ λ +(1 − s )¯ λ ( φ ) , so for each λ ∈ [ˆ λ, ¯ λ ), we have M V i ,λ ≥ δ λ · (cid:0) I V i ( φ ) − J V i ( φ ) (cid:1) − C where δ λ = − r (ˆ λ ) λ − ¯ λ ˆ λ − ¯ λ >
0. Lemma 3.1 and 3.3 claim that the minimizer of the twistedMabuchi functional is just that of the twisted Ding functional, so the twisted K¨ahler-Riccisoliton φ t,λ is the minimum of M V t ,λ . Therefore for λ ∈ [ˆ λ, ¯ λ ), we have Z M i ( θ i − h i ) e θ i ω nt i = M V i ,λ (0) ≥ M V i ,λ ( φ t i ,λ ) ≥ δ λ · (cid:0) I V i ( φ t i ,λ ) − J V i ( φ t i ,λ ) (cid:1) − C where θ i = θ M ti and h i is defined by Ric( ω t i ) = ω t i + √− ∂ ¯ ∂h i . According to Proposition2.21 [41] and Proposition 2.2, we obtain I V i ( φ t i ,λ ) ≤ Cδ − λ . If fix a small number ǫ >
0, then for any λ ∈ [1 − m − + ǫ, ¯ λ − ǫ ], we have I V i ( φ t i ,λ ) ≤ C ′ ǫ which implies || φ t i ,λ || L ∞ ≤ C ǫ by Lemma 4.5. Lemma 4.6 claims that by passing to asubsequence t ′ i ( { t ′ i } ⊂ { t i } ), φ t ′ i ,λ ◦ G t ′ i C ∞ -converges to a smooth function φ ,λ on M \ S ,which satisfies e θ M + V ( φ ,λ ) ( ω + √− ∂ ¯ ∂φ ,λ ) n = e − r ( λ ) φ ,λ Ω . This equation implies that φ ,λ is a twisted K¨ahler-Ricci soliton on M . By Theorem 3.2,we know that φ t i ,λ ◦ G t i C ∞ -converges to φ ,λ on M \ S . Proposition 4.6.
For λ ∈ [1 − m − + ǫ, ¯ λ − ǫ ] , suppose that φ t,λ are the twisted K¨ahler-Ricci solitons, then lim sup δ → max | t | = δ I ( φ t,λ ) < + ∞ . In particular || φ t,λ || L ∞ < C ǫ . Proof.
By the previous argument and Lemma 4.4, I ( φ t i ,λ ) ≤ C . Next we argue bycontradiction, then we pick s j and | s j | → I ( φ s j ,λ ) = C + 1. The sameargument as above claims that φ s j ,λ ◦ G s j C ∞ -converges to φ ,λ . Lemma 4.7 giveslim j →∞ I ( φ s j ,λ ) = C + 1 = I ( φ ,λ ) = lim i →∞ I ( φ t i ,λ ) ≤ C. This is a contradiction, so we obtain this consequence. (cid:3)
Finally, we give the following theorem.
Theorem 4.2.
For λ ∈ [1 − m − + ǫ, ¯ λ − ǫ ] , suppose that φ t,λ are the twisted K¨ahler-Riccisolitons, then φ t,λ ◦ G t C ∞ -converges to φ ,λ on M \ S . XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 23 Gromov-Hausdorff convergence under L ∞ -bound on K¨ahlerpotentials In this section we investigate the behavior of twisted K¨ahler-Ricci solitons in a Q -Gorestein smoothing family. We use the techniques in [50] [35] [15] [37] [16] [28] [22] [49][34] and [43], to show that the Gromov-Hausdorff limit as t → ω t,λ are twisted K¨ahler-Riccisolitons on M t for t ∈ ∆ ∗ and 1 − m − + ǫ ≤ λ <
1, then Theorem B [30] says that thediameter of ( M t , ω t,λ ) has a uniform upper bound only depending on r ( λ ). Lemma 6.1[50] claims that(5.6) |∇ θ ( φ t,λ ) | ω t,λ + |△ ω t,λ θ ( φ t,λ ) | ≤ C where θ ( φ t,λ ) = θ M t + V t ( φ t,λ ) and C is a positive constant only depending on | θ φ t,λ | and | V t | ω t , moreover we can choose C independent of t . By Theorem 6.2 [50], we have thenon-collapsing property, i.e. for any p t ∈ M t ,Vol (cid:0) B p t (1) , ω t,λ (cid:1) ≥ C > , where C is a positive constant independent of t . By the Gromov precompactness theorem,passing to a subsequence t i →
0, we may assume that( M t i , ω t i ,λ ) d GH −−→ ( X, d ) . The limit (
X, d ) is a compact length metric space. It has regular/singular decomposition X = R ∪ S , a point x ∈ R if and only if the tangent cone at x is the Euclidean space R n . To simplify notation, we denote ( M i , ω i,λ ) by ( M t i , ω t i ,λ ). Lemma 5.1.
The regular set R is open in the limit space ( X, d ) .Proof. If x ∈ R , then there exists r = r ( x ) > H n ( B d ( x, r )) ≥ (1 − δ )Vol( B r ),where H n denotes the Hausdorff measure and B r is a ball of radius r in 2 n -Euclideanspace. Suppose x i ∈ M i satisfying x i d GH −−→ x , then by the volume convergence theorem(Remark 5.2 [50]), Vol( B ( x i , r ) , ω i,λ ) ≥ (1 − δ )Vol( B r ) for sufficiently large i. Proposition21 [15] claims that there exists a positive constant A such that 0 < Ric( ω i,λ ) − L V i ω i,λ ≤ Aω i,λ in B ω i,λ ( x i , r ). So by the Proposition 19 [15], there exists a constant δ ′ such that B d ( x, δ ′ r ) has C ,α harmonic coordinate. This implies B d ( x, δ ′ r ) ⊂ R , furthermore R isopen with a C ,α K¨ahler metric ω and ω i,λ converges to ω in C ,α -topology. (cid:3) Since R is dense in X , so we have the following lemma. Lemma 5.2. ( X, d ) = ( R , ω ) , the metric completion of ( R , ω ) . Next, by the argument of [35], we define Γ t := M t \ G t ( S ), where S denotes the singularset of M . Define the Gromov-Hausdorff limit of Γ t Γ := (cid:8) x ∈ X | there exists x i ∈ Γ i := Γ t i such that x i → x (cid:9) . Assume that the K¨ahler potentials || φ t,λ || L ∞ is uniformly bounded, then we have Proposition 5.1. ( X, d ) is isometric to ( M \ S, ω ,λ ) , where ω ,λ is the unique twistedK¨ahler-Ricci soliton on M . Proof.
First we prove the following claim.
Claim 5.1. Γ \S is a subvariety of dimension ( n − if it is not empty.Proof. Let x ∈ Γ \S and x i ∈ Γ i such that x i d GH −−→ x . By the C ,α convergence of ω i,λ ,there are C, r > i and a sequence of harmonic coordinates in B ω i,λ ( x i , r )such that C − ω E ≤ ω i,λ ≤ Cω E where ω E is the Euclidean metric in the coordinates.Furthermore, the sequence of harmonic coordinates can be perturbed to holomorphiccoordinates on B ω i,λ ( x i , r ) according to Lemma 3.11 [44]. Since the total volume of Γ i isuniformly bounded, the local analytic set Γ i ∩ B ω i,λ ( x i , r ) have a uniform bound of degreeand so converge to an analytic set Γ ∩ B d ( x, r ). (cid:3) From the above claim we know that dim R (Γ) ≤ n −
2. By the argument of [35],( X \ Γ , ω ) is homeomorphic and locally isometric to ( M \ S, ω ,λ ). Since X is a lengthmetric space and dim R (Γ) ≤ n −
2, ( X \ Γ , ω ) is isometric to ( M \ S, ω ,λ ). So we have( X, d ) = ( X \ Γ , ω ) = ( M \ S, ω ,λ ) . (cid:3) A direct corollary is
Corollary 5.1. ( M t , ω t,λ ) converges globally to ( X, d ) in the Gromov-Hausdorff topologyas t → . Proposition 5.2. M \ S = R , the regular set of X .Proof. Since M \ S has smooth structure in X , we have M \ S ⊂ R . Next we showthe converse. We argue by contradiction. Suppose p ∈ R\ ( M \ S ), then there exists asequence of points p t ∈ Γ t such that p t d GH −−→ p . By the C ,α regularity of ( R , ω ), thereexist C, r > t and a sequence of holomorphic coordinates on B ω t,λ ( p t , r )such that C − ω E ≤ ω t,λ ≤ Cω E . Denote q = dim C (Γ t ), thenVol (cid:0) Γ t ∩ B ω t,λ ( p t , r ) (cid:1) = Z Γ t ∩ B ωt,λ ( p t ,r ) ω qt,λ ≥ Z Γ t ∩ B ωE ( C − r ) ( C − ω E ) q which has a positive lower bound. However this contradicts with the following argumentVol (cid:0) Γ t ∩ B ω t,λ ( p t , r ) (cid:1) ≤ Z Γ t ω qt,λ = Z Γ t ω qt which tends to 0 as t → (cid:3) Next we will obtain some uniform L -estimates for H ( M t , K − mM t ). For a fixed λ , usingthe same notations in [16], we denote K ♯M t = K − mM t , h ♯t,λ = h mt,λ , ω ♯t,λ = m · ω t,λ , L p,♯ ( M t ) = L p ( M t , ω ♯t,λ ) , where ω t,λ is twisted K¨ahler-Ricci soliton on each M t and h t,λ is the Hermitian metricon K − M t with its curvature Ric( h t,λ ) = r ( λ ) ω t,λ + (1 − λ ) ω F S , i.e. h t,λ = e θ t,λ ω nt,λ , where θ ( φ t,λ ) = θ M t + V t ( φ t,λ ). Let f g t,λ = e − n − θ ( φ t,λ ) g t,λ , then the estimate (5.6) implies thatthe Ricci curvature of f g t,λ has a uniform lower bound. Therefore, the Sobolev constantis uniform bounded for f g t,λ , so it is for g t,λ as f g t,λ and g t,λ are uniformly equivalent. Thesame argument of Proposition 4.1 [34] gives XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 25 (1) Let s be a holomorphic section of H ( M t , K ♯M t ), then there exist two constants C C independent of t such that || s || L ∞ ,♯ ≤ C || s || L ,♯ and ||∇ s || L ∞ ,♯ ≤ C || s || L ,♯ . (2) Assume that σ is a K ♯M t -valued (0 , L inner product is defined by Z M t | σ | h ♯t,λ ,g ♯t,λ e θ ( φ t,λ ) ( ω ♯t,λ ) n . Denote ¯ ∂ ∗ θ ( φ t,λ ) by the adjoint operator of ¯ ∂ , then there exists a constant A inde-pendent of t such that ¯ ∂ ¯ ∂ ∗ θ ( φ t,λ ) + ¯ ∂ ∗ θ ( φ t,λ ) ¯ ∂ ≥ A .The next definition comes from [12] and [42]. Definition 5.1.
Let p ∈ X and C ( Y ) be the tangent cone at p . We say that the tangentcone is good if the following hold: (1) the regular set Y reg is open in Y and smooth, (2) the distance function on C ( Y reg ) is induced by a Ricci flat K¨ahler metric, (3) for all η > , there is a Lipschitz function g on Y , equal to on a neighborhoodof the singular set S Y ⊂ Y , supported on the η -neighborhood of S Y and with ||∇ g || L ≤ η . The argument of [15] (P1001) claims that all the tangent cones are good. So by theargument of [16], we have
Theorem 5.1.
Let π : M → ∆ be a Q -Gorestein smoothing family and V be a reduc-tive vector field on M , which preserves the fibers. For λ ∈ (1 − m − , there is a twistedK¨ahler-Ricci soliton ω t,λ := ω t + √− ∂ ¯ ∂φ t,λ on each M t for t ∈ ∆ with uniformly bounded || φ t,λ || L ∞ ( t = 0) . Then ( M t , ω t,λ ) converges to ( M , ω ,λ ) in the Gromov-Hausdorff topol-ogy as t → . Remark 5.1.
The same conclusion is true if λ vary and stay bounded, i.e. λ ∈ [ λ , λ ] where − m − < λ < λ < . Existence of K¨ahler-Ricci solitons
In this section we show the main theorem of this article by using the argument ofsection 4 [41]. We define the following function: λ t := sup n λ ∈ (1 − m , (cid:12)(cid:12)(cid:12) ∃ twisted K¨ahler-Ricci solitons on M t for all κ ≤ λ o . Proposition 6.1. If ( M , V ) is K-stable, then the function λ t is lower semi-continuouson ∆ .Proof. We only deal with the lower semi-continuous at t = 0 and the other case is easierby the same argument. Suppose that λ t is not lower semi-continuous at t = 0, i.e.lim inf t → λ t = λ ∞ < λ ≤
1. Choosing an increasing sequence λ i < λ ∞ with lim i →∞ λ i = λ ∞ . For any i , the definition of λ t implies that there exists a twisted K¨ahler-Ricci soliton ω t,λ i on each M t when | t | is small enough. There is a twisted K¨ahler-Ricci soliton ω ,λ on M for each λ ∈ [ λ, λ ), where λ is defined in Proposition 4.1. According to Theorem5.1, for each i , ( M t , V t , (1 − λ i ) ω F S , ω t,λ i ) converges to ( M , V , (1 − λ i ) ω F S , ω ,λ i ) in the Gromov-Hausdorff topology. Then, using the diagonal arguments as section 3 in [15], bypassing to a subsequence we have that ( M , V , (1 − λ i ) ω F S , ω ,λ i ) converges to ( Y, e V , (1 − λ ∞ ) β, ω ) in the Gromov-Hausdorff topology as λ i → λ ∞ , where Y is a Q -Fano variety, e V is a holomorphic vector field, β is a closed positive (1 , ω is a twistedK¨ahler-Ricci soliton. Note that ( M , V ) is K-stable, so section 3 [15] (P991-992) impliesthat ( M , V , (1 − λ ∞ ) ω F S ) ∼ = ( Y, e V , (1 − λ ∞ ) β ). Theorem 3.2 claims that ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ) ∼ = ( Y, e V , (1 − λ ∞ ) β, ω ).Let Z be the space of all ( M t , V t , (1 − λ ) ω F S , ω t,λ ) with λ ∈ [ λ, λ t ). Denote Z by theclosure of Z under the Gromov-Hausdorff convergence and C by the subspace of Z\Z which consists of limits of some sequnece ( M t i , V t i , (1 − λ i ) ω F S , ω t i ,λ i ) with t i → λ i → λ ∞ . By the argument of [16], we have an injective continuous map from Z into Ch /U ( N ),where Ch denotes the Chow variety. We observe that ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ) is in C . Lemma 6.1.
We have C = (cid:8) ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ) (cid:9) Proof.
First, we claim that there is an open neighborhood U of ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ )such that C ∩ U = (cid:8) ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ) (cid:9) . Otherwise, we can choose a sequence (cid:8) ( Y i , V Y i , (1 − λ ∞ ) β Y i , ω Y i ) (cid:9) ∞ i =1 ⊂ C converging in the Gromov-Hausdorff topology to( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ). Take some sequence ( M t ij , V t ij , (1 − λ j ) ω F S , ω t ij ,λ j ) such that( Y i , V Y i , (1 − λ ∞ ) β Y i , ω Y i ) is the Gromov-Hausdorff limit as t ij → λ j → λ ∞ for each i . For any sequence { t kj k } ∞ k =1 converging to 0, we have that ( M t kjk , V t kjk , (1 − λ j k ) ω F S , ω t kjk ,λ jk )converges to ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ), which implies the functional I ( φ t kjk ,λ jk ) → I ( φ ,λ ∞ ) as k → ∞ . Thus, φ t ij ,λ j is uniformly bounded in L ∞ for all i and j . Fur-thermore, ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ) ∼ = ( Y i , V Y i , (1 − λ ∞ ) β Y i , ω Y i ) for each i accordingto Theorem 5.1.Second, define a family C α := S < | t | <α (cid:8) ( M t , V t , (1 − λ ) ω F S , ω t,λ ) | λ ∈ ( λ t − α, λ t ) (cid:9) indexed by α ∈ (0 , α → C α = C and each C α is path-connected. Lemma 4.3 [41] claims that C is connected.So we complete the proof of this lemma. (cid:3) By the definition of λ t , we let λ tends to λ t , then by [15], ( M t , V t , (1 − λ ) ω F S , ω t,λ )converges by subsequence to some limit ( X t , e V t , (1 − λ t ) β t , ω t ) such that Aut ( X t , e V t , (1 − λ t ) β t , ω t ) is non-trivial. Choose λ t i → λ ∞ and t i → t → λ t = λ ∞ ), the limiting se-quence ( X t i , f V t i , (1 − λ t i ) β t i , ω t i ) converges by subsequence to ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ )due to the structure of C . This is a contradiction with Lemma 6.1. (cid:3) Next we prove the main theorem of this article.
Theorem 6.1.
Suppose that ( M , V ) is K-stable, then there exists a K¨ahler-Ricci solitonon M .Proof. We define a setΛ := { λ ≤ | there exists a twisted K¨ahler-Ricci soliton on M for each κ ≤ λ } . By Proposition 4.1, it suffices to show that Λ is both open and closed in [ λ, XISTENCE OF K ¨AHLER-RICCI SOLITONS ON SMOOTHABLE Q-FANO VARIETIES 27
First, we prove the openness. For any λ ∈ Λ, by the definition of Λ, we have atwisted K¨ahler-Ricci soliton on M for each κ ≤ λ , so λ < λ . Thus, λ t > λ for | t | small enough since λ t is lower semi-continuous. We can choose a number ˜ λ such that λ t > ˜ λ > λ for | t | small enough. For λ ′ ∈ [ λ, ˜ λ ], the arguments of section 4 and 5 implythat ( M t , V t , (1 − λ ′ ) ω F S , ω t,λ ′ ) converges to ( M , V , (1 − λ ′ ) ω F S , ω ,λ ′ ). Thus, Λ is open.Second, we prove the closedness. Take any sequence { λ i } ∞ i =1 ⊂ Λ which strictlyincreases to λ ∞ . Since λ t is lower semi-continuous, for any i , λ t > λ i when | t | issmall enough. Furthermore, ( M , V , (1 − λ i ) ω F S , ω ,λ i ) is the Gromov-Hausdorff limitof ( M t , V t , (1 − λ i ) ω F S , ω t,λ i ) as t → M , V , (1 − λ i ) ω F S , ω ,λ i ) converges to( Y, e V , (1 − λ ∞ ) β, ω ) as λ i → λ ∞ . The condition that ( M , V ) is K-stable gives that( Y, e V , (1 − λ ∞ ) β, ω ) ∼ = ( M , V , (1 − λ ∞ ) ω F S , ω ,λ ∞ ) according to the argument of [15](P991-992). This implies that Λ is closed. (cid:3) Acknowledgements:
The author was supported by a grant from the FundamentalResearch Funds for the Central Universities. The author also thanks Yi Yao for hisenthusiastic discussion.
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