Kähler-Einstein metrics along the smooth continuity method
aa r X i v : . [ m a t h . DG ] J un K ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTHCONTINUITY METHOD
VED DATAR AND G ´ABOR SZ´EKELYHIDI
Abstract.
We show that if a Fano manifold M is K-stable with respect to spe-cial degenerations equivariant under a compact group of automorphisms, then M admits a K¨ahler-Einstein metric. This is a strengthening of the solution ofthe Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun [17], and can be used to obtain new examples of K¨ahler-Einstein mani-folds. We also give analogous results for twisted K¨ahler-Einstein metrics andKahler-Ricci solitons. Introduction
Let M be a Fano manifold of dimension n . A basic problem in K¨ahler geometryis whether M admits a K¨ahler-Einstein metric. The Yau-Tian-Donaldson conjec-ture [53, 46, 26], confirmed recently by Chen-Donaldson-Sun [17, 18, 19, 20], saysthat M admits a K¨ahler-Einstein metric if and only if it is K-stable. In general itseems to be intractable at present to check K-stability since in principle one muststudy an infinite number of possible degenerations of M to Q -Fano varieites. Onegoal of this paper is to study some situations with large symmetry groups, wherethe problem reduces to checking a finite number of possibilities. This can then beused to yield new examples of K¨ahler-Einstein manifolds.Suppose then that a compact group G acts on M by holomorphic automor-phisms. Our main theorem is the following equivariant version of the result ofChen-Donaldson-Sun. Theorem 1.
Suppose that ( M, K − M ) is K-stable, with respect to special degenera-tions that are G -equivariant. Then M admits a K¨ahler-Einstein metric. Here a G -equivariant special degeneration is a special degeneration X → C inthe sense of Tian [46], together with a holomorphic G action which commutes withthe C ∗ -action, preserves the fibers, and restricts to the given action of G on thegeneric fibers X t = M for t = 0. We also obtain an analogous result for K¨ahler-Ricci solitons, and their twisted versions; see Definition 9 for detailed definitions,and Proposition 10 for the most general result.An important special case is when G is a torus. In particular if M is a toricmanifold, and G = T n is the n -torus, then Proposition 10 implies that we only needto check special degenerations of the form X = M × C to ensure the existence of aK¨ahler-Einstein metric or K¨ahler-Ricci soliton on M . In particular this recovers theresult of Wang-Zhu [50] showing that all toric Fano manifolds admit a K¨ahler-Riccisoliton. In addition we can recover the result of Li [33] on the greatest lower boundon the Ricci curvature of toric Fano manifolds.A more interesting situation is when G = T n − , i.e. M is a complexity-one T -variety. In this case it is possible, in concrete examples, to check all G -equivariant special degenerations of M , and as a consequence we can obtain new examples ofthreefolds with K¨ahler-Einstein metrics and K¨ahler-Ricci solitons. Work in progressby Ilten-S¨uss [30] suggests that we obtain five new K¨ahler-Einstein threefolds. Toour knowledge these are the first examples where K-stability is used to obtain newK¨ahler-Einstein manifolds.Our method of proof of Theorem 1 is to use the classical continuity path(1) Ric( ω t ) = tω t + (1 − t ) α for t ∈ [0 ,
1] proposed by Aubin [6], and its analog for K¨ahler-Ricci solitons studiedby Tian-Zhu [48], and to show that if we cannot find a solution for t = 1, thenthere must be a G -equivariant destabilizing special degeneration. In particular weobtain a new proof of the result of Chen-Donaldson-Sun [20], without using metricswith conical singularities. At the same time our arguments are analogous to thosein [20], using also the adaptation of some of those ideas to the smooth continuitymethod in [42].A key advantage of the smooth continuity path is that it allows one to work ina G -equivariant setting. In contrast, in [20] one considers K¨ahler-Einstein metricssingular along a smooth divisor D ⊂ M , and such a divisor can not be G -invariantunless G is finite (see Song-Wang [39, Theorem 2.1]). The disadvantage of thesmooth continuity path is that in effect one must consider pairs ( V, χ ) of a variety V together with a possibly singular current χ , as opposed to pairs ( V, D ) of a varietyand a divisor. In [20] a destabilizing special degeneration is obtained by applyingthe Luna slice theorem, and for this we must restrict ourselves to a suitable finitedimensional variety rather than the infinite dimensional space of currents. For thisthe basic idea is to approximate a current χ by a sum of currents of integrationalong divisors.A brief outline of the paper is as follows. In Section 2 we collect some basicdefinitions and results on twisted K¨ahler-Ricci solitons. The proof of the mainresult, Proposition 10, will then be given in Section 3. We give some examplesof the applications of our results to toric manifolds and other manifolds of largesymmetry group in Section 4. In Section 5 we discuss how to adapt the methodsof [42] and [36] to obtain the partial C -estimates along the continuity methodfor solitons. A crucial point is the reductivity of the automorphism group of thelimiting variety. This essentially follows from the work of Berndtsson [11] as usedin [20], but since we did not find the exact statement that we need in the literature,we give a brief exposition in Section 6.2. Twisted K¨ahler-Ricci solitons
Suppose that W is a Q -Fano manifold, with log terminal singularities. In partic-ular a power K rW of the canonical bundle on the regular set W extends as a linebundle on W . We say that a metric h on K − W is continuous on W , if the inducedmetric on K − rW extends to a continuous metric on K − rW . Fixing an open cover { U i } and local trivializing holomorphic sections σ i of K − rV | U i ∩ W , we will write(2) | σ i | h r = e − rφ i , for continuous functions φ i on U i . We will write the metric h simply as e − φ followingthe notation in Berndtsson [11]. In particular e − φ defines a volume form on W , ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 3 given in a local chart U i by(3) e − φ = | σ i | /rh r ( σ i ∧ σ i ) − /r . The log terminal condition says that this volume form has finite volume. Wewrite ω φ for the curvature current of the metric e − φ on W , so in our local charts ω φ = √− ∂∂φ i . Since the potentials φ i are locally bounded, by Bedford-Taylor [8]we can form the wedge product ω nφ , which defines a measure on W , and also on W extending it trivially. The metric h φ is a weak K¨ahler-Einstein metric if ω φ is aK¨ahler current, and we have(4) e − φ = ω nφ . Berman and Witt-Nystr¨om [10] have studied the analogous notion of weakK¨ahler-Ricci solitons. Suppose that v is a holomorphic vector field on W , whoseimaginary part generates the action of a torus T on W (see Berman-Boucksom-Eyssidieux-Guedj-Zeriahi [9, Lemma 5.2] to see that one obtains an action on W ).A K¨ahler-Ricci soliton on ( W, v ) is a T invariant continuous metric e − φ , smooth on W with positive curvature current ω φ satisfying(5) e − φ = e θ v ω nφ . Here e θ v ω nφ is a measure defined in [10] for general φ . If φ is smooth, then θ v issimply a Hamiltonian function for the vector field v , satisfying(6) L v ω φ = √− ∂∂θ v , with the normalization(7) Z W e θ v ω nφ = Z W ω nφ = V. In particular θ v depends on φ . For continuous metrics h φ (or more general metricswith positive curvature current), the measure constructed in [10] still satisfies thenormalization (7). In addition by [10, Corollary 2.9] we have some fixed constant C (depending only on M, v ), such that(8) C − ω nφ ≤ e θ v ω nφ ≤ Cω nφ . We now use this to define the twisted analogs of K¨ahler-Ricci solitons, which arisenaturally along the continuity method. Suppose that e − ψ is another metric on K − W which in our local charts is given by plurisubharmonic functions ψ i ∈ L loc ( U i ∩ W ). Definition 2.
For t ∈ (0 ,
1) we say that the pair ( W, (1 − t ) ψ ) is klt , if in eachchart U i ∩ W the function e − ψ i is integrable, with respect to the volume form( σ i ∧ σ i ) − /r . We will on occasion write ( W, (1 − t ) ω ψ ) for the pair, where as before ω ψ is the curvature of e − ψ .Equivalently we can think of e − tφ − (1 − t ) ψ as a volume form on W with e − φ beinga continuous metric as above. The klt condition is then(9) Z W e − tφ − (1 − t ) ψ < ∞ . Definition 3.
A twisted K¨ahler-Ricci soliton on the triple ( W, (1 − t ) ψ, v ), where v is a holomorphic vector field as above, is a continuous metric e − φ such that(10) e − tφ − (1 − t ) ψ = e θ v ω nφ . V. DATAR AND G. SZ´EKELYHIDI
This equation is interpreted as an equality of measures on W , and in particular e − φ here need not be smooth on W , so e θ v ω nφ is the measure defined by Berman-Witt-Nystr¨om [10]. Note that the existence of such a metric implies that ( W, (1 − t ) ψ ) is klt . When t = 1 or v = 0, we will simply omit the corresponding term in the triple.So we can talk about a K¨ahler-Einstein metric on W , a twisted K¨ahler-Einsteinmetric on ( W, (1 − t ) ψ ), or a K¨ahler-Ricci soliton on ( W, v ). Remark 4. If W, φ, ψ are smooth, then the twisted K¨ahler-Ricci soliton equationis equivalent (up to adding a constant to φ ) to(11) Ric( ω φ ) − L v ω φ = tω φ + (1 − t ) ω ψ , which is the natural continuity path for finding K¨ahler-Ricci solitons, used by Tian-Zhu [48] for instance.Even when W is normal and φ is only continuous, it is useful to have an equationfor twisted K¨ahler-Ricci solitons in the form (11). For this the extra conditionneeded is that the measure e θ v ω nφ defines a singular metric e − τ on K W , with τ ∈ L loc . Then φ defines a twisted K¨ahler-Ricci soliton on ( W, (1 − t ) ψ, v ) if(12) ω τ = tω φ + (1 − t ) ω ψ , where ω τ is the curvature of e − τ . Note that by an argument similar to Berman-Boucksom-Eyssidieux-Guedj-Zeriahi [9, Proposition 3.8], if e − τ is only defined out-side a subset S ⊂ W with (2 n − n − ( S ) = 0,and Equation (11) holds on W \ S , then e − φ is a twisted K¨ahler-Ricci soliton. In-deed in this case e − τ extends as a singular metric with positive curvature currentover all of W (see Harvey-Polking [28, Theorem 1.2], Demailly [22]), and then (12)implies that up to modifying ψ by a constant, we must have(13) e θ v ω nφ = e − τ = e − tφ − (1 − t ) ψ , since (12) implies that f = τ − tφ − (1 − t ) ψ is a global L function with √− ∂∂f = 0on W .We need the following result, generalizing the classical results of Bando-Mabuchi [7]and Matsushima [35], which are essentially contained in Berndtsson [11], Boucksom-Berman-Eyssidieux-Guedj-Zeriahi [9], Berman-Witt-Nystr¨om [10] and Chen-Donaldson-Sun [20]. We will give an outline proof in Section 6. Proposition 5.
Suppose that e − φ , e − φ are two twisted K¨ahler-Ricci solitons on ( W, (1 − t ) ψ, v ) . Then there exists a holomorphic vector field w on W , commutingwith v and satisfying ι w ω ψ = 0 , such that the biholomorphisms F t : W → W induced by w satisfy F ∗ ( ω φ ) = ω φ . In addition L Im w ω φ = 0 . Definition 6.
For any triple ( W, (1 − t ) ψ, v ) we define the Lie algebra(14) g W,ψ,v = { w ∈ H ( T W ) : ι w ω ψ = 0 and [ v, w ] = 0 } . As before, we may omit ψ or v from the notation if t = 1 or v = 0. In particular g W = H ( T W ). We will also write g W,β = g W,ψ if β = ω ψ is the curvature of e − ψ .Using a projective embedding into P N , we can realize g W,ψ,v as a subalgebra of sl ( N + 1 , C ).Note that for example g W,ψ is trivial if ω ψ is strictly positive and t <
1. Infact Berndtsson [11, Proposition 8.2] implies that if e − ψ is integrable, then g W,ψ is ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 5 trivial. In our application, when ( W, (1 − t ) ψ ) is klt, e − (1 − t ) ψ will be integrable,but e − ψ will typically not be.Note also that the Lie group with Lie algebra g W,ψ will usually be strictly smallerthan the identity component of the group of biholomorphisms of W preserving ω ψ . The difference comes from the fact that if v is a real vector field then L v ω ψ does not imply L Jv ω ψ for the complex structure J , whereas our Lie algebra aboveis automatically closed under multiplication by √−
1. On the other hand when ω ψ = [ D ] is the current of integration along a divisor, then g W,ψ coincides with thevector fields on W parallel to D . Indeed ι v [ D ] = 0 is equivalent to v being parallelto D along the smooth part of D .The following theorem generalizes [20, Theorem 6], which in turn is a generaliza-tion of Matsushima’s theorem [35] on the reductivity of the automorphism groupof a K¨ahler-Einstein manifold. We will give the proof in Section 6. Proposition 7.
Suppose that ( W, (1 − t ) ψ, v ) admits a twisted K¨ahler-Einstein met-ric e − φ . Then g W,ψ,v is reductive. In addition if G is a group of biholomorphismsof W , fixing ω φ and v , then the centralizer ( g W,ψ,v ) G is also reductive. We finally recall some properties of the “twisted” Futaki invariant, generalizingthe log-Futaki invariant in [20] and the modified Futaki invariant of Tian-Zhu [48].For a smooth metric e − φ on K − W we define(15) Fut (1 − t ) ψ,v ( W, w ) = Fut v ( W, w ) − − tV (cid:20)Z W θ w ( e θ v − ω nφ + n Z W θ w ( ω ψ − ω φ ) ∧ ω n − φ (cid:21) , where Fut v ( W, w ) is Tian-Zhu’s modified Futaki invariant, which we write in theform(16) Fut v ( W, w ) = 1 V Z W θ w e θ v ω nφ − R W θ w e − φ R W e − φ . This is shown to be equivalent to Tian-Zhu’s definition by He [29]. One can check bydirect calculation that our definition of the twisted Futaki invariant is independentof the metric e − φ , remembering that ι w ω ψ = 0.We will on occasion write Fut (1 − t ) ω ψ ,v instead of Fut (1 − t ) ψ,v , when the curvatureof e − ψ is more natural. We will need the following: Proposition 8. If ( W, (1 − t ) ψ, v ) admits a twisted K¨ahler-Ricci soliton, then (17) Fut (1 − t ) ψ,v ( W, w ) = 0 for all w ∈ g W,ψ,v . If the twisted K¨ahler-Ricci soliton had smooth potential φ , at least on W , thenthis would follow directly from the definitions. In general we obtain the result byrelating the twisted Futaki invariant to the twisted Ding functional, and using thatthe twisted Ding functional is bounded below if there exists a twisted KR-soliton.This is analogous to an argument in [20], and the proof will be given in Section 6. V. DATAR AND G. SZ´EKELYHIDI
Twisted stability.
Suppose now that M is a smooth Fano manifold, with a holo-morphic vector field v such that Im v generates a torus T . Suppose that G is acompact group of automorphisms of M , containing T . We embed M ⊂ P N using G -invariant sections of K − mM for some m . Let α = m ω F S | M , which we can write asthe curvature of a smooth metric e − ψ α on K − M in the notation above. This metricwill then be G -invariant. It was shown by Dervan [24] that twisted K-stability isa necessary condition for the existence of a twisted KE metric on ( M, (1 − t ) α ),while a corresponding stability notion for K¨ahler-Ricci solitons was developed byBerman-Witt-Nystr¨om [10]. We can combine these ideas to obtain a stability notionfor twisted K¨ahler-Ricci solitons as follows.The vector field v on M is the restriction of a holomorphic vector field on P N ,which we will also denote by v . The imaginary part Im v corresponds to a matrixin u ( N + 1), with eigenvalues µ i , so that v has Hamiltonian function(18) θ v = P i µ i | Z i | P i | Z i | for suitable homogeneous coordinates Z i . We assume that θ v is normalized as before(i.e. e θ v has average 1 on M ).Under our embedding the group G above can be thought of as a subgroup of U ( N + 1). Suppose that we have a C ∗ -action λ ⊂ GL ( N + 1 , C ) G , generated by avector field w on P N , where GL ( N + 1 , C ) G denotes the centralizer of G . Supposethat the central fiber W = lim t → λ ( t ) · M is a Q -Fano variety. We can also takethe limit(19) β = lim t → λ ( t ) · α, which is a closed positive current on W . The C ∗ -action λ defines a special degen-eration (in the terminology of Tian [46]), and its twisted Futaki invariant is definedto be(20) Fut (1 − t ) α,v ( M, w ) = Fut (1 − t ) β,v ( W, w ) , again omitting α, β or v if t = 1 or v = 0. Definition 9.
We say that the triple ( M, (1 − t ) α, v ) is K-semistable (with re-spect to G -equivariant special degenerations), if Fut (1 − t ) α,v ( M, w ) ≥ w asabove. The triple is K-stable if in addition equality holds only when ( W, (1 − t ) β )is biholomorphic to ( M, (1 − t ) α ), i.e. the pairs are in the same GL G -orbit.The terminology more consistent with existing literature would be “twisted mod-ified K-polystable”, but we hope no confusion is caused by simply using the ter-minology “K-stable”. Dervan [24] showed that if ( M, (1 − t ) α ) admits a twistedK¨ahler-Einstein metric then it is K-stable, while Berman-Witt-Nystr¨om showedthat if ( M, v ) admits a K¨ahler-Ricci soliton, then it is K-stable in the sense of theabove definition. We expect that one can combine the arguments to show that if thetriple ( M, (1 − t ) α, v ) admits a twisted K¨ahler-Ricci soliton, then it is K-stable, butwe will not pursue that here. Our main result is a result in the converse direction,the proof of which will be given in Section 3. Proposition 10. If ( M, (1 − s ) α, v ) is K-semistable for all G -equivariant specialdegenerations, then ( M, (1 − t ) α, v ) admits a twisted K¨ahler-Ricci soliton for all t < s . In addition if ( M, v ) is K-stable, then ( M, v ) admits a K¨ahler-Ricci soliton. ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 7 Note that we also expect that if ( M, (1 − t ) α, v ) is K-stable, then ( M, (1 − t ) α, v )admits a twisted K¨ahler-Ricci soliton, however this does not quite follow from ourarguments.A key ingredient in our arguments is a comparison of the twisted and untwistedFutaki invariants and from (15) it follows that(21) Fut (1 − t ) α,v ( M, w ) = Fut v ( M, w ) − − tV (cid:20)Z W θ w ( e θ v − ω nφ + n Z W θ w ( β − ω φ ) ∧ ω n − φ (cid:21) . Recall here that (
W, β ) is the limit of the pair (
M, α ) under the C ∗ -action generatedby w . The following result builds on work in [32] and Dervan [24]. Proposition 11.
Using the same setup as above, we have the formula (22) 1 V Z W θ w h ( n + 1) ω φ − nβ i ∧ ω n − φ = max W θ w . We will give the proof below, after Lemma 12. For now note that as a conse-quence we have(23) Fut (1 − t ) α,v ( M, w ) = Fut v ( M, w ) + 1 − tV Z W (max W θ w − θ w ) e θ v ω nφ . In particular the difference is always positive, and is equal to zero only if θ w isconstant on W , i.e. if we had a trivial degeneration. Note also that the right handside is independent of the choice of metric α on M , however as discussed in [43] (andcan be seen from the proof below), if one replaces α by the current of integrationalong a divisor, leading to the notion of log K-stability used in [20], the twistedFutaki invariant might drop for special divisors.For the proof of Proposition 11, and also for later use we will need to represent α as an integral of currents of integration along divisors on M . The formula (22) isinvariant under scaling ω φ and ω ψ , and so to simplify notation we will assume thatthe cohomology classes [ α ] , [ ω φ ] coincide with the classes of the hyperplane divisors M ∩ H, W ∩ H . In particular we then have V = 1. We will also normalize theFubini-Study metric ω F S on P N to represent the same cohomology class as [ H ].Let us write P N ∗ for the dual projective space of hyperplanes. Since α is therestriction of ω F S to M , we have (see e.g. Shiffman-Zelditch [38])(24) α = Z P N ∗ [ M ∩ H ] dµ ( H ) , where dµ is simply the Fubini-Study volume form, scaled to have volume 1. Itfollows that the limit β = lim t → λ ( t ) ∗ α is given by(25) β = Z P N ∗ [ W ∩ H ] dµ ( H ) , where for each hyperplane H we wrote(26) H = lim t → λ ( t ) · H. In this formula for the limit β it is important that W is not contained in a hyper-plane, otherwise we would not necessarily have the relation(27) lim t → λ ( t ) · ( M ∩ H ) = (lim t → λ ( t ) · M ) ∩ (lim t → λ ( t ) · H ) , V. DATAR AND G. SZ´EKELYHIDI used above. It follows that(28) Z W θ w β ∧ ω n − F S = Z P N ∗ Z W ∩ H θ w ω n − F S dµ ( H ) . A key point is that there is a subspace P w ⊂ P N ∗ , depending on w , such that forall H P w the integral(29) Z W ∩ H θ w ω n − F S has the same value. The following lemma gives a formula for this integral, and inparticular shows this independence. This formula is essentially contained in [32,proof of Theorem 12], and was made more explicit by Dervan [24].
Lemma 12.
Let us normalize the Fubini-Study metric so that [ ω F S ] = [ H ] in H ( P N ) . Then there is a subspace P w ⊂ P N ∗ such that for H P w we have (30) Z W ∩ H θ w ω n − F S = 1 n Z W h ( n + 1) θ w − max W θ w i ω nF S . Proof.
Let us write R = L R k for the graded coordinate ring of W . In suitablehomogeneous coordinates the function θ w on P N is given by(31) θ w ( Z ) = P i µ i | Z i | P i | Z i | , where the µ i are the weights of the C ∗ -action λ ( t ) induced by θ w , on the linearfunctions R . For a generic hyperplane H , the limit H = lim t → λ ( t ) · H has equa-tion Z max = 0, where µ max is the largest weight (if there are several equal largestweights, then Z max can denote any of the corresponding coordinates). Indeed thisis the case for all hyperplanes not passing through the set where θ w achieves itsmaximum. This can be seen from the fact that the effect of acting by λ ( t ) as t → θ w .Denoting by S = L S k the graded coordinate ring of W ∩ H , we have S = R/Z max R , i.e. S k = R k /Z max R k − . Let us write w k for the total weight of theaction λ on R k , and w ′ k for the weight of the action on S k . From the equivariantRiemann-Roch theorem we have(32) dim R k = k n Z W ω nF S n ! + O ( k n − ) , and(33) w k = k n +1 Z W θ w ω nF S n ! + ck n + O ( k n − )for some constant c . Similarly(34) w ′ k = k n Z W ∩ H θ w ω n − F S ( n − O ( k n − ) . From the description S k = R k /Z max R k − we get(35) w ′ k = w k − w k − − µ max dim R k − = ( n + 1) k n Z W θ w ω nF S n ! − µ max k n Z W ω nF S n ! . ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 9 Combining this with (34) we get(36) Z W ∩ H u ω n − F S = 1 n Z W h ( n + 1) θ w − µ max i ω nF S . The fact that W is invariant under the action of λ ( t ) and not contained in a hy-perplane implies that max W u = µ max = max P N u . (cid:3) Proposition 11 follows from this lemma together with the formula (28). Indeed,the lemma together with (28) implies that(37) Z W θ w β ∧ ω n − F S = 1 n Z W h ( n + 1) θ w − max W θ w i ω nF S , since the set of hyperplanes in P w has measure zero. At the same time, in (22)we can replace ω φ with the restriction of ω F S to W . Note that this will changethe function θ w , but the difference of the two sides of (22) remains the same. Theformula (22) in Proposition 11 then follows immediately from (37).3. Proof of the main result
In this section we give the proof of our main result, Proposition 10. The setup isthat we have a smooth Fano manifold M with the holomorphic action of a compactgroup G . We have a G -invariant K¨ahler metric α ∈ c ( M ), and for simplicity weassume that α is the restriction of m ω F S to M , under an embedding M ⊂ P N m using a basis of sections of K − mM , for some m >
0. We are also given a vector field v on M , invariant under the action of G . In order to find a K¨ahler-Ricci soliton on( M, v ) we try to solve the equations(38) Ric( ω t ) − L v ω t = tω t + (1 − t ) α, for t ∈ [0 , t = 0 and by Tian-Zhu [48] the possible values of t form an open set. We therefore have a solution for t ∈ [0 , T ) and we need to understand the limit of a sequence of solutions as t → T .3.1. The case
T < . We first focus on the case
T <
1, and we assume that thetriple ( M, (1 − s ) ψ α , v ) is K-stable with respect to G -equivariant special degenera-tions, for some s ∈ ( T, T , i.e. we can solve our equation for t = T as well. The strategyis the same as that in [20].We first show that along a sequence t k → T , the Gromov-Hausdorff limit of( M, ω t k ) has the structure of a Q -Fano variety W , together with a metric ψ on K W , and a vector field v such that the triple ( W, (1 − T ) ψ, v ) admits a twistedK¨ahler-Ricci soliton. We then need to show that W is the central fiber of a specialdegeneration for M . One difficulty, when comparing this to the analogous resultin [20], is that we are not able to show that the pair ( W, (1 − T ) ψ ) is the centralfiber of a special degeneration for ( M, (1 − T ) ψ α ) since we are not able to use theLuna slice theorem on the infinite dimensional space of pairs consisting of a varietyand a positive current. Instead we use an argument approximating α with a convexcombination of hyperplane sections.The key ingredient to understanding the Gromov-Hausdorff limit of a sequence( M, ω t k ) is the partial C -estimate, first introduced by Tian [45]. This was es-tablished in [42] in the case when v = 0, using the method in Chen-Donaldson-Sun [19], and it was shown by Phong-Song-Sturm [36] for K¨ahler-Ricci solitons (i.e. v is non-zero, but t = 1), generalizing the work of Donaldson-Sun [27]. A modestcombination and generalization of these ideas gives the analogous result for theequation (38), and we will give a brief outline of the necessary changes in Section 5.For each t , the metric ω t introduces Hermitian inner products on H ( K − mM )for all m >
0, moreover these inner products are G -invariant (by the uniqueness ofsolutions to (38) for t < C -estimate says that we can find a uniform m , and κ >
0, independent of t , such that an orthonormal basis { s , . . . , s N m } of H ( K − mM ) satisfies(39) κ < N m X i =0 | s i | ( x ) < κ − for all x ∈ M . Let us write N = N m for this choice of m from now.Let us now write V t = H ( K − mM ) for the unitary G -representation, with metricinduced by ω t . Note that V t are equivalent G -representations, and hence theyare unitarily equivalent as well. It follows that we have G -equivariant unitarymaps f t : V → V t . In other words if we pick an orthonormal basis { s , . . . , s N } for H ( K − mM ) with respect to the metric ω , then for all t > { s ( t )0 , . . . , s ( t ) N } with respect to ω t , by applying the map f t . Usingthese bases, we have embeddings F t : M → P N , such that for s = t we have F s = ρ ◦ F t with ρ ∈ GL ( N + 1) G , i.e. ρ commutes with G . In particular the vectorfield ( F t ) ∗ v along the image F t ( M ) is induced by a fixed holomorphic vector field v on P N , since v is G -invariant.We can choose a subsequence t k → T , such that F t k ( M ) converges to a limit W ⊂ P N , and as shown in Donaldson-Sun [27], the partial C -estimate implies, upto replacing m by a multiple, that W is a normal Q -Fano variety, homeomorphicto the Gromov-Hausdorff limit Z of the sequence ( M, ω t k ). Moreover the maps F t k : M → P N converge to a Lipschitz map F T : Z → P N under this Gromov-Hausdroff convergence, such that F T : Z → W is a homeomorphism. Note that bychoosing a further subsequence we can assume that the currents ( F t k ) ∗ α convergeweakly to a current β , which is necessarily supported on W and is invariant underthe action of Im v . Let us write β as the curvature ω ψ of a singular metric e − ψ on K − W . We can similarly define a weak limit ω T of the metrics ( F t k ) ∗ ( ω t k ), which isalso supported on W . Note that if we write(40) ω t k = 1 m ( F t k ) ∗ ω F S + √− ∂∂φ k , then the partial C -estimate implies that we have bounds | φ k | , |∇ φ k | ω tk < C . Thisin particular implies that the φ k converge to a Lipschitz function φ T on ( Z, d Z ), andsince Z is homeomorphic to W , this means that φ T is continuous on W (using thetopology induced from P N ). This implies that ω T is the curvature of a continuousmetric e − φ T on K − W (recall that we might need to take a power K − mW here). Weneed the following. Proposition 13.
The triple ( W, (1 − T ) ψ, v ) admits a twisted K¨ahler-Ricci soliton,and in particular ( W, (1 − T ) ψ ) has klt singularities. In fact the twisted K¨ahler-Riccisoliton is given by the metric e − φ T .Proof. Let us decompose the Gromov-Hausdroff limit as Z = R ∪ D ∪ S . Here R is the regular set, and D is the set of points which admit a tangent cone of the form ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 11 C n − × C γ , where C γ is the standard cone with cone angle 2 πγ . See Section 5for more details. From the results of Cheeger-Colding [14] and Cheeger-Colding-Tian [15] we know that S is a closed set of Hausdorff dimension at most 2 n − F T is Lipschitz, we know that F T ( S ) is also a closed set with Hausdorffdimension at most 2 n −
4. Let us write W ′ = W \ F T ( S ), where as before W isthe regular part of the algebraic variety W . We will construct the twisted K¨ahler-Ricci soliton on W ′ . As explained in Remark 4, it is enough to show that themeasure e θ v ω nT corresponding to the metric e − φ T defines a singular metric e − τ on K W ′ with τ ∈ L loc such that its curvature satisfies(41) ω τ = T ω T + (1 − T ) ψ. To simplify notation we will identify Z with W , and so on W in addition to themetric ω F S induced by the Fubini-Study metric we have the metric d Z inducingthe same topology. For simplicity let us also write d k for the metric on M inducedby ω t k , and M k for the metric space ( M, d k ). Thus we have M k → ( W, d Z ) inthe Gromov-Hausdorff sense. The maps F k : M k → P N are compatible with theconvergence in the sense that if p k → p with p k ∈ M k and p ∈ W , then F k ( p k ) → p in P N .If p ∈ W ′ , then either p ∈ R or p ∈ D . We will only deal with the case p ∈ D since the other case is easier. We can write p = lim p k for p k ∈ M k , such that fora sufficiently small r > B d k ( p k , r ), scaled to unit size are very close inthe Gromov-Hausdorff sense to the unit ball in a cone C n − × C γ , for large k . Asdiscussed in [42], based on the ideas in [19], this implies that we have biholomor-phisms H k : Ω k → B n , where Ω k ⊂ M k contain a ball around p k of a fixed size,such that the metric e ω k = r − ω t k on B n is well approximated by the standardconical metric on B n . More precisely, we have coordinates ( u, v , . . . , v n − ) suchthat if we write(42) η γ = √− du ∧ d ¯ u | u | − γ + √− n − X i =1 dv i ∧ d ¯ v i , then for some fixed constant C (independent of k )(1) e ω k = √− ∂∂φ k with 0 ≤ φ k ≤ C , | r v k ( φ k ) | < C , where v k is the solitonvector field in this chart.(2) ω Euc < C e ω k ,(3) Given any δ > K away from { u = 0 } , we can assume(by taking r above smaller and k larger if necessary), that | e ω k − η γ | C ,α < δ on K .We will also write α k for the form α in this chart.Is is shown in [20, Proposition 22], the biholomorphisms H k : Ω k → B n con-verge to a homeomorphism H ∞ : Ω ∞ → B n , and necessarily Ω ∞ contains a ball B d Z ( p, ǫ ) ⊂ W for some small ǫ >
0, since all the sets Ω k contain balls of a uni-form size around p k . It follows that Ω ∞ also contains a ball B around p in thetopology on W induced from P N , and so H ∞ defines a holomorphic chart on W in a neighborhood of p . These charts can be used to define holomorphic maps f t k : B → F t k ( M ), biholomorphic onto their image, such that the f t k converge tothe identity map as k → ∞ . In this formulation β is given as the weak limit of f ∗ t k ( F t k ) ∗ α , which in terms of our charts amounts to saying that β is the weak limitof the forms α k . In the same vein ω T is the weak limit of ω t k = r e ω k . The metrics e ω k satisfy the equations(43) e r v k ( φ k ) ( √− ∂∂φ k ) n = e − r t k φ k − (1 − t k ) ψ k ω nEuc , for suitable local potentials ψ k of the forms α k restricted to this chart. Notethat r v k ( φ k ) is a Hamiltonian for v k with respect to ω t k . The bound ω Euc Up to choosing a subsequence, we can assume that ρ k ( H ) convergesfor all H ∈ P N ∗ .Proof. Write P N ∗ = P ( V ) for an N + 1-dimensional vector space V . Thinking ofthe ρ k as matrices, let us scale each of them in such a way that all entries are in { z : | z | ≤ } , and at least one entry equals 1. We can choose a subsequence suchthat as matrices, we have(48) lim k ρ k = ρ, ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 13 where ρ is not necessarily invertible. Let W = Ker ρ . For any x ∈ P ( V ) \ P ( W )we can then take the limit(49) lim k ρ k ( x ) = ρ ( x ) . Now let us restrict the ρ k to W , thinking of them as linear maps ρ k : W → V .Once again, taking matrix representatives, we can normalize each to have entriesin the unit disk, with at least one entry equal to 1. Just as above, up to choosing afurther subsequence, we will have a limiting, nonzero linear map ρ : W → V withkernel W ⊂ W . For x ∈ P ( W ) \ P ( W ) the limit will exist as above.Repeating this process a finite number of times we will have a subsequence ρ k such that ρ k ( x ) converges for all x ∈ P ( V ). (cid:3) It follows that we have(50) β = Z P N ∗ [ W ∩ ρ ∞ ( H )] dµ ( H ) , where as before it is important to note that W is irreducible and not contained ina hyperplane.In the spirit of Definition 6, for any current τ on P N , let us denote by g W,τ ⊂ sl ( N + 1 , C ) the space of those holomorphic vector fields v , which are tangent to W and satisfy ι v τ = 0. If τ = [ S ], the current of integration along a subvariety S ,we will write g S = g [ S ] . Note that in this case g S is simply the Lie algebra of thestabilizer of S in SL ( N + 1 , C ). Lemma 15. We can find H , . . . , H d for some d such that (51) g W,β = g W ∩ d \ i =1 g [ W ∩ ρ ∞ ( H i )] . Proof. Suppose that v is a holomorphic vector field, which does not vanish along W , and let ξ = ι ¯ v ω nF S . This is an ( n, n − ι v ξ is a non-negative( n − , n − A ⊂ T p P N is a complex ( n − ι v ξ vanishes on A only if v ∈ A .If ι v β = 0, then we have(52) Z H ∈ P ∗ Z W ∩ ρ ∞ ( H ) ι v ξ dµ = 0 , and so for almost every H we must have(53) Z W ∩ ρ ∞ ( H ) ι v ξ = 0 . In particular, for almost every H we must have v ∈ A for all tangent planes A = T p ( W ∩ ρ ∞ ( H )) at all smooth points p ∈ W ∩ ρ ∞ ( H ). It follows that ι v [ W ∩ ρ ∞ ( H )] = 0, i.e.(54) g β ⊂ g [ W ∩ ρ ∞ ( H )] . If we choose one such H , say H , it may happen that g [ W ∩ ρ ∞ ( H )] is too large,i.e. there is a w ∈ g [ W ∩ ρ ∞ ( H )] such that ι w β = 0. But we have(55) ι w β = Z H ∈ P ∗ ι w [ W ∩ ρ ∞ ( H )] dµ, so we must have a positive measure set of H for which ι w [ W ∩ ρ ∞ ( H )] = 0. Wecan thus choose an H , so that we still have(56) g β ⊂ g [ W ∩ ρ ∞ ( H )] , but g [ W ∩ ρ ∞ ( H )] ∩ g [ W ∩ ρ ∞ ( H )] is strictly smaller than g [ W ∩ ρ ∞ ( H )] . Repeating thisa finite number of times, we obtain the required result. (cid:3) It follows from this result that we can choose H ′ , . . . , H ′ l for some l such that theLie algebra of the stabilizer of the ( l + 1)-tuple ( W, W ∩ ρ ∞ ( H ′ ) , . . . , W ∩ ρ ∞ ( H ′ l ))in GL G , for the action on a product of Hilbert schemes, is equal to the G -invariantpart of g W,β , and so according to Proposition 7 it is reductive. Using a result similarto Luna’s slice theorem [34] as in [25, Proposition 1] (as in [20] as well), we cantherefore find a C ∗ -subgroup λ ⊂ GL G and an element g ∈ GL G such that(57) ( W, W ∩ ρ ∞ ( H ′ ) , . . . , W ∩ ρ ∞ ( H ′ l )) = lim t → λ ( t ) g · ( M, M ∩ H ′ , . . . , M ∩ H ′ l ) . In addition for a subset of E ⊂ P N ∗ of measure zero, if H , . . . , H K E , thenthe stabilizer of(58) ( W, W ∩ ρ ∞ ( H ′ ) , . . . , W ∩ ρ ∞ ( H ′ l ) , W ∩ ρ ∞ ( H ) , . . . W ∩ ρ ∞ ( H K ))will still be the same as that of ( W, β ), and so we can still find a corresponding C ∗ -subgroup λ and g ∈ GL G which will satisfy (57) as well as(59) W = lim t → λ ( t ) g · MW ∩ ρ ∞ ( H i ) = lim t → λ ( t ) g · ( M ∩ H i ) , for i = 1 , . . . , K. Note that all of these λ must fix W , but the λ may vary as we change the collection( H , . . . , H K ).Each of the C ∗ -actions λ is generated by a vector field w commuting with v ,with Hamiltonian function θ w . We will assume that θ w is normalized so that(60) Z W θ w ω nF S = 0 . Let us write k w k = sup W | θ w | , although note that any two norms on the finitedimensional space of such w are equivalent.Because of (50), for any ǫ > K large, and H , . . . , H K E , suchthat no N + 1 of the H i lie on a hyperplane in P N ∗ , and for all vector fields w asabove we have(61) Z W θ w β ∧ ω n − F S ≤ ǫ k w k + 1 K K X j =1 Z W ∩ ρ ∞ ( H j ) θ w ω n − F S . Applying this to the w corresponding to the C ∗ -action λ that we obtain for ( H , . . . , H K ),we have(62) Z W θ w β ∧ ω n − F S ≤ ǫ k w k + 1 K K X j =1 lim t → Z λ ( t ) · ( M ∩ H j ) θ w ω n − F S . Using Lemma 12, and the fact that no N + 1 of the H i are in a hyperplane, weobtain, using also the normalization of θ w , that(63) Z W θ w β ∧ ω n − F S ≤ (cid:18) ǫ + N CK (cid:19) k w k − K − NKn Z W max W θ w ω nF S , ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 15 for some fixed constant C . Choosing K sufficiently large (depending on ǫ ), weobtain a C ∗ -action generated by a vector field w , with Hamiltonian function θ w asabove, such that(64) Z W θ w β ∧ ω n − F S ≤ ǫ k w k − n Z W max W θ w ω nF S . Moreover this C ∗ -action satisfies W = lim t → λ ( t ) g · M , but not necessarily β =lim t → λ ( t ) g · α . Nevertheless the vector field v satisfies ι v β = 0 by construction.Since ( W, (1 − T ) β ) admits a twisted K¨ahler-Ricci soliton, we know that(65) Fut (1 − T ) β,v ( W, w ) = 0 , and so(66) Fut v ( M, w ) − − TV (cid:20)Z W θ w e θ v ω nF S + Z W θ w nβ ∧ ω n − F S (cid:21) = 0 . At the same time we are assuming that for some s > T , the triple ( M, (1 − s ) ψ, v )is K-semistable, which, using Proposition 11, implies that we have(67) Fut v ( M, w ) − − sV (cid:20)Z W θ w e θ v ω nF S − V max W θ w (cid:21) ≥ . Together (66) and (67) imply(68) s − TV Z W θ w e θ v ω nF S + (1 − s ) max W θ w + 1 − TV Z W θ w nβ ∧ ω n − F S ≥ . Using also (64) we then get(69) 0 ≤ − TV nǫ k w k + s − TV Z W ( θ w − max W θ w ) e θ v ω nF S . Since s > T and T < 1, this is a contradiction if ǫ is sufficiently small, unless k w k = 0. For this, note that there is a uniform constant c > Z W (max W θ w − θ w ) e θ v ω nF S ≥ c k w k for all possible w that we have, since these form a finite dimensional space.It follows that we must have k w k = 0, which means that θ w is constant on W .This implies that the corresponding C ∗ -action λ is trivial, and so in fact by (59)we have(71) ( W, W ∩ ρ ∞ ( H ) , . . . , W ∩ ρ ∞ ( H K )) = g · ( M, M ∩ H , . . . , M ∩ H K )for some g ∈ SL G . If follows that(72) lim k →∞ ρ k ( H i ) = ρ ∞ ( H i ) = g ( H i ) . We can assume that H , . . . , H N +1 are in general position in P N ∗ , and then each ρ k is determined by the hyperplanes ρ k ( H i ) for i = 1 , . . . , N + 1. In particular(72) then implies that ρ k → g in SL G , which in turn implies that the sequence ρ k ∈ SL G is bounded. If we write(73) 1 m ( F k ) ∗ ω F S = ω + √− ∂∂φ k for the pullbacks of the Fubini-Study metrics to M under our embeddings F k , wethen have a uniform bound | φ k | < C . The partial C -estimate implies that thenwe also have(74) ω t k = ω + √− ∂∂φ ′ k with | φ ′ k | < C ′ for a uniform constant, for the metrics ω t k along the continuitypath. It is then standard using the estimates of Yau [52] that we have uniform C l,α bounds for ω t k , and so we can obtain a solution of Equation (38) for t = T (seealso Zhu [55] for the C -estimate in the soliton case).3.2. The case T = 1 . Suppose now that T = 1, i.e. we can solve Equation (38)for all t < 1. This case is much more similar to the work of Chen-Donaldson-Sun [20], since the “current part” of the equation disappears as t → 0. The case ofK¨ahler-Ricci solitons was also studied by Jiang-Wang-Zhu [31]. We briefly describethe argument for the sake of completeness. Just as in the case T < 1, we haveembeddings F t : M → P N using suitable orthonormal bases for H ( K − mM ) withrespect to the metric ω t , for some large m . The partial C -estimate is still valid,in the K¨ahler-Einstein case by [42] based on the method in [20], and in the solitoncase due to Jiang-Wang-Zhu [31]. It follows that as before, up to increasing m andchoosing a sequence t k → F t k ( M ) → W ∈ P N to a normal Q -Fano variety, homeomorphic to the Gromov-Hausdorff limit ( Z, d Z )of the sequence ( M, ω t k ). As before, we identify ( M, α ) = ( F ( M ) , ( F ) ∗ α ) and so( F t k ( M ) , ( F t k ) ∗ α ) = ρ k · ( M, α ) for ρ k ∈ SL G . The vector field v on each F t k ( M )is induced by a fixed vector field v on P N , which is also tangent to the limit W .We can also choose a further subsequence of t k if necessary to have a weak limit( F t k ) ∗ ω t k → ω . We have the following, see [31, Corollary 1.4]. A proof can alsobe given in the spirit of the proof of Proposition 13. Proposition 16. The pair ( W, v ) admits a K¨ahler-Ricci soliton, and in fact thissoliton is given by the current ω . It follows from [10, Corollary 3.6] that the stabilizer of W in SL G is reductive,and so we can find a C ∗ -subgroup λ ∈ SL G generated by a vector field w commutingwith v , and an elements g ∈ SL G such that(75) W = lim t → λ ( t ) g · M. This is a special degeneration for M , whose central fiber is W . Since W admitsa K¨ahler-Ricci soliton, the corresponding Futaki invariant Fut v ( W, w ) = 0. Byassumption ( M, v ) is K-stable, and so W must be biholomorphic to M . This meansthat ω is a K¨ahler-Ricci soliton on M , which is what we wanted to obtain.4. Some applications In this section, we look at some applications of Theorem 1 to existence of K¨ahler-Einstein metrics on Fano manifolds with large symmetry groups. Toric manifolds A compact Kahler manifold M of complex dimension n is toric if the compact torus T n acts by isometries on M and the extension of the action to the complex torus( C ∗ ) n acts holomorphically with a free, open, dense orbit. We can then recover thefollowing theorem of Wang-Zhu [50] as a consequence of Theorem 1. ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 17 Theorem 17. There exists a K¨ahler-Ricci soliton, which is unique up to holo-morphic automorphisms, on every toric Fano manifold. As a consequence, thereexists a K¨ahler-Einstein metric on a toric Fano manifold if and only if the Futakiinvariant vanishes.Proof. Let M be a toric manifold with dim C M = n . We wish to apply Theorem 1with G = T n with a fixed identification as a subgroup of GL ( N + 1 , C ). The keyobservation is that if v is a toric vector field, then any ( C ∗ ) n -equivariant specialdegeneration of ( M, v ) is necessarily trivial. Indeed, if λ : C ∗ → GL ( N + 1 , C ) G isa test configuration and if M = lim t → λ ( t ) · M is not in the GL ( N + 1 , C )-orbitof M , then the stabilizer of M must contain a ( C ∗ ) n +1 . On the other hand, since M is irreducible and not contained in any hyperplane, the action of this stabilizeron M must also be effective. This is a contradiction since any torus acting on an n -dimensional normal variety cannot have a dimension greater than n . The upshotis that M must be bi-holomorphic to M and the test configuration is induced by atoric vector field w on M . To verify K-stability of ( M, v ), it then suffices to checkthat the modified Futaki invariant vanishes: Fut v ( M, w ) = 0, for all toric vectorfields w on M .Next, recall that any toric manifold M with an ample line bundle correspondsto a unique (up to translations) polytope P ⊂ R n defined by a finite collection ofaffine linear inequalities l j ( x ) ≥ 0. This polytope is in fact the image of the free( C ∗ ) n orbit in M under the moment map. Since M is Fano, one can normalize thepolytope so that l j (0) = 1 for all j . Any toric vector field can be written as w = P nj =1 c j z j ∂∂z j for some c ∈ R n where ( z , · · · , z n ) are the usual complex coordinateson ( C ∗ ) n . In terms of the polytope data, for a vector field v = P nj =1 a j z j ∂∂z j ,equation (16) then reduces toFut v ( M, w ) = c · R P x e a · x d x V , where V = V ol ( P ) is the volume of M . But then, as in Tian-Zhu [48], by minimizingthe functional F ( a ) = R P e a · x d x , one can find a vector a such that the integral onthe right vanishes, and hence F ut v ( M, w ) vanishes identically for the correspondingtoric vector field v . (cid:3) If M does not admit a K¨ahler-Einstein metric and α ∈ c ( M ) is a K¨ahler form,then R ( M ) = sup { t | ∃ ω ∈ c ( M ) such that Ric ( ω ) = tω + (1 − t ) α } , provides a natural obstruction. It follows from the work of the second author [44]that R ( M ) is in fact independent of the choice of α . We can then recover thefollowing result of Li [33], expressing R ( M ) in terms of the corresponding polytope. Theorem 18. Let M be toric, Fano, and P be the canonical polytope as above withbarycenter P C . Let Q be the the point of intersection of the ray − sP C , s ≥ with ∂P . If O denotes the origin, R ( M ) = | QO || QP C | Proof. By the above discussion and Proposition 10 it is enough to find the maximum t such that Fut (1 − t ) ψ ( M, w ) ≥ w where α = √− ∂∂ψ . We once again write w = P nj =1 c j z j ∂∂z j for some c ∈ R n . Then thetwisted modified Futaki invariant (equation (16)) takes the formFut (1 − t ) ψ ( M, w ) = t c · P c + (1 − t ) max x ∈ P c · x . Now let the face of the polytope containing Q be given by the vanishing of theaffine linear functional l ( x ) := u · x + 1. Note that since l (0) = 1, it follows fromelementary arguments that | QO | / | QP C | = 1 /l ( P C ). We also remark that l ( P C ) ≥ Claim: For any c ∈ R n , c · P C max x ∈ P c · x ≥ − l ( P C ) . Assuming this, for t ≤ /l ( P C ) and any holomorphic toric vector field w , it iseasily seen that F ut (1 − t ) w ( M, w ) ≥ 0, and hence R ( M ) ≥ /l ( P C ). On the otherhand, if w is a special holomorphic vector field corresponding to − u ∈ R n , thenmax x ∈ P ( − u ) · x = 1, and hence F ut (1 − t ) ψ ( M, w ) = 1 − t · l ( P C ) . This is negative when t > /l ( P C ), which implies that R ( M ) = 1 /l ( P C ), completingthe proof of the theorem. To prove the claim, we first normalize c so that max x ∈ P c · x = 1. If we now let ˜ l ( x ) = − c · x + 1, then ˜ l ( x ) ≥ x ∈ P . Moreover, since c · P C = 1 − ˜ l ( P C ) it is enough to show that l ( P C ) ≥ ˜ l ( P C ). Once again consider theray − sP C with s ≥ 0. If this does not intersect the hyperplane { ˜ l = 0 } , then clearly c · P C ≥ 0, and hence ˜ l ( P C ) ≤ ≤ l ( P C ). On the other hand, suppose the ray doesintersect the hyperplane, at say a point Q ′ . Since the polytope P lies entirely onone side of the hyperplane, we have | QP C | < | Q ′ P C | . In fact, since ˜ l (0) = l (0) = 1,˜ l ( P C ) = | Q ′ P C || Q ′ O | = | QQ ′ | + | QP C || QQ ′ | + | QO | ≤ | QP C || QO | = l ( P C ) , and the claim is proved. (cid:3) T-varieties Relaxing the toric condition, we consider Fano manifolds M with an effective ac-tion of the torus T m for some m < n = dim M . The simplest case is that of acomplexity-one action, where m = n − 1. K¨ahler-Einstein metrics on such mani-folds, in particular Fano 3-folds with 2-torus actions, was studied by S¨uss [40, 41].In particular in [41, Theorem 1.1] a list of 9 such manifolds is given with vanishingFutaki invariant, for 5 of which it was not known whether they admit a K¨ahler-Einstein metric or not. Using Theorem 1 one only needs to check T -equivariantspecial degenerations, and such degenerations can be classified using combinatorialdata. [41, Section 5] lists all such degenerations to canonical toric Fano varieties,while the more general degenerations to log-terminal toric Fanos are classified byIlten-S¨uss [30]. The conclusion is that all 9 Fano threefolds with vanishing Futakiinvariant in [41, Theorem 1.1] admit a K¨ahler-Einstein metric. Other manifolds with large symmetry group. We expect that Theorem 1 can be used to show the existence of K¨ahler-Einsteinmetrics on many other classes of Fano manifolds with large symmetry group. Oneinteresting class is that of reductive varieties, studied by Alexeev-Brion [2, 3]. Let ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 19 G be a connected compact group, T ⊂ G a maximal torus, and W the correspond-ing Weyl group. Denote by Λ the character group of T , which is a lattice in thereal vector space Λ R . To every W -invariant maximal dimensional convex latticepolytope P ⊂ Λ R one can associate a variety V P , which is a G c × G c -equivariantcompactification of G c , the action being left and right multiplication. As shownin [3] (see also Alexeev-Katzarkov [4]), the equivariant degenerations of V P cor-respond to convex, rational, W -invariant, piecewise linear functions f on P , inanalogy to the toric case studied in Alexeev [1], Donaldson [26]. If we have anequivariant special degeneration, then in particular the central fiber is irreducible,and this will only happen when f is linear on P ∩ Λ + R , where Λ + R ⊂ Λ R is a positiveWeyl chamber corresponding to a Borel subgroup of G c , containing T c . It followsthat there are only a finite number of degenerations that need to be checked inorder to apply Theorem 1.In the case when P ∩ Λ + R is a maximal set on which f is linear, then the centralfiber of the corresponding special degeneration is a horospherical variety. Theseare the homogeneous toric bundles studied by Podesta-Spiro [37], who showed thatall such Fano manifolds admit a K¨ahler-Ricci soliton. This also follows from theabove discussion together with our main result, since the polytope P can not besubdivided further, and so a horospherical variety has no non-trivial equivariantspecial degenerations, just as the toric manifolds discussed above.5. The partial C -estimate for solitons In this section we briefly outline the changes that have to be made to the argu-ments in [42], using also techniques in Zhang [54], Tian-Zhang [47] and Phong-Song-Sturm [36], to prove the partial C -estimate for the family of metrics ω t ∈ c ( M )solving(76) Ric( ω t ) − L v ω t = tω t + (1 − t ) α, where t ∈ [0 , T ) with T < 1. The case when T = 1 has been established by Jiang-Wang-Zhu [31]. Here v is a holomorphic vector field, such that Im v generates acompact torus of isometries of the metric α . In particular ω t will also be invariantunder this torus. To simplify notation, we will drop the subscript t , and so in whatfollows, ω denotes a solution of (76) for some t ∈ [0 , T ).Recall that we have the Hamiltonian function θ v of v , with respect to the metric ω , defined by(77) ι v ω = √− ∂θ v , with the normalization(78) Z M e θ v ω n = Z M ω n . From Zhu [55], and Wang-Zhu [49, Lemma 6.1] we know that we have estimates(79) | θ v | + |∇ θ v | ω + | ∆ ω θ v | < C. The Equation (76) implies that(80) Ric( ω ) − L v ω ≥ . In addition as soon as t is bounded away from 0, the volume comparison and Myerstype theorem in Wei-Wylie [51] implies that the diameter of ( M, ω ) is bounded, and we have the non-collapsing property(81) Vol( B ( p, , ω ) ≥ c > . There are two basic approaches to studying metrics satisfying this lower bound forthe Bakry-´Emery Ricci curvature, generalizing the theory of Cheeger-Colding [14]in the case when v = 0. One approach is to study the conformally related metrics e g j ¯ k = e − n − θ v g j ¯ k , where g j ¯ k is the metric with K¨ahler form ω . This approach,similar to that used in Zhang [54] and Tian-Zhang[47] (who used the Ricci potentialinstead of θ v ), effectively reduces the problem to studying non-collapsed metricswith a lower Ricci curvature bound so that the theory of Cheeger-Colding can beapplied. Indeed, in real coordinates the Ricci tensor of e g satisfies(82) e R ij = R ij + ∇ i ∇ j θ v + 12( n − ∇ i θ v ∇ j θ v − n − (cid:2) |∇ θ v | g − ∆ g θ v (cid:3) g ij , and so (80), (79) together with the fact that v is holomorphic, and so ∇ i ∇ j θ v is oftype (1 , e g has a Ricci lower bound. In addition it is clear that e g isuniformly equivalent to g . The other approach is to build up the Cheeger-Coldingtheory using the bound (80) on the Bakry-Emery Ricci curvature. This approach isexecuted by Wang-Zhu [49]. We summarize the main conclusions from these worksthat we need.If we have a sequence ( M, ω i ), satisfying (79), (80) and (81), then up to choosing asubsequence, the Riemannian manifolds ( M, g i ) converge in the Gromov-Hausdorffsense to a length space ( Z, d ). At each point p ∈ Z there exists a tangent cone C ( Y ) which is a metric cone. We can stratify the space Z as(83) S n ⊂ S n − ⊂ . . . ⊂ S = S ⊂ Z, where S k consists of those points, where no tangent cone is of the form C n − k +1 × C ( Y ).The regular part of Z is defined to be R = Z \ S , and at p ∈ R every tangentcone is C n . We also write D = S \ S . The following is analogous to Anderson’sregularity result [5], showing that we have good control of the metrics on the regularset if we also have an upper bound of the Bakry-´Emery Ricci curvature. Proposition 19. Suppose that B ( p, is a unit ball in K¨ahler manifold ( M, ω ) ,together with a holomorphic vector field v with Hamiltonian θ , satisfying bounds ofthe form (1) sup M | θ | + |∇ θ | + | ∆ θ | < K (2) 0 ≤ Ric( ω ) − L v ω ≤ Kω .There are constants δ, κ > depending on K such that if d GH ( B ( p, , B n ) < δ ,then for each q ∈ B ( p, ) , the ball B ( q, κ ) is the domain of a holomorphic coordinatesystem in which the components of ω satisfy (84) 12 δ jk < ω j ¯ k < δ jk , k ω j ¯ k k L ,p < , for all p. Proof. We use the conformal scaling e g = e − n − θ g , so that by (82) e g satisfies two-sided Ricci curvature bounds. Suppose that d GH (( B ( p, , g ) , B n ) < δ . The bound ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 21 on ∇ θ implies that if q ∈ B ( p, ) and r is sufficiently small, then(85) d GH (( B ( q, r ) , e g ) , rλB n ) < δ, for a suitable scaling factor λ (depending on the value θ ( q )).If δ is sufficiently small, then Colding’s volume convergence result [21] combinedwith Anderson’s gap theorem implies that there is a harmonic coordinate systemon the ball B ( q, rθλ, e g ) in which the metric e g is controlled in L ,p for any p . Themetrics e g and g are C -equivalent, so we also control the components of g in C .The Laplacian bound on θ then implies that we have L ,p estimates on θ so in fact g and e g are equivalent in L ,p . In particular in our harmonic coordinates (harmonicfor e g ) we control the coefficients of g in L ,p . Using that the complex structureis covariant constant, this allows us to find holomorphic coordinates on a possiblysmaller ball, in which the coefficients of g are controlled in L ,p . (cid:3) Following Chen-Donaldson-Sun, define(86) I (Ω) = inf B ( x,r ) ⊂ Ω V R ( x, r ) , where Ω is any domain in a K¨ahler manifold, and V R ( x, r ) is the ratio of volumesof the ball B ( x, r ) in Ω and the Euclidean ball rB n . If the Ricci curvature isnon-negative, the Bishop-Gromov comparison theorem and Colding’s volume con-vergence implies that if B is a unit ball in Ω, then 1 − I ( B ) controls d GH ( B, B n ),and conversely d GH ( B, B n ) controls 1 − I ( B ). In our setting, with the bound (80),a similar statement will only hold once the metrics are scaled up by a sufficientamount. We have the following. Proposition 20. Suppose that B is a unit ball in a K¨ahler manifold ( M, ω ) satis-fying (87) Ric ( ω ) − L v ω ≥ , as well as (88) sup B |∇ θ | + | ∆ θ | ≤ δ, where θ is a Hamiltonian of X . Then (89) d GH ( B, B n ) = Ψ( δ, − I ( B )) , and for any λ < , (90) 1 − I ( λB ) = Ψ( δ, d GH ( B, B n ) , − λ ) , where Ψ( ǫ , . . . , ǫ k ) denotes a function converging to zero as ǫ i → . We havesuppressed the dependence of Ψ on the dimension n .Proof. We can assume that θ (0) = 0. Use the conformal metric e g = e − n − θ g . Thenunder our assumptions we have Ric ( e g ) > − C ′ δ e g and the metric e g is very close in C to the metric g . We can then apply the volume convergence under lower Riccicurvature bounds to the metric e g . (cid:3) We now return to our original setup, of a metric ω on M satisfying(91) Ric ( ω ) − L v ω = tω + (1 − t ) α, for some t ∈ [0 , T ), and T < 1. The vector field v and background metric α is fixed.As before we can assume that the metrics are non-collapsed, and in addition theHamiltonian θ v of v satisfies(92) sup M ( |∇ θ v | + | ∆ θ v | ) ≤ K, for some fixed constant K . The square is inserted for scaling reasons. Note thatfor any point p ∈ M we can choose the θ v so that θ v ( p ) = 0. We will exploit thefact that α is a fixed metric. In particular we can assume that K is chosen suchthat on any ball of radius at most K − with respect to α we can find holomorphiccoordinates in which the coefficients of α are controlled in C .To understand the tangent cones of the Gromov-Hausdorff limit of a sequenceof metrics satisfying these conditions, we need to study very small balls in ( M, ω ),scaled up to unit size. Let ( B, η ) be a small ball in ( M, ω ) scaled to unit size, sothat η = Λ ω for some large Λ. Let w = Λ − v . Then η satisfies(93) Ric ( η ) − L w η = λη + (1 − t ) α, for some λ ∈ (0 , 1] and t ∈ (0 , T ). In addition we can choose the Hamiltonian θ w for w relative to η such that θ w (0) = 0, and(94) sup M ( |∇ θ w | η + | ∆ η θ w | ) ≤ Λ − K. The following is the generalization of Proposition 8 in [42], showing that onthe regular set the Gromov-Hausdorff limit behaves as if we had a two-sided Riccicurvature bound. Note that as in Proposition 20 we need an extra assumptionensuring that we have scaled our metrics up by a sufficient amount. Proposition 21. There is a δ > depending on K above, such that if − I ( B ) < δ ,and the scaling factor Λ > δ − then (95) α < η in B. Proof. The method of proof is the same as in [42]. Suppose that(96) sup B d x | α ( x ) | η = M, where d x is the distance of x to the boundary of B with respect to η , and supposethat the supremum is achieved at q ∈ B . If M > B (cid:18) q, d q M − / (cid:19) , scaled to unit size e B , with scaled metric e η = 4 M d − q η . Note that e η satisfies thesame estimates as η , but in addition | α | e η ≤ e B . If δ is sufficiently small, thenwe can apply Propositions 19 and 20 to find holomorphic coordinates z i on a smallball τ e B , in which the components of e η are controlled in C ,α .The metric e η satisfies(98) Ric( e η ) = L w η + λη + (1 − t ) α ≥ (4 M d − q ) − L w e η + (1 − t ) α, and for any ǫ > w satisfies |∇ θ w | η < ǫ , which implies | w | e η < M d − q ǫ . Since w is aholomorphic vector field, we obtain that in the coordinates z i , on the half ball τ B , ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 23 the components of w , along with their derivatives are bounded by (4 M d − q ǫ ) / . Itfollows that on this ball we have(99) | L w η | e η < Cǫ / (4 M d − q ) − / , for some fixed constant C . In particular if δ is chosen sufficiently small, then wewill have L w η < ǫ e η and so(100) Ric( e η ) ≥ − ǫ e η + (1 − t ) α. Using this, the rest of the proof is essentially identical to that in [42]. (cid:3) Together with Proposition 19 it follows from this that in the Gromov-Hausdorfflimit of a sequence of metrics ω satisfying (91), with t < T < 1, the regular set isopen and smooth, and the convergence of the metrics is C ,α on the regular set. Inaddition the same holds for iterated tangent cones.What remains is to study tangent cones of the form C γ × C n − , i.e. the pointsin the set D in the Gromov-Hausdorff limit. The arguments in [42, Proposition 11,12, 13] can be followed closely with a couple of remarks. First of all the resultsof Chen-Donaldson-Sun [19] on good tangent cones can be applied. The maindifference here is that a variant of the L -estimates in [27, Proposition 2.1] needsto be used, following [36, Proposition 4.1], with the Hamiltonian θ v replacing theRicci potential u . This implies that if a scaled up ball ( B, η ) as above is sufficientlyclose to the unit ball in C γ × C n − , then on a smaller ball we have holomorphiccoordinates, in which the metric η satisfies the conditions (1) , (2) , (3) in the proofof Proposition 13.An additional important fact used several times is that by Cheeger-Colding-Tian [15], no tangent cone of the form C γ × C n − can form in the Gromov-Hausdorfflimit of a sequence of K¨ahler metrics with bounded Ricci curvature. The analogousresult with the bound on Ricci curvature replaced by a bound on Ric( ω ) − L v ω was shown by Tian-Zhang [47], and it also follows from the more recent work ofCheeger-Naber [16] in the general Riemannian case. With these observations theproof of the partial C -estimate for solutions of (76) follows the argument in [42]closely.6. Reductivity of the automorphism group and vanishing of theFutaki invariant In this section we briefly outline the proofs of Proposition 5 and Proposition 7following [11],[20] and [10]. As before, let W be the normal Q -Fano variety obtainedas the Gromov-Hausdorff limit along the continuity method, and v ∈ H ( W, T W )such that Im ( v ) generates the action of a torus T on W . We let H v denote the spaceof continuous T -invariant metrics h φ = e − φ on − K W with non-negative curvature.Then the twisted Ding functional is defined as(101) D (1 − t ) ψ,v ( φ ) = − tE v ( φ ) − log (cid:16) Z W e − tφ − (1 − t ) ψ (cid:17) , where E v is defined by its variation at φ in the direction ˙ φ by(102) dds E v ( φ ) = 1 V Z W ˙ φe θ v ω nφ , as in Berman-Witt-Nystr¨om [10]. Next, we recall the definition of a geodesic inthe path of K¨ahler metrics. We let R = { s ∈ C | Re ( s ) ∈ [0 , } . Recall that a path φ s ∈ H v is called a geodesic if Φ : W × R → R defined by Φ( x, s ) = φ Re ( s ) ( x )satisfies √− ∂ ¯ ∂ s,W (Φ) ≥ √− ∂ ¯ ∂ s,W Φ) n +1 = 0 , where the ∂ ¯ ∂ is taken in both W and R directions. Then the following is provedin [11] Lemma 22. For any φ , φ ∈ H v , there exists a geodesic φ s ∈ H v connecting themsuch that || φ s ′ − φ s || L ∞ ( W ) < C | s ′ − s | The key point is that the Ding functional is convex along these geodesics. It isproved in Berman-Witt-Nystr¨om [10, Proposition 2.17] that the functional E v ( φ )is affine along geodesics and continuous up to the boundary. So the convexity ofthe Ding functional is a consequence of the following result of Bendtsson [11]. Proposition 23. Let φ s be a geodesic as above. Then the functional (103) F ( s ) = − log (cid:16) Z W e − tφ s − (1 − t ) ψ (cid:17) is convex. Moreover, if F ( s ) is affine, then there exists a holomorphic vector fields w s on W with i w s √− ∂∂ψ = 0 , and such that the flow F s satisfies F ∗ s ( √− ∂∂φ s ) = √− ∂∂φ This was proved on compact K¨ahler manifolds by Berndtsson [11] and extendedto normal varieties by Chen-Donaldson-Sun [20] when √− ∂∂ψ is the current ofintegration along a divisor (see also [9]). Though the above statement does not seemto follow directly from either of the works, the arguments can be easily adapted,and we briefly provide an outline of the proof. Proof. For ease of notation, we let τ s = tφ s + (1 − t ) ψ . Let p : W ′ → W be a log-resolution. and ω ′ be a fixed K¨ahler metric on W ′ . Since W has only log terminalsingularities, one has the following adjunction formula(104) − K W ′ = − p ∗ K W − E + ∆ , where E and ∆ are effective divisors, and ∆ = P a j E j with a j ∈ (0 , e − τ s is a smooth family of metrics on − K W , inducing a smooth familyof pull-back metrics on − p ∗ K W with curvature ω τ ′ s = √− ∂∂τ ′ s . We write L = K − W ′ ⊗ E . Then from (104) it is clear that τ ′ s = p ∗ τ s + X a j log | s j | , where s j is the defining function of E j , induces a family of singular metrics e − τ ′ s on L . Moreover, if u is a holomorphic L -valued ( n, 0) form with zero divisor E (whichis unique up to multiplication by a constant) it can be easily checked that up toscaling u by a constant, F ( s ) = − log Z W ′ u ∧ ¯ u e − τ ′ s . ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 25 Let us pretend for the moment that the metrics e − τ ′ s are smooth. Consider theequation(105) ∇ s ν s = P s (cid:16) dτ ′ s ds u (cid:17) , where ∇ s = ∂ − ∂τ ′ s ∧ · is the Chern connection of e − τ ′ s and P s is the projectiononto the orthogonal complement of L -valued holomorphic ( n, 0) forms. As arguedin [11], it can be shown that there always exists a smooth solution ν s to (105)satisfying ¯ ∂ν s ∧ ω ′ = 0. Next, the Hessian of F is given by ([11, Theorem 3.1],[20,Lemma 14])(106) || u || τ ′ s √− ∂∂ F ( s ) = Z W ′ ω ′ s ∧ ˜ u ∧ ¯˜ u e − τ ′ s + || ∂ν s || τ ′ s √− ds ∧ d ¯ s, where ˜ u = u − ds ∧ ν s and ω ′ s = √− ∂∂ s,W ′ ( τ ′ s ). This is in fact a special caseof the general positivity of direct image sheaves discovered by Bendtsson [12]. Forsmooth geodesics, the convexity follows directly from this formula.In our case the metrics τ ′ s are not smooth, and hence we first need to use aregularization. First, if we let η = ω ′ + √− ds ∧ d ¯ s , then by the approximationtheorem of Demailly [23] (see also Blocki-Kolodziej [13]) there exists a decreasingsequence of smooth metrics ρ s,ǫ ց p ∗ τ s such that √− ∂∂ s,W ′ ( ρ s,ǫ ) ≥ − Cη . Byaveraging we can also suppose that ρ s,ǫ are independent of Re ( s ) and T -invariant.To approximate τ ′ s we then let τ ′ s,ǫ = ρ s,ǫ + log h ǫ where(107) log h ǫ = X a j (log ( | s j | h j + ǫ ) − log h j )and h j is a metric on the line bundle generated by E j . Clearly e − τ ′ s,ǫ are metricson L with τ ′ s,ǫ ց τ ′ s and √− ∂∂ s,W ′ ( τ ′ s,ǫ ) > − Cη for some C > 0. Moreover, forany neighborhood U of ∆ there exists a constant C U such that √− ∂∂ s,W ′ ( τ ′ s,ǫ ) > − ǫC U η, on W ′ \ U. We then let ν s,ǫ be the solutions to (105) corresponding to τ s,ǫ . The key point nowis the following lemma of Berndtsson which guarantees uniform estimates for thesesolutions independent of s and ǫ . Lemma 24. [11, Lemmas 6.3,6.5] , [20, Lemmas 17,19] • There exists a constant C (independent of s, ǫ ) such that k ν s,ǫ k L ( τ ′ s,ǫ ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13) dτ ′ s,ǫ ds u (cid:13)(cid:13)(cid:13)(cid:13) L ( τ ′ s,ǫ ) • For every δ -neighborhood U δ of ∆ , there exists a constant c δ such that c δ → as δ → and Z U δ | ν s,ǫ | τ ′ s,ǫ ≤ c δ (cid:16) Z W ′ | ν s,ǫ | τ ′ s,ǫ + | ¯ ∂ν s,ǫ | τ ′ s,ǫ (cid:17) Note that the norms of ν s,ǫ also involve a K¨ahler metric on W ′ which we taketo be the fixed metric ω ′ . We also remark that this was proved by Berndtsson formetrics e − ξ where ξ is only upper bounded, and hence is applicable in our situationsince τ ′ s,ǫ are easily seen to be upper bounded. Once we have this uniform L estimate, the rest of the argument in [20] can be followed almost verbatim. Thatis, if we write for F ǫ ( s ) for the functional corresponding to τ ′ s,ǫ , then F ǫ ց F . Moreover, using the Hessian formula above one can show that for any r ∈ (0 , 1) on[ r, − r ] we have d F ǫ ds > − c ǫ → . This shows that F is indeed convex.Suppose now that F is affine linear. Observe that since τ ′ s,ǫ decrease to τ ′ s and τ ′ s,ǫ are uniformly Lipschitz in s , || ν s,ǫ || L ( τ ′ s,ǫ ) are uniformly bounded. Hence ν s,ǫ converges weakly in L ( τ ′ s ) to an L -valued ( n − , 0) form ν s with ¯ ∂ν s = 0.Integrating by parts, it can be shown that ν s solves (105) weakly on W ′ \{ ψ = −∞} or equivalently, ∇ s ν s − u dτ ′ s /ds is holomorphic on { ψ = ∞} , and it is in L . Butsince pluripolar sets are removable for L holomorphic forms, ∇ s ν s − u dτ ′ s /ds isalso holomorphic globally. Using the formula ¯ ∂ ∇ s ν s + ∇ s ¯ ∂ν s = ω τ ′ s ∧ ν s it followsthat(108) ω τ ′ s ∧ ν s = √− ∂ (cid:16) dτ ′ s ds (cid:17) ∧ u. A family of holomorphic vector fields w ′ s can now be defined on W ′ \ E by ι w ′ s u = ν s , so that away from E we have ι w ′ s ω τ ′ s = −√− ∂ ˙ τ ′ s . Then w s = p ∗ w ′ s is a holo-morphic vector field on W which by normality of W extends to a global time-dependent holomorphic vector field on W . Next, note that p − is a biholomor-phism when restricted to W , and ω τ s = ( p − ) ∗ ω τ ′ s . It then follows that on W o , ι w s ω τ s = −√− ∂ ˙ τ s and hence,(109) L w s ω τ s = − ∂∂s ω τ s , as currents. Moreover, it can be shown that ∂w s /∂ ¯ s = 0, and hence w s generatesa holomorphic flow F s (see [9, Lemma 5.2]). Also, note that w ′ s has uniform L bound (independent of s ) away from E , and hence the flow F s extends continuouslyto s = 0 , F is the identity. From (109) it follows that on W , ∂∂s F ∗ s ω τ s = F ∗ s (cid:16) ∂∂s ω τ s + L w s ω τ s (cid:17) = 0 . In particular F ∗ s ω τ s = ω τ on W , and hence globally on W by unique extensionof closed positive (1 , 1) currents over sets of Hausdorff co-dimensions greater thantwo. Now, if we define a holomorphic vector field W s = ∂/∂s − w s on W × R ,following the same line of argument as in [11, Lemma 4.3] we can show that ι W s √− ∂∂ s,W ( τ s ) = 0 . Again following [11]0 = ι W s ι W s √− ∂∂ s,W ( τ s ) = t ι W s ι W s √− ∂∂ s,W ( φ s ) + (1 − t ) ι w s ι w s √− ∂∂ψ. Since both the (1 , 1) currents on the right are non-negative, each has to be zero.Again, since √− ∂∂ψ ≥ 0, by Cauchy’s inequality for any (1 , 0) vector field ξ , ι ξ ι w s √− ∂∂ψ = 0, and hence ι w s √− ∂∂ψ = 0. In particular, L w s √− ∂∂ψ = 0,and hence F ∗ s √− ∂∂φ s = φ , which completes the proof of the proposition. (cid:3) ¨AHLER-EINSTEIN METRICS ALONG THE SMOOTH CONTINUITY METHOD 27 Proof of Proposition 5. Let e − φ and e − φ be two soliton metrics on ( W, (1 − t ) ψ, v ) and φ s ∈ H v be a bounded geodesic connecting φ and φ . Since solitonsare the stationary points of D (1 − t ) ψ,v , the one sided derivatives at s = 0 and s = 1 (which exist by convexity of the Ding functional) are zero. As a consequence D (1 − t ) ψ,v ( φ s ), and hence F ( s ), is affine, and by Proposition 23 there exists a familyof holomorphic vector fields w s with flow F s such that F ∗ s ω φ s = ω φ . Next, notethat φ j for j = 0 , Ric ( ω φ j ) = tω φ j + (1 − t ) √− ∂∂ψ + L v ω φ j on W . So on the one hand, since φ s are stationary points of D (1 − t ) ψ,v , ω φ s alsosatisfies (110), while on the other hand ω φ s satisfies (110) with v replaced by ( F s ) ∗ v .Hence if we set ξ s = ( F s ) ∗ v − v , then L ξ s ω φ s = 0. This implies that if h s isthe hamiltonian of ξ s with respect to ω φ s , then √− ∂∂h s = 0 and consequently v = ( F s ) ∗ v . To show the time-independence of the vector fields, arguing as in theproof of [11, Proposition 4,5], we can show that ι ( F − s ) ∗ w s − w ω φ = 0 . Since φ is bounded, and hence in particular e − φ is integrable, by Berndtsson [11,Proposition 8.2] the above equation forces ( F − s ) ∗ w s = w . This shows that thevector fields are independent of time, and in fact F s is just the flow generated by w . Finally since ι w ω φ = −√− ∂ ˙ φ and φ is real valued, Im ( w ) is also aKilling field for ω φ . This completes the proof of the proposition with w = w . Proof of Proposition 7. As shown in [20] reductivity follows from uniqueness,and we reproduce their arguments. Suppose ω is the twisted K¨ahler-Ricci solitonon the triple ( W, (1 − t ) ψ, v ), and let H be the connected group with Lie algebra g W,ψ,v naturally identified as a subgroup of SL ( N + 1 , C ). Let K ⊂ H be thesubgroup of isometries of ω with the corresponding Lie sub-algebra of g W,ψ,v givenby k W,ψ,v = { w ∈ H ( W, T , W ) : L Re ( w ) ω = 0 , ι w ω ψ = 0 , [ w, v ] = 0 } , which can naturally be identified as a sub-algebra of su ( N + 1 , C ). Moreover, sincethe trace form on su ( N + 1 , C ) given by B ( x, y ) = tr( xy ) is negative definite.it’s restriction to k W,ψ,v is a non-degenerate bilinear form, and hence k W,ψ,v is areductive Lie algebra. Next, if K c ⊂ SL ( N +1 , C ) is the connected complexificationof K , then clearly K c ⊂ H . Conversely, for any h ∈ H , it can be checked that h ∗ ω is also a twisted K¨ahler-Ricci soliton for the triple ( W, (1 − t ) ψ, v ), and henceby Proposition 4 there exists an element F ∈ K c such that h ∗ ω = F ∗ ω . But then h ◦ F − ∈ K , and hence H = K c . As a consequence g W,ψ,v = k W,ψ,v ⊗ R C , andis reductive. The same proof suitably modified shows that the centralizer g GW,ψ,v isalso reductive. Proof of Proposition 8. Suppose that e − φ is a smooth metric on K − W , and f t ∈ Aut( W ) is a one-parameter group of biholomorphisms, generated by w ∈ g W,ψ,v .In particular since f ∗ t ω ψ = ω ψ , we must have f ∗ t ( e − ψ ) = c t e − ψ for some constants c t . Similarly to [20, Lemma 12], we consider the quantity(111) I ( e − φ ) = 1 V Z W log (cid:0)R W e − φ (cid:1) − e − φ (cid:0)R W e − tφ − (1 − t ) ψ (cid:1) − e − tφ − (1 − t ) ψ ω nφ = log R W e − tφ − (1 − t ) ψ R W e − φ − − tV Z W ( φ − ψ ) ω nφ , where we note that φ − ψ is a globally defined integrable function. We have I ( f ∗ t ( e − φ )) = I ( e − φ ), and differentiating this at t = 0 we obtain (using ˙ φ = θ w ),that(112) R W θ w e − φ R W e − φ − R W tθ w e − tφ − (1 − t ) ψ R W e − tφ − (1 − t ) ψ − − tV Z W θ w ω nφ − n − tV Z W ( φ − ψ ) √− ∂∂θ w ∧ ω n − φ = 0 . Integrating by parts in the last integral, and using the definition (15) of the twistedFutaki invariant, we obtain(113) Fut (1 − t ) ψ,v ( W, w ) = tV Z W θ w e θ v ω nφ − t R W θ w e − tφ − (1 − t ) ψ R W e − tφ − (1 − t ) ψ . Note that this formula is not well defined if e − (1 − t ) ψ is not integrable, but weonly need it in that case, since by assumption ( W, (1 − t ) ψ, v ) admits a twistedK¨ahler-Ricci soliton.By the convexity of D (1 − t ) ψ,v , twisted K¨ahler-Ricci solitons minimize the twistedDing functional, we know that D (1 − t ) ψ,v is bounded below. At the same time (113)implies that(114) ddt D (1 − t ) ψ,v ( f ∗ t φ ) = − Fut (1 − t ) ψ,v ( W, w ) , and as a result the twisted Futaki invariant must vanish. Acknowledgements. We would like to thank Valery Alexeev, Robert Berman,Duong Phong, Jian Song, Jacob Sturm, and Hendrik S¨uss for helpful discussions.The second named author is supported by National Science Foundation grantsDMS-1306298 and DMS-1350696. 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