The Soliton-Ricci Flow with variable volume forms

We introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previous work. We still call this new flow the Soliton-Ricci flow. It corresponds to a forward Ricci type flow up to a gauge transformation generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times and represents the gradient flow of Perelman's $\mathcal{W}$ functional with respect to a pseudo-Riemannian structure over the space of metrics and normalized positive volume forms. We obtain an expression of the Hessian of the $\mathcal{W}$ functional with respect to such structure. Our expression shows the elliptic nature of this operator in directions orthogonal to the orbits obtained by the action of the group of diffeomorphism. In the case the initial data is K\"ahler then the Soliton-Ricci flow preserves the K\"ahler condition and the symplectic form. The space of tamed complex structures embeds naturally to the space of metrics and normalized positive volume forms via the Chern-Ricci map. Over such space the pseudo-Riemannian structure restricts to a Riemannian one. We perform a study of the sign of the restriction of the Hessian of the $\mathcal{W}$ functional over such space. This allows us to obtain a finite dimensional reduction, and thus the solution, of the well known problem of the stability of K\"ahler-Ricci solitons.


Introduction and statement of the main result
This is the first of a series of papers whose purpose is the study the following problem.
Let (X, J 0 ) be a Fano manifold. We remind that the first Chern class c 1 (X, [J 0 ]) ∈ H 2 d (X, Ê) depends only on X and the coboundary class [J 0 ] of the complex structure J.
Let also ω ∈ 2πc 1 (X, [J 0 ]) be an arbitrary J 0 -invariant Kähler form over X. We want to find under which conditions on J 0 and ω there exists a smooth complex structure J ∈ [J 0 ] and a smooth volume form Ω > 0 over X such that    ω = Ric J (Ω) , i.e. the Riemannian metric g := −ωJ, is a J-invariant Kähler-Ricci soliton. This set up represents a particular case of the Hamilton-Tian conjecture with a stronger conclusion, namely we avoid the singularities in the solution of the Kähler-Ricci soliton equation.
Proofs of the Hamilton-Tian conjecture have been posted on the arXiv server in (2013) by Tian-Zhang [Ti-Zha] in complex dimension 3 and quite recently by Chen-Wang [Ch-Wa] in arbitrary dimensions.
Our starting point of view is Perelman's twice contracted second Bianchi type identity introduced in [Per].
We remind first what this is about. Let Ω > 0 be a smooth volume form over an oriented compact and connected Riemannian manifold (X, g). We remind that the Ω-Bakry-Emery-Ricci tensor of g is defined by the formula Ric g (Ω) := Ric(g) + ∇ g d log dV g Ω .
We equip the set smooth Riemannian metrics M with the scalar product (u, v) −→ X u, v g Ω, (1.1) for all u, v ∈ H := L 2 (X, S 2 Ê T * X ). Let P * g be the formal adjoint of some operator P with respect to a metric g. We observe that the operator P * Ω g := e f P * g e −f • , with f := log dVg Ω , is the formal adjoint of P with respect to the scalar product (1.1). We define also the Ω-Laplacian operator ∆ Ω g := ∇ * Ω g ∇ g = ∆ g + ∇ g f ¬∇ g . It is also useful to introduce the Ω-divergence operator acting on vector fields as follows; div Ω ξ := d(ξ¬Ω) Ω = e f div g e −f ξ = div g ξ − g (ξ, ∇ g f ) .
We infer in particular the identity div Ω ∇ g u = −∆ Ω g u, for all functions u. We observe also the integration by parts formula − X u div Ω ξΩ = X g(∇ g u, ξ)Ω.
This operator is self-adjoint with respect to the scalar product (1.1) thanks to the identity for all symmetric 2-tensors u and v (see section 3). We define also the set of normalised volume forms V 1 := Ω > 0 | X Ω = 1 . From now on we consider that the functions h and H are defined over M × V 1 . Notice that the tangent space of M × V 1 is T M×V1 = C ∞ (X, S 2 T * X ) ⊕ C ∞ (X, Λ m T * X ) 0 , where We denote by End g (T X ) the bundle of g-symmetric endomorphisms of T X and by C ∞ (We will give a detailed proof in section 3.) We infer that the variations of the non-linear operators h and H are elliptic in restriction to the space This fact strongly suggest that the following flow represents a strictly parabolic system.
Definition 1 The Soliton-Ricci flow is the smooth curve (g t , Ω t ) t 0 ⊂ M × V 1 solution of the evolution system   ġ t = −h gt,Ωt , with H g,Ω := H g,Ω − X H g,Ω Ω.
Indeed this is the case as shown in the proof of the following basic fact Lemma 1 For every (ǧ 0 , Ω 0 ) ∈ M × V 1 there exists a unique smooth solution (g t , Ω t ) t 0 ⊂ M × V 1 of the Soliton-Ricci flow equation with initial data (ǧ 0 /λ, Ω 0 ), for some λ > 0. In the case (X, J 0 ) is a Fano variety anď g 0 J 0 ∈ 2πc 1 (X) we can choose λ = 1. In this case the Soliton-Ricci-flow represents a smooth family of Kähler structures and normalized positive volumes (J t , g t , Ω t ) t 0 uniquely determined by the evolution system           ġ t = −h gt,Ωt , that we call the Soliton-Kähler-Ricci flow.
In the Dancer-Wang Kähler-Ricci soliton case H 0,1 g,Ω (T X,J ) 0 = {0} thanks to a result in . Let H TX,J be the L 2 Ω -projector over the space H 0,1 g,Ω (T X,J ). We define now the non-negative cone We observe also that that Perelman's twice contracted second Bianchi type identity implies that the set of all Kähler-Ricci solitons inside S ω is given by KRS ω := (g, Ω) ∈ S ω | H g,Ω = 0 .
The following statement provides a finite dimensional reduction which represents the solution of the stability of Kähler-Ricci solitons problem.
In section 17 we obtain also quite general and sharp second variation formulas for Perelman's W functional with respect to variations (v, V ) ∈ g,Ω over a Kähler-Ricci soliton point which arise from variations of Kähler structures preserving the first Chern class of X.
This formulas provide a precise control of the sign of the second variation of Perelman's W functional over a Kähler-Ricci soliton point. This can be of independent interest for experts. (In particular we will see below some general consequences for the classical stability of Kähler-Einstein metrics.) For our geometric applications the most striking particular case is the one corresponding to the main theorem 1.
The highly geometric nature of the Soliton-Kähler-Ricci flow combined with the main theorem 1, suggest to the author the following version of the Hamilton-Tian conjecture (compare with the statements made in [Ti-Zha]
We explain now that a very particular consequence of our study of the stability problem provides a result on the stability in the classical sense of Kähler-Einstein manifolds. We introduce first a few basic notations.
We denote by KS the space of Kähler structures over X and we set KS 2πc1 := {(J, g) ∈ KS | gJ ∈ 2πc 1 } .
We define also the set ÃÎ J g (2πc 1 ) of symmetric variations of Kähler structures preserving the first Chern class of X as the set of elements v ∈ C ∞ X, S 2 Ê T * X such that there exists a smooth curve (J t , g t ) t ⊂ KS 2πc1 with (J 0 , g 0 ) = (J, g), g 0 = v andJ 0 = (J 0 ) T g . In section 14 we show the inclusion ÃÎ J Theorem 2 Let (X, J, g) be a Fano Kähler-Einstein manifold. Then for any v ∈ Ker ∇ * g ∩ J g,0 , hold the inequality with equality if and only if v * g ∈ H 0,1 g (T X,J ). (See sub-section 17.1 for the proof). A similar result in the case of negative or vanishing first Chern class has been proved in the remarkable paper [D-W-W2] (see also [D-W-W1]). The statement about the equality case hold also under more general assumptions (see lemma 29 in the appendix B).
In the next section we enlighte the results obtained by other authors in the long standing problem of the stability of Kähler-Ricci solitons and on the Hamilton-Tian conjecture. 2 Other works on the subject A question of central importance in complex differential geometry is the Hamilton-Tian conjecture.
Solutions of this conjecture have been posted on the arxiv server in (2013) by Tian-Zhang [Ti-Zha] in complex dimension 3 and quite recently by Chen-Wang [Ch-Wa] in general.
Since we have learned about this conjecture in 2004 we asked ourself immediately which one is the precise notion of gauge needed for the convergence. (The Kähler-Ricci flow (J 0 ,ĝ t ) t 0 needs to be modified since its formal limit (J 0 ,ĝ ∞ ) as t → +∞ is a is a Kähler-Einstein metric, but Fano manifolds do not always admit such ones!) It turns out that the Soliton-Kähler-Ricci flow introduced in this paper corresponds to a modification of the Kähler-Ricci flow via the gauge provided by the gradient of the Ricci potentials.
To the very best of our knowledge the Soliton-Kähler-Ricci flow with variable volume forms introduced in this paper does not appear nowhere in the literature.
In our previous works [Pal4] and [Pal5], we introduced also the notion of Soliton-Kähler-Ricci flow with fixed volume form. This leads to a complete different approach which leads naturally to the study of the existence of ancient solutions of the Kähler-Ricci flow and their (modified, according to [Pal4] and [Pal5]) convergence as t → −∞. This approach requires some particular geometric conditions (which imply some strong regularity) on the initial data.
The key point in [Pal4] and [Pal5] is that these conditions represent a conservative law along the Soliton-Kähler-Ricci flow with fixed volume form. These conditions imply good convexity properties for the convergence of this flow.
We review now the modifications of the Kähler-Ricci flow made by other authors. We can find two frequent approaches in the literature. One is based on the gauge transformation generated by a holomorphic vector fields with imaginary part generating an S 1 -action on the manifold (see  and [P-S-S-W2] for a very elegant construction). A Kähler-Ricci-soliton vector field provides such example.
The second approach, which has been used quite intensively in the last years is based on the gauge modification constructed via the minimizers of Perelman's W functional (see  and [Su-Wa]). As far as known the minimizers are unique only in a small neighborhood of the Kähler-Ricci soliton. Therefore the "modified Kähler-Ricci flow" in  and [Su-Wa] exists only in such small neighborhood.
For historical reasons it is important to remind that Hamilton [Ham] pointed out first that to any flow of Kähler structures with fixed complex structure corresponds an other flow of Kähler structures which preserves the symplectic form (see also Donaldson [Don] for the same remark). He suggested this approach for the study of the Kähler-Ricci flow. As far as we know he did not pursuit on this idea.
As explained in the introduction our definition of the Soliton-Ricci flow with variable volume forms was inspired to us from Perelman's twice contracted second Bianchi type identity and the strict ellipticity of the first variation of the functions h and H in the directions .
It was surprising for us to discover that the corresponding Soliton-Kähler-Ricci flow with variable volume forms (from now on we will refer only to this flow) preserves the symplectic structure.
We realized quickly the power of this fact since it allows us to apply Futaki's weighted complex Bochner identity with uniform lower bound on the first eigenvalue of the corresponding weighted Laplacian [Fu1]. The main feature of the Soliton-Kähler-Ricci flow in this paper is that it presents the jumping of the complex structure at the limit when t → +∞. This phenomenon is necessary for the existence of Kähler-Ricci solitons in general. We learned for the first time about this key phenomenon in the Pioneer work of [P-S-S-W1]. In this fundamental work the authors introduce a condition on stability (is the condition (B) in [P-S-S-W1]) wich is the key phenomenon occourring in the convergence of the Kähler-Ricci flow. We refer also to [P-S-S-W3] for further developpements.
We remind now that by definition, the stability of a critical point of a functional corresponds to determine a sign of its second variation in determinate directions.
The stability of critical metrics for natural geometric functionals was naturally born with differential geometry (see [Bes]). The main classic example is the Einstein metric. In the case of this metric the corresponding functional is the integral of the scalar curvature.
In 2003 Grigory Perelman astonished the mathematical community with his spectacular proof of the Poincaré conjecture. In this celebrated paper [Per] he introduced various entropy functionals for Ricci-solitons. Shrinking Riccisolitons corresponds to critical points of his W functional or to his entropy functional ν.
Since then the second variation of Perelman's functional W and ν has been studied quite intensively. We wish to point out that the results in this paper and in [Pal3] are of completely different nature with respect to the previous works. The reason is that in our work we compute the second variation of Perelman's W functional with respect to the pseudo-Riemannian structure G. (The work [Pal3] is a particular case.) An important fact about Kähler-Ricci solitons is that once they exist, one can obtain the Einstein condition by proving the vanishing of the Futaki invariant [Fut]. From our point of view they provide a natural and necessary generalization in order to control the Einstein condition.
The stability of Kähler-Ricci solitons is important in order to understand the convergence of the Kähler-Ricci flow. The first work on the subject is due to .
In 2009 Sun-Wang [Su-Wa] posted on the arxiv server a stability result for the Kähler-Ricci flow basing on the Lojasiewicz inequality (see [Co-Mi]). In this paper the autors use the modified flow in . The same method was used in Ache [Ach], where a uniform bound on the curvature is made. We report finally a quite recent work on the same subject by Kröncke [Kro] which combines the technical details in [Su-Wa], [Ach] and [Co-Mi] in the Riemannian set up.
The statements made in this section are based on the very best of our knowledge and understanding of the subject. We sincerely apologize to other authors in case of inaccuracies or omissions in the claims of this section.
3 Proof of the first variation formulas for the maps h and H 3.1 The first variation of the Bakry-Emery-Ricci tensor We remind (see [Pal3]) that the first variation of the Bakry-Emery-Ricci tensor with fixed volume form Ω > 0 is given by the formula where D g :=∇ g − 2∇ g , with∇ g being the symmetrization of ∇ g acting on symmetric 2-tensors. Explicitlŷ for all p-tensors α. Fixing an arbitrary time τ and time deriving at t = τ the decomposition we deduce, thanks to (3.1), the general variation formula This formula implies directly Perelman's general first variation formula for the W functional (see appendix A). We define the Hodge Laplacian (resp. the Ω-Hodge Laplacian) operators acting on q-forms as ∆ TX,g := ∇ TX,g ∇ * g + ∇ * g ∇ TX,g , ∆ Ω TX,g := ∇ TX,g ∇ * Ω g + ∇ * Ω g ∇ TX,g .
We remind also the following Weitzenböck type formula proved in [Pal4] Lemma 3 Let (X, g) be a orientable Riemannian manifold, let Ω > 0 be a smooth volume form and let A ∈ C ∞ (X, End(T X )). Then In analogy to the Ω-Hodge Laplacian we can define the Laplace type operator Using this notation we observe that for any u ∈ C ∞ (X, S 2 T * X ) hold the identities The last one follows from the equalities ∇ * Ω g u = g∇ * Ω g u * g and∇ g (gξ) = L ξ g, ξ ∈ C ∞ (X, T X ). We observe now that for any symmetric 2-tensor u the tensor R g * u is also symmetric. In fact let (e k ) k be a g(x)-orthonormal base of T X,x . Then . Furthermore if we choose the g(x)-orthonormal base (e k ) k such that u is diagonal with respect to this one, then . We observe also that the previous computation shows the identity We deduce in particular the equality (3.4) We remind that the Ω-Lichnerowicz Laplacian ∆ Ω L,g is self-adjoint with respect to the scalar product (1.1) thanks to the identity (1.3) that we show now.
We pick a g(x)-orthonormal base (e k ) k ⊂ T X,x such that v is diagonal with respect to this one at the point x. Using (3.3) we infer = R g (e l , e k , v * g e l , u * g e k ) = R g (e k , e l , u * g e k , v * g e l ) = R g * v, u g , since these identities are independent of the choice of the g(x)-orthonormal base (e k ) k ⊂ T X,x .
We observe that if µ, ζ are two germs of vector fields near x 0 such that [µ, ζ] (x 0 ) = 0 then hold the identity at the point x 0 Using this identity we infer the equalities at the point x 0 by obvious symmetries. Combining the identities obtained so far and simplifying we obtain the identitŷ which in its turn implies the required identity (3.5).
The Weitzenböck type identity in lemma 4 combined with the variation formula (3.2) implies directly the variation formula (1.4).

Proof of the first variation formula for Perelman Hfunction
We show now the variation formula (1.5). For this purpose let 0 < (g t , Ω t ) t be a smooth family and set as usual f t := log dVg t Ωt . We start time deriving the identity −∆ Ωt gt f t = div Ωt ∇ gt f t .
We compute first the variation of the Ω-divergence operator. Set u t :=Ω * t and time derive the definition identity Moreover expanding the left hand side we obtain which implies the formula We observe also the variation formulas Combining all this formulas we obtain . We expand last therm using the identity div Ω ξ = Tr Ê (∇ g ξ) − g (ξ, ∇ g f ) .
We obtain with respect to a g t (x)-orthonormal basis (e k ) k ⊂ T X,x at an arbitrary We infer the variation formula We observe next the identity thanks to the variation formula (1.4). We deduce the formula thanks to the identities Tr g (L ξ g) = 2 Tr Ê (∇ g ξ) = 2 div g ξ. (3.11) In order to show the identity (3.10) we expand with respect to a g(x)-orthonormal basis (e k ) k ⊂ T X,x the therm The first equality in (3.11) follows from the elementary identity where A T g denotes the transpose of an endomorphism A of T X with respect to g.
In conclusion combining the variation formulas (3.8), (3.9) and (3.7) we infer the variation identity and thus the required variation formula (1.5). We prove in this sub-section lemma 1.
Proof From now on we will set for notation simplicity h t ≡ h gt,Ωt , H t ≡ H gt,Ωt and H t ≡ H gt,Ωt . We observe that for any smooth curve (g t , Ω t ) t 0 ⊂ M × V 1 the identity is equivalent to the evolution equation with initial data f 0 := log dVg 0 Ω0 . Along the Soliton-Ricci flow, the equation (4.1) rewrites as We infer that the Soliton-Ricci flow equation is equivalent to the evolution system (4.2) with f 0 := log dVg 0 Ω0 . We consider now the flow of diffeomorphisms (ϕ t ) t 0 solution of with ϕ 0 = Id X and we define (ĝ t ,f t ) := ϕ * t (g t , f t ). We observe the evolution formulas We deduce thanks to the diffeomorphism invariance of the W functional that the evolution system (4.2) is equivalent to with initial data (ĝ 0 ,f 0 ) := (g 0 , f 0 ). Notice indeed that we can obtain (4.2) from (4.3) by performing the inverse gauge transformation (g t , f t ) := ψ * t (ĝ t ,f t ) with ψ t = ϕ −1 t being characterized by the evolution equation ψ 0 = Id X . In order to show all time existence and uniqueness of the solutions of the evolution system (4.3) we consider a solution of the Ricci flow (ǧ t ) t∈[0,T ) , with initial dataǧ 0 and 0 < T < +∞. Then (ĝ t ) t 0 defined bŷ satisfies the evolution equation relative to the metrics in (4.3). Then we set λ := 2T . In the case (X, J 0 ) is a Fano variety andǧ 0 J 0 ∈ 2πc 1 (X) we can choose λ = 1 since the the evolution equation ofĝ t in (4.3) represents a solution of the Kähler-Ricci flow equation. The existence and uniqueness of the solutions of the evolution equation forf t in (4.3) follows directly from standard parabolic theory with respect to Hölder spaces. Notice indeed that the presence of the integral therm W(ĝ t ,f t ) (we consider the expression involving the H 1 (X) norm of f ) does not produce any issue in this theory.
In the Fano set up we define the complex structures J t := ψ * t J 0 . Then the family (J t , g t ) t 0 represents a flow of Kähler structures since (J 0 ,ĝ t ) t 0 is also a flow of Kähler structures. The identity ϕ * t J t ≡ J 0 is equivalent to the equality i.e to the equationJ This combined with the J t -linearity of the first two therms in the right hand side of the complex decomposition implies the required characterization 2J t = [J t ,ġ * t ] of the evolution of the complex structures J t .

Monotony of Perelman's W-functional along the soliton Soliton-Kähler-Ricci flow
We observe first the following elementary fact.
Lemma 5 Let (X, J) be a Fano manifold and let g be a J-invariant Kähler metric with symplectic form ω := gJ ∈ 2πc 1 (X, [J]). Then (J, g) is a Kähler-Ricci soliton if and only if there exists a smooth volume form Ω > 0 with X Ω = 1 such that ∆ Ω g f − 2f + 2 X f Ω = 0, f := log ω n n!Ω . Proof We assume first that (J, g) is a Kähler-Ricci soliton. Then Perelman's twice contracted Bianchi type identity (1.2) implies H g,Ω = 0. This last is equivalent to the second equation of the system (S) thanks to the identity Tr g h g,Ω = 0. We show now that the solution of the system (S) implies that (J, g) is a Kähler-Ricci soliton. Indeed multiplying by ∇ g f both sides of the identity (1.2) and integrating by parts we obtain the general formula (4.4) In our case this rewrites as (4.5) thanks to the condition ω = Ric J (Ω) and the complex decomposition of the Bakry-Emery-Ricci tensor. We infer the required conclusion.
We provide now a proof of the monotony statement in lemma 2. Proof STEP I. Let (J,ĝ t ) t 0 be a solution of the Kähler-Ricci flow and observe that this equation rewrites in the equivalent form with ω t :=ĝ t J, and ω 0 := ω. We define the function and we observe the analogue of (4.1) This combined with the first equation in (4.6) implies (4.7) On the other hand time differentiating the identity ω t = Ric J (Ω t ) in (4.6) we obtain which combined with (4.7) implies for some time dependent constant C t which can be determined time deriving the integral condition XΩ t = 1. Indeed using (4.7) we obtain We infer the evolution formula with initial dataf 0 := log ω n Ωn! .
We observe now that the identity ω t = Ric J (Ω t ) in (4.6) implieŝ and thus Trĝ t hĝ t ,Ωt = 0. We deduce the equality (4.10) We infer by Cauchy's uniqueness that the evolution equation (4.8) is equivalent with the second evolution equation in (4.3). We obtain, as in the proof of lemma 1, a Soliton-Kähler-Ricci flow (J t , ω t , Ω t ) t 0 with initial data (J 0 , ω 0 , Ω 0 ) = (J, ω, Ω). We observe that thanks to (4.9) and (4.10) the Soliton-Ricci flow evolution system (4.2) writes in our case as (4.11) Time deriving the identity ω t = g t J t and using the evolution formula for the complex structure 2J t = [J t ,ġ * t ] in the Soliton-Kähler-Ricci flow equation we inferω thanks to the first equation in (4.11). We deduce ω t ≡ ω and thus the identity in time ω = Ric Jt (Ω t ).
(4.12) STEP IIa. We provide now a first proof of the monotony statement for the Soliton-Kähler-Ricci flow. The equality (4.10) rewrites as (4.13) thanks to the invariance by diffeomorphisms of W. Let and observe that the second evolution equation in (4.11) rewrites as (4.14) Time deriving the expression (4.13) and using the evolution equation (4.14) we infer thanks to the estimate λ 1 (∆ Ω g ) 2 for the first eigenvalue λ 1 (∆ Ω g ) of the weighted Laplacian ∆ Ω g in the case gJ = Ric J (Ω). (See the estimate (13.15) in the section 13.) Indeed by the variational characterization of the first eigenvalue hold the estimate with We assume now equality in (4.16). We assume also F =0 otherwise g will be a J-invariant Kähler-Einstein metric. Equality in (4.16) implies 2 = λ 1 (∆ Ω g ) and , attains the infinitum in (4.15). Thus we can apply the method of Lagrange multipliers to the functionals which imply D u0 Φ = µD u0 Ψ, i.e. ∆ Ω g u 0 = µu 0 , with µ = 2. This last is equivalent to the equation ∆ Ω g F = 2F . Then the required conclusion follows from lemma 5.
STEP IIb. We give here a different proof of the monotony statement. We remind first that Perelman's first variation formula for the W functional [Per] writes as Thus along the Soliton-Ricci flow hold the identity Then the conclusion follows from the identity (4.12) combined with the elementary lemma below.
Lemma 6 Let (X, J) be a Fano manifold, let g be a J-invariant Kähler metric with symplectic form ω := gJ ∈ 2πc 1 (X, [J]) and let Ω > 0 be a smooth volume form with X Ω = 1 such that ω = Ric J (Ω). Then with equality if and only if (J, g) is a Kähler-Ricci soliton.
Proof The condition ω = Ric J (Ω) and the complex decomposition of the BER tensor imply h g,Ω = g∂ TX,J ∇ g f, and thus Tr g h g,Ω = 0. We deduce thanks to the integral identity (4.5). The conclusion follows from the variational argument at the end of step IIa.
Remark 1 We observe that the elementary identities ∇ g f = Jω −1 df = 2ω −1 d c J f , with 2d c J f := −df · J, allow to rewrite the Soliton-Ricci flow evolution system (4.11) as (4.19) We notice also that the Soliton-Kähler-Ricci flow evolution system with initial data (J 0 , g 0 , Ω 0 ) = (J, g, Ω) such that ω := gJ = Ric J (Ω) is equivalent to the system (1.6). Indeed the argument in step I of the proof of lemma 2 shows that our Soliton-Kähler-Ricci flow is equivalent to the Kähler-Ricci flow equation (4.6) via the gauge transformation given by the diffeomorphisms ϕ t . But (1.6) is also equivalent to (4.6) via the same gauge transformation. Notice in fact the identities The corresponding identities for the transformation of the complex structure have been considered at the end of the proof of lemma 1. We infer the equivalence of our Soliton-Kähler-Ricci flow with (1.6).
Remark 2 Let (g t , Ω t ) t 0 be the Soliton-Ricci flow and set for notation simplicity W t := W(g t , Ω t ), h t := h gt,Ωt , H t := H gt,Ωt . Perelman's twice contracted differential Bianchi identity (1.2) implies Then the fundamental variation formula (1.5) implies the evolution equation along the Soliton-Ricci flow This combined with the monotony statement in lemma 2 or in [Pal1] implies the inequality along the Soliton-Kähler-Ricci flow.
5 The second variation of the W functional along the Soliton-Kähler-Ricci flow Let (J t , g t , Ω t ) t 0 be the Soliton-Kähler-Ricci flow. In the proof of step I of lemma 2 we obtained the identitẏ Time deriving this we obtain Time deriving the identity and thus the evolution formulä We observe now that the second evolution equation in the system (4.11) rewrites as thanks to (4.13). Time deriving this we infer Plunging the identity (4.20) in the variation formula (3.8) and using the first equation in the system (4.11) we obtain Thusḟ t = H t thanks to (4.14). Using (5.2) we infer (This last follows also from the general evolution formula (4.21).) Integrating by parts we obtain the identity (since X F t Ω t = 0). Plunging this identity in the evolution formula (5.1) we deduce the simple second variation formulä The Levi-Civita connection of the pseudo--Riemannian structure G In this section we compute the Levi-Civita connection of the pseudo-Riemannian structure G. This is needed for the computation of the second variation of the W functional with respect to such structure. In the computations that will follow we set for notations simplicity T := T M×V1 and we compute the first variation of G at an arbitrary point (g, Ω), In a direction (θ, Θ) ∈ T this is given by the identity where (g t , Ω t ) t∈(−ε,ε) ⊂ M × V 1 is a smooth curve with (g 0 , Ω 0 ) = (g, Ω) and (ġ 0 ,Ω 0 ) = (θ, Θ). For notation simplicity let denote u * t := g −1 t u and U * t := U/Ω t . Then hold the equality sinceġ t is also symmetric. Indeed we observe the elementary identities and thus Summing up we infer the expression of the variation of G at the point (g, Ω) in the direction (θ, Θ) We can compute now the Levi-Civita connection ∇ G = D + Γ G of the pseudo-Riemannian structure G. At a point (g, Ω) the symmetric bilinear form Expanding and arranging the therms of the right hand side we obtain This concludes the computation of the Levi-Civita connection ∇ G .
7 The second variation of the W functional with respect to the pseudo-Riemannian structure G We justify first the geometric interpretation of g,Ω provided by the identity (1.7). We observe indeed that (v, V ) ∈ T ⊥G [g,Ω], (g,Ω) if and only if which shows the required conclusion. We introduce now the operator By abuse of notations we define also defined by the same formula We observe that (3.3) implies the identity (L Ω g v) * g = L Ω g v * g . We show now the second variation formula for the W functional.
Lemma 7 The Hessian endomorphism ∇ 2 G W(g, Ω) of the W functional with respect to the pseudo-Riemannian structure G at the point (g, Ω) ∈ M × V 1 in the directions (v, V ) ∈ g,Ω is given by the expressions In particular if h g,Ω = 0 then Proof We consider a smooth curve (g t , Ω t ) t∈Ê ⊂ M×V 1 with (g 0 , Ω 0 ) = (g, Ω) and with arbitrary speed (ġ 0 ,Ω 0 ) = (v, V ). We observe that the G-covariant derivative of its speed is given by the expressions We infer Using this expressions and Perelman's first variation formula we expand the Hessian form Using the variation formulas (1.4) and (1.5) and evaluating the previous identity at time t = 0 we obtain the expression since X H g,Ω Ω = 0. Arranging symmetrically the integrand therms via the identity we infer the general expressions Then the required expression of the Hessian of W follows from the assumption (v, V ) ∈ g,Ω . If h g,Ω = 0 then the required conclusion follows from Perelman's twice contracted second Bianchi identity (1.2) which implies H g,Ω = 0.
8 The anomaly space of the pseudo-Riemannian structure G Let Isom 0 g,Ω be the identity component of the group

and let
Kill g,Ω := Lie Isom 0 We define the anomaly space of the pseudo-Riemannian structure G at an arbitrary point (g, Ω) as the vector space We will study some properties of this space. It is clear by definition that this space is generated by the vector fields ξ ∈ C ∞ (X, T X ) such that More precisely there exists the exact sequence of finite dimensional vector spaces We observe that if α = du ∈ Ker∆ Ω g then the function u satisfies the equation which is equivalent to the equation thanks to the general identity Let C ∞ Ω (X, Ê) 0 be the space of smooth functions with zero integral with respect to Ω. We set We observe that in the soliton case h g,Ω = 0 we have thanks to the identity (8.2). By duality we can consider Kill g,Ω ⊂ Ker∆ Ω g and we observe the inclusion where the symbol ⊥ g,Ω indicates the orthogonal space inside Ker∆ Ω g with respect to the scalar product (1.1) at the level of 1-forms. The previous inclusion hold true for any (g, Ω) since for any β ∈ Kill g,Ω . We infer that in the soliton case the previous exact sequence can be reduced to the sequence In order to show that the previous map is also surjective we need to show a few differential identities. We show first the Weitzenböck type formulâ (This implies in particular the identification of Î g,Ω in terms of functions). We decompose the expression∆ We decompose first the therm We fix an arbitrary point p and we choose the vector fields ξ and η such that 0 = ∇ g ξ(p) = ∇ g η(p). Let (e k ) k be a g-orthonormal local frame such that ∇ g e k (p) = 0. Then at the point p hold the identities We infer the expression Moreover and to the fact that [ξ, e k ] (p) = 0. We infer the required formula (8.5). We deduce that in the soliton case h g,Ω = 0 hold the equality We define now the Ω-Hodge Laplacian acting on scalar valued differential forms as the operator At the level of scalar valued 1-forms we observe the identities We infer thanks to the identity (8.5) that for any scalar valued 1-form α hold the Weitzenböck type formula Applying the ∇ * Ω g -operator to both sides of this identity and using the fact that (∇ * Ω g ) 2 = 0 at the level of scalar valued differential forms we obtain In the soliton case h g,Ω = 0 this implies the formula Then the identity (8.8) implies that the map is well defined. More precisely there exists the exact sequence of finite dimensional vector spaces Indeed the surjectivity follows from the isomorphism (8.3). The injectivity follows from the fact that This hold true thanks to the identity which follows from the expression For dimensional reasons we conclude the existence of the required exact sequence (We observe also that for dimensional reasons (8.4) is an equality.) 9 Properties of the kernel of the Hessian of W Lemma 8 In the soliton case h g,Ω = 0 hold the inclusion We start with a few notations. For any tensor A ∈ C ∞ (X, (T * X ) ⊗p+1 ⊗ T X ) we define the divergence type operations The once contracted differential Bianchi identity writes often as div g R g = −∇ TX ,g Ric * g . This combined with the identity ∇ TX ,g ∇ 2 We define the Ω-Lichnerowicz Laplacian ∆ Ω L,g acting on g-symmetric endomorphisms A as .
Let now (x 1 , ..., x m ) be g-geodesic coordinates centered at an arbitrary point p and set e k := ∂ ∂x k . Then the local frame (e k ) k is g(p)-orthonormal at the point p and satisfies ∇ g e k (p) = 0 for all k. We take now an arbitrary vector field ξ with constant coefficients with respect to the g-geodesic coordinates (x 1 , ..., x m ). Therefore ∇ g ξ(p) = 0. We expand the identity at the point p Commuting derivatives at the point p we obtain Taking a covariant derivative of this identity we infer Combining with the previous expression we obtain On the other hand deriving the identity thanks again to the algebraic Bianchi identity. We infer thanks to (9.1) and (8.2). Thus We observe now that the endomorphism section ∇ g u¬∇ * Ω g R g is g-anti-symmetric thanks to the identity which is a consequence of the alternating property of the (4, 0)-Riemann curvature operator. Notice indeed that the previous identity implies for all vector fields ξ, η, µ. Combining the g-symmetric and g-anti-symmetric parts in the identity (9.2) we infer the formulas for all ξ ∈ T X since the function u is arbitrary. In the case ∇ g Ric * g (Ω) = 0 we deduce the identities ∆ Ω L,g ∇ 2 g u = ∇ 2 g ∆ Ω g u and ∇ * Ω g R g = 0. More in particular in the soliton case h g,Ω = 0 the first formula reduces to the differential identity We infer the conclusion of lemma 8. This formula will be also quite crucial for the study of the sign of the second variation of the W functional at a Kähler-Ricci soliton point.
10 Invariance of under the action of the Hessian endomorphism of W We observe that Perelman's twice contracted second Bianchi type identity (1.2) rewrites as; ∇ * Ω g h g,Ω + dH g,Ω = 0. If we differentiate this over the space M × V 1 we obtain We deduce using the fundamental variation formulas (1.4) and (1.5) We infer that in the soliton case h g,Ω = 0 the map ∇ 2 G W(g, Ω) : g,Ω −→ g,Ω , is well defined. In order to investigate the general case we use a different method which has the advantage to involve less computations. Let (e k ) k be a g-orthonormal local frame of T X . For any u, v ∈ C ∞ X, S 2 T * X we define the real valued 1-form for all ξ ∈ T X . One can show that the operator is related with the torsion of the distribution . We observe now that by lemma 3 hold the identity Applying the ∇ * Ω g -operator to both sides of this identity we deduce the commutation formula We observe now that for any ψ ∈ C ∞ (X, Λ 2 T X ⊗ Ê T X ) and ξ ∈ C ∞ (X, T X ) hold the equalities We infer This combined with the expression implies the identity which rewrites also under the form thanks to (3.3) and the anti-symmetry property On the other hand the once contracted differential Bianchi type identity (9.1) rewrites as thanks to the algebraic Bianchi identity. Therefore for any ξ ∈ C ∞ (X, Using the general formula ,Ω , thanks to Perelman's twice contracted differential Bianchi type identity (1.2). Using the identity (15.1) we expand the therm Summing up we infer We observe now the identity We deduce the formula Setting (v, V ) = (h g,Ω , H g,Ω Ω) ∈ g,Ω in the previous identity we infer This shows the fundamental property (1.8) of the Soliton-Ricci flow.

The Kähler set up
In this sections we introduce a few basic notations needed in sequel. Let (X, J, g) be a compact connected Kähler manifold with symplectic form ω := gJ. Let h := g − igJ = 2gπ 1,0 J be the hermitian metric over T X,J induced by g. We remind that in the Kähler case the Chern connection of the hermitian vector bundle (T X,J , h) coincides with the Levi-Civita connection ∇ g . We set T X := T X ⊗ Ê and T * X := T * X ⊗ Ê . We observe further that the sesquiliner extension of g is a hermitian metric over T X and the -linear extension of the Levi-Civita connection ∇ g is a g -hermitian connection over the vector bundle T X ⊗ Ê . We will focus our interest on the sections of the hermitian vector bundle and we will denote by abuse of notations ∇ g ≡ ∇ g ⊗D g TX,J the g ⊗h-hermitian connection over this vector bundle. Still by abuse of notations we will use the identification ·, · ω := g ⊗ h. With these notations we define the operators by the formulas Then the formal adjoints of the operators ∂ g TX,J and ∂ TX,J acting on T X,J -valued differential forms satisfy the identities (see [Pal6]) for any α ∈ C ∞ (X, Λ q T * X ⊗ T X,J ). We remind now that with our conventions (see [Pal3]) the Hodge Laplacian operator acting on T X -valued q-forms satisfies the identity We define also the holomorphic and antiholomorphic Hodge Laplacian operators acting on T X -valued q-forms as with the usual convention ∞ · 0 = 0. This Hodge Laplacian operators coincide with the standard ones used in the literature. We remind that in the Kähler case hold the decomposition identity We observe now that the formal adjoint of the ∂ g TX,J operator with respect to the hermitian product ·, · ω,Ω := X ·, · ω Ω, (11.1) is the operator In a similar way the formal adjoint of the ∂ TX,J operator with respect to the hermitian product (11.1) is the operator With these notations we define the holomorphic and anti-holomorphic Ω-Hodge Laplacian operators acting on T X -valued q-forms as 12 The decomposition of the operator L Ω g in the Kähler case For any A ∈ End(T X ) we denote by A ′ J and by A ′′ J the J-linear, respectively the J-anti-linear parts of A. We observe that the operator defined by the formula restricts as; Indeed this properties follow from the identities for any A ∈ End(T X ). In their turn they are direct consequence of the identities In order to see (12.5) and (12.6) let (e k ) k be a g-orthonormal real basis. Using the J-invariant properties of the curvature operator we infer where η k := Je k . The fact that (η k ) k is also a g-orthonormal real frame implies (12.6). By (12.1) and (12.2) we conclude the decomposition formula We observe that the properties (12.1) and (12.2) imply also that A ∈ Ker L Ω g if and only if A ′ J ∈ Ker L Ω g and A ′′ J ∈ Ker L Ω g . We observe further that the identity (9.3) combined with the properties (12.1) and (12.2) implies the formulas (12.9) in the Kähler-Ricci soliton case. The properties (12.1) and (12.2) imply (1.11) and (1.12).

Basic complex Bochner type formulas
We need to rewiew in detail now some fact from [Fut], (see also [Pal1]). Let (X, J, g) be a compact connected Kähler manifold with symplectic form ω := gJ. We remind that the hermitian product induced by ω over the bundle Λ 1,0 J T * X satisfies the identity 2 α, β ω = Tr ω iα ∧β .
Let Ω > 0 be a smooth volume form and set as usual f := log dVg Ω . We define the Ω-weighted complex Laplace type operator acting on functions u ∈ C ∞ (X, ) as Indeed integrating by parts we obtain (Notice the equality Ω = e −f ω n /n!.) We observe in particular the identity which implies that all the eigenvalues satisfy λ j (∆ Ω g,J ) 0. For any function u ∈ C ∞ (X, ) we define the J-complex g-gradient as the real vector field; ∇ g,J u := ∇ g Re u + J∇ g Im u ∈ C ∞ (X, T X ).
With these notations hold the complex decomposition formula ∇ g,J u¬g = ∂ J u + ∂ J u. (13.2) We consider now the linear operator This is a first order differential operator. Indeed since J is g-anti-symmetric. We extend B Ω g,J over C ∞ (X, ) by complex linearity. Let also Then the identity 2∂ J = d + 2id c J implies the decomposition In other therms The following lemma is needed for the study of the operator ∆ Ω g,J . (Compare also with [Fut].) Lemma 9 Let (X, J, g) be a Kähler manifold with symplectic form ω := gJ and let Ω > 0 be a smooth volume form. Then for all u ∈ C ∞ (X, Ê) and v ∈ C ∞ (X, ) hold the complex Bochner type formulas Proof Let ξ ∈ C ∞ (X, T X ) and observe that for bi-degree reasons hold the identity Let (e k ) 2n k=1 be a local g-orthonormal frame over a neighborhood of an arbitrary point p such that ∇ g e k (p) = 0. Then at the point p hold the equalities since (Je k ) 2n k=1 is also a local g-orthonormal frame. Then the fact that [e k , Je k ] (p) = 0 implies We infer the complex Bochner type formula 2∂ * g,Ω TX,J ∂ g TX,J ξ = ∆ Ω g ξ + Ric * (g)ξ − J∇ g,J∇g f ξ. (13.5) In a similar way we obtain 2∂ * g,Ω TX,J ∂ TX,J ξ = ∆ Ω g ξ − Ric * (g)ξ + J∇ g,J∇g f ξ. (13.6) Using formulas (13.5) and (8.2) we deduce the expressions Using the first order expression of B Ω J,g we obtain We infer the complex differential Bochner type formula (13.3). In a similar way using formulas (13.6) and (8.2) we deduce Using the first order expression of B Ω J,g we obtain We infer the complex differential Bochner type formula More in general for all v ∈ C ∞ (X, ) this writes as (13.4).
Notice that for bi-degree reasons the identity (8.2) decomposes as Then we can obtain (13.7) from (13.3) and vice versa. We observe also that the complex Bochner identities (13.3), (13.4) write in the KS case as for all u ∈ C ∞ (X, Ê) and v ∈ C ∞ (X, ). Obviously the identity (13.9) still hold in the more general case Ric J (Ω) = ω. We observe now an other integration by parts formula.
Let ξ ∈ C ∞ (X, T X ), A ∈ C ∞ X, T * X,−J ⊗ T X,J and observe that the comparison between Riemannian and hermitian norms of T X -valued 1-forms (see the appendix in [Pal2]) implies Using this and multiplying both sides of (13.9) by ∇ g,J v we obtain the identity (13.10) in the case Ric J (Ω) = ω. We consider now the J-anti-linear component of the complex Hessian map; We observe that H 1 d (X, ) = 0 in the case of Fano manifolds and we remind the following well known fact Lemma 10 Let (X, J, g) be a compact connected Kähler manifold such that H 1 d (X, ) = 0. Then the map is an isomorphism of complex vector spaces.
Proof We observe first the injectivity. Using the complex decomposition (13.2) which implies the required identity.
On the other hand the identities (13.9) and (13.10) show that in the case Ric J (Ω) = ω hold the identity Ker(∆ Ω g,J − 2Á) = Ker H 0,1 g,J . (13.12) We infer the following well known result due to Futaki [Fut]. (See also [Gau] and the sub-section 21.2 in appendix B for a more more complete statement.) Corollary 1 Let (X, J) be a Fano manifold and let g be a J-invariant Kähler metric such that ω := gJ ∈ 2πc 1 (X, [J]). Let also Ω > 0 be the unique smooth volume form with X Ω = 1 such that Ric J (Ω) = ω. Then the map is well defined and it represents an isomorphism of complex vector spaces. The first eigenvalue λ 1 (∆ Ω g,J ) of the operator ∆ Ω g,J satisfies the estimate λ 1 (∆ Ω g,J ) 2, with equality in the case H 0 (X, T X,J ) = 0. Moreover if we set Kill g := Lie(Isom 0 g ) and is well defined and it represents an isomorphism of real vector spaces.
The injectivity of the map (13.13) is obvious.
Using the variational characterization of the first eigenvalue we observe; thanks to the identity 2 |∂ J u| 2 ω = |∇ g u| 2 g . We deduce that in the set-up of corollary 1 hold the estimate (13.15)

Symmetric variations of Kähler structures
We show a few fundamental facts about the space of symmetric variations of Kähler structures ÃÎ J g given by the elements v ∈ C ∞ X, S 2 Ê T * X such that there exists a smooth family (J t , g t ) t ⊂ KS with (J 0 , g 0 ) = (J, g),ġ 0 = v anḋ J 0 = (J 0 ) T g . One can observe (see [Pal3] where (v * g ) 1,0 J and (v * g ) 0,1 J denote respectively the J-linear and J-anti-linear parts of the endomorphism v * g . We remind here some lines of this basic fact. We define The fact that the (1, 1)-form v ′ J J is real implies that the identity d(v ′ J J) = 0 is equivalent to the identity ∂ J (v ′ J J) = 0. In its turn this is equivalent to the identity ∂ g TX,J A ′ = 0. We observe indeed that for all ξ, η, µ ∈ C ∞ (X, T X ⊗ Ê ) hold the equalities In order to continue the study of the space J g we need to show a few general and fundamental facts. We start with the following weighted complex Weitzenböck type formula.
Lemma 12 Let (X, J, g) be a Kähler manifold, let Ω > 0 be a smooth volume form and let A ∈ C ∞ X, T * X,−J ⊗ T X,J . Then hold the identity Proof We observe that for bi-degree reasons hold the identities ∆ Ω,−J TX,g A := ∂ TX,J ∂ * g,Ω We fix an arbitrary point p and we choose an arbitrary vector field ξ such that ∇ g ξ(p) = 0. Let (e k ) k be a g-orthonormal local frame such that ∇ g e k (p) = 0. We observe the local expression At the point p hold the identities 2∇ g,e k ∇ 0,1 g,J A(e k , ξ) = 2∇ g,e k ∇ 0,1 g,J A(ξ, e k ) = ∇ g,e k ∇ g,ξ A · e k + J∇ g,e k ∇ g,Jξ A · e k , and thus We obtain the identity at the point p, We remind that for any A ∈ C ∞ (X, End(T X )) and ξ, η ∈ C ∞ (X, T X ) hold the general formula thanks to (12.4).
Multiplying both sides of (14.2) by A and integrating by parts we infer Using the fact that ∇ 1,0 g,J A ′′ J , ∇ 0,1 g,J A ′′ J g = 0 we obtain the integral identity We observe also the following corollary.
Corollary 2 Let (X, J, g) be a Kähler manifold, let Ω > 0 be a smooth volume form and let A ∈ C ∞ X, T * X,−J ⊗ T X,J . Then hold the identities Proof It is obvious that the identity (14.2) rewrites as (14.5). In order to show (14.6) let (η k ) n k=1 be a local complex frame of T X,J in a neighborhood of a point p with ∇ g η k (p) = 0 such that the real frame (e l ) 2n l=1 , e l = η l , l = 1, . . . , n and e n+k = Jη k , k = 1, . . . , n is g-orthonormal. Then at the point p hold the equalities thanks to the general formula (14.3) and thanks to the fact that [η k , Jη k ] (p) = 0.
Using the J-linear and J-anti-linear properties of the tensors involved in the previous equality we obtain Notice indeed the identities 2 Ric * (g) = JR g (η k , Jη k ).
We conclude the required formula (14.6).
Proof We consider the decomposition A = A sm + A as , where A as and A as are respectively the g-symmetric and g-anti-symmetric parts of A. We observe the symmetries The fact that A ∈ C ∞ X, T * X,−J ⊗ T X,J implies A sm , A as ∈ C ∞ X, T * X,−J ⊗ T X,J and thus Then the identity (14.2) implies the equalities ∆ Ω,−J TX,g A as We deduce that in the case Ric J (Ω) = λω, with λ = ±1, 0, the condition A ∈ H 0,1 g,Ω (T X,J ) is equivalent to the conditions A sm , A as ∈ H 0,1 g,Ω (T X,J ). We focus now on the Fano case λ = 1. We remind the identity R g * A as = 0. (See (20.9) in the appendix.) Thus if A ∈ H 0,1 g,Ω (T X,J ) and Ric J (Ω) = ω then the integral formula (14.4) reduces to 0 = X |∇ 0,1 g,J A as | 2 g + |A as | 2 g Ω, which shows A as = 0 and thus the required conclusion of the lemma.
We obtain also the following statement (the case c 1 < 0 has been proved in [D-W-W2]): Lemma 14 Let (X, J, g) be a compact non Ricci flat Kähler-Einstein manifold. Then hold the identity Proof Using the identities (14.7) and (14.8) with Ω = CdV g we deduce that in the Kähler-Einstein case Ric(g) = λg, with λ = ±1, 0, the condition A ∈ H 0,1 g (T X,J ) is equivalent to the conditions A sm , A as ∈ H 0,1 g (T X,J ). On the other hand the identities (14.5) and (14.6) imply in the case Ω = CdV g the formula for any A ∈ C ∞ X, T * X,−J ⊗ T X,J . The fact that R g * A as = 0 implies the formula We conclude that in the Kähler-Einstein case Ric(g) = λg, with λ = ±1, 0, any A ∈ H 0,1 g (T X,J ) satisfies ∇ g A as = 0. Then the formula (14.6) with Ω = CdV g implies 0 = div g ∇ g,J• (JA as ) = Ric * (g)A as + A as Ric * (g) = 2λA as .
We deduce A as = 0 in the case λ = ±1. This shows the required conclusion.
We denote by We obtain as corollary of lemma (13) the following fundamental fact.

Corollary 3 (Decomposition of the variation of the complex structure)
Let (X, J) be a Fano manifold, let g be a J-invariant Kähler metric with symplectic form ω := gJ ∈ 2πc 1 (X, [J]) and let Ω > 0 be the unique smooth volume form with X Ω = 1 such that ω = Ric J (Ω). Then for all v ∈ J g there exists a unique ψ ∈ Λ Ω,⊥ g,J and a unique A ∈ H 0,1 g,Ω (T X,J ) such that (v * g ) 0,1 J ∂ TX,J ξ is also g-symmetric. Then formula (13.11) implies that for all η, µ ∈ C ∞ (X, T 0,1 X ) hold the identity Then the argument showing the surjectivity of the map in lemma 10 in the section 13 implies the existence of a function Ψ ∈ C ∞ Ω (X, ) 0 such that ξ = ∇ g,J Ψ.
We infer the required conclusion.
We show now the inclusion (1.21). Time deriving the condition ω t := g t J t ∈ 2πc 1 we infer {ω 0 } d = 0. Then (1.21) follows from the complex decomposition identity 15 The decomposition of the space g,Ω in the soliton case Lemma 15 Let (X, g, Ω) be a compact shrinking Ricci soliton. Then the linear map is an isomorphism of vector spaces.
Proof STEP I. We observe first that in the compact shrinking Ricci soliton case the first eigenvalue λ 1 (∆ Ω g ) of ∆ Ω g satisfies the inequality λ 1 (∆ Ω g ) > 1. Indeed multiplying both sides of the identity (8.2) with ∇ g u and integrating we Let now u ∈ C ∞ Ω (X, Ê) 0 be an eigen-function corresponding to λ 1 (∆ Ω g ) > 0. By definition u ≡ 0. Thus by the previous integral identity we deduce which implies the required estimate. STEP II. Multiplying both sides of the the identity (8.2) with g we obtain Let now (v, V ) := T g,Ω (ϕ, θ) and observe the equalities The last one follows from (15.1). We infer that the linear map T g,Ω is well defined. The fact that in the soliton case h g,Ω = 0 the differential operator ∆ Ω g − Á is invertible over C ∞ Ω (X, Ê) 0 implies the injectivity of the map T g,Ω .
In order to show the surjectivity of the map T g,Ω let (v, V ) ∈ g,Ω and define the function Then the identity implies that the tensor θ := v − ∇ g dϕ satisfies ∇ * Ω g θ = 0. We deduce the orthogonal decomposition with respect to the scalar product (1.1) with ∇ * Ω g θ = 0. We deduce the required surjectivity statement. We need to introduce a few notations. From now on we assume H 1 d (X, Ê) = 0 (this is the case of any Fano manifold) and we observe that the first projection map With the notations introduced so far we can state the following decomposition result.
Lemma 16 Let (J, g) be a KRS and let Ω > 0 be the unique smooth volume form such that gJ = Ric J (Ω) and X Ω = 1. Then the linear map with τ ∈ C ∞ Ω (X, Ê) 0 the unique solution of the equation is an isomorphism of real vector spaces. In particular the linear map Ω g,J ⊕ H 0,1 with ϕ ∈ C ∞ Ω (X, Ê) 0 the unique solution of the equation is also an isomorphism of real vector spaces.
Proof Let first v ∈ Ë g,Ω and observe that the decomposition formula (15.2) rewrites as This implies that v ∈ J g if and only if θ ∈ J g , and also v ∈ J g,0 if and only if θ ∈ J g,0 .
Let now v ∈ Ë J g,Ω (0). Then the decomposition of the variation of the complex structure in corollary 3 implies the existence of unique τ ∈ C ∞ Ω (X, Ê) 0 , ψ ∈ Λ Ω,⊥ g,J and A ∈ H 0,1 g,Ω (T X,J ) such that For bi-degree reasons the condition ∇ * Ω g θ = 0 is equivalent to the identity 0 = 2∂ * g,Ω TX,J ∂ g TX,J ∇ g τ + 2∂ * g,Ω TX,J ∂ TX,J ∇ g,J ψ. This last is equivalent to the equation thanks to the complex Bochner identities (13.8) and (13.9). We remind that if u ∈ C ∞ Ω (X, Ê) 0 satisfies ∇ g,J u = 0 then u = 0. (See the proof of the injectivity statement in lemma 10 in the section 13.) We conclude that the condition ∇ * Ω g θ = 0 is equivalent to the equation (15.3) via the decomposition (15.5) of θ.
Then the required decomposition statement concerning the space Ë J g,Ω (0) follows from the fact that the condition τ real valued is equivalent to the equation defining ψ ∈ Ω g,J . In order to see this we show first the commutation identity ∆ Ω g , B Ω g,J = 0. (15.6) Indeed using an arbitrary g-orthonormal local frame (e k ) k we obtain thanks to formula (8.2) applied to f and thanks to the fact that (∆ Ω g − 2Á)f = 0. Moreover the endomorphism J∇ 2 g f is g-anti-symmetric since in the soliton case J, ∇ 2 g f = 0. We deduce ∆ Ω g B Ω g,J u = g(∇ g ∆ Ω g u, J∇ g f ) = B Ω g,J ∆ Ω g u, thanks to formula (8.2) applied to u. We infer the identity (15.6) which implies ∆ Ω g,J , ∆ Ω g,J = 2i B Ω g,J , ∆ Ω g = 0. (15.7) Multiplying both sides of (15.3) with ∆ Ω g,J we obtain The invertible operator ∆ Ω g,J ∆ Ω g,J : C ∞ Ω (X, ) 0 −→ C ∞ Ω (X, ) 0 , is real thanks to (15.7). We deduce that the condition τ real valued is equivalent to the left hand side of (15.8) being real valued, thus equivalent to the equation defining ψ ∈ Ω g,J .
Remark 2. We observe that the linear map ∆ Ω g,J : Λ Ω,⊥ g,J −→ Λ Ω,⊥ g,J , (15.10) is well defined and it represents an isomorphisms of complex vector spaces. In fact this follows from the identity for all v ∈ Λ Ω g,J . Thus the linear map is also well defined and represents an isomorphisms of complex vector spaces. The surjectivity of the latter follows from the finiteness theorem for elliptic operators. By definition of Ω g,J we deduce the existence of the isomorphism of real vector spaces We notice also the inclusion 16 The geometric meaning of the space J g,Ω [0] We define the subspaces J g,Ω (0) := In the previous section we gave a parametrization of the space Ë J g,Ω (0), and thus of J g,Ω (0), which is fundamental for the computation of a general second variation formula for the W functional at a Kähler-Ricci soliton point. In this section we give a simpler parametrization of the sub-space J g,Ω [0] and a useful geometric interpretation of it. Indeed let We denote by TC Sω,(g,Ω) the tangent cone of S ω at an arbitrary point (g, Ω) ∈ S ω . With these notations we show the following useful fact.
We show first a quite general variation formula for the Ricci-Chern form.
Lemma 18 Let (g t , J t ) t ⊂ KS, (Ω t ) t ⊂ V 1 be two smooth families such thaṫ J t = (J t ) T gt . Then hold the first variation formula Proof In the case of a fixed volume form Ω > 0 we have the variation formula (see [Pal6]) For an arbitrary family (Ω t ) t ⊂ V 1 we fix an arbitrary time τ and we time derive at t = τ the decomposition We obtain the required variation formula.
We show now that for any point (g, Ω) ∈ S ω hold the inclusion (1.10). Indeed for any smooth curve (g t , Ω t ) t ⊂ S ω , with (g 0 , Ω 0 ) = (g, Ω) we haveġ * t = −J tJt and thus thanks to the variation formula (16.1). Then the inclusion (1.10) follows from (1.9). We can provide at this point the proof of lemma 17. Proof We remind that by the orthogonal decomposition in corollary 3 any with unique ψ v ∈ Λ Ω,⊥ g,J and A v ∈ H 0,1 g,Ω (T X,J ). Moreover the weighted complex Bochner identity (13.9) implies the equality (16.5) (for any z ∈ we write z = Re z + i Im z) and observe that (16.2) implies the identity We notice now the equalities (g,Ω) if and only if for all u ∈ C ∞ Ω (X, Ê) 0 hold the equalities If we assume (v, V ) ∈ Ì J g,Ω then ∆ Ω g I ψv = 0, which is equivalent to the condition I ψv = 0. We infer The reverse inclusion is obvious. We deduce the identity (1.13). Then the identity (1.14) follows from the inclusion (1.10).
17 The sign of the second variation of the W functional at a Kähler-Ricci soliton point Proposition 1 Let (J, g) be a KRS and let Ω > 0 be the unique smooth volume form such that gJ = Ric J (Ω) and X Ω = 1. Let also (g t , Ω t ) t∈Ê ⊂ M × V 1 be a smooth curve with (g 0 , Ω 0 ) = (g, Ω) and with (ġ 0 ,Ω 0 ) = (v, V ) ∈ J g,Ω (0). Then with the notations of lemma 16 hold the second variation formula is a non-negative self-adjoint real elliptic operator with respect to the L 2 Ω -hermitian product.
Proof STEP I. Let (g, Ω) be a shrinking Ricci soliton point and let (g t , Ω t ) t∈Ê ⊂ M × V 1 be a smooth curve with (g 0 , Ω 0 ) = (g, Ω) and with arbitrary speed (ġ 0 ,Ω 0 ) = (v, V ) ∈ g,Ω . We know from lemma 7 By the considerations in the beginning of section 10 we deduce that in the soliton case h g,Ω = 0 hold the identity for all (v, V ) ∈ g,Ω . Applying the operator ∇ * Ω g to both sides of this identity we infer For any function ϕ ∈ C ∞ Ω (X, Ê) 0 let (v, V ) := T g,Ω (ϕ, 0). Integrating by parts and using the identity (17.2) we infer the equalities Remark 1. We can also compute the integral in the previous expansion via the formula (9.3). Indeed thanks to the identity (8.2). We conclude integrating by parts Remark 2. We set Φ := (∆ Ω g −2Á)ϕ ∈ C ∞ Ω (X, Ê) 0 . Then the previous variation formula rewrites also as the last inequality follows from the variational characterization of the first eigenvalue of ∆ Ω g , which satisfies the inequality λ 1 (∆ Ω g ) > 1. STEP II. Let (v, V ) ∈ g,Ω . Using the L 2 -orthogonal decomposition (15.2) in the proof of lemma 15, we expand the integral therm We observe that X L Ω g θ, ∇ g dϕ g Ω = X ∇ * Ω g L Ω g θ, dϕ g Ω = 0, since ∇ * Ω g L Ω g θ = 0 thanks to the identity (17.1) applied to (θ, 0) ∈ g,Ω . On the other hand formula (9.3) implies We conclude the decomposition identity L Ω g ∇ g dϕ, ∇ g dϕ g + L Ω g θ, θ g Ω.
Then step I implies On the other hand using the decomposition (15.5) of θ and the decomposition formula (12.7) we infer Using the identities (12.8), (12.9) and the property (12.5) we deduce By bi-degree reasons ∇ * Ω g A = 0, which means (gA, 0) ∈ g,Ω . We infer ∇ * Ω g L Ω g A = 0 thanks to the identity (17.1). Then the property (12.2) implies by bi-degree reasons. Integrating by parts further and using the weighted complex Bochner identities (13.8), (13.9) we obtain Using the integration by parts formulas (20.4) and (20.3) in the subsection 20.2 of the appendix, we deduce We observe now that the commutation identity (15.6) implies Completing the square we obtain Using the equation (15.3) we infer We observe now that the operator B Ω g,J is L 2 Ω -anti-adjoint. This implies in particular thanks to the commutation identity (15.6). Thus thanks to the second equation in (S). Using again the second equation in (S) we expand the therm thanks to the expression (15.9). We observe further that the formula (17.4) hold thanks to the commutation identity (15.6). We conclude which implies the required formula for the variations (v, V ) ∈ J g,Ω (0). STEP III. We compute now the stability integral involving A. The trivial identity combined with the formula (14.9) implies since ∂ TX,J A = 0. Integrating by parts we deduce We show now the second variation formula corresponding to the particular case (v, V ) ∈ J g,Ω [0]. With this assumption hold the relation ϕ = −τ . Thus we rearrange the expression Proof Let E λ k (A) ⊂ H be the eigenspace of A corresponding to an eigenvalue λ k ∈ Ê 0 . Then the identity [A, B] = 0 implies that the restriction B : E λ k (A) −→ E λ k (A) is well defined and represents a non-negative selfadjoint operator. We deduce by the spectral theorem in finite dimensions the existence of an orthonormal basis (e k,l ) l∈I k ⊂ E λ k (A) such that Be k,l = µ k,l e k,l , with µ k,l ∈ Ê 0 . Moreover Ae k,l = λ k e k,l . We consider a strictly monotone increasing parametrization (λ k ) k . Then any u ∈ H writes as u = k 0 l∈I k c k,l e k,l , c k,l ∈ . In particular for u ∈ C ∞ (X, ) 0 hold the expressions The inequality in the general case u ∈ D follows from the density of the smoth functions in the graph topology of A. In order to see thatĀ 0 we observe the trivial equalities with v := u. In order to show its self-adjointness we observe also the trivial equalities We deduce the following corollary of proposition 1.

Corollary 4 In the setting of proposition 1 assume
with equality if and only if (v, V ) = (0, 0).
We notice indeed that the equality hold if and only if ϕ = 0.

The Kähler-Einstein case
In the Kähler-Einstein case the complex operator ∆ Ω g,J reduces to the real operator ∆ Ω g . Let and let Λ ⊥ g ⊂ C ∞ (X, Ê) 0 be its L 2 -orthogonal with respect to the measure dV g .
Remark 3 We consider the particular case of a smooth curve (g t , Ω t ) t ⊂ S ω with g 0 Kähler-Einstein metric. Time deriving twice the expression thanks to the Kähler-Einstein condition. Then a trivial change of variables allows to deduce our previous second variation formula in the particular case 17.2 The case of variations in the direction Ì J g,Ω Proposition 2 Let (J, g) be a KRS and let Ω > 0 be the unique smooth volume form such that gJ = Ric J (Ω) and X Ω = 1. Let also (g t , Ω t ) t∈Ê ⊂ M × V 1 be a smooth curve with (g 0 , Ω 0 ) = (g, Ω) and with (ġ 0 ,Ω 0 ) = (v, V ) ∈ Ì J g,Ω . Then with respect to the decomposition v * g = ∂ TX,J ∇ g,J ψ + A, with unique ψ ∈ Λ Ω,⊥ g,J and A ∈ H 0,1 g,Ω (T X,J ), hold the second variation formula is a non-negative self-adjoint real elliptic operator with respect to the L 2 Ω -hermitian product. Moreover if (v, V ) ∈ J g,Ω [0] then the previous formula writes as Step I. Reconsidering a computation in the poof of step II of the proposition 1 we have that for all variations v ∈ J g,[0] hold the identity Using the identity (12.9) and the property (12.5) we deduce Using the identity (17.3), integrating by parts further and using the weighted complex Bochner identity (13.9) we obtain Using the integration by parts formula (20.3) in the subsection 20.2 of the appendix we infer . STEP II. We show first the variation formula in the case (v, V ) ∈ J g,Ω [0] since the proof is simpler than in the general one. Using the expression (16.3) for the space J g,Ω [0] inside the identity (17.11) we deduce the equalities Let write ψ = ψ 1 + iψ 2 , with ψ j real valued functions. Then the condition in the expression (16.3) rewrites as We use now the condition (17.12) in the formula Using again the condition (17.12), we expand the integral thanks to the identities (15.6) and (17.13). We infer using the formula (17.4) Using again the condition (17.12) and the commutation identity (15.6) we expand the integral thanks to the fact that the operator B Ω g,J is L 2 Ω -anti-adjoint. Using again this fact and the condition (17.13) we deduce and thus X 4 |V * Ω | 2 Ω = X P Ω g,J ψ 1 · ψ 1 − P Ω g,J ψ 2 · ψ 2 Ω.
We infer the second variation formula The conclusion follows from the computation in the beginning of step III in the proof of the proposition 1. STEP III. We show now the second variation formula in the more general case of variations (v, V ) ∈ Ì J g,Ω . We observe first that the general expression of ∇ 2 G W(g, Ω) obtained at the end of the proof of lemma 7 implies that over a shrinking-Ricci-Soliton point hold the variation formula for arbitrary directions (v, V ) ∈ T M×V1 . Using now the fact that in the case (v, V ) ∈ Ì J g,Ω hold the expressions R ψ = −2V * Ω , (we use here the definitions (16.4), (16.5)) and (16.7) we obtain for all (v, V ) ∈ Ì J g,Ω . Thanks to the commutation identity (15.6) we can rewrite the identity (17.11) as Adding and subtracting 2ψ to the factor ∆ Ω g,J ψ and respectively 2ψ to the factor ∆ Ω g,J ψ, we infer We deduce the equalities Using the expression The fact that the operator B Ω g,J is L 2 Ω -anti-adjoint combined with the commutation identity (15.6) implies that B Ω g,J (∆ Ω g − 2Á) is also L 2 Ω -anti-adjoint. We deduce in particular the identity and thus the equality Using the fact that the operator B Ω g,J is L 2 Ω -anti-adjoint and the commutation identity (15.6) we can simplify in order to obtain the required variation formula.
18 Positivity of the metric G g,Ω over the space Ì J g,Ω Lemma 20 For any (g, Ω) ∈ S ω the restriction of the symmetric form G g,Ω to the vector space Ì J g,Ω , with J := g −1 ω, is positive definite.
Proof Let (u, U ) , (v, V ) ∈ Ì J g,Ω . Using the expression (16.6) for the space with unique ϕ, ψ ∈ Λ Ω,⊥ g,J and A, B ∈ H 0,1 g,Ω (T X,J ). We decompose now the term Integrating by parts and using the weighted complex Bochner formula (13.9) we transform the integral Using the integration by parts formula (20.4) in the subsection 20.2 of the appendix we deduce Adding and subtracting 2ψ to the factor ∆ Ω g,J ψ and respectively 2ψ to the factor ∆ Ω g,J ψ, we infer We deduce the general formula In particular with equality if and only if ϕ = 0 and A = 0, i.e. (u, U ) = (0, 0), thanks to the variational characterization of the first eigenvalue λ 1 ∆ Ω g,J 2 of the elliptic operator ∆ Ω g,J .
18.1 Double splitting of the space Ì J g,Ω Let H k , with H 0 = L 2 , be a Sobolev space of sections over X. For any subset S of smooth sections over X we denote with H k S its closure with respect to the H k -topology. The pseudo-Riemannian metric G g,Ω is obviously continuous with respect to the L 2 -topology. At the moment we are unable to say if the topology induced by G g,Ω over L 2 Ì J g,Ω is equivalent with the L 2 -topology. Nevertheless we can show the following basic decomposition result Corollary 6 For any (g, Ω) ∈ S ω hold the decomposition identity and let Λ Ω,⊥ g,Ê ⊂ L 2 Ω (X, Ê) 0 be its L 2 -orthogonal with respect to the measure Ω.
Then corollary 5 and its proof show that the map is an isomorphism. We notice also that the expression of the metric G g,Ω obtained at the end of the proof of lemma 20 hold true for arbitrary functions Φ and Ψ. So we put (u, U ) := χ (ϕ) and Φ = Ψ = −2iϕ in this formula. Using the fact that the operator B Ω g,J is L 2 Ω -anti-adjoint and the expression (with equality if and only if ϕ = 0). We remind now that the proof of the weighted Bochner formula (13.9) shows the identity Thus the operator is elliptic. This implies (see for example [Ebi]) that the image is closed in the L 2 -topology. We infer that the map is a topological isomorphism. We deduce that the extension in the sense of distributions (g,Ω) , of the map χ is also a topological isomorphism and a (γ g,Ω , G g,Ω )-isometry. The fact that the map is a topological isomorphism provides the estimate Then the Lax-Milgram theorem implies that the map is a topological isomorphism. (The sign * here denotes the topological dual). We infer that the restricted map is also a topological isomorphism thanks to the fact that the extended map χ is a (γ g,Ω , G g,Ω )-isometry. Applying the elementary lemma 21 below to the spaces E := L 2 X, S 2 T * X ⊕ L 2 Ω (X, Ê) 0 and V := L 2 T [g,Ω] ω ,(g,Ω) we deduce the G-orthogonal decomposition (g,Ω) , and thus Then the conclusion follows from the identity (1.13).
Lemma 21 Let E be a real Banach space, E * its topological dual and G : E×E −→ Ê be a topologically non degenerate bilinear form, i.e. G : E −→ E * is an isomorphism. If there exists a closed subspace V ⊂ E such that the restriction Proof Let j : V ֒−→E be the canonical inclusion and notice the trivial identity By assumption for any element e ∈ E there exists a unique v ∈ V such that j * (e¬G) = j * (v¬G). Thus (e − v)¬G ∈ V ⊥ . By definition the restriction G : V ⊥G −→ V ⊥ provides an isomorphism. We conclude e − v ∈ V ⊥G .
We notice that the condition V ∩ V ⊥G = {0} is equivalent to Ker(G : V −→ V * ) = {0} but in general not sufficient to insure the surjectivity of G : V −→ V * .
18.2 Triple splitting of the space Ì J g,Ω By abuse of notations we will denote by G g,Ω the scalar product over Λ Ω,⊥ g,J ⊂ C ∞ induced by the isomorphism By abuse of notations also we will consider from now on Λ Ω,⊥ g,Ê ⊂ C ∞ . We introduce the vector space and we observe that the expression (16.3) for the space J g,Ω [0] shows that the map η restricts to the isomorphism and observe that the decomposition in corollary 6 implies and thus We deduce that the map η restricts to a G-isometry (g,Ω) .
Furthermore the decomposition in corollary 6 implies also the G-orthogonal decomposition since η extends to an isomorphism We observe now the following elementary lemma.
Lemma 22 Let T : D ⊂ L 2 (X, ) −→ L 2 (X, ) be a closed densely defined L 2 Ω -self-adjoint operator such that [T, T ] = 0. Then The assumption [T, T ] = 0 implies that the restriction T : Ker T −→ Ker T is well defined. This combined with the fact that T is also L 2 Ω -self-adjoint implies that the restriction is also well defined. The inclusion Ker(T T ) ⊇ Ker T +Ker T is obvious. In order to show the reverse inclusion let u ∈ Ker(T T ), i.e. T u ∈ Ker T , and consider the decomposition u = u 1 + u 2 with u 1 ∈ Ker T and u 2 ∈ (Ker T ) ⊥ . Then T u ∈ Ker T if and only if T u 2 ∈ Ker T since T u 1 ∈ Ker T . But T u 2 ∈ Ker T if and only if T u 2 = 0 since T u 2 ∈ (Ker T ) ⊥ . We infer the reverse inclusion. Thus which implies the required conclusion.
We remind that if (g, Ω) ∈ S ω is a KS with J := g −1 ω then which allows to apply the previous lemma to the L 2 Ω -self-adjoint operator P Ω g,J . Thus The finiteness theorem for elliptic operators implies The last inclusion is obvious. The inclusion P Ω g,J C ∞ is also obvious. We conclude Lemma 23 If (g, Ω) ∈ S ω is a KS then hold the identity Proof We notice that for any ϕ ∈ J g,Ω and ψ ∈ Ç J g,Ω hold the identity thanks to (18.3).

Infinitesimal properties of the function H
We observe that lemma 5 implies; (g, Ω) ∈ S ω is a KS if and only if H g,Ω = 0. Furthermore the identity (4.18) rewrites as for all (g, Ω) ∈ S ω . We show now the following fact Lemma 24 If (g, Ω) ∈ S ω is a KS then the linear map with J := g −1 ω, is well defined and represents an isomorphism of real vector spaces.
Proof The identity 2H g,Ω = 2H g,Ω − W (g, Ω) combined with the basic variation formula (1.5) implies for all (v, V ) ∈ g,Ω over a shrinking Ricci soliton point (g, Ω). In our KS set up this last rewrites as and thus (∆ Ω g − 2Á)Λ Ω,⊥ g,J ⊆ Λ Ω,⊥ g,J . (19.4) Then the identity (19.2) shows that the map (19.1) is well defined. We will deduce that it is an isomorphism if we show that the map is an isomorphism. Indeed this is the case. The injectivity of (19.5) follows from the inclusion which hold thanks to the identity (18.1). This inclusion implies also (19.6) and thus We use now the obvious fact that is an isomorphism. Thus for any f ∈ Λ Ω,⊥ g,J ∩ C ∞ Ω (X, Ê) 0 there exists a unique u ∈ Λ Ω,⊥ g,Ê such that We decompose u = u 1 + u 2 , with u 1 ∈ Λ Ω g,J and u 2 ∈ Λ Ω,⊥ g,J . Then the inclusions (19.3) and (19.4) imply the L 2 Ω -orthogonal decomposition We deduce u 1 ∈ Λ Ω g,Ê . But u 2 ∈ Λ Ω,⊥ g,Ê thanks to the inclusion (19.6). We infer u 1 = 0 since u ∈ Λ Ω,⊥ g,Ê . Thus We obtain the surjectivity of the map (19.5) and thus the required conclusion.
Lemma 25 If (g, Ω) ∈ S ω is a KS then hold the identity with J := g −1 ω.
Proof With the notations in the proof of lemma 17, the basic variation formula (1.5) combined with the identities (16.6) and (16.7) implies that for all (v, V ) ∈ Ì J g,Ω over a shrinking Ricci soliton point (g, Ω) hold the equalities since P Ω g,J is a real operator in the KS case. Then lemma 23 implies i.e. the required conclusion.
Proof of the main theorem 1 Proof The inequality in the statement follows immediately from proposition 2. If equality hold then obviously A ∈ H 0,1 g,Ω (T X,J ) 0 and X P Ω g,J Re ψ · Re ψ Ω = 0.
Then the spectral theorem applied to the non-negative L 2 Ω -self-adjoint real elliptic operator P Ω g,J implies P Ω g,J Re ψ = 0. Thus the conclusion follows from lemma 23 and the identity (19.7). In order to show the inclusion (1.20) we observe that if (g t , Ω t ) t∈Ê ⊂ KRS ω is a smooth curve with (g 0 , Ω 0 ) = (g, Ω) and with (ġ 0 ,Ω 0 ) = (v, V ) then hold the identity H gt,Ωt ≡ 0 and thus thanks to the identity (19.7). On the other side if we set W t := W (g t , Ω t ) theṅ W t ≡ 0 and thus thanks to proposition 2 and lemma 23. We conclude the required inclusion.

We show now the following integration by parts formulas
Lemma 26 For any u, v ∈ C ∞ (X, ) hold the integration by parts identity If u ∈ C ∞ (X, Ê) then hold also the integration by parts identity Proof Using the complex decomposition (13.2) and the fact that hermitian product ·, · ω on T * X ⊗ Ê is the sesquilinear extension of the dual of g, we deduce Integrating by parts and taking the conjugate we infer the identity Replacing u with u, v with v in (20.5) we obtain (20.3). In the case u ∈ C ∞ (X, Ê) formula (20.5) implies directly (20.4).
We show now that for all α, β ∈ Λ 1,1 Indeed we consider the local expressions and we set where K = (k 1 , k 2 ), 1 ≤ k 1 < k 2 ≤ n and the same hold for L. Explicitly the coefficients Ψ K,L are given by the expression We conclude the identity 4n(n − 1) Ψ ∧ ω n−2 ω n = 16

Action of the curvature on alternating 2-forms
We observe that as in the symmetric case we can define an action of the curvature operator over alternating 2-forms as follows for any α ∈ Λ 2 T * X . The tensor R g * α is anti-symmetric. In fact let (e k ) k be a g(x)-orthonormal base of T X,x and consider the local expression α * g = A l,k e * k ⊗e l , with A l,k = −A k,l . Then thanks to the symmetry properties of the curvature form. We observe also that the previous computation shows the identity (R g * α)(ξ, η) = −R g (ξ, e k , η, α * g e k ) = −g(R g (ξ, e k )α * g e k , η) On the other hand using the algebraic Bianchi identity we obtain the equalities and thus the formula We assume further that (X, J, g) is Kähler and α is J-anti-invariant. In this case α * g = A is J-anti-linear and so is the endomorphism R g (ξ, η)α * g . We deduce (20.9) thanks to the identity (20.7).

A
Lemma 28 Let (X, J) be a Fano manifold and let g be a J-invariant Kähler metric such that ω := gJ ∈ 2πc 1 (X, [J]). Let also Ω > 0 be the unique smooth volume form with X Ω = 1 such that Ric J (Ω) = ω. A) Then the map is well defined and it represents an isomorphism of complex lie algebras.
B) The first eigenvalue λ 1 (∆ Ω g,J ) of the operator ∆ Ω g,J satisfies the estimate λ 1 (∆ Ω g,J ) 2, with equality in the case H 0 (X, T X,J ) = 0. C) If we set Kill g := Lie(Isom 0 g ) then the map is well defined and it represents an isomorphism of real vector spaces. D) The hermitian form over Ker(∆ Ω g,J − 2Á) is non-negative and let (µ j ) N j=0 ⊂ Ê 0 , µ 0 = 0, be its spectrum with respect to the L 2 Ω -product. If g is a J-invariant Kähler-Ricci soliton then hold the decomposition

Proof
Step A. In this step we show the statement A. The fact that χ is an isomorphism follows from corollary 1. We show now that χ is also a morphism of complex Lie algebras. Let For any ξ ∈ H 0 (X, T X,J ) we denote u ξ := χ −1 (ξ) ∈ Ã − and we decompose For any u, v ∈ Ã − = Ã + we set ξ := ∇ g,J u, η := ∇ g,J v and as in [Gau] we observe the identities since ξ, η are holomorphic. We infer that for some constant C 1 ∈ Ê hold the identities u [ξ,η] 1 On the other hand , since ξ is holomorphic. We infer that for some constant C 2 ∈ Ê hold the identities u [ξ,η] 2 We conclude that for all u, v ∈ Ã − hold the identity which shows that i {·, ·} ω,Ω is a complex Lie algebra product over Ã − and that the map χ is a morphism of complex Lie algebras.
Step B,C. The statements B and C follow from corollary 1 and the remarkable identity (18.1).
Step D. We show now the statement D. We observe first that for all u, v ∈ C ∞ (X, ) hold the identity Indeed thanks to the computations in step A we deduce Integrating by parts we infer thanks to the fact that B Ω g,J is L 2 The KS assumption implies the commutation identity is a well defined non-negative L 2 Ω -self-adjoint operator and let (λ j ) N j=0 ⊂ Ê 0 , λ 0 = 0 be it's spectrum. Notice also that by definition of Ã − this operator coincides with the operator −2iB Ω g,J : Ã − −→ Ã − . Thus u ∈ Ã − is an eigen-vector corresponding to the eigenvalue λ j if and only if u ∈ Ã − satisfies This rewrites as and is equivalent to the equation Notice also that the kernel of (21.3) is given by the identity We deduce the required conclusion with µ j = λ j /2.

Consequences of the Bochner-Kodaira-Nakano formula
The holomorphic and antiholomorphic Hodge Laplacian operators are related by the Bochner-Kodaira-Nakano identity. At the level of T X -valued 1-forms it reduces to the identity where ω * ≡ ω −1 ∈ C ∞ (X, Λ 1,1 J T X ∩ Λ 2 Ê T X ) is the dual element associated to ω.
If in holomorphic coordinates ω writes as The factor 1/6 in front of the last therm on the right hand side of (21.4) is due to the convention v 1 ∧ . . . ∧ v p := σ∈Sp ε σ v σ1 ⊗ · · · ⊗ v σp .
We explicit this last therm. For this purpose we observe first that for any α ∈ Λ 1,1 J T * X ⊗ E hold the identity Tr ω α = − Tr g [α(J·, · )] .
We deduce that the Bochner-Kodaira-Nakano identity rewrites at the level of T X -valued 1-forms as where A ′ J and A ′′ J are respectively the J-linear and J-anti-linear parts of A. Using the Weitzenböck type formula in lemma 3 with Ω = CdV g we infer Using the Bochner-Kodaira-Nakano identity (21.5) we deduce the formulas L Ω g A = 2∆ −J TX,g A + Ric * (g)A − A Ric * (g) − 2R g * A ′ J + ∇ g f ¬∇ g A, and thus the identities for all ξ ∈Ō(T 0,1 X,J0 ) ×d . Notice that this formula defines a priori only an element [α, β] ∈ Alt dŌ (Ō(T 0,1 X,J0 ); C ∞ (T 1,0 X,J0 )).
On the other hand if (21.9) is satisfied then the previous identity is satisfied and thus (21.10) hold true.
For any p ∈ X a coordinate chart of X in a open neighborhood U p × B of (p, 0) is given by a smooth function f : In order to produce such family θ we need to remind a few basic facts about Hodge theory.
For notations convenience we will restrict our considerations to the Fano case even if the result that will follow and its argument hold for a general compact complex manifold. For any polarized Fano manifold (X, J, ω) we define also the sub-set of Ω-divergence free tensors in C J C div J,g := µ ∈ C J | ∂ * g,Ω TX,J µ = 0 .
We denote by H 0 (T X,J ) ⊥ ∩W k (T X,J ) the L 2 g,Ω -orthogonal space to the space of holomorphic vector fields of type (1, 0) inside W k (T X,J ). For any ξ ∈ E (T X,J ) of sufficiently small norm the map e (ξ) : X −→ X defined by e (ξ) x := exp g,x (ξ x ) , is a smooth diffeomorphism. For readers convenience we provide a proof (in the Fano case) of the following fundamental result due to Kuranishi [Kur].
Let Ξ k ⊂ W 0,1 k (T X,J ) be the subset of the elements satisfying this condition. We notice that the map F : W 0,1 k (T X,J ) −→ W 0,1 k (T X,J ), is well defined and continuous thanks to the estimate ∂ * g,Ω TX,J G TX,J [µ, µ] k C 1 [µ, µ] k−1 C 1 C 2 µ 2 k .
We infer that F is also holomorphic since F − Á is a continuous quadratic form. The fact that the differential of F at the origin is the identity implies the existence of an inverse holomorphic map F −1 in a neighborhood B W k ε (0) of the origin. Restricting this to H 0,1 g,Ω (T X,J ) ∩ B W k ε (0) we deduce the existence of a holomorphic map α ∈ H 0,1 g,Ω (T X,J ) ∩ B W k ε (0) −→ µ α ∈ W 0,1 k (T X,J ), such that By construction Im (α −→ µ α ) represents a neighborhood of the origin inside Ξ k . It is clear that µ α is of class C k−n by the Sobolev embedding. We show further that µ α is smooth for a sufficiently small choice of ε. Indeed applying the Hodge Laplacian ∆ Ω,−J TX,g to both sides of the previous identity and using the equalities which rewrites also as ∆ Ω,−J TX,g µ α + 1 2 µ α * ∇ 2 g µ α = 1 2 ∇ g µ α * ∇ g µ α + 1 2 µ α * ∇ g µ α * ∇ g f , where * denotes adequate contraction operators. The fact that the C 0 -norm of µ α can be made arbitrary small for sufficiently small ε implies that the operator ∆ Ω,−J TX,g + 1 2 µ α * ∇ 2 g , is elliptic. Then the smoothness of µ α follows by standard elliptic bootstrapping. We denote by K J,g the zero set of the holomorphic map Then the set {µ α | α ∈ K J,g } covers the set of the solutions of the system (S 2 ) in a neighborhood of the origin. STEP B. We observe first that ∂ * g,Ω For notations simplicity we denote Ψ : (µ, f ) −→ µ f . With these notations the formula (21.14) writes as ∂ J e (tξ) = −∂ J e (tξ) · Ψ (0, e (tξ)) .
Time deriving this identity at t = 0 and using the fact that d dt |t=0 e (tξ) = ξ, Ψ (0, Id X ) = 0 and e (0) = Id X we obtain ∂ TX,J ξ = −D f Ψ (0, Id X ) · ξ, where D f Ψ denotes the partial Frechet derivative of Ψ in the variable f . We observe now that for any ξ ∈ W k (T X,J ) hold the decomposition formula ξ = H TX,J ξ + G TX,J ∂ * g,Ω TX,J ∂ TX,J ξ.
We conclude the identity D ξ R (0, 0) = Á and the existence of the map ϕ −→ ξ ϕ by the implicit function theorem. In local coordinates we can consider the expansion e (ξ) = Id X +ξ + O |ξ| 2 .
Thus ξ ϕ is smooth if ϕ is smooth by elliptic regularity.
This result combined with the decomposition statement in corollary 6 implies the inclusions (1.15) and (1.16) in the introduction of the paper.