Asma Jbilou
email: asmajbilou@yahoo.fr

Complex Hessian Equations on Some Compact Kähler Manifolds

, k = 1 trivially). We solve by the continuity method the corresponding complex elliptic k-th Hessian equation, more difficult to solve than the Calabi-Yau equation (k = m), under the assumption that the holomorphic bisectional curvature of the manifold is non-negative, required here only to derive an a priori eigenvalues pinching.

1 The theorem All manifolds considered in this article are connected. Let (M, J, g, ω) be a compact connected Kähler manifold of complex dimension m ≥ 3. Fix an integer 2 ≤ k ≤ m -1. Let ϕ : M → R be a smooth function and let us consider the (1, 1)-form ω = ω + i∂ ∂ϕ and the associated 2-tensor g defined by g(X, Y ) = ω(X, JY ). Consider the sesquilinear forms h and h on T 1,0 defined by h(U, V ) = g(U, V ) and h(U, V ) = g(U, V ). We denote by λ(g -1 g) the eigenvalues of h with respect to the hermitian form h. By definition, these are the eigenvalues of the unique endomorphism A of T 1,0 satisfying :

h(U, V ) = h(U, AV ) ∀U, V ∈ T 1,0 (1)
Calculations infer that the endomorphism A writes :

A : T 1,0 → T 1,0 U i ∂ i → A j i U i ∂ j = g j l gi lU i ∂ j
A is a self-adjoint/hermitian endomorphism of the hermitian space (T 1,0 , h), therefore λ(g -1 g) ∈ R m . Let us consider the following cone :

Γ k = {λ ∈ R m /∀1 ≤ j ≤ k, σ j (λ) > 0}
, where σ j denotes the j-th elementary symmetric function.

Definition 1.1. ϕ is said to be k-admissible if and only if λ(g -1 g) ∈ Γ k .

In this article, we prove the following theorem :

Theorem 1 (The σ k Equation). Let (M, J, g, ω) be a compact connected Kähler manifold of complex dimension m ≥ 3 with non-negative holomorphic bisectional curvature, and let f : M → R be a function of class C ∞ satisfying M e f ω m = m k M ω m . There exists a unique function ϕ : M → R of class C ∞ such that :

1. M ϕ ω m = 0 2. ωk ∧ ω m-k = e f ( m k ) ω m (E k )
Moreover the solution ϕ is k-admissible.

This result was announced in a note in the Comptes Rendus de l'Académie des Sciences de Paris published online in December 2009 [START_REF] Jbilou | Equations hessiennes complexes sur des variétés kählériennes compactes[END_REF]. The curvature assumption is used, in Section 6.2 only, for an a priory estimate on λ(g -1 g) as in [1, p. 408] and it should be removed (as did Aubin for the case k = m in [START_REF] Th | Equations du type Monge-Ampère sur les variétés kählériennes compactes[END_REF]). For the analogue of (E k ) on C m , the Dirichlet problem is solved in [START_REF] Li | On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian[END_REF][START_REF] Vinacua | Nonlinear elliptic equations and the complex Hessian[END_REF] and a Bedford-Taylor type theory, for weak solutions of the corresponding degenerate equations, is addressed in [START_REF] Blocki | Weak solutions to the complex Hessian equation[END_REF]. Thanks to Julien Keller, we learned of an independent work [START_REF] Hou | Complex Hessian equation on Kähler manifold[END_REF] aiming at the same result as ours, with a different gradient estimate and a similar method to estimate λ(g -1 g), but no proofs given for the C 0 and the C 2 estimates.

Let us notice that the function f appearing in the second member of the equation (E k ) satisfies necessarily the normalisation condition M e f ω m = m k M ω m . Indeed, this results from the following lemma :

Lemma 1.2. M ωk ∧ ω m-k = M ω m
Proof. See [13, p. 44].

Let us write the equation (E k ) differently.

Lemma 1.3. ωk ∧ ω m-k = σ k (λ(g -1 g)) ( m k ) ω m
Proof. Let P ∈ M . It suffices to prove the equality at P in a g-normal g-adapted chart z centered at P . In such a chart g i j (0) = δ ij and gi j (0) = δ ij λ i (0), so at z = 0, ω = idz a ∧ dz ā and ω = iλ a (0) dz a ∧ dz ā.

Complex Hessian Equations on Some Compact Kähler Manifold 3

Thus :

ωk ∧ ω m-k = a iλ a (
(m -k)! λ a1 (0) • • • λ a k (0) ω m m! ωk ∧ ω m-k = (m -k)! m! k! σ k (λ 1 (0), ..., λ m (0)) ω m = σ k (λ(g -1 g)) m k ω m
Consequently, the equation (E k ) writes :

σ k λ(g -1 g) = e f (E k )
Let us remark that E m corresponds to the Calabi-Yau equation det(g) det(g) = e f , when E 1 is just a linear equation in Laplacian form. Since the endomorphism A is hermitian, the spectral theorem provides an horthonormal basis for T 1,0 of eigenvectors e 1 , ..., e m : Ae i = λ i e i , λ = (λ 1 , ..., λ m ) ∈ Γ k . At P ∈ M in a chart z, we have M at ∂1,...,∂m A P = [A i j (z)] 1≤i,j≤m , thus σ k λ(A P ) = σ k λ([A i j (z)] 1≤i,j≤m ) . In addition, A j i = g j l gi l = g j l(g i l + ∂ i lϕ) = δ j i + g j l∂ i lϕ, so the equation writes locally :

σ k λ([δ j i + g j l∂ i lϕ] 1≤i,j≤m ) = e f (E k )
Let us notice that a solution of this equation (E k ) is necessarily kadmissible [13, p. 46]. Let us define f k (B) = σ k (λ(B)) and F k (B) = ln σ k (λ(B)) where B = [B j i ] 1≤i,j≤m is a hermitian matrix. The function f k is a polynomial in the variables B j i , specifically : f k (B) = |I|=k B II (sum of the principal minors of order k of the matrix B ). Equivalently the equation (E k ) writes :

F k ([δ j i + g j l∂ i lϕ] 1≤i,j≤m ) = f (E k )
It is a nonlinear elliptic second order PDE of complex Monge-Ampère type. We prove the existence of a k-admissible solution by the continuity method.

2 Derivatives and concavity of F k

Calculation of the derivatives at a diagonal matrix

The first derivatives of the symmetric polynomial σ k are given by :

∀1 ≤ i ≤ m, ∂σ k ∂λi (λ) = σ k-1,i (λ) where σ k-1,i (λ) := σ k-1 | λi=0 . For 1 ≤ i = j ≤ m, let us denote σ k-2,ij (λ) := σ k-2 | λi=λj =0 and σ k-2,ii (λ) = 0.
The second derivatives of the polynomial σ k are given by :

∂ 2 σ k ∂λi∂λj (λ) = σ k-2,ij (λ).
We calculate the derivatives of the function f k : H m (C) → R, where H m (C) denotes the set of hermitian matrices, at diagonal matrices using the formula :

f k (B) = 1≤i1<...<i k ≤m σ∈S k ε(σ) B i σ(1) i1 ... B i σ(k) i k = 1 k! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k B j1 i1 ... B j k i k where (2) ε i1...i k j1...j k =      1 if i 1 , .
.., i k distincts and j 1 , ..., j k even permutation of i 1 , ..., i k -1 if i 1 , ..., i k distincts and j 1 , ..., j k odd permutation of i 1 , ..., i k 0 else These derivatives are given by [13, p. 48] :

∂f k ∂B j i diag(b 1 , ..., b m ) = 0 if i = j σ k-1,i (b 1 , ..., b m ) if i = j (3) if i = j ∂ 2 f k ∂B j j ∂B i i diag(b 1 , ..., b m ) = σ k-2,ij (b 1 , ..., b m ) ∂ 2 f k ∂B i j ∂B j i diag(b 1 , ..., b m ) = -σ k-2,ij (b 1 , ..., b m ) (4)
and all the other second derivatives of

f k at diag(b 1 , ..., b m ) vanish.
Consequently, the derivatives of the function

F k = ln f k : λ -1 (Γ k ) ⊂ H m (C) → R at diagonal matrices diag(λ 1 , ..., λ m ) with λ = (λ 1 , ..., λ m ) ∈ Γ k , where λ -1 (Γ k ) = {B ∈ H m (C)/λ(B) ∈ Γ k },
are given by :

∂F k ∂B j i diag(λ 1 , ..., λ m ) = 0 if i = j σ k-1,i (λ) σ k (λ) if i = j (5) if i = j ∂ 2 F k ∂B i j ∂B j i diag(λ 1 , ..., λ m ) = - σ k-2,ij (λ) σ k (λ) ∂ 2 F k ∂B j j ∂B i i diag(λ 1 , ..., λ m ) = σ k-2,ij (λ) σ k (λ) - σ k-1,i (λ)σ k-1,j (λ) σ k (λ) 2 ∂ 2 F k ∂B i i ∂B i i diag(λ 1 , ..., λ m ) = - σ k-1,i (λ) 2 σ k (λ) 2 (6)
and all the other second derivatives of F k at diag(λ 1 , ..., λ m ) vanish.

The invariance of F k and of its first and second differentials

The function

F k : λ -1 (Γ k ) → R is invariant under unitary similitudes : ∀B ∈ λ -1 (Γ k ), ∀U ∈ U m (C), F k (B) = F k ( t U BU ) (7)
Differentiating the previous invariance formula [START_REF] Caffarelli | The Dirichlet problem for nonlinear second order elliptic equations, III : Functions of the eigenvalues of the Hessian[END_REF], we show that the first and second differentials of F k are also invariant under unitary similitudes :

∀B ∈ λ -1 (Γ k ), ∀ζ ∈ H m (C), ∀U ∈ U m (C), (dF k ) B .ζ = (dF k )t U BU .( t U ζU ) (8) et ∀B ∈ λ -1 (Γ k ), ∀ζ ∈ H m (C), ∀Θ ∈ H m (C), ∀U ∈ U m (C), (d 2 F k ) B .(ζ, Θ) = (d 2 F k )t U BU .( t U ζU, t U ΘU ) (9)
These invariance formulas allow to come down to the diagonal case, when it is useful.

Concavity of F k

We prove in [START_REF] Jbilou | Equations hessiennes complexes sur des variétés kählériennes compactes[END_REF] the concavity of the functions u•λ and more generally

u • λ B when u ∈ Γ 0 (R m )
: λ -1 (Γ k ) → R, B → F k (B) = ln σ k λ(B)
is concave (this holds for all k ∈ {1, ..., m}).

Proof. The function F k is of class C 2 , so its concavity is equivalent to the following inequality :

∀B ∈ λ -1 (Γ k ), ∀ζ ∈ H m (C) m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i (B) ζ j i ζ s r ≤ 0 (10) Let B ∈ λ -1 (Γ k ), ζ ∈ H m (C) and U ∈ U m (C) such that t U BU = diag(λ 1 , ..., λ m ). We have λ = (λ 1 , ..., λ m ) ∈ Γ k . Let us denote ζ = t U ζU ∈ H m (C). S : = m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i (B) ζ j i ζ s r = (d 2 F k ) B .(ζ, ζ) so by the invariance formula (9) = (d 2 F k )t U BU .( t U ζU, t U ζU ) = m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i (diag(λ 1 , ..., λ m )) ζj i ζs r = m i =j=1 - σ k-2,ij (λ) σ k (λ) ζj i ζi j = ζj i + m i =j=1 σ k-2,ij (λ) σ k (λ) - σ k-1,i (λ)σ k-1,j (λ) σ k (λ) 2 =:cij ζi i ζj j + m i=1 - (σ k-1,i (λ)) 2 σ k (λ) 2 ( ζi i ) 2 = m i,j=1 - σ k-2,ij (λ) σ k (λ) | ζj i | 2 + m i,j=1 c ij ζi i ζj j (11)
But

c ij = ∂ 2 (lnσ k ) ∂λi∂λj (λ), and ζi i ∈ R, so m i,j=1 c ij ζi i ζj j ≤ 0 by concavity of ln σ k at λ ∈ Γ k [7, p. 269]. In addition, σ k-2,ij (λ) > 0 since λ ∈ Γ k , consequently m i,j=1 - σ k-2,ij (λ) σ k (λ)
| ζj i | 2 ≤ 0, which shows that S ≤ 0 and achieves the proof.

The proof of uniqueness

Let ϕ 0 and ϕ 1 be two smooth k-admissible solutions of the equation

(E k ) such that M ϕ 0 ω m = M ϕ 1 ω m = 0. For all t ∈ [0, 1], let us consider the function ϕ t = t ϕ 1 +(1-t) ϕ 0 = ϕ 0 +t ϕ with ϕ = ϕ 1 -ϕ 0 . Let P ∈ M , and let us denote h P k (t) = f k ([δ j i + g j l(P )∂ i lϕ t (P )]). We have h P k (1) -h P k (0) = 0 which is equivalent to 1 0 h P ′ k (t) dt = 0. But : h P ′ k (t) = m i,j=1 m ℓ=1 ∂f k ∂B ℓ i ([δ j i + g j l(P )∂ i lϕ t (P )]) g ℓ j (P ) =:α t ij (P ) ∂ i j ϕ(P )
Complex Hessian Equations on Some Compact Kähler Manifold 7

Therefore we obtain :

Lϕ(P ) := m i,j=1
a ij (P ) ∂ i j ϕ(P ) = 0 with a ij (P ) =

1 0 α t ij (P ) dt
We show easily that the matrix [a ij (P )] 1≤i,j≤m is hermitian [13, p. 53]. Besides the function ϕ is continuous on the compact manifold M so it assumes its minimum at a point m 0 ∈ M , so that the complex Hessian matrix of ϕ at the point m 0 namely

[∂ i j ϕ(m 0 )] 1≤i,j≤2m is positive-semidefinite. Lemma 3.1. For all t ∈ [0, 1], λ(g -1 gϕt )(m 0 ) ∈ Γ k , namely the func- tions (ϕ t ) t∈[0,1] are k-admissible at m 0 . Proof. Let us denote W := {t ∈ [0, 1]/λ(g -1 gϕt )(m 0 ) ∈ Γ k }.
≤ q ≤ k, we have σ q λ(g -1 gϕt )(m 0 ) -σ q λ(g -1 gϕ0 )(m 0 ) = h m0 q (t) -h m0 q (0) = t 0 h m ′ 0 q (s) ds. Let us prove that h m ′ 0 q (s) ≥ 0 for all s ∈ [0, t[. Fix s ∈ [0, t[, the quantity h m ′ 0 q (
s) is intrinsic so it suffices to prove the assertion in a particular chart at m 0 . Now at m 0 in a g-unitary gϕsadapted chart at m 0 :

h m ′ 0 q (s) = m i,j,ℓ=1 ∂f q ∂B j i ([δ j i + g j l(m 0 )∂ i lϕ s (m 0 )]) g j l(m 0 )∂ i lϕ(m 0 ) = m i=1 ∂σ q ∂λ i λ(g -1 gϕs )(m 0 ) ∂ i īϕ(m 0 ) But λ(g -1 gϕs )(m 0 ) ∈ Γ k ⊂ Γ q since s ∈ [0, t[⊂ W, then ∂σq ∂λi λ(g -1 gϕs )(m 0 ) > 0 for all 1 ≤ i ≤ m. Besides, ∂ i īϕ(m 0 ) ≥ 0 since the matrix [∂ i j ϕ(m 0 )] 1≤i,j≤m
is positive-semidefinite. Therefore, we infer that h m ′ 0 q (s) ≥ 0. Consequently, we obtain that σ q λ(g -1 gϕt )(m 0 ) ≥ σ q λ(g -1 gϕ0 )(m 0 ) > 0 (since ϕ 0 is k-admissible). This holds for all 1 ≤ q ≤ k, we deduce then that λ(g -1 gϕt )(m 0 ) ∈ Γ k which proves that t ∈ W. This is a contradiction, we infer then that W = [0, 1].

We check easily that the hermitian matrix [a ij (m 0 )] 1≤i,j≤m is positive definite [13, p. 54] and deduce then the following lemma since the map P → a ij (P ) =

1 0 m ℓ=1 ∂f k ∂B ℓ i ([δ j i + g j l(P )∂ i lϕ t (P )]) g ℓ j (P ) dt is continuous on a neighbourhood of m 0 : Lemma 3.2.
There exists an open ball B m0 centered at m 0 such that for all P ∈ B m0 the hermitian matrix [a ij (P )] 1≤i,j≤m is positive definite.

Consequently, the operator L is elliptic on the open set B m0 . But the map ϕ is C ∞ , assumes its minimum at m 0 ∈ B m0 and satisfies Lϕ = 0, then by the Hopf maximum principle [START_REF] Hebey | Introduction à l'analyse non linéaire sur les variétés[END_REF], we deduce that ϕ(P ) = ϕ(m 0 ) for all P ∈ B m0 . Let us denote S := {P ∈ M/ϕ(P ) = ϕ(m 0 )}. This set is non-empty and it is a closed set. Let us prove that S is an open set : let m be a point of S, so ϕ(m) = ϕ(m 0 ) then the map ϕ assumes its minimum at the point m. Therefore, by the same proof as for the point m 0 , we infer that there exists an open ball B m centered at m such that ∀P ∈ B m ϕ(P ) = ϕ(m) so ∀P ∈ B m ϕ(P ) = ϕ(m 0 ) then B m ⊂ S, which proves that S is an open set. But the manifold M is connected, then S = M namely ϕ(P ) = ϕ(m 0 ) for all P ∈ M . Besides M ϕω m = 0, therefore we deduce that ϕ ≡ 0 on M namely that ϕ 1 ≡ ϕ 0 on M , which achieves the proof of uniqueness.

The continuity method

Let us consider the one parameter family of equations

(E k,t ), t ∈ [0, 1] F k [ϕ t ] := F k ([δ j i + g j l∂ i lϕ t ] 1≤i,j≤m ) = tf + ln m k M ω m M e tf ω m At (E k,t ) The function ϕ 0 ≡ 0 is a k-admissible solution of (E k,0 ) : σ k λ([δ j i + g j l∂ i lϕ 0 ] 1≤i,j≤m ) = m
k , and satisfies M ϕ 0 ω m = 0. For t = 1, A 1 = 1 so (E k,1 ) corresponds to the equation (E k ). Let us fix l ∈ N, l ≥ 5 and 0 < α < 1, and let us consider the non-empty set (containing 0) :

T l,α := {t ∈ [0, 1]/(E k,t ) have a k-admissible solution ϕ ∈ C l,α (M ) such that M ϕω m = 0} (12)
The aim is to prove that 1 ∈ T l,α . For this we prove, using the connectedness of [0, 1], that T l,α = [0, 1].

4.1 T l,α is an open set of [0, 1]
This arises from the local inverse mapping theorem and from solving a linear problem. Let us consider the following sets :

Sl,α := {ϕ ∈ C l,α (M ), M ϕω m = 0}
S l,α := {ϕ ∈ Sl,α , k-admissible for g} Sl,α is a vector space and S l,α is an open set of Sl,α . Using this notations, the set T l,α writes T l,α := {t ∈ [0, 1]/∃ϕ ∈ S l,α solution of (E k,t )}.

Lemma 4.1. The operator

F k : S l,α → C l-2,α (M ), ϕ → F k [ϕ] = F k [δ j i + g j l∂ i lϕ] 1≤i
,j≤m , is differentiable and its differential at a point ϕ ∈ S l,α , dF kϕ ∈ L Sl,α , C l-2,α (M ) , is equal to :

dF kϕ .ψ = m i,j=1 ∂F k ∂B j i [δ j i + g j l∂ i lϕ] g j l ∂ i lψ ∀ψ ∈ Sl,α
Proof. See [13, p. 60].

Proposition 4.2. The nonlinear operator F k is elliptic on S l,α .

Proof. Let us fix a function ϕ ∈ S l,α , and check that the nonlinear operator F k is elliptic for this function ϕ. This goes back to show that the linearization at ϕ of the nonlinear operator F k is elliptic. By the lemma 4.1, this linearization is the following linear operator :

dF kϕ .v = m i,o=1 m j=1 ∂F k ∂B j i [δ j i + g j ō ∂ iō ϕ] 1≤i,j≤m × g j ō ∂ iō v (13)
In order to prove that this linear operator is elliptic, it suffices to check the ellipticity in a particular chart, for example at the center of a gnormal gϕ -adapted chart. At the center of such a chart :

dF kϕ .v = m i,o=1 ∂F k ∂B o i diagλ(g -1 g) ∂ iō v = m i=1 σ k-1,i λ(g -1 g) σ k λ(g -1 g) ∂ i īv (14)
But for all i ∈ {1, ..., m} we have

σ k-1,i λ(g -1 g) σ k λ(g -1 g)
> 0 on M since λ(g -1 g) ∈ Γ k , which proves that the linearization is elliptic and achieves the proof.

Let us denote F k the operator :

F k [ϕ] := f k [δ j i + g j l∂ i lϕ] 1≤i,j≤m (15)
As F k , the operator F k : S l,α → C l-2,α (M ) is differentiable and elliptic on S l,α of differential :

dF kϕ .ψ = m i,j=1 ∂f k ∂B j i [δ j i + g j l∂ i lϕ] g j l ∂ i lψ ∀ψ ∈ Sl,α
Let us denote a ϕ the matrix [δ j i + g j l∂ i lϕ] 1≤i,j≤m and calculate this linearization in a different way, by using the expression (2) of f k :

F k [ϕ] = f k (a ϕ ) = 1 k! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k (a ϕ ) j1 i1 ... (a ϕ ) j k i k (16)
Complex Hessian Equations on Some Compact Kähler Manifold 10

Thus :

dF kϕ .v = d dt F k [ϕ + tv] |t=0 = d dt 1 k! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k (a ϕ+tv ) j1 i1 ... (a ϕ+tv ) j k i k |t=0 = 1 k! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k g j1 s∂ i1 sv (a ϕ ) j2 i2 ... (a ϕ ) j k i k + 1 k! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k (a ϕ ) j1 i1 g j2 s∂ i2 sv ... (a ϕ ) j k i k + ... + 1 k! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k (a ϕ ) j1 i1 ... (a ϕ ) j k-1 i k-1 g j k s∂ i k sv = 1 (k -1)! 1≤i1,...,i k ,j1,...,j k ≤m ε i1...i k j1...j k (a ϕ ) j1 i1 ... (a ϕ ) j k-1 i k-1 g j k s∂ i k sv by symmetry = m i,j=1 1 (k -1)! 1≤i1,...,i k-1 ,j1,...,j k-1 ≤m ε i1...i k-1 i j1...j k-1 j (a ϕ ) j1 i1 ... (a ϕ ) j k-1 i k-1 =:C i j (aϕ) ∇ j i v (17)
We infer then the following proposition :

Proposition 4.3. The linearization dF k of the operator F k is of divergence type :

dF kϕ = ∇ i C i j (a ϕ ) ∇ j
Proof. By [START_REF] Vinacua | Nonlinear elliptic equations and the complex Hessian[END_REF] we have :

dF kϕ .v = m i,j=1 C i j (a ϕ ) ∇ j i v = m i=1 ∇ i m j=1 C i j (a ϕ ) ∇ j v - m j=1 m i=1 ∇ i (C i j (a ϕ )) ∇ j v (18)
Moreover :

m i=1 ∇ i (C i j (a ϕ )) = 1 (k -2)! m i=1 1≤i1,...,i k-1 ,j1,...,j k-1 ≤m ε i1...i k-1 i j1...j k-1 j (a ϕ ) j1 i1 ... (a ϕ ) j k-2 i k-2 ∇ i (a ϕ ) j k-1 i k-1 (19)
Complex Hessian Equations on Some Compact Kähler Manifold 11

But ∇ i (a ϕ ) j k-1 i k-1 = ∇ i δ j k-1 i k-1 + ∇ j k-1 i k-1 ϕ = ∇ j k-1
ii k-1 ϕ, then :

m i=1 ∇ i (C i j (a ϕ )) = 1 (k -2)! m i=1 1≤i1,...,i k-1 ,j1,...,j k-1 ≤m ε i1...i k-1 i j1...j k-1 j (a ϕ ) j1 i1 ... (a ϕ ) j k-2 i k-2 ∇ j k-1 ii k-1 ϕ (20)
Besides, the quantity ∇

j k-1 ii k-1 ϕ is symmetric in i, i k-1 (indeed, ∇ j k-1 ii k-1 ϕ- ∇ j k-1 i k-1 i ϕ = R j k-1 sii k-1 ∇ s ϕ and R j k-1 sii k-1 = 0 since g is Kähler), and ε i1...i k-1 i j1...j k-1 j is antisymmetric in i, i k-1 , it follows then that m i=1 ∇ i (C i j (a ϕ )) = 0, consequently dF kϕ .v = m i=1 ∇ i m j=1 C i j (a ϕ ) ∇ j v .
From the proposition 4.3, we infer easily [13, p. 62] the following corollary :

Corollary 4.4. The map F : S l,α → Sl-2,α , ϕ -→ F (ϕ) = F k [ϕ] - m k
is well defined and differentiable of differential

dF ϕ = dF kϕ = ∇ i C i j (a ϕ ) ∇ j ∈ L( Sl,α , Sl-2,α ).
Now, let t 0 ∈ T l,α and let ϕ 0 ∈ S l,α be a solution of the corresponding equation

(E k,t0 ) : F (ϕ 0 ) = e t0f A t0 -m k . Lemma 4.5. dF ϕ0 : Sl,α -→ Sl-2,α is an isomorphism. Proof. Let ψ ∈ C l-2,α (M ) with M ψv g = 0.
Let us consider the equation :

∇ i C i j (a ϕ0 ) ∇ j u = ψ (21)
We have

C i j (a ϕ0 ) ∈ C l-2,α (M ) and the matrix [C i j (a ϕ0 )] 1≤i,j≤m = ∂f k ∂B j i [δ j i + g j l∂ i lϕ 0 ] 1≤i,j≤m
is positive definite (since F k is elliptic at ϕ 0 ), then by the theorem 4.7 of [3, p. 104] on the operators of divergence type, we deduce that there exists a unique function u ∈ C l,α (M ) satisfying M uv g = 0 which is solution of (21), and then solution of : dF ϕ0 u = ψ. Thus, the linear continuous map dF ϕ0 : Sl,α -→ Sl-2,α is bijective and its inverse is continuous by the open map theorem, which achieves the proof.

We deduce then by the local inverse mapping theorem that there exists an open set U of S l,α containing ϕ 0 and an open set

V of Sl-2,α containing F (ϕ 0 ) such that F : U → V is a diffeomorphism. Now,
then e tf A t -m k -e t0f A t0 -m k C l-2,α (M )
is small enough so that e tf A t -m k ∈ V , thus there exists ϕ ∈ U ⊂ S l,α such that F (ϕ) = e tf A t -m k and consequently there exists ϕ ∈ C l,α (M ) of vanishing integral for g which is solution of (E k,t ). Hence t ∈ T l,α . We conclude therefore that T l,α is an open set of [0, 1].

4.2 T l,α is a closed set of [0, 1] : the scheme of the proof This section is based on a priori estimates. Finding this estimates is the most difficult step of the proof. Let (t s ) s∈N be a sequence of elements of T l,α that converges to τ ∈ [0, 1], and let (ϕ ts ) s∈N be the corresponding sequence of functions : ϕ ts is C l,α , k-admissible, has a vanishing integral and is a solution of

F k ([δ j i + g j l∂ i lϕ ts ] 1≤i,j≤m ) = t s f + ln(A ts ) (E k,ts )
Let us prove that τ ∈ T l,α . Here is the scheme of the proof :

1. Reduction to a C 2,β (M ) estimate : if (ϕ ts ) s∈N is bounded in a C 2,β (M ) with 0 < β < 1, the inclusion C 2,β (M ) ⊂ C 2 (M, R) be- ing compact, we deduce that after extraction (ϕ ts ) s∈N converges in C 2 (M, R) to ϕ τ ∈ C 2 (M, R).
We show by tending to the limit that ϕ τ is a solution of (E k,τ ) (it is then necessarily k-admissible), of vanishing integral for g. We check finally by a nonlinear regularity theorem [5, p. 467] that ϕ τ ∈ C ∞ (M, R), which allows us to deduce that τ ∈ T l,α (see [13, p. 64-67] for details).

2. We show that (ϕ ts ) s∈N is bounded in C 0 (M, R) : first of all we prove a positivity lemma 5.4 for the equation (E k,t ), inspired by the one's of [8, p. 843] (for k = m), but in a very different way, required since the k-positivity of ωts is weaker with k < m, using a polarization method of [6, p. 1740] (cf. 5.2) and a Gårding inequality 5.3; we infer then from this lemma a fundamental inequality 5.5 as the proposition 7.18 of [3, p. 262]. We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. We deal with this C 0 estimate in section 5.

3. We establish the key point of the proof namely a C 2 a priori estimate (section 6).

4. With the uniform ellipticity at hand (consequence of the previous step), we obtain the needed C 2,β (M ) estimate by the Evans-Trudinger theory (section 7).

5 The C 0 a priori estimate

The positivity lemma

Our first three lemmas are based on the ideas of [6, section 2].

Lemma 5.1. Let π be a real (1 -1)-form, it then writes π = ip a b dz a ∧ dz b, with p a b = p(∂ a , ∂b) where p is the symmetric tensor p(U, V ) = π(U, JV ); hence :

∀ℓ ≤ m π ℓ ∧ ω m-ℓ = ℓ!(m -ℓ)! m! σ ℓ λ[g -1 p] ω m
Proof. The same proof as the lemma 1.3.

We consider for 1 ≤ ℓ ≤ m the map

f ℓ = σ ℓ • λ : H m → R
where H m denotes the R-vector space of hermitian square matrices of size m. f ℓ is a real polynomial of degree ℓ and in m 2 real variables. Moreover, it is I hyperbolic (cf. [START_REF] Gårding | An inequality for hyperbolic polynomials[END_REF] for the proof) and it satisfies

f ℓ (I) = σ ℓ (1, • • • , 1) = m ℓ > 0.
Let fℓ be the totally polarized form of f ℓ . This polarized form fℓ :

H m × • • • × H m ℓ times
→ R is characterized by the following properties :

• fℓ is ℓ-linear.

• fℓ is symmetric.

• ∀B ∈ H m fℓ (B, • • • , B) = f ℓ (B).
Using these notations, we infer from lemma 5.1 that at the center of a g-unitary chart (this guarantees that the matrix g -1 p is hermitian), we have :

π ℓ ∧ ω m-ℓ = ℓ!(m -ℓ)! m! f ℓ g -1 p ω m (22)
By transition to the polarized form in this equality we obtain the following lemma :

Lemma 5.2. Let 1 ≤ ℓ ≤ m and π 1 , ..., π ℓ be real (1 -1)-forms. These forms write π α = i(p α ) a b dz a ∧ dz b, with (p α ) a b = p α (∂ a , ∂b) where p α is the symmetric tensor p α (U, V ) = π α (U, JV ). Then, at the center of a g-unitary chart we have :

π 1 ∧ • • • ∧ π ℓ ∧ ω m-ℓ = ℓ!(m -ℓ)! m! fℓ g -1 p 1 , ..., g -1 p ℓ ω m
Proof. See [13, p. 71].

The theorem 5 of Gårding [START_REF] Gårding | An inequality for hyperbolic polynomials[END_REF] applies to f ℓ with 2 ≤ ℓ ≤ m :

Lemma 5.3 (The Gårding Inequality for f ℓ ). Let 2 ≤ ℓ ≤ m, for all y 1 , ..., y ℓ ∈ Γ(f ℓ , I), fℓ (y 1 , ..., y ℓ ) ≥ f ℓ (y 1 ) 1 ℓ • • • f ℓ (y ℓ ) 1 ℓ
Let us recall that Γ(f ℓ , I) is the connected component of {y ∈ H m / f ℓ (y) > 0} containing I. The same proof as [4, p. 129-130] implies that :

Γ(f ℓ , I) = y ∈ H m / ∀1 ≤ i ≤ ℓ f i (y) > 0 = y ∈ H m / λ(y) ∈ Γ ℓ = λ -1 (Γ ℓ ) (23)
Note that the Gårding inequality (lemma 5.3) holds for

Γ(f ℓ , I) = y ∈ H m / ∀1 ≤ i ≤ ℓ f i (y) ≥ 0 .
Let us now apply the previous lemmas in order to prove the following positivity lemma inspired by the one's of [8, p. 843] (for k = m); let us emphasize that the proof is very different since the k-positivity is weaker :

Lemma 5.4 (Positivity Lemma). Let α be a real 1-form on M and j ∈ {1, ..., k -1} then the function

f : M → R defined by f ω m = t Jα ∧ α ∧ ω m-1-j ∧ ωj is nonnegative. Proof. Let 1 ≤ j ≤ k -1, then 2 ≤ ℓ = j + 1 ≤ k. Let α be a real 1-form, it then writes α = α a dz a + α a dz ā. Let π 1 = t Jα ∧ α, hence π 1 (∂ a , ∂b) = α(J∂ a ) α(∂b) -α(J∂b) α(∂ a ) = i α a α b -(-i) α b α a = 2i α a α b . Similarly, we prove that π 1 (∂ a , ∂ b ) = π 1 (∂ ā, ∂b) = 0, conse- quently π 1 = i 2α a α b =:p a b dz a ∧ dz b. Besides, set π 2 = ... = π j+1 = ω = i ga b dz a ∧ dz b. Now, let
x ∈ M and φ be a g-unitary chart centered at x. Using the lemma 5.2, we infer that at x in the chart φ :

t Jα ∧ α ∧ ωj ∧ ω m-(j+1) = π 1 ∧ ... ∧ π j+1 ∧ ω m-(j+1) = (m -j -1)!(j + 1)! m! fj+1 (g -1 p, g -1 g, ..., g -1 g) ω m (24)
But, at x, g -1 g = g ∈ Γ(f j+1 , I) and g -1 p = p ∈ Γ(f j+1 , I). Indeed, λ(g -1 g) ∈ Γ k since ϕ is k-admissible and Γ k ⊂ Γ j+1 . Moreover, the hermitian matrix [2α a α b ] 1≤a,b≤m is positive-semidefinite since for all ξ ∈ C m , we have m a,b=1 2α a α b ξ a ξ b = 2 | m a=1 α a ξ a | 2 ≥ 0, we then deduce that for all 1 ≤ i ≤ j + 1, we have at x, f i (g -1 p) = σ i (λ(g -1 p)) ≥ 0. Finally, we infer by the Gårding inequality that at x in the chart φ we have :

fj+1 (g -1 p, g -1 g, ..., g -1 g) ≥ f j+1 (g -1 p) 1 j+1 f j+1 (g -1 g) j j+1 ≥ 0 (25)
which proves the positivity lemma.

The fundamental inequality

The C 0 a priori estimate is based on the following crucial proposition which is a generalization of the proposition 7.18 of [3, p. 262] : Proposition 5.5. Let h(t) be an increasing function of class C 1 defined on R, and let ϕ be a C 2 k-admissible function defined on M , then the following inequality is satisfied :

M m k -f k (g -1 g) h(ϕ) ω m ≥ 1 2m m k M h ′ (ϕ) | ∇ϕ | 2 g ω m Proof. We have the equality M [ m k -f k (g -1 g)] h(ϕ) ω m = m k M h(ϕ)(ω m - ωk ∧ ω m-k ). Besides, since Λ 2 M is commutatif ω m -ωk ∧ ω m-k = (ω -ω) ∧ (ω m-1 + ω m-2 ∧ ω + ... + ω m-k ∧ ωk-1 ) =:Ω , namely ω m -ωk ∧ ω m-k = -1 2 dd c ϕ∧Ω, then M [ m k -f k (g -1 g)] h(ϕ) ω m = -1 2 m k M dd c ϕ∧ (h(ϕ)Ω). But d d c ϕ ∧ h(ϕ)Ω = dd c ϕ ∧ h(ϕ)Ω + (-1) 1 d c ϕ ∧ d(h(ϕ)Ω), and d(h(ϕ)Ω) = h ′ (ϕ)dϕ ∧ Ω + (-1) 0 h(ϕ) dΩ =0 since ω and ω are closed so dd c ϕ ∧ h(ϕ)Ω = d c ϕ ∧ h ′ (ϕ)dϕ ∧ Ω + d(something).
In addition by Stokes' theorem, M d(something) = 0, therefore :

M m k -f k (g -1 g) h(ϕ) ω m = - 1 2 m k M h ′ (ϕ) d c ϕ ∧ dϕ ∧ Ω = 1 2 m k M h ′ (ϕ) (-d c ϕ) ∧ dϕ ∧ ω m-1 T1 + k-1 j=1 M h ′ (ϕ) (-d c ϕ) ∧ dϕ ∧ ω m-1-j ∧ ωj T2 (26)
Let us prove that T 2 ≥ 0 (using the positivity lemma) and that

T 1 = 1 m M h ′ (ϕ) | ∇ϕ | 2 g ω m .
Let us apply the positivity lemma to dϕ : the function

f : M → R defined by f ω m = t Jdϕ ∧ dϕ ∧ ω m-1-j ∧ ωj is nonnegative for all 1 ≤ j ≤ k -1. But t Jdϕ = -d c
t Jdϕ ∧ dϕ ∧ ω m-1 = a1,...,am-1∈{1,••• ,m-1} 2 by 2 = i m | dϕ | 2 g (dz m ∧ dz m) ∧ (dz a1 ∧ dz ā1 ) ∧ ... ∧ (dz am-1 ∧ dz ām-1 ) = a1,...,am-1∈{1,••• ,m-1} 2 by 2 = 1 | dϕ | 2 g i m (dz 1 ∧ dz 1) ∧ ... ∧ (dz m ∧ dz m) = (m -1)! | dϕ | 2 g ω m m! = 1 m | ∇ϕ | 2 g ω m (27)
Thus

T 1 = 1 m M h ′ (ϕ) | ∇ϕ | 2 g ω m , consequently M [ m k -f k (g -1 g)] h(ϕ) ω m ≥ 1 2 m k T 1 = 1 2m m k M h ′ (ϕ) | ∇ϕ | 2
g ω m , which achieves the proof of the proposition.

The Moser iteration technique

We conclude the proof using the Moser's iteration technique exactly as for the equation of Calabi-Yau. Let us apply the proposition to ϕ ts in order to obtain a crucial inequality (the inequality (IN1) below) from which we will infer the a priori estimate of ϕ ts C 0 . Let p ≥ 2 be a real number. The function ϕ ts is C 2 admissible. Let us consider the function h(u)

:= u | u | p-2 : R → R. This function is of class C 1 and h ′ (u) =| u | p-2 +u(p -2)u | u | p-4 = (p -1) | u | p-2 ≥
0, so h is increasing. Therefore we infer by the previous proposition that :

p -1 2m m k M | ϕ ts | p-2 | ∇ϕ ts | 2 v g ≤ M m k -f k (g -1 g) ϕ ts | ϕ ts | p-2 v g (28) Besides, | ∇ | ϕ ts | p 2 | 2 = 2g a b∂ a | ϕ ts | p 2 ∂b | ϕ ts | p 2 = 2g a b p 2 ϕ ts | ϕ ts | p 2 -2 2 ∂ a ϕ ts ∂bϕ ts = p 2 4 | ϕ ts | p-2 | ∇ϕ ts | 2
, so the previous inequality writes :

M | ∇ | ϕ ts | p 2 | 2 v g ≤ mp 2 2(p -1) m k M m k -f k (g -1 g) ϕ ts | ϕ ts | p-2 v g (
| ϕ ts | p m m-1 = | ϕ ts | p 2 2 2m m-1 ≤ Cste M | ∇ | ϕ ts | p 2 | 2 + M | ϕ ts | p 2 .2 (IN2)
where Cste is independent of p. Besides, f k (g -1 g) is uniformly bounded, indeed :

| f k (g -1 g) |= e tsf m k V ol(M ) M e tsf v g ≤ m k e 2ts f ∞ ≤ m k e 2 f ∞ (
| ϕ ts | p m m-1 ≤ Cste ′ × p M | ϕ ts | p-1 + M | ϕ ts | p ) (p ≥ 2) (IN4) where Cste ′ is independent of p. Suppose that Cste ′ ≥ 1.
Using the Green's formula and the inequalities of Sobolev-Poincaré (IN 2) and of Hölder, we prove [13, p. 76] following [START_REF] Th | Some Nonlinear Problems in Riemannian Geometry[END_REF] these L q estimates : Lemma 5.6. There exists a constant µ such that for all 1 ≤ q ≤ 2m m-1 , ϕ ts q ≤ µ Suppose without limitation of generality that µ ≥ 1. Now, we deduce from the previous lemma and the inequality (IN 4), by induction, these more general L p estimates using the same method as [START_REF] Th | Some Nonlinear Problems in Riemannian Geometry[END_REF] : Lemma 5.7. There exists a constant C 0 such that for all p ≥ 2,

ϕ ts p ≤ C 0 (δ m-1 C p) -m p , with δ = m m-1 and C = Cste ′ (1 + M ax(1, V ol(M ) 1 2
)) ≥ 1 where Cste ′ is the constant of the inequality (IN4).

By tending to the limit p → +∞ in the inequality of the previous lemma, we obtain the needed C 0 a priori estimate :

Corollary 5.8. ϕ ts C 0 ≤ C 0 6
The C 2 a priori estimate

6.1 Strategy for a C 2 estimate
First, we will look for a uniform upper bound on the eigenvalues λ([δ j i + g j l∂ i lϕ t ] 1≤i,j≤m . Secondly, we will infer from it the uniform ellipticity of the continuity equation (E k,t ) and a uniform gradient bound. Third step, with the uniform ellipticity at hand, we will derive a one-sided estimate on pure second derivatives and finally get the needed C 2 bound.

Eigenvalues upper bound 6.2.1 The functional

Let t ∈ T l,α and let ϕ t : M → R be a C l,α k-admissible solution of (E k,t ) satisfying M ϕ t ω m = 0. Consider the following functional :

B : U T 1,0 → R (P, ξ) → B(P, ξ) = hP (ξ, ξ) -ϕ t (P )
where U T 1,0 is the unit sphere bundle associated to (T 1,0 , h), and g is related to g by : ω = ω + i∂ ∂ϕ t . B is continuous on the compact set U T 1,0 so it assumes its maximum at a point (P 0 , ξ 0 ) ∈ U T 1,0 . In addition, for fixed P ∈ M , ξ ∈ U T 1,0 P → hP (ξ, ξ) is continuous on the compact subset U T 1,0 P (the fiber); therefore it attains its maximum at a unit vector ξ P ∈ U T 1,0 P and by the min-max principle we can choose as ξ P the direction of the largest eigenvalue of A P , λ max (A P ). Specifically, we have :

Lemma 6.1 (Min-Max Principle). hP (ξ P , ξ P ) = max ξ∈T 1,0 P ,h P (ξ,ξ)=1 hP (ξ, ξ) = λ max (A P )
For fixed P , we have max h P (ξ,ξ)=1 B(P, ξ) = B(P, ξ P ) = λ max (A P )ϕ t (P ), therefore max (P,ξ)∈U T 1,0 B(P, ξ) = max P ∈M B(P, ξ P ) = B(P 0 , ξ 0 ) ≤ B(P 0 , ξ P0 ), hence :

max (P,ξ)∈U T 1,0 B(P, ξ) = B(P 0 , ξ P0 ) = λ max (A P0 ) -ϕ t (P 0 ) (29)
At the point P 0 , consider e P0 1 , ..., e P0 m an h P0 -orthonormal basis of (T 1,0 P0 , h P0 ) made of eigenvectors of A P0 that satisfies the following properties :

1. h P0 -orthonormal : [h ij (P 0 )] 1≤i,j≤m = I m . 2. hP0 -diagonal : [ hij (P 0 )] 1≤i,j≤m = M atA P0 = diag(λ 1 , ..., λ m ), λ ∈ Γ k . 3. λ max (A P0 ) is achieved in the direction e P0 1 = ξ P0 : A P0 (ξ P0 ) = λ max (A P0 )ξ P0 = λ 1 ξ P0 and λ 1 ≥ ... ≥ λ m .
In other words, a basis satisfying :

1. [g i j (P 0 )] 1≤i,j≤m = I m . 2. [g i j (P 0 )] 1≤i,j≤m = M atA P0 = diag(λ 1 , ..., λ m ), λ ∈ Γ k . 3. λ max (A P0 ) = λ 1 ≥ ... ≥ λ m .
Let us consider a holomorphic normal chart (U 0 , ψ 0 ) centered at P 0 such that ψ 0 (P 0 ) = 0 and

∂ i | P0 = e P0
i for all i ∈ {1...m}.

Auxiliary local functional

From now on, we work in the chart (U 0 , ψ 0 ) so constructed at P 0 . The map P → g 1 1(P ) is continuous on U 0 and is equal to 1 at P 0 so there exists an open subset U 1 ⊂ U 0 such that g 1 1(P ) > 0 for all P ∈ U 1 . Let B 1 be the functional :

B 1 :U 1 → R P → B 1 (P ) = g1 1(P ) g 1 1(P ) -ϕ t (P )
We claim that B 1 assumes a local maximum at P 0 . Indeed, we have at each P ∈ U 1 : g1 1(P ) g 1 1(P ) = gP (∂1,∂1) g P (∂1,∂1) = hP (∂1,∂1) h P (∂1,∂1) = hP

∂1 |∂1| h P , ∂1
|∂1| h P ≤ λ max (A P ) (see lemma 6.1), thus : B 1 (P ) ≤ λ max (A P ) -ϕ t (P ) ≤ λ max (A P0 ) -ϕ t (P 0 ) = B 1 (P 0 ).

Differentiating the equation

For short, we drop henceforth the subscript t of ϕ t . Let us differentiate the equation (E k,t ) at P , in a chart z :

t ∂1f = dF k [δ j i +g j l (P )∂ i l ϕ(P )] 1≤i,j≤m .[∂1(g j l∂ i lϕ)] 1≤i,j≤m = m i,j=1 ∂F k ∂B j i [δ j i + g j l∂ i lϕ] ∂1g j l ∂ i lϕ + g j l ∂1 i lϕ (30)
Complex Hessian Equations on Some Compact Kähler Manifold 19

Differentiating once again, we find :

t ∂ 1 1f = m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i [δ j i + g j l∂ i lϕ] ∂ 1 g sō ∂ rō ϕ + g sō ∂ 1rō ϕ ∂1g j l ∂ i lϕ + g j l ∂1 i lϕ + m i,j=1 ∂F k ∂B j i [δ j i + g j l∂ i lϕ] ∂ 1 1g j l ∂ i lϕ + ∂1g j l ∂ 1i lϕ + ∂ 1 g j l ∂1 i lϕ + g j l ∂ 1 1i lϕ (31)
Using the above chart (U 1 , ψ 0 ) at the point P 0 , normality yields :

g j l = δ jℓ , ∂ α g i l = 0 and ∂ α g i l = 0. Furthermore [δ j i +g j l∂ i lϕ] = [δ j i +∂ i j ϕ] = [g i j ] = diag(λ 1 , ..., λ m ).
In this chart, we can simplify the previous expression, we get then at P 0 :

t ∂ 1 1f = m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i (diag(λ 1 , ..., λ m )) ∂ 1rs ϕ ∂1 i j ϕ + m i,j=1 ∂F k ∂B j i (diag(λ 1 , ..., λ m )) ∂ 1 1g j ī ∂ i īϕ + ∂ 1 1i j ϕ (32)
Besides, ∂ 1 1g j ī = ∂1 -g j sg o ī∂ 1 g os , so still by normality, we obtain at P 0 that : ∂ 1 1g j ī = -g j sg o ī∂ 1 1g os = -∂ 1 1g i j -R 1 1i j . Therefore we get :

t ∂ 1 1f = m i,j=1 ∂F k ∂B j i (diag(λ 1 , ..., λ m )) ∂ 1 1i j ϕ -R 1 1i j ∂ i īϕ + m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i (diag(λ 1 , ..., λ m )) ∂ 1rs ϕ ∂1 i j ϕ (33)

Using concavity

Now, using the concavity of ln σ k [START_REF] Caffarelli | The Dirichlet problem for nonlinear second order elliptic equations, III : Functions of the eigenvalues of the Hessian[END_REF], we prove as for the proposition 2.1 that the second sum of (33) is negative [13, p. 84]. This is not a direct consequence of the concavity of the function F k since the matrix [∂ 1i j ϕ] 1≤i,j≤m is not hermitian. Lemma 6.2.

S := m i,j,r,s=1 ∂ 2 F k ∂B s r ∂B j i (diag(λ 1 , ..., λ m )) ∂ 1rs ϕ ∂1 i j ϕ ≤ 0
Hence, from (33) combined with lemma 6.2 we infer :

t ∂ 1 1f ≤ m i=1 σ k-1,i (λ) σ k (λ) ∂ 1 1i īϕ -R 1 1i ī ∂ i īϕ (34)

Differentiation of the functional B 1

Let us differentiate twice the functional B 1 :

B 1 (P ) = g1 1(P ) g 1 1(P ) -ϕ(P )

∂īB 1 = ∂īg 1 1 g 1 1 -g1 1∂īg 1 1 (g 1 1) 2 -∂īϕ ∂ i īB 1 = ∂ i ī g1 1 g 1 1 - ∂ i g 1 1 ∂īg 1 1 + ∂ i g1 1 ∂īg 1 1 + g1 1∂ i īg 1 1 (g 1 1) 2 + 2g 1 1∂ i g 1 1∂īg 1 1 (g 1 1) 3 -∂ i īϕ
Therefore at P 0 , in the above chart (U 1 , ψ 0 ) we find

∂ i īB 1 = ∂ i ī(g 1 1 + ∂ 1 1ϕ) -λ 1 ∂ i īg 1 1 -∂ i īϕ = R 1 1i ī + ∂ 1 1i īϕ -λ 1 R 1 1i ī -∂ i īϕ.
Let us define the operator :

L := m i,j=1 ∂F k ∂B j i ([δ j i + g j l∂ i lϕ] 1≤i,j≤m ) ∇ j i (35)
Thus, we have at P 0 :

L(B 1 ) = m i=1 σ k-1,i (λ) σ k (λ) ∂ 1 1i īϕ + (1 -λ 1 )R 1 1i ī -∂ i īϕ (36)
Combining ( 36) with (34), we get rid of fourth derivatives :

t ∂ 1 1f -L(B 1 ) ≤ m i=1 σ k-1,i (λ) σ k (λ) R 1 1i ī(λ 1 -1 -λ i + 1) + m i=1 σ k-1,i (λ) σ k (λ) (λ i -1) (37)
Since B 1 assumes its maximum at P 0 , we have at P 0 that L(B 1 ) ≤ 0. So we are left with the following inequality at P 0 :

0 ≥ m i=2 σ k-1,i (λ) σ k (λ) (-R 1 1i ī)(λ 1 -λ i ) - m i=1 σ k-1,i (λ) σ k (λ) λ i + m i=1 σ k-1,i (λ) σ k (λ) + t ∂ 1 1f (38)
Curvature assumption : Henceforth, we will suppose that the holomorphic bisectional curvature is nonnegative at any P ∈ M . Thus in a holomorphic normal chart centered at P we have R aāb b(P ) ≤ 0 for all 1 ≤ a, b ≤ m. This holds in particular at P 0 in the above chart ψ 0 . This assumption will be used only to derive an a priori eigenvalues pinching and is not required in the other sections.

Back to the inequality (38), we have σ k (λ) > 0 and σ k-1,i (λ) > 0 since λ ∈ Γ k , and under our curvature assumption (-R 1 1i ī) ≥ 0 for all i ≥ 2.

Besides, λ i ≤ λ 1 for all i therefore :

m i=2 σ k-1,i (λ) σ k (λ) (-R 1 1i ī)(λ 1 -λ i ) ≥ 0.
So we can get rid of the curvature terms in (38) and infer from it the inequality :

0 ≥ - m i=1 σ k-1,i (λ) σ k (λ) λ i + m i=1 σ k-1,i (λ) σ k (λ) + t ∂ 1 1f
(39)

A λ 1 's upper bound

Here, we require elementary identities satisfied by the σ ℓ 's [START_REF] Lin | On some inequalities for elementary symmetric functions[END_REF] namely :

∀ 1 ≤ ℓ ≤ m σ ℓ (λ) = σ ℓ,i (λ) + λ i σ ℓ-1,i (λ) (40) ∀ 1 ≤ ℓ ≤ m m i=1 σ ℓ-1,i (λ) λ i = ℓ σ ℓ (λ) (41) so in particular m i=1 σ k-1,i (λ) σ k (λ) λ i = k (42) ∀ 1 ≤ ℓ ≤ m m i=1 σ ℓ,i (λ) = (m -ℓ) σ ℓ (λ) (43)
so in particular

m i=1 σ k-1,i (λ) σ k (λ) = (m -k + 1) σ k-1 (λ) σ k (λ) (44)
Consequently, (39) writes :

q k := (m -k + 1) k σ k-1 (λ) σ k (λ) ≤ 1 - t k ∂ 1 1f So q k ≤ 1 + 1 k | ∂ 1 1f |. But at P 0 , | ∇ 2 f | 2 g = 2 g ac g d b ∇ a bf ∇ cd f + ∇ ad f ∇ cbf = 2 m a,b=1 | ∂ a bf | 2 + | ∂ ab f | 2 , then | ∂ 1 1f |≤| ∇ 2 f | g , and consequently q k ≤ 1 + 1 k f C 2 (M ) =: C 1 .
In other words, there exists a constant C 1 independent of t ∈ [0, 1] such that :

q k ≤ C 1 ( 45
)
To proceed further, we recall :

Lemma 6.3 (Newton Inequalities). ∀ℓ ≥ 2, ∀λ ∈ R m , σ ℓ (λ) σ ℓ-2 (λ) ≤ (ℓ -1)(m -ℓ + 1) ℓ(m -ℓ + 2) [σ ℓ-1 (λ)] 2
Let us use Newton inequalities to relate q k to σ 1 . Since for 2 ≤ ℓ ≤ k and λ ∈ Γ k we have σ ℓ (λ) > 0, σ ℓ-1 (λ) > 0 and σ ℓ-2 (λ) > 0 (σ 0 (λ) = 1 by convention), Newton inequalities imply then that : λ) , or else q ℓ-1 ≤ q ℓ , consequently

(m-ℓ+2) (ℓ-1) σ ℓ-2 (λ) σ ℓ-1 (λ) ≤ (m-ℓ+1) ℓ σ ℓ-1 (λ) σ ℓ (
Complex Hessian Equations on Some Compact Kähler Manifold 22

q k ≥ q k-1 ≥ ... ≥ q 2 = (m-1) σ1(λ) 2 σ2(λ)
. By induction, we get σ 1 (λ) ≤ ℓ!(m-ℓ)! (m-1)! σ ℓ (λ) (q ℓ ) ℓ-1 for all 2 ≤ ℓ ≤ k. In particular :

σ 1 (λ) ≤ k!(m -k)! (m -1)! σ k (λ) (q k ) k-1 (46) But σ k (λ) ≤ e 2 f ∞ m
k , combining this with ( 45) and ( 46) we obtain at P 0 that :

σ 1 (λ) ≤ m e 2 f ∞ (C 1 ) k-1 =: C 2 (47)
Hence we may state :

Theorem 2. There exists a constant C 2 > 0 depending only on m, k, f ∞ and f C 2 such that : ∀1 ≤ i ≤ m λ i (P 0 ) ≤ C 2 .
Combining this result with the C 0 a priori estimate ϕ t C0 ≤ C 0 , immediately yields the following :

Theorem 3. There exists a constant C ′ 2 > 0 depending only on m, k, f C 2 and C 0 such that : ∀P ∈ M, ∀1 ≤ i ≤ m, λ i (P ) ≤ C 2 + 2C 0 =: C ′ 2 .

Uniform pinching of the eigenvalues

We infer automatically the following pinchings of the eigenvalues :

Proposition 6.4. ∀1 ≤ i ≤ m -(m -1)C 2 ≤ λ i (P 0 ) ≤ C 2 Proposition 6.5. ∀P ∈ M, ∀1 ≤ i ≤ m, -(m -1)C ′ 2 ≤ λ i (P ) ≤ C ′ 2 6.

Uniform ellipticity of the continuity equation

To prove the next proposition on uniform ellipticity, we require some inequalities satisfied by the σ ℓ 's.

Lemma 6.6 (Maclaurin Inequalities). ∀1 ≤ ℓ ≤ s ∀λ ∈ Γ s σs(λ) ( m s ) 1 s ≤ σ ℓ (λ) ( m ℓ ) 1 ℓ
Proposition 6.7 (Uniform Ellipticity). There exists constants E > 0 and F > 0 depending only on m, k, f ∞ and C 2 such that :

E ≤ σ k-1,1 (λ) ≤ ... ≤ σ k-1,m (λ) ≤ F where λ = λ(P 0 ). Proof. We have ∂σ k ∂λ1 = σ k-1,1 (λ) ≤ ... ≤ σ k-1,m (λ) ≤ m-1 k-1 (C 2 ) k-1 =:
F where, indeed, the constant F so defined depends only on m, k and C 2 . Let us look for a uniform lower bound on σ k-1,1 (λ), using the identity :

σ k (λ) = λ 1 σ k-1,1 (λ) + σ k,1 (
λ). We distinguish two cases :

• Case 1 : σ k,1 (λ) ≤ 0. When so, we have σ k (λ) ≤ λ 1 σ k-1,1 (λ) therefore σ k-1,1 (λ) ≥ σ k (λ) λ1 . But σ k (λ) ≥ e -2 f ∞ m k and 0 < λ 1 ≤ C 2 , hence σ k-1,1 (λ) ≥ e -2 f ∞ ( m k ) C2 . • Case 2 : σ k,1 (λ) > 0. For 1 ≤ j ≤ k -1, σ j (λ 2 , ..., λ m ) = σ j,1 (λ) > 0 since j + 1 ≤ k and λ ∈ Γ k . Besides σ k (λ 2 , ..., λ m ) = σ k,1 (λ) > 0 by hypothesis, therefore (λ 2 , ..., λ m ) ∈ Γ k,1 = {β ∈ R m-1 /∀1 ≤ j ≤ k σ j (β) > 0}.
From the latter, we infer by Maclaurin inequalities :

σ k (λ2,...,λm) ( m-1 k ) 1 k ≤ σ k-1 (λ2,...,λm) ( m-1 k-1 ) 1 k-1 or else σ k,1 (λ) ( m-1 k ) 1 k ≤ σ k-1,1 (λ) ( m-1 k-1 ) 1 k-1 thus we have σ k,1 (λ) ≤ m-1 k σ k-1,1 (λ) ( m-1 k-1 ) 1+ 1 k-1
, consequently :

σ k (λ) = λ 1 σ k-1,1 (λ) + σ k,1 (λ) ≤ λ 1 σ k-1,1 (λ) + m -1 k σ k-1,1 (λ) m-1 k-1 1+ 1 k-1 ≤ σ k-1,1 (λ) λ 1 + m-1 k m-1 k-1 σ k-1,1 (λ) m-1 k-1 1 k-1
Here, let us distinguish two subcases of case 2 :

-If σ k-1,1 (λ) > m-1
k-1 then we have the uniform lower bound that we look for.

-Else σ k-1,1 (λ) ≤ m-1 k-1 , thus σ k-1,1 (λ) ( m-1 k-1 ) 1 k-1 ≤ 1 therefore σ k (λ) ≤ σ k-1,1 (λ) λ 1 + ( m-1 k ) ( m-1 k-1 ) = σ k-1,1 (λ) (λ 1 + m k -1), then we get σ k-1,1 (λ) ≥ σ k (λ) λ1+ m k -1 ≥ e -2 f ∞ ( m k ) C2+ m k -1 . Consequently σ k-1,1 (λ) ≥ min e -2 f ∞ ( m k ) C2 , m-1 k-1 , e -2 f ∞ ( m k ) C2+ m k -1 or fi- nally : σ k-1,1 (λ) ≥ min m-1 k-1 , e -2 f ∞ ( m k ) C2+ m k -1
=: E, where the constant E so defined depends only on m, k, f ∞ and C 2 .

Similarly we prove : Proposition 6.8 (Uniform Ellipticity). There exists constants E 0 > 0 and F 0 > 0 depending only on m, k, f ∞ and C ′ 2 such that :

∀P ∈ M, ∀1 ≤ i ≤ m, E 0 ≤ σ k-1,i (λ(P )) ≤ F 0 .

Gradient uniform estimate

The manifold M is Riemannian compact and ϕ t ∈ C 2 (M ) so by the Green's formula :

ϕ t (P ) = 1 V ol(M ) M ϕ t (Q)dv g (Q) + M G(P, Q)△ϕ t (Q)dv g (Q)
where G(P, Q) is the Green's function of the Laplacian △. By differentiating locally under the integral sign we obtain :

∂ u i ϕ t (P ) = M △ϕ t (Q) (∂ u i ) P G(P, Q)dv g (Q)
. We infer then that, at P in a holomorphic normal chart, we have :

| (∇ϕ t ) P |≤ √ 2m M | △ϕ t (Q) | | ∇ P G(P, Q) | dv g (Q) (48)
Now, using the uniform pinching of the eigenvalues, we prove easily the following estimate of the Laplacian : Lemma 6.9. There exists a constant C 3 > 0 depending on m and

C ′ 2 such that △ϕ t ∞,M ≤ C 3 .
Combining lemma 6.9 with (48), we deduce that : [3, p. 109], there exists constants C and C ′ such that :

| (∇ϕ t ) P |≤ √ 2m C 3 M | ∇ P G(P, Q) | dv g (Q). Besides, classically
| ∇ P G(P, Q) |≤ C d g (P, Q) 2m-1 and M 1 d g (P, Q) 2m-1 dv g (Q) ≤ C ′
We thus obtain the following result : Proposition 6.10. There exists a constant C 5 > 0 depending on m, C ′ 2 and (M, g) such that : ∀P ∈ M | (∇ϕ t ) P |≤ C 5 .

Specifically, we can choose

C 5 = √ 2m C 3 C C ′ .

Second derivatives estimate

Our equation is of type :

F (P, [∂ u i u j ϕ] 1≤i,j≤2m ) = v P ∈ M (E)

The functional

Consider the following functional :

Φ : U T → R (P, ξ) → (∇ 2 ϕ t ) P (ξ, ξ) + 1 2 | (∇ϕ t ) P | 2 g
where U T is the real unit sphere bundle associated to (T M, g). Φ is continuous on the compact set U T so it assumes its maximum at a point (P 1 , ξ 1 ) ∈ U T .

6.5.2

Reduction to finding a one-sided estimate for (∇

2 ϕ t ) P1 (ξ 1 , ξ 1 )
If we find a uniform upper bound for (∇

2 ϕ t ) P1 (ξ 1 , ξ 1 ), since | ∇ϕ t | ∞ ≤ C 5
, we readily deduce that there exists a constant C 6 > 0 such that :

(∇ 2 ϕ t ) P (ξ, ξ) ≤ C 6 ∀(P, ξ) ∈ U T
Fix P ∈ M . Let (U P , ψ P ) be a holomorphic g-normal g-adapted chart centered at P , namely [g i j (P )] 1≤i,j≤m = I m , ∂ ℓ g i j (P ) = 0 and [g i j (P )] 1≤i,j≤m = [diag(λ 1 (P ), ..., λ m (P ))].

Since | ∂ x j | g = √ 2,
we obtain ∂ x j x j ϕ t (P ) = 2 (∇ 2 ϕ t ) P (

∂ x j √ 2 , ∂ x j √
2 ) ≤ 2C 6 and similarly ∂ y j y j ϕ t (P ) = 2 (∇ 2 ϕ t ) P (

∂ y j √ 2 , ∂ y j √
2 ) ≤ 2C 6 for all 1 ≤ j ≤ m. Besides, we have ∂ x j x j ϕ t (P ) + ∂ y j y j ϕ t (P ) = 4 ∂ j j ϕ t (P ) = 4(λ j (P ) -1) ≥ -4[(m -1)C ′ 2 + 1], therefore we obtain :

∂ x j x j ϕ t (P ) ≥ -4[(m -1)C ′ 2 + 1] -2C 6 =: -C 7 and ∂ y j y j ϕ t (P ) ≥ -C 7 ∀1 ≤ j ≤ m (49)
Let us now bound second derivatives of mixed type ∂ u r u s ϕ t (P ). Let for all 1 ≤ i, j ≤ 2m. Therefore we deduce that :

1 ≤ r = s ≤ 2m. Since | ∂ x r ±∂ x s | g = 2, we have : (∇ 2 ϕ t ) P ( ∂ x r ±∂ x s 2 , ∂ x r ±∂ x s 2 ) = 1 4 ∂ x r x r ϕ t (P )+ 1 4 ∂ x s x s ϕ t (P )± 1 2 ∂ x r x s ϕ t (P ) ≤ C 6 , which yields ±∂ x r x s ϕ t (P ) ≤ 2 C 6 -1 2 ∂ x r x r ϕ t (P )-
| (∇ 2 ϕ t )(P ) | 2 g = 1 4 1≤i,j≤2m (∂ u i u j ϕ t (P )) 2 ≤ m 2 (2 C 6 + C 7 ) 2 (50)
Theorem 4 (Second Derivatives Uniform Estimate). There exists a constant C 8 > 0 depending only on m, C ′ 2 and C 6 such that :

∀P ∈ M | (∇ 2 ϕ t ) P | g ≤ C 8 .
This allows to deduce the needed uniform C 2 estimate :

ϕ C 2 (M,R) ≤ C 0 + C 5 + C 8
(51)

Chart choice

For fixed P ∈ M , ξ ∈ U T P → (∇ 2 ϕ t ) P (ξ, ξ) is continuous on the compact subset U T P (the fiber), therefore it assumes its maximum at a unit vector ξ P ∈ U T P . Besides, (∇ 2 ϕ t ) P is a symmetric bilinear form on T P M , so by the min-max principle we have : (∇ 2 ϕ t ) P (ξ P , ξ P ) = max ξ∈T P M,g(ξ,ξ)=1 (∇ 2 ϕ t ) P (ξ, ξ) = β max (P ), where β max (P ) denotes the largest eigenvalue of (∇ 2 ϕ t ) P with respect to g P , furthermore we can choose as ξ P the direction of the largest eigenvalue β max (P ). For fixed P , we now have : max ξ∈T P M,g P (ξ,ξ)=1 Φ(P, ξ) = Φ(P, ξ P ) = (∇ 2 ϕ t ) P (ξ P , ξ P ) + 1 2 | (∇ϕ t ) P | 2 g = β max (P ) + 1 2 | (∇ϕ t ) P | 2 g , consequently max (P,ξ)∈U T Φ(P, ξ) = max P ∈M Φ(P, ξ P ) = Φ(P 1 , ξ 1 ) ≤ Φ(P 1 , ξ P1 ), hence max (P,ξ)∈U T Φ(P, ξ) = Φ(P 1 , ξ P1 ) = β max (P 1 ) + 1 2 | (∇ϕ t ) P1 | 2 g . At the point P 1 , consider ε P1 1 , ..., ε P1 2m a (real) basis of (T P1 M, g P1 ) that satisfies the following properties :

• [g ij (P 1 )] 1≤i,j≤2m = I 2m . • [(∇ 2 ϕ t ) ij (P 0 )] 1≤i,j≤2m = diag(β 1 , ..., β 2m ). • β 1 = β max (P 1 ) ≥ β 2 ≥ ... ≥ β 2m .
Complex Hessian Equations on Some Compact Kähler Manifold 26

Let (U ′ 1 , ψ 1 ) be a C ∞ g-normal real chart at P 1 obtained from a holomorphic chart z 1 , ..., z m by setting (u 1 , ..., u 2m ) = (x 1 , ..., x m , y 1 , ..., y m ) where z j = x j + iy j (namely [g ij (P 1 )] 1≤i,j≤2m = I 2m and ∂ u ℓ g ij = 0 for all 1 ≤ i, j, ℓ ≤ 2m) satisfying ψ 1 (P 1 ) = 0 and ∂ u i | P1 = ε P1 i , so that ∂ u 1 | P1 is the direction of the largest eigenvalue β max (P 1 ).

Auxiliary local functional

From now on, we work in the real chart (U ′ 1 , ψ 1 ) so constructed at

P 1 . Let U 2 ⊂ U ′
1 be an open subset such that g 11 (P ) > 0 for all P ∈ U 2 , and let Φ 1 be the functional :

Φ 1 : U 2 → R P → Φ 1 (P ) = (∇ 2 ϕ t ) 11 (P ) g 11 (P ) + 1 2 | (∇ϕ t ) P | 2 g
We claim that Φ 1 assumes its maximum at P 1 . Indeed, (∇ 2 ϕt)11(P )

g11(P ) = (∇ 2 ϕ) P (∂ u 1 ,∂ u 1 ) g P (∂ u 1 ,∂ u 1 ) = (∇ 2 ϕ) P ( ∂ u 1 |∂ u 1 |g , ∂ u 1 |∂ u 1 |g ) ≤ β max (P ), so Φ 1 (P ) ≤ β max (P )+ 1 2 | (∇ϕ t ) P | 2 g ≤ β max (P 1 ) + 1 2 | (∇ϕ t ) P1 | 2 g = Φ 1 (P 1
) proving our claim. Let us now differentiate twice in the real direction ∂ u 1 the equation :

F P, [∂ u i u j ϕ] 1≤i,j≤2m = v (E)
At the point P , in a chart u, we obtain :

∂ u 1 v = ∂F ∂u 1 [ϕ] + 2m i,j=1 ∂F ∂r ij [ϕ] ∂ u 1 u i u j ϕ (52)
Differentiating once again :

∂ u 1 u 1 v = ∂ 2 F ∂u 1 ∂u 1 [ϕ] + 2m i,j=1 ∂ 2 F ∂r ij ∂u 1 [ϕ] ∂ u 1 u i u j ϕ + 2m i,j=1 ∂ 2 F ∂u 1 ∂r ij [ϕ] + 2m e,s=1 ∂ 2 F ∂r es ∂r ij [ϕ] ∂ u 1 u e u s ϕ ∂ u 1 u i u j ϕ + 2m i,j=1 ∂F ∂r ij [ϕ] ∂ u 1 u 1 u i u j ϕ (53)
But at the point P 1 , for our function F (P, r) = F k [δ j i + 1 4 g j l(P )(r iℓ + r (i+m)(ℓ+m) + ir i(ℓ+m) -ir (i+m)ℓ )] 1≤i,j≤m , we have ∂ 2 F ∂rij ∂u 1 [ϕ] = 0 since ∂ u 1 g sq (P 1 ) = 0. Hence, we infer that :

∂ u 1 u 1 v = ∂ 2 F ∂u 1 ∂u 1 [ϕ] + 2m i,j,e,s=1 ∂ 2 F ∂r es ∂r ij [ϕ] ∂ u 1 u e u s ϕ ∂ u 1 u i u j ϕ + 2m i,j=1

∂F ∂r ij

[ϕ] ∂ u 1 u 1 u i u j ϕ (54)

Using concavity

The function F is concave with respect to the variable r. Indeed :

F (P, r) = F k [δ j i + 1 4 g j l(P )(r iℓ + r (i+m)(ℓ+m) + ir i(ℓ+m) -ir (i+m)ℓ )] 1≤i,j≤m = F k (g -1 (P )r) where r = [g i j (P ) + 1 4 (r ij + r (i+m)(j+m) + ir i(j+m) -ir (i+m)j )] 1≤i,j≤m = F k (g - 1
2 (P )rg - 1
2 (P ) ∈Hm(C) ) = F k ρ P (r) where ρ P (r) := δ j i + 1 4 m ℓ,s=1 g - 1
2 (P ) iℓ g - 1
2 (P ) sj (r ℓs + r (ℓ+m)(s+m) + ir ℓ(s+m) -ir (ℓ+m)s ) 1≤i,j≤m (55)
but, for a fixed point P the function r ∈ S 2m (R) → ρ P (r) ∈ H m (C) is affine (where S 2m (R) denotes the set of symmetric matrices of size 2m), we deduce then that the composition F (P, .)

= F k •ρ P is concave on the set {r ∈ S 2m (R)/ρ P (r) ∈ λ -1 (Γ k )} = ρ -1 P λ -1 (Γ k )
, which proves our claim. Hence, since the matrix [∂ u 1 u i u j ϕ] 1≤i,j≤m is symmetric, we obtain that :

2m i,j,e,s=1 ∂ 2 F ∂r es ∂r ij [ϕ] ∂ u 1 u e u s ϕ ∂ u 1 u i u j ϕ ≤ 0 (56)
Consequently :

∂ u 1 u 1 v -∂ u 1 u 1 F [ϕ] ≤ 2m i,j=1 ∂F ∂r ij [ϕ] ∂ u 1 u 1 u i u j ϕ (57)
Let us now calculate the quantity ∂ u 1 u 1 F [ϕ] (at P 1 ). Since ∂ u 1 g sq (P 1 ) = 0, we have :

∂ 2 F ∂u 1 ∂u 1 (P 1 , D 2 ϕ(P 1 )) = m s=1 σ k-1,s (λ(P 1 )) σ k (λ(P 1 )) × ∂ u 1 u 1 g ss (P 1 ) ∂ ss ϕ(P 1 ) (58)
But at P 1 , ∂ u 1 u 1 g ss = -g sō g qs ∂ u 1 u 1 g qō and [g i j ] 1≤i,j≤m = 2I m , then ∂ u 1 u 1 g ss = -4∂ u 1 u 1 g ss so that ∂ u 1 u 1 g ss = -∂ u 1 u 1 g ss -∂ u 1 u 1 g (s+m)(s+m) . Moreover Γ u r u j u s = 1 2 ∂ u j g os + ∂ u s g oj -∂ u o g js g or , thus : ∂ u i Γ u r u j u s = 1 2 ∂ u i u j g rs +∂ u i u s g rj -∂ u i u r g js . Similarly, we have at P 1 : ∂ u i Γ u s u j u r = 1 2 ∂ u i u j g rs + ∂ u i u r g sj -∂ u i u s g jr .
Consequently, we deduce that :

∂ u i u j g rs = ∂ u i Γ u r u j u s + ∂ u i Γ u s u j u r . Hence, we have at P 1 : ∂ u 1 u 1 g ss = Besides, we have Γ u 1 u j u 1 = 1 2 ∂ u j g s1 + ∂ u 1 g sj -∂ u s g j1 g s1 , therefore we deduce that ∂ u i Γ u 1 u j u 1 = 1 2 ∂ u i u j g s1 +∂ u i u 1 g sj -∂ u i u s g j1 g s1 +0 = 1 2 ∂ u i u j g 11 , namely ∂ u i u j g 11 = 2 ∂ u i Γ u 1 u j u 1 .
Moreover, we have at P 1 :

∂ u i u j | (∇ϕ) P | 2 g =∂ u i u j 2m r,s=1 g rs ∂ u r ϕ ∂ u s ϕ = 2m r,s=1 ∂ u i u j g rs ∂ u r ϕ ∂ u s ϕ + g rs ∂ u i u j u r ϕ ∂ u s ϕ + g rs ∂ u j u r ϕ ∂ u i u s ϕ + g rs ∂ u i u r ϕ ∂ u j u s ϕ + g rs ∂ u r ϕ ∂ u i u j u s ϕ = 2m r,s=1 ∂ u i u j g rs ∂ u r ϕ ∂ u s ϕ + 2 2m s=1 ∂ u i u j u s ϕ ∂ u s ϕ + 2 2m s=1 ∂ u i u s ϕ ∂ u j u s ϕ (64) But at P 1 , ∂ u i u j g rs = -∂ u i u j g rs , in addition at this point ∂ u i u j g rs = ∂ u i Γ u r u j u s + ∂ u i Γ u s u j u r ,
therefore we obtain at P 1 in the chart ψ 1 :

∂ u i u j | (∇ϕ) P | 2 g = -2 2m r,s=1 ∂ u i Γ u r u j u s ∂ u r ϕ ∂ u s ϕ + 2 2m s=1 ∂ u i u j u s ϕ ∂ u s ϕ + 2 2m s=1 ∂ u i u s ϕ ∂ u j u s ϕ (65)
Henceforth, and in order to lighten the notations, we use ∂ i instead of ∂ u i and Γ s ij instead of Γ u s u i u j , so we have :

∂ ij Φ 1 =∂ ij11 ϕ -∂ ij Γ s 11 ∂ s ϕ -∂ j Γ s 11 ∂ is ϕ -∂ i Γ s 11 ∂ js ϕ -2 ∂ i Γ 1 j1 (∇ 2 ϕ) 11 (P 1 ) - 2m r,s=1 ∂ i Γ r js ∂ r ϕ ∂ s ϕ + 2m s=1 ∂ ijs ϕ ∂ s ϕ + 2m s=1 ∂ is ϕ ∂ js ϕ (66)
Let us now consider the second order linear operator :

L = 2m i,j=1 ∂F ∂r ij [ϕ] ∂ ij (67)
Since the functional Φ 1 assumes its maximum at the point P 1 , we have L(Φ 1 ) ≤ 0 at P 1 in the chart ψ 1 . Besides, combining the inequalities Complex Hessian Equations on Some Compact Kähler Manifold 30

(60) and (66), we obtain :

2 LΦ 1 ≤0 -∂ 11 v ≥ 2m i,j=1 ∂F ∂r ij [ϕ] ∂ ij11 ϕ -∂ ij Γ s 11 ∂ s ϕ -∂ j Γ i 11 ∂ ii ϕ -∂ i Γ j 11 ∂ jj ϕ -2 ∂ i Γ 1 j1 (∇ 2 ϕ) 11 (P 1 ) - 2m r,s=1 ∂ i Γ r js ∂ r ϕ ∂ s ϕ + 2m s=1 ∂ ijs ϕ ∂ s ϕ + δ j i (∂ ii ϕ) 2 -∂ 11ij ϕ + 1 2 m s=1 σ k-1,s (λ(P 1 )) σ k (λ(P 1 )) ∂ 1 Γ s 1s + ∂ 1 Γ s+m 1(s+m) ∂ ss ϕ + ∂ (s+m)(s+m) ϕ (68)
The fourth derivatives simplify. Moreover, we have at

P 1 : ∂ s v = ∂F ∂u 1 [ϕ]+ 2m i,j=1 ∂F ∂rij [ϕ] ∂ sij ϕ with ∂F ∂u 1 (P 1 , D 2 ϕ(P 1
)) = 0, consequently :

0 ≥∂ 11 v + 2m s=1 ∂ s v ∂ s ϕ + 2m i,j=1 ∂F ∂r ij [ϕ] -2 ∂ i Γ 1 j1 (∇ 2 ϕ) 11 (P 1 ) -∂ i Γ j 11 ∂ jj ϕ -∂ j Γ i 11 ∂ ii ϕ - 2m s=1 ∂ ij Γ s 11 ∂ s ϕ - 2m r,s=1 ∂ i Γ r js ∂ r ϕ ∂ s ϕ + δ j i (∂ ii ϕ) 2 + 1 2 m s=1 σ k-1,s (λ(P 1 )) σ k (λ(P 1 )) ∂ 1 Γ s 1s + ∂ 1 Γ s+m 1(s+m) ∂ ss ϕ + ∂ (s+m)(s+m) ϕ (69)
Let us now express the quantities ∂ i Γ

∂ i Γ 1 j1 = 1 3 (R j11i + R ji11 =0 ) = 1 3 R j11i ∂ i Γ j 11 = 1 3 (R 1j1i + R 1i1j ) = 2 3 R 1j1i ∂ j Γ i 11 = 2 3 R 1i1j ∂ i Γ r js = 1 3 (R jrsi + R jisr ) ∂ 1 Γ s 1s = 1 3 (R 1ss1 + R 11ss =0 ) = 1 3 R 1ss1 ∂ 1 Γ s+m 1(s+m) = 1 3 R 1(s+m)(s+m)1 ∂ ij Γ s 11 = 1 4 ∇ i R 1j1s + ∇ i R 1s1j + ∇ j R 1s1i + ∇ j R 1i1s - 1 12 ∇ s R 1i1j + ∇ s R 1j1i = 1 2 ∇ i R 1s1j + ∇ j R 1s1i - 1 6 ∇ s R 1i1j
We then obtain :

0 ≥∂ 11 v + 2m s=1 ∂ s v ∂ s ϕ + 2m i,j=1

∂F ∂r ij

[ϕ] -2 3 R j11i (∇ 2 ϕ) 11 (P 1 )

- 2 3 R 1j1i ∂ jj ϕ - 2 3 R 1i1j ∂ ii ϕ - 2m s=1 1 2 ∇ i R 1s1j + 1 2 ∇ j R 1s1i - 1 6 ∇ s R 1i1j ∂ s ϕ - 2m r,s=1 1
∂F ∂r ij [ϕ] δ ij (β i ) 2 - 2 3 R j11i β 1 - 2 3 R 1j1i β j - 2 3 R 1i1j β i - 1 3 2m r,s=1 (R jrsi +R jisr ) ∂ r ϕ ∂ s ϕ- 1 2 2m s=1 ∇ i R 1s1j +∇ j R 1s1i - 1 3 ∇ s R 1i1j ∂ s ϕ + 1 6 m i=1 σ k-1,i (λ(P 1 )) σ k (λ(P 1 )) × R 1ii1 + R 1(i+m)(i+m)1 β i + β i+m
But for F [ϕ] = F k [δ j i + g j l∂ i lϕ] 1≤i,j≤m since ∂ ss ϕ = 1 4 (∂ u s u s + ∂ u s+m u s+m ), we obtain at P 1 in the chart ψ 1 that :

∂F ∂r ij

[ϕ] = Then :

∀1 ≤ i = j ≤ 2m ∂F ∂r ij [ϕ] = 0 ∀1 ≤ i ≤ m ∂F ∂r ii
[ϕ] = ∂F ∂r (i+m)(i+m)

[ϕ] = 1 4

∂F k ∂B i i diag(λ(P 1 )) = 1 4
σ k-1,i λ(P 1 ) σ k λ(P 1 )

>0 since λ(P1)∈Γ k Hence :

0 ≥ ∂ 11 v + 2m s=1 ∂ s v ∂ s ϕ + 2m i=1 ∂F ∂r ii [ϕ] (β i ) 2 + 2 3 R 1i1i (β 1 -2 β i ) + 1 3 2m r,s=1 R iris ∂ r ϕ ∂ s ϕ - 2m s=1 ∇ i R 1s1i - 1 6 ∇ s R 1i1i ∂ s ϕ + 1 6 m i=1 σ k-1,i (λ(P 1 )) σ k (λ(P 1 )) R 1ii1 + R 1(i+m)(i+m)1 β i + β i+m (72)
But at P 1 in the chart ψ 1 , R 2 g = g ai g bj g cr g ds R abcd R ijrs = Hence at P 1 in the chart ψ 1 , we obtain :

-

-t ∂ 11 f -t 2m s=1 ∂ s f ∂ s ϕ ≤ f C 2 (M ) (1 + 2m C 5 ) (76)
Besides :

2m i=1 ∂F ∂r ii [ϕ] = m i=1

∂F ∂r ii

[ϕ] + ∂F ∂r (i+m)(i+m)

[ϕ] = 1 2 m i=1 σ k-1,i λ(P 1 ) σ k λ(P 1 ) (77)

Consequently, we obtain :

f C 2 (M ) (1+2m C
Let us now estimate | β i | for 1 ≤ i ≤ m using β 1 . We follow the same method as for the proof of theorem 4. For all (P, ξ) ∈ U T , we have the inequality (∇ 2 ϕ t ) P (ξ, ξ) ≤ β 1 + 1 2 (C 5 ) 2 , then at P in a holomorphic g-normal g-adapted chart ψ P , namely a chart such that [g i j (P )] 1≤i,j≤m = I m , ∂ ℓ g i j (P ) = 0 and [g i j (P )] 1≤i,j≤m = diag λ 1 (P ), ..., λ m (P ) , we deduce that for all j ∈ {1, ..., m} :

∂ x j x j ϕ t (P ) = 2 (∇ 2 ϕ t ) P ∂ x j √ 2 , ∂ x j √ 2 ≤ 2β 1 + (C 5 ) 2
and ∂ y j y j ϕ t (P )

≤ 2β 1 + (C 5 ) 2 (79) m R g 2 (C 5 ) 2 +C 9 -f C 2 (M ) (1 + 2m C 5 ) (91)
Set :

I := 80 3 m 2 e 4 f ∞ F 0 E 0 R g > 0 J := 4 m 2 e 4 f ∞ F 0 E 0 7 6 C 5 ∇R g + 2 3 m (C 5 ) 2 R g + 5 3 2 (C 5 ) 2 + C 9 R g + 4 m k e 2 f ∞ E 0 f C 2 (M ) (1 + 2m C 5 ) > 0
Complex Hessian Equations on Some Compact Kähler Manifold 36

The previous inequality writes then :

(β 1 ) 2 -I β 1 -J ≤ 0 (92)
The discriminant of this polynomial of second order is equal to △ = I 2 + 4 J > 0, which gives an upper bound for β 1 .

7 A C 2,β a priori estimate

We infer from the C 2 estimate, a C 2,β estimate using a classical Evans-Trudinger theorem [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]; which achieves the proof of theorem 1 (see [13, p. 109-117] for details).

  0) dz a ∧ dz ā k ∧ b i dz b ∧ dz b m-k = (a1,...,a k )∈{1,••• ,m} distinct integers (b1,...,b m-k )∈{1,••• ,m}\{a1,...,a k } distinct integers i m λ a1 (0) • • • λ a k (0) (dz a1 ∧ dz ā1 )∧... ∧ (dz a k ∧ dz āk ) ∧ (dz b1 ∧ dz b1 ) ∧ ... ∧ (dz b m-k ∧ dz b m-k ) Now a 1 , ..., a k , b 1 , ..., b m-k are m distinct integers of {1, • • • , m} and 2-forms commute therefore, ωk ∧ ω m-k = (a1,...,a k )∈{1,••• ,m} distinct integers (b1,...,b m-k )∈{1,••• ,m}\{a1,...,a k } distinct integers λ a1 (0) • • • λ a k (0) i m (dz 1 ∧ dz 1) ∧ ... ∧ (dz m ∧ dz m) ...,a k )∈{1,••• ,m} distinct integers

  ϕ and h is an increasing function, then T 2 ≥ 0. Let us now calculate T 1 . Fix x ∈ M and let us work in a g-unitary chart centered at x and satisfying dϕ |dϕ|g = dz m +dz m √ 2 at x. We have then ω = idz a ∧ dz ā at x and t Jdϕ ∧ dϕ = i | dϕ | 2 g dz m ∧ dz m, therefore :

  IN1) Let us infer from the inequality (IN1) an other inequality (the inequality (IN4) below) that is required for the proof. It follows from the continuous Sobolev embedding H 2 1 (M ) ⊂ L 2m m-1 (M ) that :

  IN3) Using the inequalities (IN 1), (IN 2), (IN 3) and p 2 2(p-1) ≤ p we obtain :

3 ( 3 R) 6 . 5 . 7 1 )

 336571 R jrsi + R jisr ) ∂ r ϕ ∂ s ϕ + δ j i (∂ ii ϕ) 1,s (λ(P 1 )) σ k (λ(P 1 )) 1 1ss1 + R 1(s+m)(s+m)1 ∂ ss ϕ + ∂ (s+m)(s+m) ϕ (70The uniform upper bound of β 1 = (∇ 2 ϕ) P1 (ξ 1 , ξBy the uniform estimate of the gradient we have :| ∂ j ϕ t |≤ C 5 for all 1 ≤ j ≤ 2m. Moreover, at P 1 in the chart ψ 1 : [(∇ 2 ϕ) ij (P 1 )] 1≤i,j≤2m = [∂ ij ϕ(P 1 )] 1≤i,j≤2m = diag(β 1 , ..., β 2m ). Consequently :

∂

  (r ss + r (s+m)(s+m) ) ∂r ij(71)

2ma

  ,b,c,d=1 (R abcd ) 2 then | R abcd |≤ R g for all a, b, c, d ∈ {1, ..., 2m}, consequently : P 1 in the chart ψ 1 , we have : ∇R 2 g = g el g ai g bj g cr g ds ∇ e R abcd ∇ l R ijrs = 2m e,a,b,c,d=1 (∇ e R abcd ) 2 , so | ∇ e R abcd |≤ ∇R g for all e, a, b, c, d ∈ Complex Hessian Equations on Some Compact Kähler Manifold 33 {1, ..., 2m}, therefore :

  1 2 ∂ x s x s ϕ t (P ) hence as well | ∂ x r x s ϕ t (P ) |≤ 2 C 6 + C 7 . Similarly we prove that at P , in the above chart ψ P , we have : | ∂ y r y s ϕ t (P ) |≤ 2 C 6 + C 7 for all 1 ≤ r = s ≤ m and | ∂ x r y s ϕ t (P ) |≤ 2 C 6 +C 7 for all 1 ≤ r, s ≤ m. Consequently : | ∂ u i u j ϕ t (P ) |≤ 2 C 6 +C 7

  1 j1 , ∂ i Γ j 11 , ∂ j Γ i 11 , ∂ i Γ r js and ∂ ij Γ s 11 using the components of the Riemann curvature (at the point P 1 in Complex Hessian Equations on Some Compact Kähler Manifold 31 the normal chart ψ 1 ) :

  t∂ 11 f -t But | ∂ 11 f (P 1 ) |≤ f C 2 (M ) , | ∂ s f (P 1 ) |≤ f C 2 (M ) and | ∂ s ϕ |≤ C 5for all s then :

					2m s=1	∂ s f ∂ s ϕ ≥	2m i=1	∂F ∂r ii	[ϕ] (β i ) 2 +	2 3	R 1i1i (β 1 -2 β i )
	+	1 6	m i=1	σ k-1,i (λ(P 1 )) σ k (λ(P 1 ))	× R 1ii1 + R 1(i+m)(i+m)1	β i + β i+m
	+	2m i=1	∂F ∂r ii	[ϕ] × -	4 3	m 2 (C 5 ) 2 R g -	7 3	m C 5 ∇R g	(75)

u 1 u s -2 ∂ u 1 Γ u s+m u 1 u s+m . Besides, ∂ ss ϕ = 1 4 (∂ u s u s ϕ+∂ u s+m u s+m ϕ), which infers that at P 1 :

Consequently, the inequality (57) becomes :

6.5.6 Differentiation of the functional Φ 1

We differentiate twice the functional Φ 1 :

Hence, at P 1 in the chart ψ 1 , we obtain :

) Let us now calculate the different terms of this formula (at P 1 in the chart ψ 1 ) :

Since λ j (P ) ≥ -(m -1)C ′ 2 , we infer the following inequalities :

Consequently :

Hence we infer that :

, consequently we obtain :

Thus :

Besides :

Hence :