HOLOMORPHIC MORSE INEQUALITIES ON Q

HOLOMORPHIC MORSE INEQUALITIES ON

-CONVEX MANIFOLDS

after the work of Thierry Bouche

by Jean-Pierre DEMAILLY

Universit´e Joseph Fourier Grenoble I, Institut Fourier, BP 74,

Laboratoire associ´e au C.N.R.S. n^o188, F-38402 Saint-Martin d’H`eres

1. Statement of results and applications.

This note is a report about a recent work of Thierry Bouche (Univ. Joseph Fourier, Grenoble). Bouche [Bo] obtained an extension of the holomorphic Morse inequalities proved in Demailly [D3] for strongly q–convex manifolds.

LetX be a complex manifold of dimensionn. We assume thatX is strongly q–convex in the sense of Andreotti-Grauert [A-G], i.e. that there exists a C^∞ exhaustion function ψ on X such that the complex Hessian id^′d^′′ψ has at least n−q+1 positive eigenvalues outside a compact subset ofX. In this case, Andreotti- Grauert’s theorem [A-G] asserts that all cohomology groupsH^m(X,F) with values in a coherent analytic sheaf are finite dimensional for m>q .

LetL−→X be a holomorphic line bundle with a hermitian metric of class C^∞ and E −→ X a holomorphic vector bundle of rank r . Let D = D^′ +D^′′

be the Chern connection of L and c(L) = D² the associated curvature form. We denote by X(m, L) the set of points of X where the real (1,1)–form ic(L) is non degenerate and has exactly m negative eigenvalues. We also set

X(6m, L) = [

ν6m

X(ν, L) , X(>m, L) = [

ν>m

X(ν, L) .

Then the cohomology groups H^m(X, E⊗L^k) are finite dimensional for m > q , and the following theorem gives an asymptotic estimate of dimH^m(X, E⊗L^k) as k tends to +∞ .

1.1. Theorem. — Assume that X is strongly q–convex and that the curvature form ic(L) has at least n− p + 1 nonnegative eigenvalues outside a compact subset of X . Then for all m > p + q − 1 the following asymptotic inequalities hold :

(a_m) Weak Morse inequalities : dimH^m(X, E⊗L^k)6rkⁿ

n!

Z

X(m,L)

(−1)^m i

2πc(L)n

+ o(kⁿ) . (b_m) Strong Morse inequalities:

X

m6ν6n

(−1)^ν−mdimH^ν(X, E⊗L^k)6rkⁿ n!

Z

X(>m,L)

(−1)^m i

2πc(L)n

+ o(kⁿ) .

Observe that (a_m) is obtained simply by adding (b_m) and (b_m+1). The special case when X is compact (p = q = 0) has been first proved in [D3]. Then (a_m) and (b_m) are valid for all m > −1 . As the strong Morse inequalities (b₀) , (b₋₁) are identical except for the sign of their terms, we see that the case m= 0 is an asymptotic equality ; this equality is in fact a weak form of the Riemann- Roch-Hirzebruch formula for the Euler-Poincar´e characteristic χ(X, E ⊗ L^k) . Subtracting inequality (b₂) from (b₀) , we get the lower bound

(1.2)

dimH⁰(X, E⊗L^k)−dimH¹(X, E⊗L^k)>rkⁿ n!

Z

X(61,L)

i

2πc(L)n

+ o(kⁿ) . This result is used in [D3] in order to obtain a new proof of the Grauert- Riemenschneider conjecture [G-R], which had been solved in the affirmative by Y.T. Siu [S2] soon before. More precisely, we get the following stronger version of the conjecture.

1.3. Theorem. — LetXbe a connectedn–dimensional compact manifold.

If X carries a hermitian line bundle L such thatR

X(61,L) ic(L)n

>0, then X is Moiˇsezon.

Let us recall that the algebraic dimension a(X) is by definition the trans- cendence degree of the field of meromorphic functions of X . This degree always verifies 0 6 a(X) 6 n and X is said to be a Moiˇsezon variety if a(X) = n . Inequality (1.2) shows that dimH⁰(X, L^k) > ckⁿ for some constant c > 0 and k > k₀; by a standard method of Poincar´e, Serre [Se] and Siegel [Si], it follows that the field of meromorphic functions generated by quotients of sections in H⁰(X, L^k) , k ∈N , has transcendence degree >n , hence a(X) =n .

Another consequence of Morse inequalities is a general a priori estimate for Monge-Amp`ere operator (id^′d^′′)ⁿ on q–convex manifolds.

1.4. Theorem. — Let X be a strongly q–convex manifold and ϕ a C^∞ function onX, weaklyp-convex outside a compact subsetK ⊂X . For06ν 6n,

let Gν be the set of points of X where id^′d^′′ϕ is non degenerate and admits ν negative eigenvalues. Then for all m>p+q−1 one has

X

m6ν6n

Z

Gν

(−1)^m(id^′d^′′ϕ)ⁿ>0 .

This result has been first obtained by Y.T. Siu [S4] for q–convex domains Ω in a Stein manifold X . Siu’s method is rather sophisticated and rests merely upon the compact case of theorem 1.1 : X is “approximated” by affine algebraic manifolds X_alg , and the proof is then reduced to an application of Morse inequalities on a compactification of X_alg . The general statement given above appears in fact as a straightforward consequence of the q–convex case of theorem 1.1 : take for L the trivial bundle X×C endowed with the curvature form defined by the weight e^−ϕ . Then ic(L) = id^′d^′′ϕ . Since H^m(X, L^k) = H^m(X,C) is independent of k and finite dimensional for m>q , theorem 1.1 (b) implies

kⁿ X

m6ν6n

Z

Gν

(−1)^m(id^′d^′′ϕ)ⁿ >constant for all k >k₀ and m> p+q−1 , whence the result.

Following Th. Bouche [Bo], we give now an account of the main ideas occurring in the proof of theorem 1.1. We refer to [Bo] and to [D3] for the details.

2. The Bochner technique.

The cohomology groups H^m(X, E ⊗ L^k) are computed by means of the Dolbeault complex D^′′ : C_0,•^∞(X, E ⊗L^k) , where C_s,t^∞(X, E) denotes the space of C^∞ sections of the bundle Λ^s,tT^⋆X ⊗E . This complex is isomorphic to the complex C_n,•^∞ (X,Ee⊗L^k) with Ee = ΛⁿT X , thus after changing E into Ee , we can work with forms of bidegree (n, m) instead of (0, m) (note that the estimates of theorem 1.1 do not depend on E except for the rank r).

Assume that the bundleE is endowed with aC^∞ hermitian metric, and let D =D^′+D^′′ , c(E) = D² be the corresponding Chern connection and curvature form. Let ω be a complete hermitian metric onX (to be chosen later in a suitable way), h?,?i the pointwise inner product of differential forms on X with values in E and hh?,?ii the global L² inner product obtained after integration with respect to the volume form dV =ωⁿ/n! . The corresponding norms are denoted |?| and

||?||, and Λ stands as usual for the (pointwise) adjoint of the wedge multiplication by ω . Finally, we denote by δ=δ^′+δ^′′ the formal adjoint ofD .

If A and B are endomorphisms of respective degrees a, b of the graded module C_•^∞(X, E) , we set [A, B] = AB − (−1)^abBA . The holomorphic and conjugate-holomorphic Laplace-Beltrami operators are then defined by

∆^′ = [D^′, δ^′] , ∆^′′ = [D^′′, δ^′′] .

These operators are related by a formula of Weitzenb¨ock type, known as the Bochner-Kodaira-Nakano identity ; we refer to [D1] for the explicit formulation given below.

2.1. Proposition. — Let τ = [Λ, d^′ω] , let τ^⋆ be its pointwise adjoint,

∆^′_τ = [D^′+τ, δ^′+τ^⋆] and T_ω = [Λ,[Λ,₂ⁱd^′d^′′ω]]−[d^′ω,(d^′ω)^⋆] . Then

∆^′′ = ∆^′_τ + [ic(E),Λ] +Tω .

Now, assume that ψ is an exhaustion function on X that is strongly q–

convex on X\K and that ic(L) has at least (n−p+ 1) nonnegative eigenvalues on X \ K , for some compact set K in X . In order to take into account the q–convexity of X in formula 2.1, we multiply the metric of L by a weight e^−χ◦ψ where χ is a convex increasing function. The resulting hermitian line bundle is denoted L_χ . Formula 2.1 is now applied to the bundle E⊗L^k_χ , whose curvature form is c(E⊗L^k_χ) =c(E) +kc(L_χ)⊗Id_E . For every compactly supported smooth form with values in E⊗L^k one gets

(2.2) hh∆^′′u, uii=hh∆^′_τu, uii+ Z

X

kh[ic(L_χ),Λ]u, ui+hΘu, ui dV

where Θ = [ic(E),Λ] +T_ω is a C^∞ endomorphism independent of χ and k . Now, we havehh∆^′′u, uii =||D^′′u||²+||δ^′′u||² and a similar formula for ∆^′_τ , in particular hh∆^′_τu, uii is nonnegative.

For every c∈R , we consider the sublevel sets Xc ={x∈X; ψ(x)< c} .

Select c₀ such thatK ⊂X_c0 . One can choose a hermitian metricω⁰ onX in such a way that the eigenvalues γ₁⁰6. . .6γ_n⁰ of id^′d^′′ψ with respect to ω⁰ verify

−1/n6γ₁⁰ 6. . .6γ_q−1⁰ 61 and γ_q⁰ =. . .=γ_n⁰ = 1 on X\Xc0 ; this can be achieved by taking ω⁰ equal to id^′d^′′ψ on a C^∞ subbundle of T X of rank n−q + 1 on which id^′d^′′ψ is positive, and ω⁰ large on the orthogonal complement. We select ω = e^ρω₀ where ρ is a function increasing fast enough at infinity so thatω is complete. Let α be the lowest eigenvalue ofic(L) with respect to ω . The eigenvalues β₁ 6. . .6β_n of

ic(L_χ) =ic(L) +id^′d^′′(χ◦ψ)>ic(L) +χ^′◦ψ id^′d^′′ψ with respect to ω verify the inequality

βj >α+χ^′◦ψ γ_j⁰/e^ρ . Let u be a (n, m)–form. A standard computation gives (2.3) h[ic(L_χ),Λ]u, ui>(β₁+. . .+β_m)|u|² . Moreover, we have

β₁+. . .+β_m >mα+e^−ρχ^′◦ψ(γ₁⁰+. . .+γ_m⁰ )>α+ 1

ne^−ρχ^′◦ψ on X\X_c0

because γ_j⁰ > −1/n and γ_m⁰ = 1 (m > q) . It follows that one can choose χ increasing very fast in such a way that (2.2) and (2.3) imply

(2.4) hh∆^′′u, uii>k Z

X\Xc0

A|u|²dV −C Z

Xc0

|u|²dV ,

where A >1 is a function tending to +∞ at infinity on X and where C is a large constant. The completeness of the metricω shows that inequality (2.4) is still valid for any u in the domain of ∆^′′ acting on the Hilbert space L²_n,m(X, E⊗L^k_χ) .

2.5. Lemma. — When χ increases sufficiently fast at infinity, the space H^n,m(X, E⊗L^k_χ) of ∆^′′–harmonic forms inL²_n,m(X, E⊗L^k_χ)is isomorphic to the cohomology group H^m(X,ΛⁿT^⋆X⊗E⊗L^k) for all k ∈Nand m>q .

AsAtends to +∞in inequality (2.4), Rellich’s lemma easily shows that ∆^′′

has a compact resolvent. Hence the spectrum of ∆^′′ is discrete and its eigenspaces are finite dimensional. When X is compact, lemma 2.5 is true for all m > 0 and for any choice of χ (a very well-known fact of Hodge theory). The q–convex case is slightly more involved; it rests essentially on the above finiteness result and on an approximation theorem of Runge type deduced from (2.4) (cf. for example [D2]).

3. Witten’s complex.

We come now to one of the main ideas introduced by E. Witten [Wi]. This idea led Witten to an analytic new proof of the standard Morse inequalities for Betti numbers of a compact differentiable manifold. In our case, we consider the Dolbeault complex instead of the De Rham complex, but the method is similar by many aspects.

Since D^′′ commutes with ∆^′′ , the sequence of eigenspaces E^m(λ) ⊂ L²_n,m(X, E⊗L^k_χ) corresponding to a given eigenvalueλ is asubcomplexof the Dol- beault complexD^′′ :C_n,•^∞ (X, E⊗L^k) . Moreover we haveD^′′δ^′′+δ^′′D^′′ = ∆^′′ =λId onE^•(λ) , thus _λ¹δ^′′ is a homotopy operator forD^′′ andE^•(λ) is acyclic forλ 6= 0 . For any λ >0 , let

(3.1) H^m_X,k(λ) , m>q ,

be the sum of the eigenspaces of ∆^′′ corresponding to eigenvalues 6 kλ (the reason for this choice will become apparent later). By what we have said at the end of §2, the sequence H^m_X,k(λ) , m > q , is a finite dimensional subcomplex of the Dolbeault complex. Its cohomology groups are equal to the desired groups H^m(X,ΛⁿT^⋆X⊗E⊗L^k) , except perhaps form=q . In order to get the correct group for m=q, one possibility is to add an extra term

H^q−1_X,k(λ) =δ^′′

cycles in H^q_X,k(λ) .

The following elementary lemma of homological algebra is then involved.

3.2. Lemma. — Let 0 −→ C^{0 d}

0

−→ C^{1 d}

1

−→ · · · −→ Cⁿ −→^dn 0 be a complex of finite dimensional vector spaces of dimensions c^q over a field K . Let h^q = dimH^q(C^•) . Then for every index m the following “strong Morse

inequalities” hold :

h^m−h^m+1+· · ·+ (−1)^n−mhⁿ 6c^m−c^m+1+· · ·+ (−1)^n−mcⁿ .

In our case, C^m = H^m_X,k(λ) and H^m(C^•) = H^m(X,ΛⁿT^⋆X ⊗E ⊗L^k) for m > q . The proof of theorem 1.1 is therefore reduced to finding an asymptotic estimation of the dimensions dimH^m_X,k(λ) . The first step is to show that this can be done after replacing X by relatively compact open sets Ω⊂⊂X .

For a domain Ω⊂⊂X , we consider the quadratic form Q^m_Ω,k(u) = 1

k Z

Ω

|D^′′u|²+|δ^′′u|² dV

associated to _k¹∆^′′ with Dirichlet conditions on∂Ω ; this means that the domain of Q^m_Ω,k(u) is the Sobolev space W₀¹(Ω,Λ^n,mT^⋆X⊗E⊗L^k) of sections which are in L²(Ω) so as their first derivatives, and which can be approximated in this topology by smooth sections with compact support in Ω . We denote by H_Ω,k^m (λ) the direct sum of all eigenspaces of Q^m_Ω,k corresponding to eigenvalues6λ(i.e.6kλ for ∆^′′).

3.3. Lemma. — For everyλ >0andε >0, there exists a domainΩ⊂⊂X and an integer k0 such that

dimH^m_Ω,k(λ)6dimH^m_X,k(λ)6dimH^m_Ω,k(λ+ε) for k >k₀ .

Proof. — As we work on a complete manifold, the eigenvalues of ¹_k∆^′′ on L²_m,n(X, E⊗L^k_χ) are the same as those of the quadratic formQ^m_X,k . The left hand inequality is then a consequence of the minimax principle, because the domain of Q^m_Ω,k is contained in that of Q^m_X,k (the sections of W₀¹(Ω) can be extended by 0 on X \Ω).

For the other inequality, we proceed in several steps. Let u ∈ H^m_X,k(λ) . Then

hh∆^′′u, uii=||D^′′u||²+||δ^′′u||² 6kλ||u||² , and a combination of this estimate with (2.4) yields

k Z

X\Xc0

A|u|²dV −C Z

Xc0

|u|²dV 6kλ Z

X

|u|²dV .

Consider the compact set P = X_c0 ∪ {A(x)6 a} . Then A(x) > a on X \P and

we find Z

X\P

|u|²dV 6 C+kλ ka

Z

X

|u|²dV . If we choose a large enough, we get R

X\P |u|²dV 6 ε||u||² . Let ϕ ∈ D(X) be a cut-off function equal to 1 on P and Ω⊂⊂X an open set containing the support of ϕ . The sectionϕu is in the domain of Q^m_Ω,k and we have

Q^m_Ω,k(ϕu) = 1 k

Z

Ω

|d^′′ϕ∧u+D^′′u|²+|δ^′′u−d^′ϕ u|² dV 6(1 +ε)Q^m_X,k(u) + C^′

k (1 + 1/ε)||u||²

6

(1 +ε)λ+ C^′

k (1 + 1/ε)

||u||² .

As ||ϕu||² >R

P |u|²dV >(1−ε)||u||² , we infer Q^m_Ω,k(ϕu)61 +ε

1−ελ+ C(ε) k

||ϕu||² .

If ε is replaced by a smaller number and k taken large enough, we obtain Q^m_Ω,k(v) 6 (λ+ ε)||v||² when v describes the space ϕH^m_X,k(λ) . The right hand inequality in lemma 3.3 follows by the minimax principle.

In order to handle properly the quadratic form Q^m_Ω,k , we need another calculation derived from the Bochner-Kodaira-Nakano identity (2.2) (cf. [D3]) : (3.4) 2Q^m_Ω,k(u) =

Z

Ω

1

k|∇u+Su|²− hV u, ui+ 1

khΘu, ui dV ,

where ∇ is the canonical hermitian connection on the bundle Λ^n,mT^⋆X⊗E⊗L^k_χ deduced from the Chern connections on ΛT^⋆X ,E and Lχ . The operators S and Θ are hermitian endomorphisms which act only on the component Λ^n,mT^⋆X⊗E and are independent of k . Finally V is a curvature endomorphism acting only on the component Λ^n,mT^⋆X . Let (ξ₁, . . . , ξ_n) , (e₁, . . . , e_r) and l be orthonormal frames of T X , E , L_χ respectively, and assume that ic(L_χ) is diagonalized in (ξ_j) . One finds

u= X

|J|=m,16h6r

u_J,hξ₁^⋆∧. . .∧ξ_n^⋆ ∧ξ^⋆_J ⊗e_h⊗l^k , hV u, ui= X

|J|=m,16h6r

(β∁J −βJ)|uJ,h|² , (3.5)

where β₁, . . . , β_n are the eigenvalues of ic(L_χ) and β_J =P

j∈Jβ_j .

4. A spectral theorem for Schr¨ odinger operators.

The main tool for the proof of Morse inequalities is a spectral theorem which describes very precisely the asymptotic distribution of eigenvalues for quadratic forms similar to Q^m_Ω,k .

LetM be an–dimensional Riemannian manifold of classC^∞andF −→M , L −→M hermitian vector bundles of respective ranks t and 1 . We assume that F, L are endowed with hermitian connections ∇_E , ∇_L , and we let ∇_k be the resulting connection on F⊗L^k . Finally let V andS be hermitian endomorphisms of F; we still denote by V and S their extensions V ⊗Id , S⊗Id to F ⊗L^k . For any domain Ω⊂⊂M , we consider the quadratic forms

(4.1) QΩ,k(u) = Z

Ω

1

k|∇ku+Su|²− hV u, ui

dV , u∈W₀¹(Ω, F ⊗L^k) with Dirichlet condition on ∂Ω . For every real number λ , we denote by N_Ω,k(λ) the number of eigenvalues of Q_Ω,k which are 6λ; these eigenvalues are of course counted with multiplicity. We shall see soon that N_Ω,k(λ) can be estimated in terms of the curvature of L and of the potential V .

The curvature formB =ic(L) =i∇²_L is a real 2–form on X; at each point x ∈M , this form can be written

(4.2) B(x) = X

16j6s

B_j(x)ξ_2j−1^⋆ ∧ξ_2j^⋆ , s6n/2

where (ξ^⋆₁, . . . , ξ_n^⋆) is an orthonormal basis of T_x^⋆M and B₁(x) 6 . . .6 B_s(x) >0 the modules of the non zero eigenvalues of B(x) , viewed as an antisymmetric endomorphism on T_xM . Let

(4.3) ν_B(λ) = 2^s−nπ^−n/2

Γ ⁿ₂ −s+ 1B₁. . . B_s X

(p¹,...,ps)∈N^s

λ− X

16j6s

(2p_j+ 1)B_j(n/2)−s

+ .

Here a+ means the maximum ofaand 0 , with the special convention that a⁰₊= 0 for a 60 anda⁰₊ = 1 fora >0 . It is clear that ν_B(λ) is a non decreasing function of λ that is continuous on the left side; one can also check that ν_B(x)(λ) is lower semi-continuous in x . Let ν_B(λ) = lim_ε→0+ν_B(λ) be the associated increasing function continuous on the right side. The asymptotic eigenvalue distribution of Q_Ω,k is then given by the following fundamental theorem.

4.4. Theorem. — LetV₁(x), . . . , V_t(x)be the eigenvalues of the endomor- phism V(x)∈End(F_x) . Then

lim inf

k→+∞ k^−n/2NΩ,k(λ)> X

16j6t

Z

Ω

νB(Vj +λ)dV , lim sup

k→+∞

k^−n/2N_Ω,k(λ)6 X

16j6t

Z

Ω

ν_B(V_j +λ)dV .

Assume that∂Ω has measure zero. As the integrals are increasing functions of λ and as the upper bound is the limit of the lower bound at λ+ 0 , we easily conclude :

4.5. Corollary. — There is a countable subset D ⊂R such that

N_Ω,k(λ)∼k^n/2 X

16j6t

Z

Ω

ν_B(V_j +λ)dV , ∀λ ∈R\ D .

The proof of theorem 4.4 is made in several steps. The first step is a localization procedure originated with the work of H. Weyl. Weyl [We] introduced the localization procedure and the minimax principle to get the asymptotic estimate of the distribution of eigenvalues of linear partial differential equations.

Assume that Ω is partitioned into a union of small cubes plus some extra set of small measure. The intuition is that for the Dirichlet problem with zero boundary value, the wave length of all eigenfunctions tends to 0 ask tends to infinity and the Dirichlet problem looks more and more like the union of the Dirichlet problems for the small cubes. If the side of the cubes is well chosen (k^−1/3 is a suitable scale)

the problem can be approximated by a simpler one in which the potential V and the curvature B are constant. At this point, an expansion of u as a Fourier series shows that the variables can be separated. Our Dirichlet problem is then reduced to the 1–dimensional harmonic oscillator problem, or to the Dirichlet problem for the operator−d²/dx² on an interval. In these cases, the eigenvalues and eigenfunctions can be computed explicitly. The general estimate follows.

To get the eigenvalue distribution ofQ^m_Ω,k in (3.4), we apply corollary 4.5 to the bundles F = Λ^n,mT^⋆X×E , L=L_χ and note that dim^RX = 2n. When the eigenvalues β_j are ordered in such a way that |β₁|>. . .>|β_s|>0 =β_s+1 =. . .= β_n , we get B_j = |β_j| . By (3.5), the eigenvalues V_j are the quantities β_∁J −β_J when J describes all multi-indices of length |J| = m , counted with multiplicity r . Hence the eigenvalue distribution N_Ω,k^m (λ) of Q^m_Ω,k(λ) is given by

N_Ω,k^m (λ)∼rkⁿ X

|J|=m

Z

Ω

ν_B(2λ+β_∁J −β_J)dV , ∀λ∈R\ D .

We use this estimate when λ tends to 0+ . Each integral converges to the limit R

Ων_B(β_∁J −β_J)dV and we have ν_B(β_∁J −β_J) = 2^s−2nπ⁻ⁿ

Γ(n−s+ 1)|β₁. . . β_s|×

X

(p¹,...,ps∈N^s)

β_∁J −β_J − X

16j6s

(2p_j + 1)|β_j|n−s

+ .

The expression between ( )₊ is always 6 0 , and it can be zero only when p1 = . . . =ps = 0 , βj 60 for j ∈J and βj > 0 for j ∈ ∁J . By our conventions lim_λ→0+ (λ)⁰₊ = 1 and the result is non zero only whens =n. The above function is therefore equal to zero in all cases except when β_j <0 for j ∈J and β_j >0 for j ∈∁J; then

ν_B(β_∁J −β_J) = (2π)⁻ⁿ|β₁. . . β_n| .

This situation can only happen at a point x∈ X(m, Lχ) , and the corresponding multi-index J is then unique. We find therefore

λ→0+lim lim

k→+∞ k⁻ⁿN_Ω,k^m (λ) =r Z

X(m,Lχ)

(2π)⁻ⁿ|β1. . . βn|dV

= r n!

Z

X(m,Lχ)

(−1)^m i

2πc(L_χ)n

dV .

This shows that theorem 1.1 is valid for all m > q when the curvature form ic(L) is replaced by ic(Lχ) . Let K ⊂⊂ X be the exceptional compact set where the convexity properties of ψ and L are not satisfied. Take a convex increasing function χ such that χ= 0 on ]− ∞, c0] with Xc0 ⊃K . Then

ic(L_χ) =ic(L) +id^′d^′′(χ◦ψ)

coincides withic(L) on K and has at leastn−(p−1)−(q−1) positive eigenvalues on X\K , because ic(L) andid^′d^′′ψ are both positive on subspaces of respective codimensions p−1, q−1 in T X . Hence X(m, L_χ) =X(m, L) for m>p+q−1 and ic(L_χ) =ic(L) on these sets. Theorem 1.1 is proved.

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