An eigenvalue estimate for the ∂ ¯ -Laplacian associated to a nef line bundle
aa r X i v : . [ m a t h . C V ] S e p AN EIGENVALUE ESTIMATE FOR THE ¯ ∂ -LAPLACIAN ASSOCIATED TO A NEF LINEBUNDLE Jingcao Wu
Abstract.
We study the ¯ ∂ -Laplacian on forms taking values in L k , a high power of a nef line bundle on a compact complex man-ifold, and give an estimate of the number of the eigenforms whosecorresponding eigenvalues smaller than or equal to λ . In partic-ular, the λ = 0 case gives an asymptotic estimate for the orderof the corresponding cohomology groups. It helps to generalizethe Grauert–Riemenschneider conjecture. At last, we discuss the λ = 0 case on a pseudo-effective line bundle. Introduction
Let X be a compact complex manifold of dimension n , and let L bea holomorphic line bundle on X . Fix Hermitian metrics on X and L respectively, the classic geometry theory allows us to define the adjointoperator ¯ ∂ ∗ of the ¯ ∂ -operator acting on L -valued forms as well as thecorresponding Laplacian operator∆ = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂. The theory of the elliptic partial differential operator applies on thisLaplacian operator, and it has a good counterpart in geometry due toHodge’s theorem. More specifically, the classic Hodge’s theorem saysthat any class [ α ] in the Dolbeault cohomology group H p,q ( X, L ) ownsa unique harmonic representative ˜ α , i.e. ˜ α ∈ [ α ] and ∆ ˜ α = 0. Inother word, if we denote the space of harmonic L -valued ( p, q )-formsby H p,q ( X, L ), we have H p,q ( X, L ) ≃ H p,q ( X, L ) . It is worth to mention that when L is ample, it leads to excellent resultsin geometry, such as the hard Lefschetz theorem.In this paper, L is taken as a nef line bundle, and we are interestedin the quantity of eigenforms of the Laplacian on L . The work in thisaspect dates back to [1]. Indeed, if we denote the linear space of the L -valued ( n, q )-eigenforms of ∆, with corresponding eigenvalues smallerthan or equal to λ by H n,q λ ( X, L ) , t is given in [1] an asymptotic estimate for h n,q λ ( L k ⊗ E ) = dim H n,q λ ( X, L k ⊗ E )in the case that L is a semi-positive line bundle and E is a line bundleon X . Also it is shown in [1] through an example (Proposition 4.2)that this estimate is sharp.In this paper, we will prove a similar result when L is nef. We firstintroduce a canonical way to define the Laplacian on a nef line bundlein the text as well as the eigenform space H p,q λ (Definition 2.1, Sect.2).Let h n,q λ ( L k ⊗ E ) = dim H n,q λ ( X, L k ⊗ E ) . Then we define the so-called modified ideal sheaf (Definition 2.2,Sect.2). We give a brief introduction here for readers’ convenience.Notice that when L is nef, for any ε >
0, there exits a smooth metric h ε on L such that i Θ L,h ε > − εω . Here ω is a Hermitian metric on X .So there is an L -limit h of h ε (after passing to a subsequence) with i Θ L,h >
0. In the rest part, h will always refer to such a metric.Now we furthermore assume that h has analytic singularities. In thiscase the associated multiplier ideal sheaf can be computed as followsin [8]: I ( h ) = µ ∗ O ˜ X ( X ( ρ j − ⌊ aλ j ⌋ ) D j ) . Here µ : ˜ X → X is a log-resolution, and ρ j , a, λ j are involved realnumbers. ⌊ aλ j ⌋ means the round down. The precise meaning of thesenotations will be clarified in the text. The modified ideal sheaf is thendefined as I ( h ) := µ ∗ O ˜ X ( X ( −⌈ aλ j ⌉ ) D j ) . ⌈ aλ j ⌉ refers to the round up. We will also denote it by I ( L ). It is nothard to see that I ( L ) = I ( ψ )for another function induced by h . For any vector bundle E we define H p,q λ ( L k ⊗ E ⊗ I ( L k )) := { α ∈ H p,q λ ( L k ⊗ E ); Z X | α | e − ψ < ∞} . Then we prove that
Theorem 1.1.
Let L be a nef line bundle on X , and let E be a vectorbundle. Assume that h has analytic singularities. Take q > and m > . Then, if λ k , (1) h n,q λ ( L k ⊗ E ⊗ I ( L k )) C ( λ + 1) q k n − q . If k λ , then (2) h n,q λ ( L k ⊗ E ⊗ I ( L k )) Ck n . ince E is allowed to be an arbitrary vector bundle, we see by sub-stituting E ⊗ Ω pX ⊗ K − X for E , that the same asymptotic estimate alsoholds for the numbers h p,q λ .This kind of estimate has an important application in geometry,especially the λ = 0 case. In fact, H n,q ( X, L k ⊗ E ) is just the space ofthe harmonic L k ⊗ E -valued ( n, q )-forms on X , which is isomorphic tothe Dolbeault cohomology group H n,q ( X, L k ⊗ E ) . This isomorphism will be proved in next section (Proposition 2.1,4.1).It can be seen as a singular version of Hodge’s theorem. In particular, H n,q ( X, L k ⊗ E ⊗ I ( L k )) is just the image of the natural morphism i n,q : H n,q ( X, L k ⊗ E ⊗ I ( L k )) → H n,q ( X, L k ⊗ E ⊗ I ( L k )) . Based on this isomorphism, we eventually get an asymptotic estimateas follows.
Theorem 1.2.
Let L be a nef line bundle on X , and let E be a vectorbundle. Take q > . We have the following conclusions:1. Assume that h is bounded, then h ,q ( L k ⊗ E ) Ck n − q .
2. Assume that h has analytic singularities and i Θ L,h has at least n − s + 1 strictly positive eigenvalues at every point x ∈ X . Then dim Imi n,q Ck n − q for q > s .3. Assume that X is K¨ahler, and h has analytic singularities. Then dim Imi n,q Ck n − q . The asymptotic estimate for the order of the cohomology group is acomplicated problem in complex geometry. There are various work inthis aspect. Here we only list a few of them. The first result is that h ,q ( L k ) ∼ o ( k n ) , which is due to Siu [23, 24] when solving the Grauert–Riemenschneiderconjecture [16]. Later Demailly also gives that h ,q ( L k ⊗ E ) ∼ o ( k n )for a nef line bundle L and a vector bundle E on a compact K¨ahlermanifold based on his holomorphic Moser inequality [8]. Also we re-mark here that it can be optimized to h ,q ( L k ⊗ E ) ∼ O ( k n − q )when X is projective. Moreover, Matsumura [19, 20] generalizes it as h ,q ( L k ⊗ E ⊗ I ( h k )) ∼ O ( k n − q ) , here ( L, h ) is a pseudo-effective line bundle and E is a vector bundleon a projective manifold. Here I ( h ) refers to the multiplier ideal sheaf.Recently, this result has been extended to a compact complex manifoldby [27] with additional requirement that h has algebraic singularities.We remark here that [27] also considers the asymptotic estimate for h n,q λ ( L k ⊗ E ) when L is a semi-positive line bundle and E is a vectorbundle.Our result also provides such type of estimate on a compact complexmanifold, which is new.Similar with [1], we can use this estimate to solve the extensionproblem. As a result, we provide a generalization of the Grauert–Riemenschneider conjecture [16]. Theorem 1.3 (Generalization of the Grauert–Riemenschneider con-jecture) . Let X be a compact complex manifold, and let L be a nef linebundle on X . Assume that one of the following situations occurs.1. h is bounded.2. h has analytic singularities and i Θ L,h has at least n − strictlypositive eigenvalues at every point x ∈ X . i ,q is either injective orsurjective.3. X is K¨ahler, and h has analytic singularities. i ,q is eitherinjective or surjective.Then L is big iff ( L ) n > . The original Grauert–Riemenschneider conjecture says that if thereis a semi-positive line bundle L on a compact complex manifold X suchthat the curvature i Θ L > X must be Moishezon.It is well-known that X is a Moishezon manifold iff there exists a bigline bundle on X . So this conjecture actually says that if i Θ L is semi-positive and strictly positive on an open subset, L will be big. Thisconjecture has been solved by [6, 23].We also remark here that when X is a projective manifold, the con-clusion of Theorem 1.3 is a well-known result in algebraic geometry.Moreover, it has been extended to a K¨ahler manifold in [10] throughthe holomorphic Moser inequality. The method here is totally different.Theorem 1.3 has good applications. We only mention tow of them.Firstly, it partially solves Demailly–P˘aun’s conjecture posed in [10]. Conjecture 1.1 (Demailly–P˘aun) . Let X be a compact complex man-ifold. Assume that X possesses a nef cohomology class [ α ] of type (1 , such that R X α n > . Then X is in the Fujiki class C . Obviously, Theorem 1.3 partially confirms this conjecture when α isintegral.Next, we shall give a Nadel-type vanishing theorem. Theorem 1.4.
Let X be a compact K¨ahler manifold, and let L be anef line bundle. Assume that h has analytic singularities and provides nalytic Zariski decomposition. Let m be the dimension of the pole-setof h . Then we have H q ( X, K X ⊗ L ⊗ I ( L )) = 0 for q > max { n − κ ( L ) , m } . The proof of theorem 1.4 is similar with the main result in [19], butthe conclusion here is independent. Also we use two examples (Sect.2.4) to show that the requirement in Theorem 1.4 is not too demanding.Notice that in [3], it is shown another Nadel-type vanishing theoremsaying that if (
L, φ ) is pseudo-effective, then H q ( X, K X ⊗ L ⊗ I ( φ )) = 0for q > n − nd( L, φ ). Here nd(
L, φ ) is the numerical dimension of L associated with φ defined in [3]. Notice that we have κ ( L ) nd( L ) , where nd( L ) is the numerical dimension of L defined by intersectiontheory (of course without any specified metric). These two types ofnumerical dimension do not coincide well. Indeed the example in [11]shows that there do exists the case that nd( L ) > nd( L, φ min ) with φ min the minimal singular metric on L . So it seems to me that there is noobvious relation between nd( L, φ ) and κ ( L ). Therefore it is not clearcurrently that whether the work in [3] implies Theorem 1.4. Also weremark here that there is no obvious relationship between the work(Theorem 1.8) of [27] and Theorem 1.4.In final, we make a discussion on the eigenform space H n,q λ ( X, L )with λ = 0 for a pseudo-effective line bundle ( L, h ). More specifically,we will define H n,q ( X, L ) (Definition 5.1, Sect.5) and prove a singularversion of Hodge’s theorem (Proposition 5.1, Sect.5) when L is merelypseudo-effective.The plan of this paper is as follows. In Sect.2 we give a brief in-troduction on all the required materials including the nef line bundle,the modified ideal sheaf, Bergman kernel for the space H n,q λ , Siu’s ∂ ¯ ∂ -Bochner formula and so on. In Sect.3 we prove a submeanvalue in-equality for forms in H n,q λ and complete the proof of Theorem 1.1. InSect.4 we relate Theorem 1.1 to the asymptotic estimate for the co-homology group and give some applications. In the final section, weconsider the λ = 0 case for a pseudo-effective line bundle. Acknowledgment.
The author want to thank Prof. Bo Berndtsson, whointroduced and carefully explained this problem to him. Also the au-thor thanks Prof. Jixiang Fu for his suggestion and encouragement. . Preliminary
Nef line bundle.
Firstly, we briefly recall the multiplier idealsheaf. Let L → X be a line bundle on a compact complex manifold X .Let S := { h i } be a family of smooth metrics on L with weight functions { φ i } , such that R X e − φ i → ∞ as i tends to ∞ , and R V e − φ i C for someopen subset V of X . Then the Nadel-type multiplier ideal sheaf [21](or dynamic multiplier ideal sheaf) at x ∈ X can be defined as I ( S ) x := { f ∈ O X,x ; Z U | f | h i C as i → ∞} , where U is a local coordinate neighborhood of x .On the other hand, if h is a singular metric on L with the weightfunction φ , then its static multiplier ideal sheaf I ( h ) is defined in [8]by I ( h ) x := { f ∈ O X,x ; | f | h is integrable around x } . These two types of multiplier ideal sheaves coincide well when L is anef line bundle. Indeed, if L is nef, by definition there exists a familyof smooth metrics S = { h ε } such that i Θ L,h ε > − εω for any ε > ω is a Hermitian metric on X fixed before. Let φ ε be the weightfunction of h ε , then it is quasi-plurisubharmonic. Therefore { φ ε } islocally bounded in L -norm, hence relatively compact. So we can finda subsequence { φ ε i } converging to a limit φ in L -norm. In particular, i Θ L,φ >
0. Let h be the corresponding metric. In the rest part ofthis paper, h will always refers to this metric if not specified. Now if f ∈ I ( S ) x , we have Z U | f | e − φ = Z U lim i →∞ | f | h εi = lim i →∞ Z U | f | h εi < ∞ by dominate convergence theorem. It means that f ∈ I ( φ ) x . On theother hand, if g ∈ I ( φ ) x , it is easy to see that g ∈ I ( S ) x as well. Insummary, we have I ( S ) = I ( φ )when L is nef, and briefly denote it by I ( L ). It is also the start point ofour work. For more information about the multiplier ideal sheaf (thedynamic one and the static one), one could refer to [8, 21].Next we shall present a canonical way to define the Laplacian opera-tor associated to a nef line bundle L . First, by definition of the nef linebundle we have a family of smooth metrics S = { h ε } on L with weightfunctions φ ε . We take its convergent subsequence and still denote itby { h ε } . In particular, we have h ε h ε for any 0 ε ε , and the L -limit is denoted by h with weight function φ .Fix a Hermitian metric ω on X . Since h ε is a smooth metric on L ,we can define the Laplacian operator ∆ ε corresponds to ω and h ε in he usual sense. Now for any test L -valued ( p, q )-form α , we define theLaplacian operator associated to h by∆ α := lim ε → ∆ ε α in the sense of L -topology. It is easy to verify that the limit exitsif and only if h ε converges to h in L -norm, while the later has beenguaranteed. ∆ possesses some basic properties of the classic Laplacianoperator, such as ∆ ¯ ∂ = ¯ ∂ ∆ and self-adjointness, i.e. < ∆ α, β > h = < α, ∆ β > h for any α, β . It is just some basic calculation, so we omit the detailshere.There is one issue to be concerned. ∆ α may not be a smooth formeven if α is. So we need to carefully define the eigenvalue and eigenform.First, given two L -valued smooth ( p, q )-forms α, β on X , we say thatthey are Dolbeault cohomological equivalent (it may not be a standardconvention), if there exists an L -valued smooth ( p, q − γ suchthat α = β + ¯ ∂γ . It is easy to verify that it’s an equivalence relationship.We briefly denote it by β ∈ [ α ] and vice versa. In particular, if α or β is ¯ ∂ -closed, the Dolbeault cohomological equivalence just means thatthey belong to the same Dolbeault cohomology class. Now we have thefollowing definition. Definition 2.1.
Let α be an L -valued ( p, q )-form on X . Assume thatfor every ε ≪
1, there exists a Dolbeault cohomological equivalentrepresentative α ε ∈ [ α ] such that(1) ∆ ε α ε = µα ε . Here we ask that µ is independent of ε ;(2) α ε → α in L -norm.Then we call α an eigenform of the Laplacian operator ∆ with eigen-value µ . We simply denote it by ∆ α = µα .The eigenform space of ∆ is defined as H p,q λ ( X, L, ∆ ) := { α ∈ A p,q ( X, L ); ∆ α = µα and µ λ } . We are especially interested in the λ = 0 case since it corresponds tothe Dolbeaut cohomology group. In fact, given a Dolbeault cohomologyclass [ α ] ∈ H p,q ( X, L ), we have a unique ∆ ε -harmonic representative α ε ∈ [ α ] for every ε by Hodge’s theorem. Moveover, since the harmonicrepresentative minimizes the norm, we have k α ε k h ε k α k h ε . Assumethat k α k h ε C for all ε (which means that [ α ] ∈ H p,q ( X, L ⊗ I ( L ))).Then we can find a convergent subsequence of { α ε } with limit ˜ α , and˜ α ∈ [ α ]. Therefore ˜ α is an eigenform of the Laplacian operator ∆ witheigenvalue 0 by definition. In other word, we could say that ˜ α is ∆ -harmonic. We remark here that in general ˜ α is merely an L -bounded( p, q )-form. But if p = n , ˜ α must be smooth. Indeed, it is not hard to ee that ˜ α = c n − q (lim ε → ∗ α ε ) ∧ ω q . Using the Kodaira–Akizuki–Nakano formula, lim ε → ∗ α ε is a ¯ ∂ -closed( n − q, -harmonic form α must be ¯ ∂ -closed by def-inition, so it naturally defines a cohomology class[ α ] ∈ H p,q ( X, L ) . Obviously, if α, β are two different ∆ -harmonic forms, [ α ] = [ β ] in H p,q ( X, L ). In summary we have eventually proved a singular versionof Hodge’s theorem.
Proposition 2.1 (A singular version of Hodge’s theorem, I) . Let X bea compact complex manifold, and let L be a nef line bundle on X . Let ∆ be the Laplacian operator defined before. Then we have H n,q ( X, L, ∆ ) ⊂ H n,q ( X, L ) , H n,q ( X, L ⊗ I ( L ) , ∆ ) ≃ H n,q ( X, L ⊗ I ( L )) . (3) Proof.
Based on the discussions above, it only remains to show thatlim ε → ∗ α ε is ¯ ∂ -closed. Let’s recall the Kodaira–Akizuki–Nakano for-mula, which says k ¯ ∂α k h + k ¯ ∂ ∗ α k h = k ( ∂ h ) ∗ α k h + k ∂ h α k h + ( i [Θ L,h , Λ] α, α ) h for any α ∈ A n,q ( X, L ) and smooth metric h on L . Let h = h ε and α = α ε , we have k ¯ ∂α ε k h ε + k ¯ ∂ ∗ α ε k h ε = k ( ∂ h ε ) ∗ α ε k h ε + k ∂ h ε α ε k h ε + ( i [Θ L,h ε , Λ] α ε , α ε ) h ε . Since α ε is ∆ ε -harmonic, the left hand is zero. Take the limit withrespect to ε , we get that0 = lim ε → ( k ( ∂ h ε ) ∗ α ε k h ε + k ∂ h ε α ε k h ε ) + ( i [Θ L,h , Λ] α , α ) h ) . Since k ( ∂ h ε ) ∗ α ε k h ε , k ∂ h ε α ε k h ε and i Θ L,h are non-negative, we eventu-ally get thatlim ε → k ( ∂ h ε ) ∗ α ε k h ε = lim ε → k ∂ h ε α ε k h ε = ( i [Θ L,h , Λ] α , α ) h ) = 0 . In particular, lim ε → ( ∂ h ε ) ∗ α ε = ∗ ¯ ∂ (lim ε → ∗ α ε ) = 0 . It exactly implies that lim ε → ∗ α ε is ¯ ∂ -closed. (cid:3) The proof of Proposition 2.1 also shows that the ∆ -harmonic rep-resentative minimizes the L -norm defined by h .As is shown before, an element of H n,q ( X, L, ∆ ) must be ¯ ∂ -closed.For a general λ , we will see (in the proof of the main result) that the α ∈ H n,q λ ( X, L, ∆ ) with ¯ ∂α = 0 also plays an important role in theestimate of the number h n,q λ . hen X is K¨ahler, one could even parallel extend the other proper-ties in Hodge theory to this situation. However, it is not the theme ofthis paper, so we will leave it for the future.2.2. The modified ideal sheaf.
We will introduce a notion called themodified ideal sheaf in this subsection. Remember that for a singularmetric ϕ on L with analytic singularities, its (static) multiplier idealsheaf can be computed precisely. Indeed, suppose that ϕ ∼ a log( | f | + · · · + | f N | )near the poles. Here f i is a holomorphic function. We define S to bethe sheaf of holomorphic functions h such that | h | e − ϕa C . Then onecomputes a smooth modification µ : ˜ X → X of X such that µ ∗ S isan invertible sheaf O ˜ X ( − D ) associated with a normal crossing divisor D = P λ j D j , where D j is the component of the exceptional divisor of˜ X . Now, we have K ˜ X = µ ∗ K X + R , where R = P ρ j D j is the zerodivisor of the Jacobian function of the blow-up map. After some simplecomputation shown in [8], we will finally get that I ( ϕ ) = µ ∗ O ˜ X ( X ( ρ j − ⌊ aλ j ⌋ ) D j ) , where ⌊ aλ j ⌋ denotes the round down of the real number aλ j .Now we have the following definition. Definition 2.2.
Let h be a singular metric on L with weight function ϕ . Assume that ϕ has analytic singularities. Fix the notations asbefore, the modified ideal sheaf is defined as I ( ϕ ) := µ ∗ O ˜ X ( X ( −⌈ aλ j ⌉ ) D j ) . Here ⌈ aλ j ⌉ denotes the round up of the real number aλ j .Let τ j = ( λ j + ρ j a if aλ j is an integer λ j + ρ j +1 a if aλ j is not an integer . Then µ ∗ O ˜ X ( X ( −⌈ aλ j ⌉ ) D j ) = µ ∗ O ˜ X ( X ( ρ j − ⌊ aτ j ⌋ ) D j ) . Let g j be the generator of D j on a local coordinate ball V k . We definea function ψ k = µ ∗ ( a P τ j log( P | g j | )) on µ ( V k ). It is easy to verifythat I ( ϕ ) = I ( ψ k ) . Let g jik be the transition function of O ˜ X ( D j ) between V k and V i , then ψ k = ψ i + µ ∗ ( a X τ j log( X | g jik | )) . Since g jik is a nowhere vanishing holomorphic function, µ ∗ ( a X τ j log( X | g jik | )) s bounded. So being L -bounded against ψ k is equivalent to be L -bounded against ψ i for any i, k . Then after gluing all the ψ k togetherto be ψ via a partition of unity, we have I ( ϕ ) = I ( ψ ) . As a result, I ( ϕ ) is an ideal sheaf.We list a few basic properties of the modified ideal sheaf here. Proposition 2.2.
Let ϕ be a singular metric with analytic singulari-ties. Then (1) I ( ϕ ) ⊂ I ( ϕ ) , and I ( ϕ ) = I ( ϕ ) iff ϕ is defined by a normalcrossing divisor. (2) If f ∈ I ( ϕ ) , | f | e − ϕ is bounded. More precisely, | f | e − ϕ ∼ π ∗ (Π j | g j | ⌈ aλ j ⌉− aλ j ) ) near the poles of ϕ . Remember that g j is the generator of D j .Hence | f | e − ϕ actually vanishes near the poles of ϕ unless ϕ has algebraic singularities. (3) I ( ϕ ) = O X iff ϕ is bounded.Proof. (1) The first assertion is obvious. Observe that I ( ϕ ) = I ( ϕ ) iff τ j = λ j for all j , iff ρ j = 0 and aλ j is an integer for all j . If so, a mustbe a rational number. On the other hand, ρ j comes from the Jacobiandivisor, ρ j = 0 means that the log-resolution µ is trivial, hence ϕ isdefined by a normal crossing divisor E = P σ j E j on X . Namely, ϕ = a X σ j | f j | if we denote the generator of E j by f j . Notice that ϕ is a metric of L ,we actually have O X ( E ) = L and a = 1.(2) By definition, µ ∗ ( | f | e − ϕ ) ∼ Π j | g j | ⌈ aλ j ⌉− aλ j ) near the poles, hence the desired estimate. Moreover, | f | e − ϕ = 0 atthe poles iff ⌈ aλ j ⌉ = aλ j . In this situation, a is a rational number andthat’s the last assertion.(3) is a direct consequence of (2). (cid:3) Let’s pause for a second. We can exclude the situation that ϕ hasalgebraic singularities here since it has been discussed in [27]. So fromnow on, | f | e − ϕ will always vanish near the poles if f ∈ I ( ϕ ).Note that although I ( ϕ ) = I ( ψ )for some function ψ , the modified ideal sheaf behaves differently fromthe static multiplier ideal sheaf in many aspects. The following super-additivity is an interesting evidence. roposition 2.3 (Superadditivity) . Let ϕ , ϕ be two singular metricswith analytic singularities. Then I ( ϕ ) · I ( ϕ ) ⊂ I ( ϕ + ϕ ) . Proof.
Suppose that ϕ ∼ a log( | f | + · · · + | f N | ) , and ϕ ∼ b log( | g | + · · · + | g M | ) , where f i , g j are holomorphic functions. We define S , W to be thesheaves of holomorphic functions h, k such that | h | e − ϕ a C and | k | e − ϕ b C respectively. Let µ : ˜ X → X be a log-resolution such that µ ∗ S , µ ∗ W are invertible sheaves O ˜ X ( − D ) , O ˜ X ( − D ) of the normal crossing di-visors D = P λ i D i , D = P τ i D i , where D j is the component of theexceptional divisor of ˜ X . Thus we get that I ( ϕ ) = µ ∗ O ˜ X ( X ( −⌈ aλ i ⌉ ) D i ) , I ( ϕ ) = µ ∗ O ˜ X ( X ( −⌈ bτ i ⌉ ) D i ) , and I ( ϕ + ϕ ) = µ ∗ O ˜ X ( X ( −⌈ aλ i + bτ i ⌉ ) D i ) . Then the conclusion follows easily from the fact that ⌈ aλ i ⌉ + ⌈ bτ i ⌉ > ⌈ aλ i + bτ i ⌉ . (cid:3) We prove a Koll´ar-type injectivity theorem to finish this subsection.The more discussion about the modified ideal sheaf can be found inour papers [29].Remember that ψ is a function on X induced by h . Let H n,q ( X, L ⊗ I ( L )) := { α ∈ H n,q ( X, L ); Z X | α | e − ψ < ∞} . Proposition 2.4 (A Koll´ar-type injectivity theorem) . Assume that X is a compact K¨ahler manifold. Let s be a section of some multiple L k − such that s ∈ H ( X, L k − ⊗ I ( L k − )) . Then the following map H n,q ( X, L ⊗ I ( L )) ⊗ s −→ H n,q ( X, L k ⊗ I ( L k )) induced by tensor with s is injective. roof. By Proposition 2.1, we have H n,q ( X, L ⊗ I ( L )) ⊂ H n,q ( X, L ⊗ I ( L )) H n,q ( X, L k ⊗ I ( L k )) ⊂ H n,q ( X, L k ⊗ I ( L k ))respectively.By Gongyo–Matsumura’s injectivity theorem [15],[ α ] ∈ H n,q ( X, L ⊗ I ( L ))maps as [ sα ] injectively into H n,q ( X, L k ⊗ I ( L k )). One verifies thesection s here must satisfy the condition in their theorem.Now we consider the α ∈ H n,q ( X, L ⊗ I ( L )) . Since s ∈ H ( X, L k − ⊗ I ( L k − )) ,sα ∈ H n,q ( X, L k ⊗ I ( L k )) by superadditivity (Proposition 2.3). Theproof is finished. (cid:3) One refers to [13, 17, 18, 20] for the history of Koll´ar’s injectivitytheorem.2.3.
Bergman kernel for the space H n,q λ . The estimate of the num-bers h n,q λ is based on an observation about the Bergman kernel. TheBergman kernel at x ∈ X is defined as the function B ( x ) = X | α j ( x ) | , where { α j } is an orthonormal basis for H n,q λ , and the norm is the point-wise norm defined by the metrics h and ω on L and X . More precisely, B ( x ) is the pointwise trace on the diagonal of the true Bergman kernel,defined as the reproducing kernel for H n,q λ .The relevance of B ( x ) for our problem lies in the formula Z X B ( x ) = h n,q λ , which is evident since each term in the definition of B ( x ) contributesa 1 to the integral. On the other hand, B ( x ) is intimately related tothe solution of the extremal problem S ( x ) = sup | α ( x ) | k α k , where the supremum is taken over all α in H n,q λ . Indeed, the followinglemma is classical in Bergman’s theory of reproducing kernels. Let E be a Hermitian vector bundle of rank N on a manifold X . Let V be a subspace of the space of continuous global sections of E whosecoefficients are in L ( X ), and let { α j } be an orthonormal basis for V .Define B ( x ) and S ( x ) with { α j } and space V same as before. Then wehave emma 2.1. S ( x ) B ( x ) N S ( x ) . In particular, Z X S ( x ) dim( V ) N Z X S ( x ) . The proof can be found in [1].Theorem 1.1 therefore follows if we can prove a submeanvalue in-equality that estimates the value of a form α ∈ H n,q λ at any point x ∈ X by its L -norm.2.4. Siu’s ∂ ¯ ∂ -Bochner formula. The ∂ ¯ ∂ -Bochner formula for an L -valued ( n, q )-form is first developed by Siu in [22] on a compact K¨ahlermanifold, then it is extended to a general compact complex manifoldin [1]. Furthermore, it is generalized in [27] to a version suitable for aline bundle L tensoring with a vector bundle E . For our purpose, weonly present the latest version in [27] here. Proposition 2.5.
Let ( X, ω ) be a compact complex manifold. Let E and L be holomorphic vector bundle of rank r and line bundle respec-tively. Let α be an L ⊗ E -valued ( n, q ) -form. If α is ¯ ∂ -closed, thefollowing inequality holds: (4) i∂ ¯ ∂T α ∧ ω q − > ( − Re < ∆ α, α > + < i Θ L ⊗ E ∧ Λ α, α > − c | α | ) ω n . The constant c is zero if ¯ ∂ω q − = ¯ ∂ω q = 0 . Here T α = c n − q ∗ α ∧ ∗ ¯ αe − φ . The proof can be found in [27].If we denote γ = ∗ α , α can be expressed as α = c n − q γ ∧ ω q , and we moreover have ∗ γ = ( − n − q c n − q γ ∧ ω q . Then | α | ω n = T α ∧ ω q , so the norm of α is given by the trace of T α .2.5. Two canonical metrics.
In this subsection, we present two ex-amples of singular metrics that provide analytic Zariski decompositionfor the modified ideal sheaf. Let L be a line bundle on a compact com-plex manifold X . In [25], Siu introduces a special singular metric φ siu as follows. For a basis { s kj } N k j =1 of H ( X, L k ), the metric φ k is definedby φ k := 1 k log N k X j =1 | s kj | . Take a convergent sequence { ε k } , and the Siu-type metric φ siu on L isthen defined by φ siu := log ∞ X k =1 ε k e φ k . ertainly φ siu is pseudo-effective and provides an analytic Zariski de-composition. Proposition 2.6 (Analytic Zariski decomposition) . For all k > , wehave H ( X, L k ⊗ I ( h k siu )) = H ( X, L k ⊗ I ( h k siu )) = H ( X, L k ) . Apart from h siu , one could also consider the metric h min with minimalsingularity. Definition 2.3.
Let L be a pseudo-effective line bundle. Consider twoHermitian metrics h , h on L with curvature i Θ L,h j > h (cid:22) h , and say that h is less singular than h , ifthere exits a constant C > h Ch .2. We will write h ∼ h , and say that h , h are equivalent withrespect to singularities, if there exists a constant C > C − h h Ch .The above definition is motivated by the following observation. Lemma 2.2. ( [12] ) For every pseudo-effective line bundle L , there exitsup to equivalence of singularities a unique Hermitian metric h min withminimal singularities such that i Θ L,h min > . Certainly we have H ( X, L k ⊗ I ( h k min )) = H ( X, L k ) for all k > . Moreover, if h min has analytic singularities, we have H ( X, L k ⊗ I ( h k min )) = H ( X, L k ) for all k > . A submeanvalue inequality for the ∆ -eigenforms andthe estimate of h n,q λ This section is devoted to prove a submeanvalue inequality. Theargument here is borrowed from [1] with necessary adjustment. Hereand in the later part of this paper, L is assumed to be a nef line bundleon a compact complex manifold X unless specified, and H p,q λ ( X, L ⊗ E, ∆ )is the eigenform space defined in Sect.2, which is briefly denoted by H p,q λ ( X, L ⊗ E ). E is a vector bundle.Fix a point x in X and choose local coordinates, z = ( z , ..., z n ) near x such that z ( x ) = 0 and the metric form ω on X satisfying ω = i ∂ ¯ ∂ | z | =: β at the point x . The next proposition is the crucial step in this argument. roposition 3.1. With the same notations as in Sect.2, let α ∈ H n,q λ ( X, L k ⊗ E ⊗ I ( L k )) satisfy ¯ ∂α = 0 . Then for r < λ − / and r < c , Z | z |
0. Moreover, the limit of (asubsequence of) { h ε } exits, and is denoted by h . We apply Proposition2.5 to ( L k , h kε ). The expression < i Θ L k ⊗ E ∧ Λ α, α > ω n can be estimatedfrom below by a constant c (1 − kε ) times | α | h kε , so we get(5) i∂ ¯ ∂T α,ε ∧ ω q − > ( − < ∆ ε α, α > − c ′ (1 + kε ) | α | h kε ) ω n . Here T α,ε = c n − q ∗ α ∧ ∗ ¯ αe − φ ε , where φ ε is the weight function of h ε .For r small, put σ ( r, ε ) = Z | z |
0) + 2 r σ ( r, / λ ( r, ε tends to zero. Then it follows the same analytic technique as in[1], we conclude our desired result. (cid:3) With the help of this proposition, the problem is now reduced on aball with radius r . It is left to estimate the weighted L ∞ -norm of α ∈ H n,q λ ( X, L k ⊗ E )(i.e. sup | z |
0, we may assume the localtrivialization is chosen so that the metric φ ε on L has the form φ ε = X µ j | z j | + o ( | z | ) . For any α we express it in terms of the trivialization and local coordi-nates and put α ( k ) ( z ) = α ( z/ √ k ) , so that α ( k ) is defined for | z | < k is large enough. We also scale theLaplacian by putting k ∆ ( k ) ε α ( k ) = (∆ ε α ) ( k ) . It is not hard to see that if ∆ is defined by the metric ψ on F k , then∆ ( k ) is associated to the line bundle metric ψ ( z/ √ k ). In particular, if F k = L k and ψ = kφ ε , then ∆ ( k ) ε is associated to X µ j | z j | + o (1) , and hence converges to a k -independent elliptic operator. Obviously,the same thing happens even if we substitute L k by L k ⊗ E for a vectorbundle E . It therefore follows from G˚arding’s inequality together withSobolev estimates that | α (0) | h kε C ( Z | z | < | α ( k ) | h kε ω n + Z | z | < | (∆ ( k ) ε ) m α ( k ) | h kε ω n ) , if m > n/
2. Remember that when 0 ∈ { h = ∞} , | α (0) | h kε = 0, we canfind a C independent of ε such that | α (0) | h kε C ( Z | z | < | α ( k ) | h kε ω n + Z | z | < | (∆ ( k ) ε ) m α ( k ) | h kε ω n ) n the other hand, when 0 / ∈ { h = ∞} , h is a smooth metric at 0.Therefore using G˚arding’s inequality together with Sobolev estimatesagainst h , we can find a C such that | α (0) | h k C ( Z | z | < | α ( k ) | h k ω n + Z | z | < | (∆ ( k )0 ) m α ( k ) | h k ω n ) . Then combine the two estimates above together, we have(9) | α (0) | h k C ( Z | z | < | α ( k ) | h k ω n + Z | z | < | (∆ ( k )0 ) m α ( k ) | h k ω n )Now Z | z | < | α ( k ) | h k ω n = k n Z | z | < / √ k | α | h k ω n and Z | z | < | ∆ ( k )0 α ( k ) | h k ω n = k n − m Z | z | < / √ k | (∆ ) m α | h k ω n . As a result,(10) | α (0) | h k C ( k n Z | z | < / √ k | α | h k ω n + k n − m Z | z | < / √ k | (∆ ) m α | h k ω n ) . Do the normalization so that the L -norm of α with respect to h isone. By Proposition 2.2, (1) and Proposition 3.1 we have k n Z | z | < / √ k | α | h k ω n Ck n − q ( λ + 1) q , and k n − m Z | z | < / √ k | (∆ ) m α | h k ω n Ck n − q ( λ + 1) q ( λ/k ) m Ck n − q ( λ + 1) q . Combine these two inequalities with (10), we have thus proved the firstpart of Proposition 3.2. The second statement is much easier. We nowapply (10) to the scaling α ( λ ) instead, and get immediately that | α (0) | h k Cλ n . (cid:3) We now have all the ingredients for the proof of Theorem 1.1.
Proof of Theorem 1.1.
Let first Z n,q λ be the subspace of H n,q λ consistingof all the ¯ ∂ -closed forms. We apply Lemma 2.1 with L k ⊗ E -valued( n, q )-forms. The estimate for S ( x ) furnished by Proposition 3.2 to-gether with Proposition 2.1 then immediately gives Theorem 1.1 for Z n,q λ . We now claim that(11) h n,q λ dim Z n,q λ + dim Z n,q +1 λ , hich completes the proof since the estimate for dim Z n,q +1 λ is betterthan our desired estimate for h n,q λ . The claim is not complicated andwas first proved in [1]. We present here for readers’ convenience. Let α is an eigenform of ∆ , so that∆ α = µα. If we decompose α = α + α where α is ¯ ∂ -closed and α is orthogonal(with respect to h ) to the space of ¯ ∂ -closed forms, then the α j ’s with j = 1 , commutes with ¯ ∂ , so ¯ ∂ ∆ α = 0 and < ∆ α , η > h = < α , ∆ η > h = 0if ¯ ∂η = 0. Hence ∆ α j = (∆ α ) j = µα j for j = 1 ,
2. Now we decompose H n,q λ = Z n,q λ ⊕ ( H n,q λ ⊖ Z n,q λ ) . Since ¯ ∂ maps H n,q λ ⊖ Z n,q λ injectively into Z n,q +1 λ , (11) follows and theproof of Theorem 1.1 is complete. (cid:3) Since a semi-positive line bundle will always satisfy the condition inTheorem 1.1, we remark here that the example (Proposition 4.2) in [1]also shows that the order of magnitude given in Theorem 1.1 can notbe improved in general.4.
Applications in geometry
In order to prove Theorem 1.2, we first extend the singular Hodge’stheorem to the modified ideal sheaf. Since I ( L ) ⊂ I ( L ), there exits anatural morphism i n,q : H n,q ( X, L k ⊗ I ( L k )) → H n,q ( X, L k ⊗ I ( L k )) . Then the Hodge’s theorem can be stated as follows.
Proposition 4.1 (A (weak) singular version of Hodge’s theorem, II) . Let X be a compact complex manifold, and let L be a nef line bundleon X . Let ∆ be the Laplacian operator defined before. Assume that h has analytic singularities. We have the following conclusions:1. Suppose i Θ L,h has at least n − s + 1 positive eigenvalues at everypoint x ∈ X . Then H n,q ( X, L k ⊗ I ( L k )) = Imi n,q for q > s and k large enough.2. Suppose X is K¨ahler. Then H n,q ( X, L k ⊗ I ( L k )) = Imi n,q for all q and k . We need to prove a lemma first. emma 4.1. Assume that i Θ L,h has at least n − s + 1 positive eigen-values at every point x ∈ X . If α ∈ H n,q ( X, L k ⊗ I ( L )) with q > s ,then its ∆ -harmonic representative ˜ α satisfies that | ˜ α | h | α | h near the poleswhen k > k for some k large enough. In particular, if X is K¨ahler,the assumption for i Θ L,h is not needed and k = 1 .Proof. Let ϕ be the weight function of h , and let Z be the pole-set of ϕ . We claim that there exits an open subset V := { φ < } defined bysome quasi-plurisubharmonic function φ on X with Z ⊂ V , satisfyingthat Z V | ˜ α | h Z V | α | h . It leads to the conclusion. In fact, if | ˜ α | h is bigger than | α | h atsome point x ∈ V , | ˜ α | h > | α | h on an open subset of V since | α | h isuniformly bounded. Moreover, we can shrink V by substituting φ + C for φ . It is easy to get a contradiction to the integral inequality abovewhen V is small enough.Now it remains to prove the claim. First we assume that X is K¨ahler,and the proof is divided into four steps.1. Let’s recall two formulas. The first one is the generalized Kodaira–Akizuki–Nakano formula proved in [26]. Let ψ be a smooth real-valuedfunction on X , and let h be a smooth metric on L . Then we have k√ η ( ¯ ∂ + ¯ ∂ψ ∧ ) α k h + k√ η ¯ ∂ ∗ α k h = k√ η ( ∂ h − ∂ψ ∧ ) ∗ α k h + k√ η∂ h α k h + ( iη [Θ L,h + ∂ ¯ ∂ψ, Λ] α, α ) h (12)for any α ∈ A n,q ( X, L ) and η = e ψ .The second formula can also be found in [26]: k√ η ( ∂ψ ∧ ) ∗ α k h = k√ η ¯ ∂ψ ∧ α k h + k√ η ( ¯ ∂ψ ∧ ) ∗ α k h . (13)Now apply (12) with ψ = 1 and h = h ε , we have k ¯ ∂ ∗ ˜ α k h ε = k ( ∂ h ε ) ∗ ˜ α k h ε + k ∂ h ε ˜ α k h ε + ( i [Θ L,h ε , Λ] ˜ α, ˜ α ) h ε . Take the limit with respect to ε ,lim k ¯ ∂ ∗ ˜ α k h ε = lim k ¯ ∂ ∗ α ε k h ε = 0 , where α ε is the harmonic representative of α with respect to h ε . More-over, since i Θ L,h >
0, we havelim k ( ∂ h ε ) ∗ ˜ α k h ε = lim k ∂ h ε ˜ α k h ε = lim( i [Θ L,h ε , Λ] ˜ α, ˜ α ) h ε = 0 .
2. We fix a smooth metric ϕ ε , and take ϕ min ,ε be the metric withminimal singularity. Namely, i Θ L,ϕ ε + i∂ ¯ ∂ϕ min ,ε > ϕ min ,ε (cid:22) ϕ . The definition for the singular metric with minimal singularity can befound in Sect.2.5. Notice that although the weight function ϕ min ,ε will epend on ε , the curvature current i Θ L,ϕ min := i Θ L,ϕ ε + i∂ ¯ ∂ϕ min ,ε isindependent of ε and positive. Take the limit with respect to ε , wefinally get i Θ L,ϕ min = i Θ L,ϕ + i∂ ¯ ∂φ for some φ . Since ϕ min ,ε (cid:22) ϕ , the pole-set of ϕ is included in V := {− φ < } . There is one issue here. If ϕ min ,ε ∼ ϕ , we should replace ϕ min byanother singular metric σ . Namely, we consider a singular metric σ such that i Θ L,σ > σ (cid:23) ϕ strictly. Accordingly, we will get i Θ L,σ = i Θ L,ϕ + i∂ ¯ ∂φ . At this time, the pole-set of ϕ is included in V := { φ < } .3. Apply (12) again with ψ = φ i for i = 1 , ε , we get that k√ η ( ¯ ∂φ i ∧ ˜ α ) k h = k√ η ( − ∂φ i ∧ ) ∗ ˜ α k h + ( iη [Θ L,ϕ + ∂ ¯ ∂φ i , Λ] ˜ α, ˜ α ) h . One may wonder whether (12) is suitable for a singular weight function φ i here. Indeed, by [12] one can always approximate φ i by a family ofsmooth metrics { φ δ } . Apply (12) to φ δ and take the limit with respectto δ , we will finally get our desired equality. Later, when applyingformula (13) on a singular weight function, we use this technique againwithout pointing out.Combine with (13), we obtain that( ¯ ∂φ i ∧ ) ∗ ˜ α = 0 .
4. Observe that if we restrict the arbitrary metrics ω, h on X, L toan open subset U = { χ < } , and define the corresponding L -normon U , we have(14) ( ¯ ∂β, γ ) = ( β, ¯ ∂ ∗ γ ) + [ β, ( ¯ ∂χ ∧ ) ∗ γ ] . for any smooth α, β . Setting τ := dS/ | dχ | and [ α, ( ¯ ∂χ ∧ ) ∗ β ] is definedas [ α, ( ¯ ∂χ ∧ ) ∗ β ] := Z ∂U ( α, ( ¯ ∂χ ∧ ) ∗ β ) τ. In particular, if ( ¯ ∂χ ∧ ) ∗ β = 0 for any β , we have( ¯ ∂α, β ) = ( α, ¯ ∂ ∗ β ) . Apply (14) with h = h , χ = φ i and γ = ˜ α , we finally get that0 = lim ε → Z V i ( ¯ ∂ ∗ ˜ α, β ) h ε = Z V i ( ˜ α, ¯ ∂β ) h or any β . We are ready to prove the inequality in the claim. Consider˜ α + ¯ ∂β for any β , we have Z V i k ˜ α + ¯ ∂β k h = Z V i k ˜ α k h + Z V i k ¯ ∂β k h + 2Re Z V i ( ˜ α, ¯ ∂β ) h = Z V i k ˜ α k h + Z V i k ¯ ∂β k h . It means ˜ α minimizes the L -norm on V i , hence the desired inequality.The proof of the claim is finished.Now we deal with the second situation. Since ( X, ω ) is not necessaryto be K¨ahler, we will use the Kodaira–Akizuki–Nakano formula for non-K¨ahler manifold proved in [6].Let τ be the operator of type (1 ,
0) defined by τ = [Λ , ∂ω ], and let∆ ∂,τ = ( ∂ h + τ )( ∂ h + τ ) ∗ + ( ∂ h + τ ) ∗ ( ∂ h + τ )be the ∂ -Laplaican twisted by τ . Then we have∆ ¯ ∂ = ∆ ∂,τ + [ i Θ L,h , Λ] + T ω . Here T ω is an operator of order 0 depending only on the torsion of theHermitian metric ω : T ω = [Λ , [Λ , i ∂ ¯ ∂ω ]] − [ ∂ω, ( ∂ω ) ∗ ] . Use this formula to replace formula (12), then the same argumentas before will lead to the desired inequality after we have shown that[ i Θ L,h , Λ] + T ω is a positive operator.So it is left to prove that [ i Θ L,h , Λ] + T ω is positive. Since i Θ L,h hasat least n − s + 1 positive eigenvalues, the computation in Theorem 5.1of [9] shows that there exits a Hermitian metric ω ε such that([ i Θ L,h , Λ ω ε ] α, α ) ω ε > ( q − s + 1 − ε ( s − | α | for any L -valued ( n, q )-form α . Choosing ε = 1 /s and q > s , theright hand side will be > (1 /s ) | α | . Take k large enough such that[ i Θ L k ,h , Λ] + T ω is positive, and the proof is complete. (cid:3) Next we prove Proposition 4.1
Proof od Proposition 4.1.
Let L n,q (2) ( X, L ) ϕ be the space of all the L -valued ( n, q )-form that is L -bounded against ϕ . L n,q (2) ( X, L ) ψ is similar. LetIm ¯ ∂ := Im( ¯ ∂ : L n,q (2) ( X, L ) ϕ → L n,q (2) ( X, L ) ϕ ) , and Im ¯ ∂ := Im( ¯ ∂ : L n,q (2) ( X, L ) ψ → L n,q (2) ( X, L ) ψ ) . er ¯ ∂ and Ker ¯ ∂ are defined similarly. Recall that there are followingorthogonal decompositions [14]:Ker ¯ ∂ = Im ¯ ∂ M H n,q ( X, L ⊗ I ( L ) , ∆ )and Ker ¯ ∂ = Im ¯ ∂ M (Ker ¯ ∂ ∩ Ker ¯ ∂ ∗ ψ ) . By ¯ ∂ ∗ ψ we refer to the formal adjoint operator of ¯ ∂ with respect to the L -norm defined by ϕ . One may wonder that are these decompositionsstill valid for singular metrics. Indeed, we can approximate them bysmooth metrics then take the limit. On the other hand, it is easy toverify that(15) Ker ¯ ∂ ∩ L n,q (2) ( X, L ) ψ = Ker ¯ ∂ . Now the cohomology group can be expressed as H n,q ( X, L ⊗ I ( ϕ )) ≃ Ker ¯ ∂ Im ¯ ∂ = H n,q ( X, L ⊗ I ( L ) , ∆ ))and H n,q ( X, L ⊗ I ( ϕ )) ≃ Ker ¯ ∂ Im ¯ ∂ = Ker ¯ ∂ ∩ Ker ¯ ∂ ∗ ψ . Therefore the morphism i n,q can be rewritten as i n,q : Ker ¯ ∂ ∩ Ker ¯ ∂ ∗ ψ → H n,q ( X, L ⊗ I ( L ) , ∆ )) , hence its image equalsKer ¯ ∂ ∩ Ker ¯ ∂ ∗ ψ Im ¯ ∂ = Ker ¯ ∂ / Im ¯ ∂ Im ¯ ∂ = Ker ¯ ∂ Im ¯ ∂ = Ker ¯ ∂ ∩ L n,q (2) ( X, L ) ψ Im ¯ ∂ = H n,q ( X, L ⊗ I ( L ) , ∆ )) ∩ L n,q (2) ( X, L ) ψ . We use the fact that Im ¯ ∂ ⊂ Im ¯ ∂ to get the second equality. The third equality comes from formula (15)and the last equality is due to Lemma 4.1. The proof is finished. (cid:3) Problem 4.1.
We are willing to know whether we have H n,q ( X, L ⊗ I ( L )) ≃ H n,q ( X, L ⊗ I ( L )) . We are ready to prove Theorem 1.2 now. roof of Theorem 1.2. Apply Theorem 1.1 with λ = 0, we then have h n,q ( L k ⊗ E ⊗ I ( L k )) Ck n − q . Then the conclusion follows from Proposition 4.1 after we substitute E ⊗ K − X for E . (cid:3) We shall list some applications of Theorem 1.2. The first applicationwill be on the extension problem of the holomorphic sections. In fact,we can prove a more general version of the Grauert–Riemenschneiderconjecture.
Proof of Theorem 1.3.
The first case is simple, so we omit it here.In the second situation, we have h ,q ( L k ⊗ I ( L k )) Ck n − q for q > i ,q is injective. We use the Riemann–Roch formulainvolving the ideal sheaf. Then we have h ( L k ⊗ I ( L k )) = χ ( X, L k ⊗ I ( L k )) + h ( L k ⊗ I ( L k )) + O ( k n − )= χ ( X, L k ) − χ ( V, L k ) + h ( L k ⊗ I ( L k )) + O ( k n − )= k n ( L ) n n ! − k l ( L ) l l ! + h ( L k ⊗ I ( L k )) + O ( k n − ) . Here V = V ( I ( L k )) and l = dim V . Notice that H ( X, L k ⊗ I ( L k )) ⊂ H ( X, L k ) ,L is big if and only if ( L ) n >
0. The surjectivity case is similar.The same argument applies in the third situation, and the proof iscomplete. (cid:3)
We prove a vanishing theorem to finish this section.
Proof of Theorem 1.4.
Firstly, we claim that if H n,q ( X, L ⊗ I ( L )) isnon-zero, h ( X, L k − ⊗ I ( L k − )) dim H n,q ( X, L k ⊗ I ( L k )) . In fact, let { s j } be a basis of H ( X, L k − ⊗ I ( L k − )). Then for any α ∈H n,q ( X, L ⊗ I ( L )), { s j α } is linearly independent in H n,q ( X, L k ⊗ I ( L k ))by Proposition 2.4. It leads to the inequality.Now suppose that H n,q ( X, L ⊗ I ( L )) is non-zero for q > n − κ ( L ).We have h ( X, L k − ) = h ( X, L k − ⊗ I ( L k − )) dim H n,q ( X, L k ⊗ I ( L k )) . The first equality comes from the assumption that h provides an an-alytic Zariski decomposition, and the second inequality is due to theclaim. By the definition of Iitaka dimension κ ( L ), we havelim sup k →∞ h ( X, L k − )( k − κ ( L ) > . t means that lim sup k →∞ dim H n,q ( X, L k ⊗ I ( L k ))( k − κ ( L ) > . On the other hand, we havedim H n,q ( X, L k ⊗ I ( L k )) Ck n − q by Theorem 1.2, so n − q > κ ( L ). It contradicts to the assumptionthat q > n − κ ( L ). Hence H n,q ( X, L ⊗ I ( L )) = Imi n,q = 0for q > n − κ ( L ). Consider the cohomology long exact sequence of theshort exact sequence0 → I ( L ) → I ( L ) → I ( L ) / I ( L ) → . Notice that supp( I ( L ) / I ( L )) = { h = ∞} , the conclusion followsimmediately from the fact that Imi n,q = 0 and H q ( X, I ( L ) / I ( L )) = 0when q > m . (cid:3) The pseudo-effective case
In this section, we will discuss the situation that L is merely pseudo-effective.5.1. The harmonic forms.
As we have shown before, the ingredientto define a Laplacian operator as well as the associated eigenform fora singular metric φ is to approximate it by a family of smooth metrics { φ ε } . The difference for a pseudo-effective line bundle is that we can dosuch an approximation, only on an open subvariety Y ⊂ X . However,it seems to be enough, at least to define the harmonic L -valued ( n, q )-forms.Now let ( L, φ ) be a pseudo-effective line bundle on a compact com-plex manifold X . Assume that there exits a holomorphic section s of L k for some integer k , such that sup X | s | k φ < ∞ . Fix a Hermitianmetric ω on X . Then by Demailly’s approximation [12], we can find afamily of metrics { φ ε } on L with the following properties:(a) φ ε is smooth on X − Z ε for a subvariety Z ε ;(b) φ φ ε φ ε holds for any 0 < ε ε ;(c) I ( φ ) = I ( φ ε ); and(d) i Θ L,φ ε > − εω .Thanks to the proof of the openness conjecture by Berndtsson [2],one can arrange φ ε with logarithmic poles along Z ε according to theremark in [12]. Moreover, since the norm | s | k φ is bounded on X , theset { x ∈ X | ν ( φ ε , x ) > } for every ε > Z := { x | s ( x ) = 0 } by property (b). Here ν ( φ ε , x ) refers to the Lelongnumber of φ ε at x . Hence, instead of (a), we can assume that(a’) φ ε is smooth on X − Z , where Z is a subvariety of X independentof ε . ow let Y = X − Z . We use the method in [5] to construct acomplete Hermitian metric on Y as follows. Since Y is weakly pseudo-convex, we can take a smooth plurisubharmonic exhaustion function ψ on X . Define ˜ ω = ω + l i∂ ¯ ∂ψ for l ≫
0. It is easy to verify that ˜ ω isa complete Hermitian metric on Y and ˜ ω > ω .Let L n,q (2) ( Y, L ) φ ε , ˜ ω be the L -space of L -valued ( n, q )-forms u on Y with respect to the inner product given by φ ε , ˜ ω . Then we have theorthogonal decomposition(16) L n,q (2) ( Y, L ) φ ε , ˜ ω = Im ¯ ∂ M H n,qφ ε , ˜ ω ( L ) M Im ¯ ∂ ∗ φ ε where H n,qφ ε , ˜ ω ( L ) = { α | ¯ ∂α = 0 , ¯ ∂ ∗ φ ε α = 0 } . We give a brief explanation for decomposition (16). Usually Im ¯ ∂ isnot closed in the L -space of a noncompact manifold even if the metricis complete. However, in the situation we consider here, Y has thecompactification X , and the forms on Y are bounded in L -norms.Such a form will have good extension properties. Therefore the set L n,q (2) ( Y, L ) φ ε , ˜ ω ∩ Im ¯ ∂ behaves much like the space Im ¯ ∂ on X , which issurely closed. The complete explanation can be found in [13, 28].Now we have all the ingredients for the definition of ∆ φ -harmonicforms. We denote the Lapalcian operator on Y associated to ˜ ω and φ ε by ∆ ε . Definition 5.1.
Let α be an L -valued ( n, q )-form on X with bounded L -norm with respect to ω, φ . Assume that for every ε ≪
1, thereexists a Dolbeault cohomological equivalent class α ε ∈ [ α | Y ] such that(1) ∆ ε α ε = 0 on Y ;(2) α ε → α | Y in L -norm.Then we call α a ∆ φ -harmonic form. The space of all the ∆ φ -harmonicforms is denoted by H n,q ( X, L ⊗ I ( φ ) , ∆ φ ) . We will show that Definition 5.1 is compatible with the usual defi-nition of ∆ φ -harmonic forms for a smooth φ by proving the followingHodge-type isomorphism. Notice that here we furthermore assume that( X, ω ) is K¨ahler.
Proposition 5.1 (A singular version of Hodge’s theorem, III) . Let ( X, ω ) be a compact K¨ahler manifold. ( L, φ ) is a pseudo-effective linebundle on X . Assume that there exists a section s of some multiple L k such that sup X | s | kφ < ∞ . Then the following isomorphism holds: (17) H n,q ( X, L ⊗ I ( φ ) , ∆ φ ) ≃ H n,q ( X, L ⊗ I ( φ )) . In particular, when φ is smooth, α ∈ H n,q ( X, L, ∆ φ ) if and only if α is ∆ φ -harmonic in the usual sense. roof. We use the de Rham–Weil isomorphism H n,q ( X, L ⊗ I ( φ )) ∼ = Ker ¯ ∂ ∩ L n,q (2) ( X, L ) h,ω Im ¯ ∂ to represent a given cohomology class [ α ] ∈ H n,q ( X, L ⊗ I ( φ )) by a ¯ ∂ -closed L -valued ( n, q )-form α with k α k φ,ω < ∞ . We denote α | Y simplyby α Y . Since ˜ ω > ω , it is easy to verify that | α Y | φ ε , ˜ ω dV ˜ ω | α | φ ε ,ω dV ω , which leads to inequality k α Y k φ ε , ˜ ω k α k φ ε,ω with L -norms. Hence byproperty (b), we have k α Y k φ ε , ˜ ω k α k φ,ω which implies α Y ∈ L n,q (2) ( Y, L ) φ ε , ˜ ω . By decomposition (16), we have a harmonic representative α ε in H n,qφ ε, ˜ ω ( L ) , which means that ∆ ε α ε = 0 on Y for all ε . Moreover, since a harmonicrepresentative minimizes the L -norm, we have k α ε k φ ε , ˜ ω k α Y k φ ε , ˜ ω k α k φ,ω . So we can take the limit ˜ α of (a subsequence of) { α ε } such that˜ α ∈ [ α Y ] . It is left to extend it to X .Indeed, by (the proof of) Proposition 2.1 in [28], there is an injectivemorphism, which maps ˜ α to a ¯ ∂ -closed L -valued ( n − q, Y with bounded L -norm. We formally denote it by ∗ ˜ α . The canonicalextension theorem applies here and ∗ ˜ α extends to a ¯ ∂ -closed L -valued( n − q, X , which is denoted by S q ( ˜ α ) in [28]. Furthermore,it is shown by Proposition 2.2 in [28] that ˆ α := c n − q ω q ∧ S q ( ˜ α ) is an L -valued ( n, q )-form with ˆ α | Y = ˜ α. Therefore we finally get an extension ˆ α of ˜ α . By definition,ˆ α ∈ H n,q ( X, L ⊗ I ( φ ) , ∆ φ ) . We denote this morphism by i ([ α ]) = ˆ α .On the other hand, for a given α ∈ H n,q ( X, L ⊗ I ( φ ) , ∆ φ ), by def-inition there exists an α ε ∈ [ α Y ] with α ε ∈ H n,qφ ε, ˜ ω ( L ) for every ε . Inparticular, ¯ ∂α ε = 0. So all of the α ε together with α Y define a commoncohomology class [ α Y ] in H n,q ( Y, L ⊗ I ( φ )). It is left to extend thisclass to X .We use the S q again. It maps [ α Y ] to S q ( α Y ) ∈ H ( X, Ω n − qX ⊗ L ⊗ I ( φ )) . Furthermore, c n − q ω q ∧ S q ( α Y ) ∈ H n,q ( X, L ⊗ I ( φ )) ith [( c n − q ω q ∧ S q ( α Y )) | Y ] = [ α Y ]. Here we use the fact that ω is aK¨ahler metric. We denote this morphism by j ( α ) = [ c n − q ω q ∧ S q ( α Y )].It is easy to verify that i ◦ j = id and j ◦ i = id. The proof is finished. (cid:3) Remark . When φ has analytic singularities, we can use the samemethod as in Proposition 4.1 to extend this isomorphism to I ( φ ). Remark . In [19], a similar result (Lemma 3.2) has been shown fora line bundle (
L, h ) such that h is smooth outside a subvariety and i Θ L,h >
0. Our result, which benefits a lot form their work, generalizesit.Although Proposition 5.1 only holds for the ( n, q )-form, we remarkthat the estimate for h n,q ( L k ⊗ E ) is enough to get the estimate for allof the dim H p,q ( L k ⊗ E ) since we can substitute E by E ⊗ Ω pX ⊗ K − X .So it remains to prove an estimate for h n,q ( L k ⊗ E ).5.2. The estimate for the order of the cohomology group.
Re-member that our method highly depends on a submeanvalue inequality(Proposition 3.2). However, there is a gap when proving such an in-equality on an open subset.In fact, in order to prove Proposition 3.2, we first use the ∂ ¯ ∂ -Bochnerformula to reduce it to a ball with radius r (Proposition 3.1). If wewant to get a similar inequality on Y , first we need to equip Y with acomplete Hermitian metric ˜ ω . At this time, it is sort of like to estimate(18) Z | z | < | F ||∇ h ε | ˜ ω n . Here F is the multiplier and describes the local rescalings of infinitesi-mally small coordinate charts. When the first derivative ∇ h ε becomeslarge as the point approaching Z and ε tending zero, to make the L -norm bounded, we have to enlarge the coordinate in that direction atthat point. It is the same as collapsing the manifold along that direc-tion at that point. When we fix our sight on the manifold, ∇ h ε blowsup, but when we fix our sight on ∇ h ε , the manifold collapses. As aresult, the integral (18) is hard to control. So it is still an open questionto get an estimate for h n,q λ ( L k ⊗ E ). References [1] Berndtsson, B.: An eigenvalue estimate for the ¯ ∂ -Laplacian. J. Diff. Geom. ,295-313 (2002)[2] Berndtsson, B.: The openness conjecture and complex Brunn-Minkowski in-equalities. Complex geometry and dynamics, Abel Symp., 10, Springer, Cham(2015)[3] Cao, J.: Numerical dimension and a Kawamata–Viehweg–Nadel-type vanish-ing theorem on compact K¨ahler manifolds. Compos. Math. , 1869-1902(2014)[4] Demailly, J.-P., Ein, L., Lazarsfeld, R.: A subadditivity property of multiplierideals. Michigan Math. J. , 137-156 (2000)
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