The harmonic theory on a vector bundle with singular Hermitian metrics and positivity
aa r X i v : . [ m a t h . DG ] F e b THE HARMONIC THEORY ON A VECTOR BUNDLEWITH SINGULAR HERMITIAN METRICS ANDPOSITIVITY
Jingcao Wu
Abstract.
Let E be a holomorphic vector bundle endowed witha singular Hermitian metric H . In this paper, we develop theharmonic theory on ( E, H ). Then we extend several canonicalresults of J. Koll´ar and K. Takegoshi to this situation. In the end,we generalise Nakano’s vanishing theorem. Introduction
This is a continuation of our work [39] about the singular Hermitianmetric on a holomorphic vector bundle.Let E be a holomorphic vector bundle of rank r + 1 over a com-pact K¨ahler manifold ( Y, ω ) of dimension n . Let X := P ( E ∗ ) be theprojectivsed bundle with the natural projection π : X → Y and thetautological line bundle O E (1) := O X (1). Let Ω be a K¨ahler metric on X . We proposed an alternative definition for the singular Hermitianmetric H on E in [39], and showed that this type of singular metric dif-fers slightly with the one defined in [36]. Moreover, it has good natureto help us to define the Griffiths and Nakano positivities. The goal ofthis paper is to develop the harmonic theory on ( E, H ).We first briefly recall the canonical harmonic theory when H issmooth. In this context, the adjoint operator ¯ ∂ ∗ of the ¯ ∂ operatorwith respect to the L -norm k · k H,ω is defined as¯ ∂ ∗ := ∗ ∂ H ∗ , where ∗ is the Hodge ∗ -operator defined by ω and ∂ H is the (1 , H . Then the ¯ ∂ -Laplacianoperator [19], defined as (cid:3) = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ : A p,q ( Y, E ) → A p,q ( Y, E ) , is a self-adjoint elliptic operator. Here A p,q ( Y, E ) is the collection of allthe smooth E -valued ( p, q )-forms on Y . Thus, the eigenform of (cid:3) witheigenvalue zero is called a harmonic form, and the harmonic space isdefined as H p,q ( Y, E ) := { α ∈ A p,q ( Y, E ); (cid:3) α = 0 } . The celebrated Hodge’s theorem states that H p,q ( Y, E ) ≃ H p,q ( Y, E ) , here H p,q ( Y, E ) is the Dolbeault cohomology group [19].Now H is not necessary to be smooth. The first step is to approxi-mate it with a family of smooth metrics. When E is a line bundle, ithas been done in [7, 12]. We generalise their work to the higher rankvector bundle as follows: Proposition 1.1.
Suppose that E is equipped with a singular Hermit-ian metric H such that i Θ O E (1) ,ϕ > v for some smooth real (1 , -form v on X . Here ϕ is the metric on O E (1) corresponding to H . Moreover,assume that there exits a section ξ of some multiple O E ( k ) such that sup X | ξ | kϕ < ∞ . Then, for each positive real number ε and positiveinteger l , there is a (singular) Hermitian metric H lε on S l E such that (a) H lε is smooth on Y ′ , where Y ′ is an open subvariety of Y inde-pendent of l and ε ; (b) the sequence of metrics { ϕ ε } on O E (1) that defines H lε convergeslocally uniformly, decreasingly to ϕ on π − ( Y ′ ) . Equivalently, H lε converges locally uniformly, increasingly to S l H on Y ′ forevery l ; (c) I ( ϕ ) = I ( ϕ ε ) for all ε ; (d) for every ε > , i Θ O E (1) ,ϕ ε > v − ε Ω . If ( E, H ) is strongly positive in the sense of Nakano (see Definition2.4), we moreover have (e) for every relatively compact subset Y ′′ ⊂⊂ Y ′ and every l , i Θ S l E,H lε > − C l εω ⊗ Id S l E over Y ′′ in the sense of Nakano (see Definition 2.1) for a con-stant C l . Here S l H is the natural metric on the l -th symmetric product S l E induced by H . One could find many similarities between the stronglyNakano positivity and the min { n, r + 1 } -nefness defined in [4] basedon Proposition 1.1, (e), but we will not give any further discussions inthis paper.Next, for two E -valued ( n, q )-forms α, β (not necessary to be ¯ ∂ -closed), we say they are cohomologically equivalent if there exits an E -valued ( n, q − γ such that α = β + ¯ ∂γ . We denote by α ∈ [ β ]this equivalence relationship. Now H is approximated by { H ε } . Since H ε is smooth on Y ′ , the associated Laplacian (cid:3) ε is well-defined. Theharmonic form associated to H is defined as follows: Definition 1.1.
Let α be an E -valued ( n, q )-form on Y such that the L -norm against H is bounded. Assume that for every ε , there existsa cohomological equivalent class α ε ∈ [ α | Y ′ ] such that(a) (cid:3) ε α ε = 0 on Y ′ ;(b) α ε → α | Y ′ in the L -norm against H . hen we call α a (cid:3) -harmonic form, and denote it by (cid:3) α = 0. Noticethat this equality is taken in the sense of L -topology. The space of allthe (cid:3) -harmonic forms is denoted by H n,q ( Y, E ( H ) , (cid:3) ) . The analytic sheaf E ( H ) is defined in [4] as E ( H ) y := { u ∈ E y ; k u k H,ω is integrable in some neighbourhood of y } for a given point y ∈ Y . Although it has already been proved in [4, 24]that E ( H ) is coherent [22] in several situations, we would like to presenta general version that describes the coherence in Proposition 2.3. Inparticular, E ( H ) will always be coherent in our context in the view ofProposition 2.3. Although the (cid:3) -harmonic space is defined only on Y ′ , we can still prove the following proposition when ( E, H ) is stronglyNakano positive (see Definition 2.4).
Proposition 1.2 (A singular version of Hodge’s theorem) . Assumethat ( E, H ) is strongly positive in the sense of Nakano. The followingisomorphism holds: (1) H n,q ( Y, E ( H ) , (cid:3) ) ≃ H q ( Y, K Y ⊗ E ( H )) . In particular, when H is smooth, we have E ( H ) = E . Thus, α ∈ H n,q ( Y, E, (cid:3) ) if and only if α is harmonic in the usual sense. The group H q ( Y, K Y ⊗ E ( H )) is interpreted as a cohomology groupassociated with a coherent sheaf K Y ⊗ E ( H ) as is explained in [22].Moreover we obtain the following regularity property due to the canon-ical Bochner technique. Proposition 1.3.
Assume that ( E, H ) is a (singular) Hermitian vectorbundle that is strongly positive in the sense of Nakano. Let α be an E -valued ( n, q ) -form whose L -norm against H is bounded. Then (1) if α is (cid:3) -harmonic, ¯ ∂ ( ∗ α ) = 0 . In particular, ∗ α is holomor-phic. (2) if α is a weak solution of (cid:3) α = 0 , α must be smooth. Here ∗ refers to the Hodge ∗ -operator defined by ω . Based on theharmonic theory constructed above, we then generalise several canon-ical results of Koll´ar [28, 29] and Takegoshi [37] as follows: Theorem 1.1.
Let f : Y → Z be a fibration between two compactK¨ahler manifolds. Let n = dim Y and m = dim Z . Suppose that ( E, H ) is a (singular) Hermitian vector bundle over Y that is strongly positivein the sense of Nakano. Moreover, assume that H | Y z is well-defined forevery z ∈ Z . Then the following theorems hold: Decomposition Theorem . The Leray spectral sequence [19] E p,q = H p ( Z, R q f ∗ ( K Y ⊗ E ( H ))) ⇒ H p + q ( Y, K Y ⊗ E ( H )) degenerates at E . As a consequence, it holds that dim H i ( Y, K Y ⊗ E ( H )) = X p + q = i dim H p ( Z, R q f ∗ ( K Y ⊗ E ( H ))) for any i > . II Torsion freeness Theorem . For q > the sheaf homomorphism L q : f ∗ (Ω n − q ⊗ E ( H )) → R q f ∗ ( K Y ⊗ E ( H )) induced by the q -times left wedge product by ω admits a splitting sheafhomomorphism S q : R q f ∗ ( K Y ⊗ E ( H )) → f ∗ (Ω n − q ⊗ E ( H )) with L q ◦ S q = id . In particular, R q f ∗ ( K Y ⊗ E ( H )) is torsion free [26] for q > andvanishes if q > n − m . Furthermore, it is even reflexive if E ( H ) = E . III Injectivity Theorem . Let ( L, h ) be a (singular) Hermitianline bundle over Y . Recall that Y ′ is the open subvariety appeared inProposition 1.1. Assume the following conditions: (a) the singular part of h is contained in Y − Y ′ ; (b) i Θ L,h > γ for some real smooth (1 , -form γ on Y ; (c) for some rational δ ≪ , the Q -twisted bundle E < − δL > | Y z is strongly positive in the sense of Nakano for every z .For a (non-zero) section s of L with sup Y | s | h < ∞ , the multiplicationmap induced by the tensor product with sR q f ∗ ( s ) : R q f ∗ ( K Y ⊗ E ( H )) → R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h )) is well-defined and injective for any q > . IV Relative vanishing Theorem . Let g : Z → W be a fibrationto a compact K¨ahler manifold W . Then the Leray spectral sequence: R p g ∗ R q f ∗ ( K Y ⊗ E ( H )) ⇒ R p + q ( g ◦ f ) ∗ ( K Y ⊗ E ( H )) degenerates. Theorem II can also been seen as a singular version of the hard Lef-schetz theorem [19]. The definition of the Q -twisted bundle in TheoremIII can be found in [30]. Except that, ( E ⊗ L )( H ⊗ h ) is interpreted asthe following sheaf:( E ⊗ L )( H ⊗ h ) y := { u ∈ ( E ⊗ L ) y ; k u k H ⊗ h,ω is integrablein some neighbourhood of y } . It is also coherent by Proposition 2.3.In the end we discuss various vanishing theorems. heorem 1.2. Let f : Y → Z be a fibration between two compactK¨ahler manifolds. I Nadel-type vanishing Theorem . Let ( L, h ) be an f -big linebundle, and let ( E, H ) be a vector bundle that is strongly positive inthe sense of Nakano. Assume that H | Y z is well-defined for every z .Then R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h )) = 0 for every q > . II Nakano-type vanishing Theorem . Assume that ( E, H ) isstrongly strictly positive in the sense of Nakano. Then H q ( Y, K Y ⊗ S l E ( S l H )) = 0 for every l, q > . III Griffiths-type vanishing Theorem . Assume that ( E, H ) isstrictly positive in the sense of Griffiths (see Definition 2.4). Then H q ( Y, K Y ⊗ S l ( E ⊗ det E )( S l ( H ⊗ det H ))) = 0 for every l, q > . The definition for an f -big line bundle can be found in [15]. Noticethat when E is a line bundle, the strong Nakano strict positivity isequivalent to the bigness. At this time Theorem II is just Nadel’svanishing theorem [34]. Furthermore, ( E, H ) is strongly positive inthe sense of Nakano if and only if (
E, H ) is positive in the sense ofGriffiths, if and only if (
E, H ) is pseudo-effective. In this situation, thetopics related to the content of Theorem 1.1 and 1.2 are fully studiedin recent years. See [13, 14, 15, 16, 18, 31, 32, 33] and the referencestherein for more details. Our work benefits a lot from them.
Acknowledgment.
The author wants to thank Prof. Jixiang Fu, whobrought this problem to his attention and for numerous discussionsdirectly related to this work.2.
Preliminary
Set up.
In the rest of this paper, we will use the following set up.(I) Let (
Y, ω ) and Z be compact K¨ahler manifolds with dim Y = n and dim Z = m . Let f : Y → Z be a fibration, which is a surjectiveholomorphic map with connected fibres. Let ( E, H ) be a holomorphicvector bundle over Y of rank r +1, endowed with a (singular) Hermitianmetric H (see Sect.2.3.). Let X := P ( E ∗ ) be the projectivsed bundlewith the natural projection π : X → Y and tautological line bundle O E (1) := O X (1). In particular, let ϕ be the (singular) metric on O E (1)corresponding to H . Let Ω be a K¨ahler metric on X .(II) For every point y ∈ Y , we take a local coordinate y = ( y , ..., y n )around y . Fix a holomorphic frame { u , ..., u r } of E , the local coordi-nates of E , X and O E (1) are ( y, U = ( U , ..., U r )), ( y, w = ( w , ..., w r ))and ( y, w, ξ ) respectively. For every point z ∈ Z , we take a local coor-dinate z = ( z , ..., z m ) around z . Moreover, when z is a regular point f f , the system of local coordinate around Y z := f − ( z ) can be takenas f : Y → Z ( z, ( y m +1 , ..., y n )) z. We use the following conventions often.(i) Denote by c d = i d for any non-negative integer d . Denote dy = dy ∧ · · · ∧ d ¯ y n and dV Y = c n dy ∧ d ¯ y. It is similar in the other systems of local coordinate.(ii) Let G be a smooth Hermitian metric on E , we write G i = ∂G∂y i , G ¯ j = ∂G∂ ¯ y j , G α = ∂G∂U α , G ¯ β = ∂G∂ ¯ U β to denote the derivative with respect to y i , ¯ y j (1 i, j n ) and U α , ¯ U β (0 α, β r ) . The higher order derivative is similar.2.2.
Hermitian geometry revisit.
We collect some elementary factshere from [20, 25, 27]. Let G be a smooth Hermitian metric on E . Thenit induces a dual metric G ∗ on E ∗ hence a smooth metric ψ on O E (1).The curvature associated with G is represented asΘ E,G = X Θ α ¯ βi ¯ j dy i ∧ d ¯ y j ⊗ u α ⊗ u ∗ β with Θ α ¯ βi ¯ j = − G α ¯ βi ¯ j + X δ,γ G γ ¯ δ G α ¯ δi G γ ¯ β ¯ j . Now fix a point y ∈ Y , we can always assume that { u , ..., u r } is anorthonormal basis with respect to G at y . The Griffiths and Nakanopositivities [20] is defined as follows: Definition 2.1.
Keep notations before,(1) E is called (strictly) positive in the sense of Griffiths at y , if forany complex vector z = ( z , ..., z n ) and section u = P U α u α of E , X i Θ α ¯ βi ¯ j U α ¯ U β z i ¯ z j is (strictly) positive.(2) E is called (strictly) positive in the sense of Nakano at y , if forany n -tuple ( u = P U α u α , ..., u n = P U nα u α ) of sections of E , X i Θ α ¯ βi ¯ j U iα ¯ U jβ is (strictly) positive. Equivalently, i Θ E,G is a (strictly) positiveoperator on
T Y ⊗ E . obayashi proposed an intuitive way in [25, 27] to characterise theGriffiths positivity. More precisely, let { (log G ) ¯ βα } be the inverse ma-trix of { (log G ) α ¯ β } . Then for the holomorphic vector field ∂∂y i on Y , itshorizontal lift to X is defined as δδy i := ∂∂y i − X α,β (log G ) ¯ βα (log G ) ¯ βi ∂∂w α . The dual basis of { δδy i , ∂∂w α } will be { dy i , δw α := dw α + X i,β (log G ) ¯ βi (log G ) ¯ βα dy i } . Let Ψ := X iK α ¯ βi ¯ j w α ¯ w β G dy i ∧ d ¯ y j and ω F S := X i ∂ log G∂w α ∂ ¯ w β δw α ∧ δ ¯ w β , where K α ¯ βi ¯ j := − G α ¯ βi ¯ j + X δ,γ G γ ¯ δ G α ¯ δi G γ ¯ β ¯ j . It is easy to verify that they are globally defined (1 , X .Then the celebrated theorem given by Kobayashi says that Proposition 2.1 (Kobayashi, [25, 27]) . (2) i∂ ¯ ∂ψ = − Ψ + ω F S . As a consequence, we have
Proposition 2.2 (Kobayashi) . ( E, G ) is positive in the sense of Grif-fiths if and only if ( O E (1) , ψ ) is positive. The following definition is useful.
Definition 2.2 ([5, 9, 10]) . c ( ψ ) i ¯ j := < δδy i , δδy j > i∂ ¯ ∂ψ is called the geodesic curvature of ψ in the direction of i, j .Let a αi := − P β (log G ) ¯ βα (log G ) ¯ βi , we have c ( ψ ) i ¯ j = (log G ) i ¯ j − X α a αi a α ¯ j . Therefore the relationship between c ( ϕ ) i ¯ j and Ψ will be(3) − Ψ = X ic ( ψ ) i ¯ j dz i ∧ d ¯ z j . In particular, { c ( ψ ) i ¯ j } defines a Hermitian form, which is positive ifand only if Ψ is negative. .3. Singular Hermitian metric.
We recall the definition of the sin-gular Hermitian metric on E in [39]. Definition 2.3. (1) Consider the L -bounded function ϕ on X . Wedefine Y ϕ := { y ∈ Y ; ϕ | X y is well-defined } . (2) Fix a smooth metric h on O E (1) as the reference metric, wedefine H ( X ) := { ϕ ∈ L ( X ); on each X y with y ∈ Y ϕ , ϕ | X y is smooth and( i Θ O X (1) ,h + i∂ ¯ ∂ϕ ) | X y is strictly positive } . (3) A singular Hermitian metric on E is a map H with form that H ϕ ( u, u ) = Z X y | u | h e − ϕ ω rϕ,y r ! , where ϕ ∈ H ( X ). Here we use the fact that π ∗ O E (1) = E .We have presented in [39] that this type of singular metric differsslightly with the one defined in [36] and has good nature. In particu-lar, we have successfully defined the Griffiths and Nakano positivitiesconcerning ( E, H ϕ ). More precisely, Definition 2.4.
Let H ϕ be a (singular) Hermitian metric on E , andlet φ be the corresponding metric on O E (1). (Notice that φ may notequal to ϕ .) Let ( L, h ) be a (singular) Hermitian line bundle over Y .Let q be a rational number. Then(1) ( E, H ϕ ) is (strictly) positive in the sense of Griffiths, if i∂ ¯ ∂φ is(strictly) positive on X .(2) ( E < qL >, H ϕ , h ) is (strictly) positive in the sense of Griffiths,if i∂ ¯ ∂φ + qπ ∗ c ( L, h ) is (strictly) positive on X .(3) ( E, H ϕ ) is (strictly) strongly positive in the sense of Nakano, ifthe Q -twisted vector bundle( E < − r + 2 det E >, H ϕ )is (strictly) positive in the sense of Griffiths.(4) ( E < qL >, H ϕ , h ) is (strictly) strongly positive in the sense ofNakano, if the Q -twisted vector bundle( E < qr + 2 L − r + 2 det E >, H ϕ , h )is (strictly) positive in the sense of Griffiths.The definition for a Q -twisted bundle can be found in [30]. Onerefers to [39] for a detailed discussion for these positivities. .4. Multiplier ideal sheaf.
Remember that in the line bundle situ-ation, the multiplier ideal sheaf [34] is a powerful tool which establishesthe relationship between the algebraic and analytic nature of this linebundle. It would be valuable to extend this notion to a vector bundle.There are several attempts. The analytic sheaf E ( H ) is defined in [4]is a good candidate. The definition is as follows: E ( H ) y := { u ∈ E y ; k u k H,ω is integrable in some neighbourhood of y } for a given point y ∈ Y . Moreover, it is proved in [4, 24] that E ( H ) iscoherent [22] when ( E, H ) possesses certain positivity. We also provethe coherence in a general setting here.
Proposition 2.3.
Let ( E, H ) be a singular Hermitian vector bundleover Y . Let ϕ be the corresponding metric on O E (1) . Assume that i∂ ¯ ∂ϕ > γ for a real (1 , -form γ on X . Then E ( H ) is coherent.Proof. The proof follows the same method as [4, 8].Since coherence is a local property, we can assume without lossof generality that Y = U is a domain in ( C n , z = ( z , ..., z n )) and E = U × C r +1 . Let L ( U, C r +1 ) H be the square integrable C r +1 -valuedholomorphic functions with respect to H on U . It generates a coherentsubsheaf F ⊂ O U ( E ) = O U ⊕ · · · ⊕ O U | {z } r +1 as an O U -module [22]. It is clear that F ⊂ E ( H ); in order to prove theequality, we need only check that F y + E ( H ) y · m s +1 U,y = E ( H ) y for everyinteger s , in view of Nakayama’s lemma [1]. Here m U,y is the maximalideal of O U,y .Let f ∈ E ( H ) y be a germ that is defined in a neighbourhood V of y and let ρ be a cut-off function with support in V such that ρ = 1in a smaller neighbourhood of y . We solve the equation ¯ ∂u = ¯ ∂ ( ρf )by means of L -estimates (see the proof of Proposition 2.4) against e − n + s ) log | z − z ( y ) |− ψ H , where ψ is a smooth strictly plurisubharmonicfunction satisfying certain positivity as is shown in the proof of Propo-sition 2.4. Notice that here we use the curvature condition i∂ ¯ ∂ϕ > γ to guarantee the existence of such a ψ . We then get a solution u such that R U | u | H | z − z ( y ) | n + s ) < ∞ . Thus F = ρf − u is holomorphic, F ∈ L ( U, C r +1 ) H and f y − F y = u y ∈ E ( H ) y · m s +1 U,y . This proves the equality hence the coherence. (cid:3) de Rham–Weil isomorphism.
The de Rham–Weil isomorphismabout the L -cohomology group is essentially used in this paper, so weprovide here a detailed proof. This proof includes an L -estimate fora singular Hermitian vector bundle, which seems to be of independent nterest. The readers who are familiar with this isomorphism couldskip this part. Assume that E ( H ) is coherent, hence the cohomologygroup H q ( Y, K Y ⊗ E ( H )) is well-defined [19]. Moreover, we have Proposition 2.4 (The de Rham–Weil isomorphism) . Fix a Stein cov-ering U := { U j } Nj =1 of Y . We have (4) H q ( Y, K Y ⊗ E ( H )) ≃ Ker( ¯ ∂ : A n,q ( Y, E ( H )) → A n,q +1 ( Y, E ( H )))Im( ¯ ∂ : A n,q − ( Y, E ( H )) → A n,q ( Y, E ( H ))) . Here we use A n,q ( U, E ( H )) to refer to all of the ( n, q ) -forms on an openset U whose coefficients are in E ( H ) . In other words, α ∈ A n,q ( U, E ( H )) if α is an E -valued ( n, q ) -form such that Z U k α k H,ω < ∞ . Proof.
The proof follows the same method as [34] except that we willessentially use the Koszul complex without mentioning it. Fix a Steincovering U := { U j } Nj =1 of Y . Then we have the isomorphism between H q ( Y, K Y ⊗ E ( H )) and the ˇCech cohomology group ˇ H q ( U , K Y ⊗ E ( H )).For simplicity, we put U j j ...j q := U j ∩ · · · ∩ U j q . The class [ α ] maps to a q -cocycle α = { α j ...j q } that satisfies α j ...j q ∈ H n, ( U j ...j q , E ( H )) and δα = 0 , where δ is the coboundary operator of the ˇCech complex. Then by usinga partition { ρ j } of unity associated to U , we define α := { α j ...j q − } by α j ...j q − := P ρ j α jj ...j q − . By the construction, we have δα = α and δ ¯ ∂α = ¯ ∂δα = ¯ ∂α = 0. Notice that¯ ∂α = { ¯ ∂α j ...j q − } = { X ( ¯ ∂ρ j ) · α jj ...j q − } , we have that ¯ ∂α is a ( q − ∂α j ...j q − ∈ A n, ( U j ...j q − , E ( H )) . From the same argument, we can obtain α with δα = ¯ ∂α . More-over, ¯ ∂α is a ( q − ∂α j ...j q − ∈ A n, ( U j ...j q − , E ( H )) and δ ¯ ∂α = 0 . By repeating this process, we finally obtain α q = { α j } . Then ¯ ∂α j de-termines the ( n, q )-form on U j with ¯ ∂α j ∈ A n,q ( U j , E ( H )). Moreover,since δ ¯ ∂α j = 0, they patch together to give a (0 , q )-form α q ∈ A n,q ( X, E ( H )) ith ¯ ∂α q = 0. From the argument above, we have obtained the (well-defined) mapˇ H q ( U , E ( H )) → Ker( ¯ ∂ : A n,q ( X, E ( H )) → A n,q +1 ( X, E ( H )))Im( ¯ ∂ : A n,q − ( X, E ( H )) → A n,q ( X, E ( H ))) . Now we see that this map is actually an isomorphism by using the L -estimate on each Stein subset U j ...j k for k = 0 , ..., q . For every β ∈ Ker( ¯ ∂ : A n,q ( X, E ( H )) → A n,q +1 ( X, E ( H ))) , we define β := { β j } by β j := β | U j . By the L -estimate on U j against H , we obtain β = { β j } such that¯ ∂β = β , k β k H Uj : = X j Z U j k β j k H C k β k H Uj . Here C is an independent constant.We give a short explanation for this L -estimate. We admit Propo-sition 1.1 for the time being. Then there exits a regularising sequence { H ε } which is smooth on an open subvariety Y ′ ⊂ U j . Moreover, since U j is a Stein open subset of C n , we can even make Y ′ = U j . Then,at each point y ∈ U j , we may then choose a coordinate system whichdiagonalizes simultaneously the hermitians forms ω ( y ) and i Θ E,H ε , insuch a way that ω ( y ) = i X dy j ∧ d ¯ y j and i Θ E,H ε = i X λ εj dy j ∧ d ¯ y j . Since β is an ( n, q )-form, at point y we have < [ i Θ E,H ε , Λ] β , β > ω > ( λ ε + · · · + λ εq ) | β | , where Λ is the adjoint operator of ω ∧ · . On the other hand, since U j isStein, we can always find a smooth strictly plurisubharmonic function ψ , such that ( E, e − ψ H ) is strictly strongly positive in the sense ofNakano on U j . By Proposition 1.1, (e), we then obtain that i Θ E,e − ψ H ε = i X τ εj dz j ∧ d ¯ z j such that τ εj > C ′ for a universal positive constant C ′ . Now apply the L -estimate [8] against e − ψ H ε , we obtain β = { β j } such that¯ ∂β = β , k β k H ε,Uj : = X j Z U j k β j k H ε C k β k H ε,Uj . Recall that β = { β | U j } with β ∈ A n,q ( X, E ( H )), solim ε → k β k H ε,Uj < ∞ . ake the limit of the inequality before with respect to ε , we then obtainthe desired estimate.By the construction, we have ¯ ∂δβ = δ ¯ ∂β = δβ = 0. Moreover, δβ = { β j | U j j − β j | U j j } ∈ A n,q − ( U j j , E ( H )) . The explicit meaning of this inclusion should be β j | U j j − β j | U j j ∈ A n,q − ( U j j , E ( H )) , but we abuse the notation here and in the rest part. Therefore by thesame method, we can obtain β = { β j j } such that¯ ∂β = δβ , k β k H Uj j : = X j ,j Z U j j k β j j k H C k β k H Uj . Similarly, ¯ ∂δβ = δ ¯ ∂β = δδβ = 0 and δβ ∈ A n,q − ( U j j j , E ( H )) . By repeating this process, we finally obtain β i such that ¯ ∂β i = δβ i − and δβ i ∈ A n, ( U j ...j i , E ( H )). Since ¯ ∂δβ i = δ ¯ ∂β i = δδβ i − = 0, weactually have δβ i ∈ H n, ( U j ...j i , E ( H )). Hence we have obtained j : Ker( ¯ ∂ : A n,q ( Y, E ( H ))) → A n,q +1 ( Y, E ( H ))) → ˇ H q ( U , K Y ⊗ E ( H )) . We claim that Im( ¯ ∂ : A n,q − ( Y, E ( H )) → A n,q ( Y, E ( H ))) maps to thezero space under j , hence it is easy to see that the maps i, j togethergive an isomorphism H q ( Y, K Y ⊗ E ( H )) ≃ Ker( ¯ ∂ : A n,q ( Y, E ( H )) → A n,q +1 ( Y, E ( H )))Im( ¯ ∂ : A n,q − ( Y, E ( H )) → A n,q ( Y, E ( H ))) . Now we prove the claim by diagram chasing. We only prove the casethat i = 1 and i = 2, the general follows the same method. For every β ∈ Im( ¯ ∂ : A n, ( Y, E ( H )) → A n, ( Y, E ( H ))), we define β := { β j } by β j := β | U j . On the other hand, there exits an E -valued ( n, γ on Y such that β = ¯ ∂γ and k γ k H Uj < ∞ for every j . So we can take β = { β j } := { γ | U j } . The morphism j in this situation can be simplywritten as j ( β ) = δβ . Since γ is globally defined hence δβ = 0. Wehave successfully proved that j ( β ) = 0 when i = 1.Next we consider the β ∈ Im( ¯ ∂ : A n, ( Y, E ( H )) → A n, ( Y, E ( H ))),we define β := { β j } by β j := β | U j . On the other hand, there exitsan E -valued ( n, γ on Y such that β = ¯ ∂γ and k γ k H Uj < ∞ for every j . So we can take β = { β j } := { γ | U j } . Now apply the -estimate on U j against H , we obtain γ = { γ j } such that¯ ∂γ = β , k γ k H Uj : = X j Z U j k γ j k H C k β k H Uj . Let β = { β j j } = δγ . Since ¯ ∂β = ¯ ∂δγ = δ ¯ ∂γ = δβ and k β k H Uj j < ∞ , the morphism j is j ( β ) = δβ = δδγ = 0 at thistime. We have successfully proved that j ( β ) = 0 when i = 2. (cid:3) Bochner technique.
This subsection is devoted to introduce theclassic Bochner technique in harmonic theory. It mainly comes from[8, 19, 37] and the references therein. One should pay attention thatthe content of this subsection works for a K¨ahler manifold (
Y, ω ) thatis not necessary to be compact.Let (
E, G ) be a holomorphic vector bundle with the smooth Hermit-ian metric G . Then G as well as ω defines an L -norm k · k G,ω on thespace A ∗ ( Y, E ) = ⊕ A p,q ( Y, E ) . Let ∗ be the Hodge ∗ -operator, let e ( θ ) be the left wedge productacting by the form θ ∈ A p,q ( Y ) and Let ∂ G be the (1 , E associated with G . Then the adjoint operatorsof ¯ ∂, ∂ G , e ( θ ) with respect to k · k G,ω are denoted by ¯ ∂ ∗ , ∂ ∗ G and e ( θ ) ∗ ,respectively. In particular, e ( ω ) is also denoted by L and e ( ω ) ∗ isdenoted by Λ. The Laplacian operators are defined as follows: (cid:3) G = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂, ¯ (cid:3) G = ∂ G ∂ ∗ G + ∂ ∗ G ∂ G . Then the Bochner formula is as follows:
Proposition 2.5.
For any α ∈ A p,q ( Y, E ) and positive smooth function η on Y with χ := log η , we have (cid:3) G = ¯ (cid:3) G + [ i Θ E,G , Λ] , (cid:3) e − χ G = ¯ (cid:3) e − χ G + [ i (Θ E,G + ∂ ¯ ∂χ ) , Λ] and k√ η ( ¯ ∂ + e ( ¯ ∂χ )) α k G,ω + k√ η ¯ ∂ ∗ α k G,ω = k√ η ( ∂ ∗ G − e ( ∂χ ) ∗ ) α ) k G,ω + k√ η∂ G α k G,ω + < ηi [Θ E,G + ∂ ¯ ∂ϕ, Λ] α, α > G,ω , (5) when the integrals above are finite. The following formula is due to the K¨ahler property of ω : Proposition 2.6 (Donnelly and Xavier’s formula, [11]) . For any α ∈ A n,q ( Y, E ) nd smooth function χ on Y , we have [ ¯ ∂, e ( ¯ ∂χ ) ∗ ] + e ( ∂χ ) ∂ ∗ G = ie ( ∂ ¯ ∂ϕ )Λ and k e ( ∂χ ) ∗ α k G,ω = k e ( ¯ ∂χ ) α k G,ω + k e ( ¯ ∂χ ) ∗ α k G,ω . (6) when the integrals above are finite. The regularising technique
This section is devoted to prove Proposition 1.1. Recall that in[7, 12], such an approximation was already made for a line bundle asfollows:
Proposition 3.1 (Theorem 2.2.1, [12]) . Let ( L, ϕ ) be a (singular) Her-mitian line bundle over Y such that i Θ L,ϕ > v for a real smooth (1 , -form v . There exits a family of singular metrics { ϕ ε } ε> with the fol-lowing properties: (a) ϕ ε is smooth on Y − Z ε for a subvariety Z ε ; (b) ϕ ε > ϕ ε > ϕ holds for any < ε ε ; (c) I ( ϕ ) = I ( ϕ ε ) ; and (d) i Θ L,ϕ ε > v − εω . Thanks to the openness property of the multiplier ideal sheaf [21],one can arrange h ε with logarithmic poles along Z ε according to theremark in [12]. Now we furthermore assume that there exists a section ξ of some multiple L k such that sup Y | ξ | h k < ∞ . Then the set { y ∈ Y ; ν ( h ε , y ) > } for every ε > Z := { y ∈ Y ; ξ ( y ) = 0 } by property (b). Here ν ( h ε , y ) refers to the Lelong number [8] of h ε at y . Hence, instead of (a), we can assume that(a’) h ε is smooth on Y − Z and has logarithmic poles along Z , where Z is a subvariety of Y independent of ε .Now we are ready to prove Proposition 1.1. Proof of Proposition 1.1.
Take a K¨ahler form Ω on X . By Proposition3.1 and the remark after that, there exists a family of singular metrics { ϕ ε } with the following properties:(a) ϕ ε is smooth on X ′ for an open subvariety X ′ independent of ε ;(b) ϕ ε > ϕ ε > ϕ holds for any 0 < ε ε ;(c) I ( ϕ ) = I ( ϕ ε ); and(d) i Θ O E (1) ,ϕ ε > v − ε Ω.Moreover, since ξ ∈ O E ( k ), π ∗ ξ ∈ S k E by the canonical isomorphism π ∗ ( O E ( k )) = S k E. s a result, π ( X ′ ) = π ( X − { ξ = 0 } )= Y − { π ∗ ξ = 0 } =: Y ′ , which is an open subvariety of Y . In particular, it is easy to verify that ϕ ε ∈ H ( X ) for every ε . Therefore it defines a singular metric H lε ( U, U ) := Z X y | U | e − lϕ ε ω rϕ ε ,y r !on S l E (see Definition 2.2) for every positive integer l and U ∈ S l E .Here we use the fact that π ∗ O E ( l ) = S l E . It remains to prove thedesired properties.(a), (c) and (d) are obvious. Recall that S l H can be rewritten as S l H ( U, U ) := Z X y | u | e − lϕ ω rϕ,y r !by [39]. On the other hand, both ϕ and ϕ ε are smooth on X y with y ∈ Y ′ , we immediately conclude that ϕ ε converges locally uniformlyand decreasingly to ϕ on X ′ . Hence H lε → S l H locally uniformly andincreasingly on Y ′ . (b) is proved.In the end, we prove (e) under the assumption that ( E, H ) is stronglypositive in the sense of Nakano. Indeed, by definition i Θ O E (1) ,ϕ − ir + 2 π ∗ Θ det E, det H is positive. Moreover, i Θ O E (1) ,ϕ is also positive by Theorem 1.3 in [39].Hence i Θ O E ( r + l +1) , ( r + l +1) ϕ − iπ ∗ Θ det E, det H is positive for all positive integer l . Consequently, i Θ O E ( r + l +1) , ( r + l +1) ϕ ε − iπ ∗ Θ det E, det H > − C l ε Ω . Now on Y ′′ we apply Berndtsson’s curvature formula in [2, 3], whichsays for any line bundle ( L, ψ ) over X , the curvature of π ∗ ( K X/Y ⊗ L )associated with the L -metric satisfies that X < i Θ i ¯ j s i , s j > > X Z X y ic ( ψ ) i ¯ j s i ∧ ¯ s j e − ψ . Here { s i } is n -tuple of sections of π ∗ ( K X/Y ⊗ L ). Let( L, ψ ) = ( O E ( r + l + 1) ⊗ π ∗ det E ∗ , ( r + l + 1) ϕ ε − π ∗ φ ) , where φ is the weight function of det H . At this time, the direct imageequals to S l E , the L -metric is just H lε and the associated curvaturesatisfies(7) X < i Θ i ¯ j s i , s j > > − C l ε ′ X Z X y ic (Ω) i ¯ j s i ∧ ¯ s j e − ( r + l +1) ϕ ε + π ∗ φ , here c (Ω) i ¯ j is defined as (see Definition 2.2) c (Ω) i ¯ j := < δδy i , δδy j > Ω . The estimate (7) is due to formula (3), the fact that i Θ O E ( r + l +1) , ( r + l +1) ϕ ε − iπ ∗ Θ det E,φ > − C l ε Ωand some elementary computation. Since y ∈ Y ′′ ⊂ Y ′ , ϕ is smoothalong X y . Therefore X Z X y ic (Ω) i ¯ j s i ∧ ¯ s j e − ( r + l +1) ϕ ε + π ∗ φ is uniformly bounded with respect to ε . It obviously implies that i Θ S l E,H lε > − C l ε ′′ ω ⊗ Id S l E in the sense of Nakano. Moreover, from the proof we see that if i Θ O E (1) ,ϕ − ir +2 π ∗ Θ det E, det H > εω , which means that ( E, H ) is stronglystrictly positive in the sense of Nakano, we can even arrange the thingthat i Θ S l E,H lε > C l (1 − ε ′ ) ω ⊗ Id S l E in the sense of Nakano. (cid:3) Then throughout the whole paper, for a given singular metric H on E , ϕ will always refer to its corresponding metric on O E (1). { H kε } and { ϕ ε } will always be the regularising sequence provided by Proposition1.1. In particular, H kε is smooth on an open subvariety Y ′ , whereas ϕ ε is smooth on X ′ with X ′ = π − ( Y ′ ). When k = 1, H ε will be simplydenoted by H ε . 4. Harmonic theory
Global theory.
Firstly, we use the method in [6] to construct acomplete K¨ahler metric on Y ′ as follows. Since Y ′ is weakly pseudo-convex, we can take a smooth plurisubharmonic exhaustion function ψ on Y ′ . Define ˜ ω l = ω + l i∂ ¯ ∂ψ for l ≫
0. It is easy to verify that ˜ ω l isa complete K¨ahler metric on Y ′ and ˜ ω l > ˜ ω l > ω for l l .Let L n,q (2) ( Y ′ , E ) H ε , ˜ ω l be the L -space of E -valued ( n, q )-forms α on Y ′ with respect to the inner product given by H ε , ˜ ω l . Then we have theorthogonal decomposition [17](8) L n,q (2) ( Y ′ , E ) H ε , ˜ ω l = Im ¯ ∂ M H n,qH ε , ˜ ω l ( E ) M Im ¯ ∂ ∗ H ε where Im ¯ ∂ = Im( ¯ ∂ : L n,q − ( Y ′ , E ) H ε , ˜ ω l → L n,q (2) ( Y ′ , E ) H ε , ˜ ω l ) , H n,qH ε , ˜ ω l ( E ) = { α ∈ L n,q (2) ( Y ′ , E ) H ε , ˜ ω l ; ¯ ∂α = 0 , ¯ ∂ ∗ H ε α = 0 } , and Im ¯ ∂ ∗ H ε = Im( ¯ ∂ ∗ H ε : L n,q +1(2) ( Y ′ , E ) H ε , ˜ ω l → L n,q (2) ( Y ′ , E ) H ε , ˜ ω l ) . e give a brief explanation for decomposition (8). Usually Im ¯ ∂ is notclosed in the L -space of a noncompact manifold even if the metricis complete. However, in the situation we consider here, Y ′ has thecompactification Y , 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 ′ , E ) H ε , ˜ ω l ∩ Im ¯ ∂ behaves much like the spaceIm( ¯ ∂ : L n,q − ( Y, E ) H,ω → L n,q (2) ( Y, E ) H,ω )on Y , which is surely closed. The complete explanation can be foundin [14, 38].Now we have all the ingredients for the definition of (cid:3) -harmonicforms. We denote the Lapalcian operator on Y ′ associated to ˜ ω l and H ε by (cid:3) l,ε . Recall that for two E -valued ( n, q )-forms α, β (not necessaryto be ¯ ∂ -closed), we say they are cohomologically equivalent if thereexits an E -valued ( n, q − γ such that α = β + ¯ ∂γ . We denoteby α ∈ [ β ] this equivalence relationship. Definition 4.1 (=Definition 1.1) . Let α be an E -valued ( n, q )-form on Y with bounded L -norm with respect to H, ω . Assume that for every l ≫ , ε ≪
1, there exists a representative α l,ε ∈ [ α | Y ′ ] such that(1) (cid:3) l,ε α l,ε = 0 on Y ′ ;(2) α l,ε → α | Y ′ in L -norm.Then we call α a (cid:3) -harmonic form. The space of all the (cid:3) -harmonicforms is denoted by H n,q ( Y, E ( H ) , (cid:3) ) . We will show that Definition 4.1 is compatible with the usual defini-tion of (cid:3) -harmonic forms for a smooth H by proving the Hodge-typeisomorphism, i.e. Proposition 1.2. Proof of Proposition 1.2.
By Proposition 2.4, we have H q ( Y, K Y ⊗ E ( H )) ≃ Ker( ¯ ∂ : A n,q ( Y, E ( H )) → A n,q +1 ( Y, E ( H )))Im( ¯ ∂ : A n,q − ( Y, E ( H )) → A n,q ( Y, E ( H ))) . Hence a given cohomology class [ α ] ∈ H q ( Y, K Y ⊗ E ( H )) is representedby a ¯ ∂ -closed E -valued ( n, q )-form α with k α k H,ω < ∞ . We denote α | Y ′ simply by α Y ′ . Since ˜ ω l > ω , it is easy to verify that | α Y | H ε , ˜ ω l dV ˜ ω l | α | H ε ,ω dV ω , which leads to inequality k α Y k H ε , ˜ ω l k α k H ε,ω with L -norms. Henceby property (b), we have k α Y k H ε , ˜ ω l k α k H,ω which implies α Y ∈ L n,q (2) ( Y ′ , E ) H ε , ˜ ω l . By decomposition (8), we have a harmonic representative α l,ε in H n,qH ε, ˜ ωl ( E ) , hich means that (cid:3) l,ε α l,ε = 0 on Y ′ for all l, ε . Moreover, since aharmonic representative minimizes the L -norm, we have k α l,ε k H ε , ˜ ω l k α Y k H ε , ˜ ω l k α k H,ω . So there exists a limit ˜ α of (a subsequence of) { α l,ε } such that˜ α ∈ [ α Y ′ ] . It is left to extend it to Y .Indeed, if we admit Proposition 1.3 for the time being, ∗ ˜ α will be aholomorphic E -valued ( n − q, Y ′ . In particular, since the ∗ -operator preserves the L -norm, the L -norm of ∗ ˜ α is also bounded.Now apply the canonical L -extension theorem [35], we obtain a holo-morphic extension of ∗ ˜ α on Y , which is still denoted by ∗ ˜ α . We denotethis morphism by S q : H q ( Y, K Y ⊗ E ( H )) → H ( Y, Ω n − qY ⊗ E )[ α ]
7→ ∗ ˜ α. Let ˆ α := c n − q ω q ∧ ∗ ˜ α , we then obtain an E -valued ( n, q )-form on Y .It is easy to verify that ˆ α | Y ′ = ˜ α . In summary, we have successfullydefined a morphism i : H q ( Y, K Y ⊗ E ( H )) → H n,q ( Y, E ( H ) , (cid:3) )[ α ] ˆ α. On the other hand, for a given α ∈ H n,q ( Y, E ( H ) , (cid:3) ), by definitionthere exists an α l,ε ∈ [ α Y ′ ] with α l,ε ∈ H n,qH ε, ˜ ωl ( E ) for every l, ε such thatlim α l,ε = α Y ′ . In particular, ¯ ∂α l,ε = 0. So all of the α l,ε together with α Y ′ define a common cohomology class [ α Y ′ ] in H n,q ( Y ′ , E ( H )). It isleft to extend this class to Y .We use the S q again. It maps [ α Y ′ ] to S q ( α Y ′ ) ∈ H ( Y, Ω n − qY ⊗ E ) . Then c n − q ω q ∧ S q ( α Y ′ ) ∈ H q ( Y, K Y ⊗ E ( H ))with [( c n − q ω q ∧ S q ( α Y ′ )) | Y ′ ] = [ α Y ′ ] as a cohomology class in H n,q ( Y ′ , E ( H )) . We denote this morphism by j : H n,q ( Y, E ( H ) , (cid:3) ) → H q ( Y, K Y ⊗ E ( H )) 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) We now prove Proposition 1.3 to finish this subsection. roof of Proposition 1.3. (1) Since α is (cid:3) -harmonic, there exists an α l,ε ∈ [ α Y ′ ] with α l,ε ∈ H n,qH ε, ˜ ωl ( E ) for every l, ε such that lim α l,ε = α Y ′ .In particular, ¯ ∂α l,ε = ¯ ∂ ∗ H ε α l,ε = 0. Apply Proposition 2.5 on Y ′ with η = 1, we have0 = k ¯ ∂α l,ε k H ε , ˜ ω l + k ¯ ∂ ∗ H ε α l,ε k H ε , ˜ ω l = k ∂ ∗ H ε α l,ε k H ε , ˜ ω l + k ∂ H ε α l,ε k H ε , ˜ ω l + < i [Θ E,H ε , Λ] α l,ε , α l,ε > H ε , ˜ ω l . (9)Remember that i Θ E,H ε > − εω ⊗ Id E in the sense of Nakano, < i [Θ E,H ε , Λ] α l,ε , α l,ε > H ε , ˜ ω l > − ε ′ ˜ ω l by elementary computation. Now take the limit on the both sides offormula (9) with respect to l, ε , we eventually obtain thatlim k ∂ ∗ H ε α l,ε k H ε , ˜ ω l = lim k ∂ H ε α l,ε k H ε , ˜ ω l = lim < i [Θ E,H ε , Λ] α l,ε , α l,ε > H ε , ˜ ω l = 0 . In particular, 0 = lim ∂ ∗ H ε α l,ε = ∗ ¯ ∂ ∗ lim α l,ε = ∗ ¯ ∂ ∗ α in L -topology. Equivalently, ¯ ∂ ∗ α = 0 on Y ′ in analytic topology, henceis a holomorphic E -valued ( n − q, Y ′ . On the other hand,since ∗ α has the bounded L -norm on Y ′ , it extends to the whole spaceby classic L -extension theorem [35]. In other word, ¯ ∂ ∗ α = 0 actuallyholds on Y , hence ∗ α is an E -valued holomorphic ( n − q, Y .(2) Apply the same argument before, we have α = c n − q ω q ∧ ∗ α .Therefore α must be smooth. (cid:3) Local theory.
In practice, we will also deal with the manifoldwith boundary, hence the local theory is also needed. Let V be abounded domain with smooth boundary ∂V on ( Y, ω ). Moreover, thereis a smooth plurisubharmonic exhaustion function r on V . In particu-lar, V = { r < } and dr = 0 on ∂V . The volume form dS of the realhypersurface ∂V is defined by dS := ∗ ( dr ) / | dr | ω . Let G be a smoothHermitian metric on E . Let L p,q (2) ( V, E ) G,ω be the space of E -valued( p, q )-forms on V which are L -bounded with respect to G, ω . Setting τ := dS/ | dr | ω we define the inner product on ∂V by[ α, β ] G := Z ∂V < α, β > G τ for α, β ∈ L p,q (2) ( V, E ) G,ω . Then by Stokes’ theorem we have the follow-ing: < ¯ ∂α, β > G = < α, ¯ ∂ ∗ β > G +[ α, e ( ¯ ∂r ) ∗ β ] G < ∂ G α, β > G = < α, ∂ ∗ G β > G +[ α, e ( ∂r ) ∗ β ] G (10) here ¯ ∂ ∗ , ∂ ∗ are the adjoint operators defined on Y (see Sect.2.6). Inparticular, if e ( ¯ ∂r ) ∗ α = 0, the Bochner formula (Proposition 2.5) on V will be (cid:3) G = ¯ (cid:3) G + [ i Θ E,G , Λ] , (cid:3) e − χ G = ¯ (cid:3) e − χ G + [ i (Θ E,G + ∂ ¯ ∂χ ) , Λ] and k√ η ( ¯ ∂ + e ( ¯ ∂χ )) α k G,ω + k√ η ¯ ∂ ∗ α k G,ω = k√ η ( ∂ ∗ G − e ( ∂χ ) ∗ ) α ) k G,ω + k√ η∂ G α k G,ω + < ηi [Θ E,G + ∂ ¯ ∂ϕ, Λ] α, α > G,ω +[ ∂ ∗ G α, e ( ∂r ) ∗ α ] G (11)where η is a positive smooth function on Y with χ := log η .We then define the space of harmonic forms on V by H n,q ( V, E ( G ) , r, ω ) := { α ∈ L n,q (2) ( ¯ V , E ) G,ω ; ¯ ∂α = ¯ ∂ ∗ α = e ( ¯ ∂r ) ∗ α = 0 } . Now return back to our setting: (
E, H ) is a singular Hermitian vectorbundle that is strongly positive in the sense of Nakano. By Proposition1.1, there exists a regularising sequence { H ε } such that i Θ E,H ε > − εω ⊗ Id E in the sense of Nakano. Take V ⊂ Y ′ . Using the same notations as inSect.4.1, the harmonic space with respect to H is defined as H n,q ( V, E ( H ) , r ) := { α ∈ L n,q (2) ( V, E ) H,ω ; there exits α l,ε ∈ [ α ] such that α l,ε ∈ H n,q ( V, E ( G ε ) , r, ˜ ω l ) and α l,ε → α in L -limit } . We then generalise the work in [37] here.
Proposition 4.1.
Assume that ( E, H ) is strongly positive in the senseof Nakano (so that E ( H ) is coherent by Proposition 2.3). Then wehave the following conclusions: (1) Assume α ∈ L n,q (2) ( Y, E ) H,ω satisfied e ( ¯ ∂r ) ∗ α = 0 on V . Then α satisfies ¯ ∂α = lim ¯ ∂ ∗ H ε α = 0 on V if and only if ¯ ∂ ∗ α = 0 and lim < ie (Θ H ε + ∂ ¯ ∂r )Λ α, α > H ε = 0 on V . (2) H n,q ( V, E ( H ) , r ) is independent of the choice of exhaustion func-tion r . (3) H n,q ( V, E ( H ) , r ) ≃ H q ( V, K Y ⊗ E ( H )) . (4) For Stein open subsets V , V in V such that V ⊂ V , the re-striction map H n,q ( V , E ( H ) , r ) → H n,q ( V , E ( H ) , r ) is well-defined, and further it satisfies the following commuta-tive diagram: H n,q ( V , E ( H ) , r ) i (cid:15) (cid:15) S qV / / H ( V , Ω n − qY ⊗ E ( H )) (cid:15) (cid:15) H n,q ( V , E ( H ) , r ) S qV / / H ( V , Ω n − qY ⊗ E ( H )) . roof. The proof uses the same argument as Theorems 4.3 and 5.2 in[37] with minor adjustment. So we only provide the necessary details.(1) Let G = e − r H and G ε = e − r H ε . If ¯ ∂α = lim ¯ ∂ ∗ H ε α = 0, thenlim ¯ ∂ ∗ G ε α = 0 and so lim (cid:3) G ε α = 0. By formula (11) we obtainlim( k ∂ ∗ H ε α k G ε + < ie (Θ E,G ε + ∂ ¯ ∂r )Λ α, α > G ε +[ ie ( ∂ ¯ ∂r )Λ α, α ] G ε ) = 0on V . Since < ie (Θ E,H ε + ∂ ¯ ∂r )Λ α, α > G ε > − εω and[ ie ( ∂ ¯ ∂r )Λ α, α ] G ε > , the equality above implies that ∗ ¯ ∂ ∗ α = lim < ie (Θ E,H ε + ∂ ¯ ∂r )Λ α, α > G ε = lim[ ie ( ∂ ¯ ∂r )Λ α, α ] G ε = 0 . Equivalently,¯ ∂ ∗ α = lim < ie (Θ E,H ε + ∂ ¯ ∂r )Λ α, α > H ε = 0 . The necessity is proved.Now assiume that ¯ ∂ ∗ α = 0 and lim < ie (Θ H ε + ∂ ¯ ∂r )Λ α, α > H ε = 0.Since r is plurisubharmonic and lim < ie (Θ H ε )Λ α, α > H ε >
0, we havelim < ie ( ∂ ¯ ∂r )Λ α, α > H ε = lim < ie (Θ H ε )Λ α, α > H ε = 0. By formula(11) we have ¯ ∂α = lim ¯ ∂ ∗ H ε α = 0.(2) Let τ be an arbitrary smooth plurisubharmonic function on V .Donnelly and Xavier’s formula (6) implies that ¯ ∂e ( ¯ ∂τ ) ∗ α = ie ( ∂ ¯ ∂ )Λ α if α ∈ H n,q ( V, E ( H ) , r ) . Therefore < ie ( ∂ ¯ ∂τ )Λ α, α > e τ H ε = < ¯ ∂e ( ¯ ∂τ ) ∗ α, α > e τ H ε = < e ( ¯ ∂τ ) ∗ α, ¯ ∂ ∗ e τ H ε α > e τ H ε = < e ( ¯ ∂τ ) ∗ α, ¯ ∂ ∗ H ε α > e τ H ε −k e ( ¯ ∂τ ) ∗ α k e τ H ε . Take the limit with respect to ε , we then obtain that < ie ( ∂ ¯ ∂τ )Λ α, α > e τ H = −k e ( ¯ ∂τ ) ∗ α k e τ H . Notice that τ is plurisubharmonic, we actually have < ie ( ∂ ¯ ∂τ )Λ α, α > e τ H = k e ( ¯ ∂τ ) ∗ α k e τ H = 0 . Combine with (1), we eventually obtain that H n,q ( V, E ( H ) , r ) = H n,q ( V, E ( H ) , r + τ )for any smooth plurisubharmonic τ , hence the desired conclusion.(3) is similar with Proposition 1.2, and we omit its proof here.(4) is intuitive due to the discussions in the global setting. In par-ticular, S qV i with i = 1 , (cid:3) . The main theorem
This section is devoted to prove Theorem 1.1.
Theorem 5.1 (=Theorem 1.1) . Let f : Y → Z be a fibration betweentwo compact K¨ahler manifolds. Let n = dim Y and m = dim Z . Sup-pose that ( E, H ) is a (singular) Hermitian vector bundle over Y that isstrongly positive in the sense of Nakano. Moreover, assume that H | Y z is well-defined for every z ∈ Z . Then the following theorems hold: I Decomposition Theorem . The Leray spectral sequence [19] E p,q = H p ( Z, R q f ∗ ( K Y ⊗ E ( H ))) ⇒ H p + q ( Y, K Y ⊗ E ( H )) degenerates at E . As a consequence, it holds that dim H i ( Y, K Y ⊗ E ( H )) = X p + q = i dim H p ( Z, R q f ∗ ( K Y ⊗ E ( H ))) for any i > . II Torsion freeness Theorem . For q > the sheaf homomorphism L q : f ∗ Ω n − q ⊗ E ( H ) → R q f ∗ ( K Y ⊗ E ( H )) induced by the q -times left wedge product by ω admits a splitting sheafhomomorphism S q : R q f ∗ ( K Y ⊗ E ( H )) → f ∗ Ω n − q ⊗ E ( H ) with L q ◦ S q = id . In particular, R q f ∗ ( K Y ⊗ E ( H )) is torsion free [26] for q > andvanishes if q > n − m . Furthermore, it is even reflexive if E ( H ) = E . III Injectivity Theorem . Let ( L, h ) be a (singular) Hermitianline bundle over Y . Recall that Y ′ is the open subvariety appeared inProposition 1.1. Assume the following conditions: (a) the singular part of h is contained in Y − Y ′ ; (b) i Θ L,h > γ for some real smooth (1 , -form γ on Y ; (c) for some rational δ ≪ , the Q -twisted bundle E < − δL > | Y z is strongly positive in the sense of Nakano for every z .For a (non-zero) section s of L with sup Y | s | h < ∞ , the multiplicationmap induced by the tensor product with sR q f ∗ ( s ) : R q f ∗ ( K Y ⊗ E ( H )) → R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h )) is well-defined and injective for any q > . IV Relative vanishing Theorem . Let g : Z → W be a fibrationto a compact K¨ahler manifold W . Then the Leray spectral sequence: R p g ∗ R q f ∗ ( K Y ⊗ E ( H )) ⇒ R p + q ( g ◦ f ) ∗ ( K Y ⊗ E ( H )) degenerates. roof. I . Let { U, r U } be a finite Stein covering of Z with smooth strictlyplurisubharmonic exhaustion function r U . Let H n,q ( f − ( U ) , E ( H ) , f ∗ r U )be the harmonic space defined in Sect.4.2. Then the data {H n,q ( f − ( U ) , E ( H ) , f ∗ r U ) , i } with the restriction morphisms i : H n,q ( f − ( U ) , E ( H ) , f ∗ r U ) → H n,q ( f − ( U ) , E ( H ) , f ∗ r U ) , ( U , r U ) ⊂ ( U , r U ), yields a presheaf [22] on Z by Proposition 4.1,(4). We denote the associated sheaf by f ∗ H n,q ( E ( H )). Since R q f ∗ ( K Y ⊗ E ( H ))is defined as the sheaf associated with the presheaf U → H q ( f − ( U ) , K Y ⊗ E ( H )) , the sheaf f ∗ H n,q ( E ( H )) is isomorphic to R q f ∗ ( K Y ⊗ E ( H )) by Propo-sition 4.1, (3).Let C p,q be the space of p -cochains associated to { U } with valuesin f ∗ H n,q ( E ( H )). Then {C p,q , δ } is a complex with the coboundaryoperator δ whose cohomology group H p ( C ∗ ,q ) is isomorphic to E p,q = H p ( Y, R q f ∗ ( K Y ⊗ E ( H ))) . Since the differential d := δ + ¯ ∂ is identically zero, (recall that anelement α ∈ f ∗ H n,q ( E ( H )) must satisfy ¯ ∂α = 0), d : E p,q → E p +2 ,q − is also so which implies the degeneration of the Leray spectral sequenceat E , i.e. E p,q = E p,q ∞ . II . Recall that in the proof of Proposition 1.2, we’ve defined twomorphisms S q : H q ( Y, K Y ⊗ E ( H )) → H ( Y, Ω n − qY ⊗ E ( H ))[ α ]
7→ ∗ ˜ α and L q : H ( Y, Ω n − qY ⊗ E ( H )) → H q ( Y, K Y ⊗ E ( H )) β [ c n − q ω q ∧ β ]such that L q ◦ S q = Id. These two morphisms lift to the direct imagesas S q : R q f ∗ ( K Y ⊗ E ( H )) → f ∗ (Ω n − qY ⊗ E ( H )) ,L q : f ∗ (Ω n − qY ⊗ E ( H )) → R q f ∗ ( K Y ⊗ E ( H )) . Here we abuse the notation. In particular, L q ◦ S q = Id. As a result, R q f ∗ ( K Y ⊗ E ( H )) is splitting embedded into f ∗ (Ω n − qY ⊗ E ( H )). Obvi-ously, f ∗ (Ω n − qY ⊗ E ( H )) is an O Z -submodule of f ∗ (Ω n − qY ⊗ E ), whereas f ∗ (Ω n − qY ⊗ E ) is torsion free (even reflexive) by [23]. We then conclude hat R q f ∗ ( K Y ⊗ E ( H )), as an O Z -submodule of f ∗ (Ω n − qY ⊗ E ), is alsotorsion free.When E ( H ) = E , the reflexivity is also inherited since it is a splittingembedding. III . It is enough to prove that for an arbitrary point z ∈ Z , ⊗ s : H q ( f − ( z ) , K Y ⊗ E ( H )) → H q ( f − ( z ) , K Y ⊗ ( E ⊗ L )( H ⊗ h ))is injective. Equivalently, we should prove that ⊗ s : H n,q ( f − ( z ) , E ( H )) → H n,q ( f − ( z ) , ( E ⊗ L )( H ⊗ h ))is well-defined and injective by Proposition 1.2.Let α ∈ H n,q ( f − ( z ) , E ( H )), by Proposition 4.1, (1) with r = 1, wehave ¯ ∂ ∗ α = lim < ie (Θ H ε )Λ α, α > H ε = 0. Let { h ε } be a regularis-ing sequence of h (Proposition 1.1), hence { H ε ⊗ h ε } is a regularisingsequence of H ⊗ h . In particular, H ε ⊗ h ε is smooth on Y ′ ∩ f − ( z ).Therefore ¯ ∂ ∗ ( sα ) = s ¯ ∂ ∗ α = 0and on Y ′ ∩ f − ( z ) we havelim < ie (Θ E ⊗ L,H ε ⊗ h ε )Λ sα, sα > H ε ⊗ h ε = lim | s | h ε lim < ie (Θ E ⊗ L,H ε ⊗ h ε )Λ α, α > H ε = | s | h lim < ie (Θ L,h ε ⊗ Id E )Λ α, α > H ε δ | s | h lim < ie (Θ E,H ε )Λ α, α > H ε =0 . The inequality is due to the assumption that
E < − δL > is stronglypositive in the sense of Nakano. Now we apply the Bochner formula(Proposition 2.5) on ( E ⊗ L, H ε ⊗ h ε ) and take the limit, eventually weobtain that lim k ¯ ∂ ∗ H ε ⊗ h ε ( sα ) k H ε ⊗ h ε = 0 , and ¯ ∂ ( sα ) = 0 . On the other hand, in any local coordinate neighbourhood V in f − ( z ), Z V | sα | H ⊗ h sup Y | s | h Z V | α | H < ∞ . The discussions above together imply that sα ∈ H n,q ( f − ( z ) , ( E ⊗ L )( H ⊗ h ))by Proposition 4.1, (1). Therefore ⊗ s : H n,q ( f − ( z ) , E ( H )) → H n,q ( f − ( z ) , ( E ⊗ L )( H ⊗ h ))is well-defined. The injectivity is obvious. V is a direct application of I . For a given point w ∈ W , by I theLeray spectral sequence E p,q = H p ( g − ( w ) , R q f ∗ ( K Y ⊗ E ( H ))) ⇒ H p + q (( g ◦ f ) − ( w ) , K Y ⊗ E ( H ))degenerates at E . Therefore the Leray spectral sequence: R p g ∗ R q f ∗ ( K Y ⊗ E ( H )) ⇒ R p + q ( g ◦ f ) ∗ ( K Y ⊗ E ( H ))degenerates. (cid:3) Vanishing theorem
We should prove Theorem 1.2 in the end.
Theorem 6.1 (=Theorem 1.2) . Let f : Y → Z be a fibration betweentwo compact K¨ahler manifolds. I Nadel-type vanishing Theorem . Let ( L, h ) be an f -big linebundle, and let ( E, H ) be a vector bundle that is strongly positive inthe sense of Nakano. Assume that H | Y z is well-defined for every z .Then R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h )) = 0 for every q > . II Nakano-type vanishing Theorem . Assume that ( E, H ) isstrongly strictly positive in the sense of Nakano. Then H q ( Y, K Y ⊗ S l E ( S l H )) = 0 for every l, q > . III Griffiths-type vanishing Theorem . Assume that ( E, H ) isstrictly positive in the sense of Griffiths. Then H q ( Y, K Y ⊗ S l ( E ⊗ det E )( S l ( H ⊗ det H ))) = 0 for every l, q > . Recall that L is f -big if the Iitaka dimension [15] κ ( Y z , L ) = dim Y − dim Z for every z ∈ Z . Proof. I . We claim that if L is f -big, there exists a singular metric h on L such that h | Y z is well-defined and i Θ L,h | Y z is strictly positive forevery z ∈ Z .Indeed, by definition for every z ∈ Z there exists a singular metric h z on L | Y z such that i Θ L,h z is strictly positive [8]. Now take a smoothmetric h on L and define a singular metric on L as follows: h := h δ ⊗ h − δz for y ∈ Y z . It is easy to verify that h satisfies the desired property when δ is smallenough. In particular, i Θ L,h > γ for some real smooth (1 , Y . Let A be a sufficiently ample line bundle over Z . Then H q ( Y z , K Y ⊗ ( E ⊗ L )( H ⊗ h ) ⊗ A ) = 0 for q > R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h ) ⊗ A ) = 0 for q > . n the other hand, it is easy to verify that A satisfies the conditionsin Theorem 1.1, III, therefore R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h )) → R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h ) ⊗ A )is injective. As a result, R q f ∗ ( K Y ⊗ ( E ⊗ L )( H ⊗ h )) = 0 for q > II . By Proposition 1.2, it is enough to prove that H n,q ( Y, S l E ( S l H ) , (cid:3) ) = 0 . Since (
E, H ) is strongly strictly positive in the sense of Nakano,( S l E, H lε )is strictly positive in the sense of Nakano on Y ′ for every ε by Propo-sition 1.1, (e). Indeed, in the proof of (e) it is easy to see that ifthe positivity of ( E, H ) is strict, the regularising sequence { H lε } willsatisfies that(e’) for every relatively compact subset Y ′′ ⊂⊂ Y ′ and every l , i Θ S l E,H lε > C l (1 − ε ) ω ⊗ Id S l E over Y ′′ in the sense of Nakano.Now we apply Bochner’s formula (5) to ( S l E, H lε ) on Y ′ . Noticethat Y − Y ′ is a closed subvariety hence has real codimension >
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