L 2 -extension theorems for jet sections of nef holomorphic vector bundles on compact Kähler manifolds and rational homogeneous manifolds, I
aa r X i v : . [ m a t h . AG ] D ec L -EXTENSION THEOREMS FOR JET SECTIONS OF NEFHOLOMORPHIC VECTOR BUNDLES ON COMPACT K ¨AHLERMANIFOLDS AND RATIONAL HOMOGENEOUS MANIFOLDS, I. QILIN YANG
Abstract.
In this paper we study holomorphic vector bundles with singularHermitian metrics whose curvature are Hermitian matrix currents. We obtainan extension theorem for holomorphic jet sections of nef holomorphic vectorbundle on compact K¨ahler manifolds. Using it we prove that Fano manifoldswith strong Griffiths nef tangent bundles are rational homogeneous spaces.
Contents
1. Introduction 12. Hermitian matrix current 43. Existence theorem for ¯ ∂ -equations and notions of nef vector bundles 83.1. Hermitian vector bundles 83.2. Twisted Bochner-Kodaira-Nakano identity 93.3. Existence thorem for ¯ ∂ -equation 103.4. Singular Hermitian metric on holomorphic vector bundle 113.5. Nef holomorphic vector bundle 144. L -Extension theorems 214.1. Intrinsic norm on jet sections 224.2. Extension theorems for jet sections on bounded Stein domains 234.3. Extension theorems on compact K¨ahler manifolds 355. Proof of the main theorem 41References 431. Introduction
In [Mor] and [SY], Mori, Siu and Yau, using different methods, proved theHartshorne-Frankel conjecture, which says that a compact K¨ahler manifold withpositive bisectional curvature is biholomorphic to the complex projective space. Abeautiful theorem established later by Mok in [Mok], which states that a simplyconnected irreducible compact K¨ahler manifold with non-negative bisectional cur-vature such that the Ricci curvature is positive at one point is isometric to thecomplex projective space or the irreducible compact Hermitian symmetric spaceof rank ≥ . Using the splitting theorem of Howard-Smyth-Wu, we get a completeclassification of all compact simply connected K¨ahler manifolds of nonnegative holo-morphic bisectional curvature.In [CP], Campana and Peternell initiate the classification of projective mani-folds whose tangent bundles are nef, a more degenerate curvature condition thannon-negative bisectional curvature. Recall that a holomorphic line bundle L on a Mathematics Subject Classification.
Key words and phrases.
Nef vector bundle, Hermitian matrix current, L - extension theorem,Holomorphic jet section, Campana-Peternell conjecture. rojective manifold M is called nef if the intersection number L · C ≥ C ⊂ M, and a holomorphic vector bundle on M is called nef if theassociated tautological line bundle O P ( E ∗ ) (1) is a nef line bundle. If we denote thefirst Chern class of L by c ( L ) then L · C = ´ C c ( L ) when C is smooth. Hence aholomorphic vector bundle over a projective manifold with nonnegative bisectionalcurvature is a nef vector bundle, but the converse is not always true. Later in[DPS], Demailly, Campana and Peternell generalized the definition of nefness forholomorphic vector bundles on projective manifolds to arbitrary compact complexmanifolds. They proved that the Albanese morphism of a compact K¨ahler manifoldwith nef tangent bundle is a submersion and whose fibers are Fano manifolds withnef tangent bundles. Since by Mok’s theorem any Fano manifold with nonnegativeholomorphic bisectional curvature is a Hermitian symmetric space, in view of this,Campana and Peternell [CP] formulated a corresponding conjecture claims that aFano manifold with nef tangent bundle is a rational homogeneous space, i.e., quo-tient of a semisimple Lie group by a parabolic subgroup. They showed that theirconjecture is true up to dimension four. Later many special cases are checked tobe true, we will not report them here even there are some very important idealsand techniques found during recent years. We refer readers to the survey [Sur] fora detailed report of related works.In proving Hartshorne-Frankel conjecture, Mori, Siu and Yau used the charac-terization of projective space established by Kobayashi and Ochiai. Mok’s proof ofhis famous theorem also used the Berger-Simons theorem on the characterization ofirreducible symmetric spaces of rank ≥ . However, until now we haven’t a unifiedtheorem which characterize all of the rational homogeneous manifolds at the sametime. This is a main difficult when we are in search of a proof of the Campana-Peternell conjecture. In [BR], Borel and Remmert established a beautiful structuretheorem of compact homogeneous K¨ahler manifolds which says that they are di-rect product of complex tori and rational homogeneous manifolds. Since the Fanomanifolds are simply connected K¨ahler manifolds, if we could prove they are holo-morphical homogeneous then they are rational homogeneous. The homogeneity, arelaxed characterization, is much easier to deal with than giving a straight way anda precise characterization of the rational homogeneous manifolds.How to prove a compact complex manifold is homogeneous? A natural way isto decide its holomorphic automorphism group (which is a complex Lie group byBochner-Montgoemery’s theorem) and prove the automorphism group’s action ishomogeneous. However it is usually very difficult to calculate the holomorphic au-tomorphism group of a complex manifold. passing to infinitesimal level, if we couldprove it has many holomorphic vector fields, then its holomorphic automorphismgroup, whose Lie algebra could be identified with the set of global holomorphicvector fields, would be large enough to act transitively on it.The action of the diffeomorphism group of a real manifold is alway transitivesince we could glue the real local vector fields to global vector fields in a smoothway. Every complex manifold has a Stein open covering and every Stein opensubset have many holomorphic vector fields. However it is very difficult to patchup analytically the local holomorphic vector fields to get global holomorphic vectorfields. This is due to the rigidity of holomorphic objects. But the rigidity mayhave an advantage in some special cases, where it could propagate form local toglobal. We also called this phenomenon the extension or analytical continuation ofa holomorphic object. The corresponding theorems are called extension theorems,and was extensively studied for many years and has a long history.Siu obtained many important extension theorems and explained related tech-niques in the monograph [SBook] , see also his ICM lecture [Siu80] for a report. Itis well known ([Siu74]) that we could extend a L -bounded holomorphic functionform a sub manifold to an open neighborhood, however, we did not know how largethe scale of the open neighborhood is until the work of Ohsawa-Takegoshi appeared n [OT]. Different form the older extension theorems, the Ohsawa-Takegoshi’s ex-tension theorem gains more information including controlling the L norm of theextended holomorphic functions in a precise way, hence it found many applications.A surprise applications is to prove the deformational invariance of plurigenera ofthe algebraic variety of general type by Siu in [Siu98], he successfully extendedholomorphic object from an open submanifold to a family of compact complexmanifold. This is usually impossible as we know even an open complex manifoldhave many holomorphic holomorphic functions but none of them could extend tothe compactified complex manifold.A careful check the assumptions in the Campana-Peternell conjecture, we meetthe following difficulties. Firstly, we have no extension theorems of the Ohsawa-Takegoshi type for holomorphic tangent bundle of high dimensional complex mani-fold. Though the extension theorems of the Ohsawa-Takegoshi type are extensivelystudied in recent years, almost all are concentrate in line bundles case. Secondly,the nefness assumption of the tangent bundle as a curvature condition is imposed infact on the tautological line bundle of projective bundle of the dual of the tangentbundle, rather directly on the tangent bundle itself. Hence we need to constructHermitian metrics on the tangent bundle only under the assumption that the tauto-logical line bundle have a singular Hermitian metric whose curvature is semi positivein the sense of distribution. It is not so difficult to construct Hermitian metric onthe tautological line bundle form those on the tangential bundle and the inverseprocess is not known up to present. In this paper we will study a weaker version ofCampana-Peternell conjecture by changing the assumption of curvature conditions,imposing them directly on the tangential bundle. Accordingly, as a solution of thefirst difficult, we will study extension theorems for higher rank holomorphic vectorbundles with singular Hermitian metric.What curvature condition imposed on a holomorphic vector bundle which is closeto that the tautological line bundle is nef? We know nefness is a generalization ofampleness: a holomorphic vector bundle is called ample if the corresponding tau-tological line bundle is ample. A vector bundle is ample is equivalent to that it isGriffith positive. Hence we have a weaker version of Campana-Peternell conjecturethat a Fano manifold with “Griffiths nef” (Definition 3.10) tangential bundle isrational homogeneous, here the new definition is a nef version of Griffiths positiv-ity, just like nef line bundle is a generalization a Griffiths positive line bundle orequivalently, an ample line bundle. In this paper we add more condition on thepositivity on the tangent bundle, call the Griffiths strongly nef positivity and provethe following main result: Main Theorem.
A Fano manifold whose tangent bundle is strongly Griffiths nefis a rational homogeneous space .The main tool used in proving our main theorem is a L -extension theorem oncompact K¨ahler manifold, which we will give in Section 4 after a series preliminarywork from Section 2 to Section 3. We will study holomorphic vector bundles withsingular Hermitian metrics, to define their curvatures we need to study Hermitianmatrix current. In Section 2 we establish a representation theorem for Hermit-ian matrix current, and study the positivity of the Hermitian matrix currents. InSection 3 we introduce the singular Hermitian metrics via an example and givethe definitions of Griffiths nef and Nakano nef holomorhic vector bundles, we clar-ify the relations among Griffith nefness, Nakano nefness, and nefness defined in[DPS](called nef in usual sense). Finally we gives Bochner-Kodaira identity forholomorphic vector bundles with singular Hermitian metrics and existence theoremfor ¯ ∂ and approximate ¯ ∂ -equations which are extensively used in the next section forstudying extension theorem. In Section 4 we establish Ohsawa-Takegoshi type ex-tension theorems for holomorphic jet sections with finite isolated support of Nakanosemi positive vector bundles and strong Nakano nef vector bundles over bounded tein domains and compact K¨ahler manifolds respectively. Finally in Section 5 weprove that a compact K¨ahler manifold with strong Griffiths nef tangent bundle ishomogeneous, and as a corollary we prove the Main theorem.2. Hermitian matrix current
Singular Hermitian metric on a holomorphic line bundle L over a complex man-ifold was systematically studied by Siu and Demailly, we may refer to the ICMlectures [Siu02] and [De06] for a report of related works. If we write the metric of L locally as h = e − ϕ , where ϕ is only a locally integrable function. It is locallya PSH function if L is a nef or more generally a pseudoeffective line bundle. TheChern curvature of L is Θ h = id ′ d ′′ ϕ. Here the derivatives are in the sense of dis-tribution and Θ h is a closed positive current. In this paper we will consider similarconstructions for higher rank holomorphic vector bundles, i.e., study holomorphicvector bundles with measurable metrics. The curvatures are the weak differentialsof metrics with measurable coefficients, we need to consider them in the sense ofdistribution. We refer readers to [Siu74], [BT] and [DBook] for the theory of closedpositive currents. In this section we will give the definitions of weak derivatives of aHermitian matrix function and study the holomorphic vector bundle with singularmetrics whose curvatures are positive Hermitian matrix currents.Let M be a complex manifold of complex dimension n and E a smooth complexvector bundle of complex rank r on M. In this paper we use C k ( M, E ) to denotethe set of E -valued sections with compact supports and whose derivatives up to k -order are continuous; and L p ( M, E )(resp. L ploc ( M, E )) to denote the set of E -valued sections which are L p (resp. locally L p ) integrable; and L ∞ ( M, E )(resp. L ∞ loc ( M, E )) to denote the set of E -valued sections which are bounded (resp. locallybounded); and W k,p ( M, E ) to denote the set of sections s ∈ L p ( M, E ) whose weakderivatives up to k -order are also in L p ( M, E ) . If E is Hermitian vector bundle, we use Her(E) to denote the sub-bundle ofEnd ( E ) , consisting of Hermitian transformations on each fibre of E, and Her + (E)the subset of Her(E) whose elements are everywhere non-negative definite Hermit-ian transformations. We use C k ( M, Her(E)) to denote the set of Hermitian trans-formation with matrix representation H = ( h α ¯ β ) r × r such that every entry h α ¯ β iscontinuous differentiable up to k -order. Similarly we can define L p ( M, Her(E)) and L ploc ( M, Her(E)) for any 1 ≤ p ≤ ∞ . If E is a trivial bundle we denote, for example, C k ( M, Her(E)) by C k ( M, Her( C r )) . Let T M be the holomorphic tangent bundle of M, T ∗ M and T ∗ M denote the holo-morphic and antiholomorphic cotangent bundles of M respectively. Let Λ p,q T ∗ M bethe tensor product bundle ∧ p T ∗ M ⊗ ∧ p T ∗ M . Equip the space C k ( M, Λ p,q T ∗ M ) withthe following topology induced by semi-norms: write u ∈ C k ( M, Λ p,q T ∗ M ) in localholomorphic coordinate ( z , · · · , z n ) as u = X | I | = p, | J | = q u I ¯ J ( z ) dz I ∧ d ¯ z J , where dz I denotes dz i ∧ · · · ∧ dz i p , while multi-index I is a short writing for( i , · · · , i p ) and | I | = i + · · · + i p . The semi-norms p kK on u for any compactsubset K ⊂ M, are defined by p kK ( u ) = sup x ∈ K max | α |≤ k max | I | = p, | J | = q | ∂ α u I ¯ J ( x ) | , (1)where α = ( α , · · · , α n ) , and x = ( x , · · · , x n ) are real coordinate of M, and ∂ α = ∂ | α | /∂x α · · · ∂x α n n . With respect to these semi-norms C k ( M, Λ p,q T ∗ M ) is acomplete locally convex topological space. Let D ′ k ( M, Λ p,q T ∗ M ) be the space of ontinuous linear functionals on C k ( M, Λ n − p,n − q T ∗ M ) , its elements are called the k -order currents of bidegree ( p, q ) . The space C k ( M, Λ p,q T ∗ M ⊗ C r ) of C r -valued differential forms has a naturallydefined topology induced by the topology on C k ( M, Λ p,q T ∗ M ) , defined by the semi-norms p kK ( u ) = max ≤ j ≤ r p kK ( u j )for any u = ( u , · · · , u r ) ∈ C k ( M, Λ p,q T ∗ M ⊗ C r ) with u j ∈ C k ( M, Λ p,q T ∗ M ) . A matrix current of order k and bidegree ( p, q ) on M is a bilinear function C k ( M, Λ f,g T ∗ M ⊗ C r ) × C k ( M, Λ s,t T ∗ M ⊗ C r ) → C , ( u, v ) U ( u, v ) , where f + s = n − p and g + t = n − q, satisfying (I). U is bilinear linear, i.e., U ( au, bv ) = abU ( u, v )for any a, b ∈ C ; and (II). U is continuous, which means for any compact subset K ⊂ M, there exists a positive constant C such that | U ( u, v ) | ≤ C X r + s = k p rK ( u ) p sK ( v ) . (2)Given a matrix current U we may get a sesquelinear linear function ˆ U defined byˆ U ( u, v ) = U ( u, ¯ v ) . We called ˆ U a sesquelinear linear current . We can also inversethe process to obtain a matrix current form a sesquelinear linear current. Since theset of order k and bidegree ( p, q )- matrix currents is the same as the set of order k and bidegree ( p, q )- sesquelinear linear currents, we denote them by the samesymbol D ′ kp,q ( M, End ( C r )) . Given a sesquelinear linear current H ∈ D ′ kp,p ( M, End ( C r )) , its conjugate ¯ H : C k ( M, Λ f,g T ∗ M ⊗ C r ) × C k ( M, Λ s,t T ∗ M ⊗ C r ) → C is defined by ¯ H ( ξ, η ) = H ( ξ, η );its transpose H t : C k ( M, Λ s,t T ∗ M ⊗ C r ) × C k ( M, Λ f,g T ∗ M ⊗ C r ) → C is defined by H t ( ξ, η ) = H ( η, ξ ) . A sesquelinear linear current H of order k and bidegree ( p, p ) iscalled a Hermitian matrix current of order k and bidegree ( p, p ) if it is Hermitiansymmetric, i.e., ¯ H = H t . The set of order k and bidegree ( p, p )- Hermitian matrixcurrents is denoted by D ′ kp,p ( M, Her( C r )) . Example 2.1.
A Hermitian matrix current of bidegree (0 ,
0) is called a generalizedHermitian matrix function . Let L loc ( M, Her + (T M )) be the set of Hermitian metrics H = ( h α ¯ β ) n × n on M, where h α ¯ β is locally defined function and h α ¯ β ∈ L loc (Ω)is locally integrable for any open subset Ω ⊂ M. Then H defines a generalizedHermitian matrix function. For example for ξ, η ∈ C k ( M, T M ) with notion η t =( η , · · · , η n ) the transpose of the column vector η , let dµ denote the Lebesguemeasure then ξdµ ∈ C k ( M, Λ n,n T ∗ M ⊗ T M ) , The function ´ M η t H ¯ ξdµ is clearlysesquelinear, continuous, and symmetric. Example 2.2.
Let U = ( U αβ ) r × r be a matrix of currents of of k -order and bide-gree ( p, q ) , and each entry U αβ ∈ D ′ k ( M, Λ p,q T ∗ M ) is a ( p, q )-form with k -orderdistribution coefficients. Define(3) U ( u, v ) = ˆ M u t U v, where u = ( u , · · · , u r ) t ∈ C k ( M, Λ f,g T ∗ M ⊗ C r ) v = ( v , · · · , v r ) t ∈ C k ( M, Λ s,t T ∗ M ⊗ C r )with n = f + r + p = g + s + q, and u α ∈ C k ( M, Λ f,g T ∗ M ) and v α ∈ C k ( M, Λ α,β T ∗ M )for α = 1 , · · · , r, and for brevity we don’t write out the wedges in (3) and u t U v = P α,β u α ∧ U αβ ∧ v β . It is easy to check that U is a matrix current of order k and idegree ( p, q ) . If ( H αβ ) is Hermitian symmetric matrix of currents of k -order andbidegree ( p, p ) , that is if ¯ H αβ = H βα for all α, β, then(4) H ( u, v ) = ˆ M u t H ¯ v, defines a Hermitian matrix current H. Proposition 2.3.
A matrix current U ∈ D ′ kp,q ( M, End ( C r )) is a matrix of currentsof order k and bidegree ( p, q ) . Proof.
Fix u ∈ C k ( M, Λ f,g T ∗ M ⊗ C r ) then U ( u, · ) : C k ( M, Λ s,t T ∗ M ⊗ C r ) → C is alinear function satisfying | U ( u, v ) | ≤ C u p kK ( v )for any compact subset K ⊂ M. By (2), C u is a fixed constant depending only u. Hence U ( u, · ) is a k -order continuous function on C k ( M, Λ p,q T ∗ M ⊗ C r ) . Write v = ( v , · · · , v r ) with v j ∈ C k ( M, Λ p,q T ∗ M ) . Identify the vectors( v , , · · · , , · · · , (0 , · · · , , v r )with the forms v , · · · , v r then U ( u, v ) = P rβ =1 U ( u, v β ) and we may consider U ( u, v β ) as a linear function on C k ( M, Λ s,t T ∗ M ) . Hence U ( u, v ) = r X β =1 ˆ M U β ( u ) ∧ v β , (5)where U β ( u ) are k -order currents of degree ( n − r, n − s ) . Write u = ( u , · · · , u r )we may get U ( u, v ) = r X α,β =1 U ( u α , v β ) = r X β =1 ˆ M U β ( u α ) ∧ v β , (6)Now fix v β we consider each term F β ( u α ) := ´ M U β ( u α ) ∧ v β as a linear functionon C k ( M, Λ f,g T ∗ M ) . Then | F β ( u α ) | ≤ C v β p kK ( u α ) , where C v β is a constant depending only on v β . For any multi-indices
I, J with | I | = p and | J | = q, take u α = h dz P ∧ d ¯ z Q ∈ C k ( M, Λ f,g T ∗ M ) and v β = h dz R ∧ d ¯ z S ∈ C k ( M, Λ s,t T ∗ M ) such that ( P ∪ R ) ∪ I = ( Q ∪ S ) ∪ J = { , · · · , n } . Denote h = h h dz ∧ · · · ∧ dz n ∧ d ¯ z ∧ · · · ∧ d ¯ z n ∈ C k ( M, Λ n,n T ∗ M ) , and set U αβ,IJ ( h ) = Sgn( IJP QRS ) ˆ M U β ( u α ) ∧ v β = U ( u α , v β ) , where Sgn( IJP QRS ) is the signature of the permutation (1 , · · · , n, · · · , n ) ( P, Q, I, J, R, S ) . Then there exist constants C and ¯ C such that | U αβ,IJ ( h ) | ≤ C X r + s = k p rK ( h ) p sK ( h ) ≤ ¯ Cp kK ( h )for any compact subset K ⊂ M. Hence U αβ,IJ is a k -order distribution. Let U αβ = X | I | = p, | J | = q U αβ,IJ dz I ∧ d ¯ z J ∈ D ′ k ( M, Λ p,q T ∗ M ) . ow it is easy to check that F β ( u α ) = ˆ M u α ∧ U αβ ∧ v β . (7)By (5),(6),(7) we know U ( u, v ) = ´ M u t U v and U is a matrix of currents of order k and bidegree ( p, q ) . (cid:3) Corollary 2.4.
A Hermitian matrix current H ∈ D ′ kp,p ( M, End ( C r )) is a Hermit-ian matrix of currents of order k and bidegree ( p, p ) . The weak differentials of a matrix current U ∈ D ′ kp − ,q ( M, End ( C r )) , consideredas a bilinear current, are defined by(8) d ′ U ( ξ, η ) = − ( − f + g U ( d ′ ξ, η ) + ( − p + q U ( ξ, d ′ η )(9) d ′′ U ( ξ, η ) = − ( − f + g U ( d ′′ ξ, η ) + ( − p + q U ( ξ, d ′′ η )for any ξ ∈ C k ( M, Λ f,g T ∗ M ⊗ C r ) and η ∈ C k ( M, Λ s,t T ∗ M ⊗ C r ) with f + s = n − p and g + t = n − q. If U ∈ D ′ kp − ,q ( M, End ( C r )) is a sesquelinear current, its weakdifferential are defined by(10) d ′ U ( ξ, η ) = − ( − f + g U ( d ′ ξ, η ) + ( − p + q U ( ξ, d ′′ η )(11) d ′′ U ( ξ, η ) = − ( − f + g U ( d ′′ ξ, η ) + ( − p + q U ( ξ, d ′ η )for any ξ ∈ C k ( M, Λ f,g T ∗ M ⊗ C r ) and η ∈ C ∞ ( M, Λ t,s T ∗ M ⊗ C r ) with f + s = n − p and g + t = n − q. Set dU = d ′ U + d ′′ U. A (Hermitian) matrix current U is called closed if dU = 0 . Proposition 2.5.
For any matrix current U ∈ D ′ p,q ( M, End ( C r )) , d ′ U = d ′′ U = d U = 0 . Proof.
For any ξ ∈ C k ( M, Λ f,g T ∗ M ⊗ C r ) and η ∈ C k ( M, Λ s,t T ∗ M ⊗ C r ) with f + s = n − p − g + t = n − q. ( d ′ U )( ξ, η ) = − ( − f + g ( d ′ U )( d ′ ξ, η ) + ( − p + q − ( d ′ U )( ξ, d ′ η )= − ( − f + g + p + q U ( d ′ ξ, d ′ η ) − ( − f + g + p + q − U ( d ′ ξ, d ′ η ) , hence d ′ = 0 . In the same way we have d ′′ U = d U = 0 . (cid:3) Identify a matrix current U ∈ D ′ p,q ( M, End ( C r )) with a matrix ( U αβ ) r × r ofcurrents. For any ξ ∈ C k ( M, Λ f,g T ∗ M ⊗ C r ) and η ∈ C k ( M, Λ s,t T ∗ M ⊗ C r ) with f + r ≤ n − p and g + t ≤ n − q, define U ( ξ, η ) = ξ t U η, then U ( ξ, η ) ∈ D ′ k ( M, Λ a,b T ∗ M ) is a current of k -order and bidegree ( a, b ) with f + r + a = n − p and g + t + b = n − q. Definition 2.6.
A Hermitian matrix ( p, p ) -current H ∈ D ′ kp,p ( M, Her( C r )) is calledpositive if for any non zero ξ ∈ C k ( M, C n ) the current H ( ξ, ξ ) is a positive ( p, p ) -current; which means for every choice of smooth (1 , -forms α , · · · , α p on M thedistribution H ( ξ, ξ ) ∧ iα ∧ ¯ α ∧ · · · ∧ iα p ∧ ¯ α p is a positive measure on M. We denote the set of positive Hermitian matrix currentsof k -order by D ′ kp,p ( M, Her + ( C r )) . xample 2.7. Let δ ( z j ) denote the Dirac delta functions and H = (cid:18) i ( | z | + 1) δ ( z ) dz ∧ d ¯ z z δ ( z ) δ ( z ) dz ∧ dz ¯ z δ (¯ z ) δ (¯ z ) d ¯ z ∧ d ¯ z i ( | z | + 1) δ ( z ) dz ∧ d ¯ z (cid:19) (12)Then H ∈ D ′ , ( C , Her + ( C )) is a positive (1 , ξ = ( ξ , ξ ) t ∈ C ( C , C ) H ( ξ, ξ ) = | ξ (0 , z ) | idz ∧ d ¯ z + | ξ ( z , | idz ∧ d ¯ z is a continuous (1 , , α = α dz + α dz we know H ( ξ, ξ ) ∧ ( iα ∧ ¯ α ) = ( | ξ (0 , z ) | | α | + | ξ ( z , | | α | ) idz ∧ d ¯ z ∧ idz ∧ d ¯ z is a positive measure.3. Existence theorem for ¯ ∂ -equations and notions of nef vectorbundles Hermitian vector bundles.
Let M be a compact K¨ahler manifold of di-mension n with a K¨ahler form ω, and let E be a holomorphic vector bundle ofrank r on M with a smooth Hermitian metric H . Let ( E ∗ , H ∗ ) be the dual vectorbundle. Let C ∞ ( M, Λ p,q T ∗ M ⊗ E ) be the space of E -valued smooth ( p, q )-forms,and C k ( M, Λ p,q T ∗ M ⊗ E ) be the space of E -valued smooth ( p, q )-forms with com-pact support. Let ∗ : C ∞ ( M, Λ p,q T ∗ M ⊗ E ) → C ∞ ( M, Λ n − q,n − p T ∗ M ⊗ E ) be theHodge star-operator with respect to ω . For any f ∈ C ∞ ( M, Λ p,q T ∗ M ⊗ E ) and g ∈ C ∞ ( M, Λ a,b T ∗ M ⊗ E ), we define f ∧ H ¯ g ∈ C ∞ ( M, Λ p + b,q + a T ∗ M ⊗ E ) as follow-ing. We take a local trivialization of E on an open subset U ⊂ X , and we regard f = ( f , . . . , f r ) t as a row vector with ( p, q )-forms f j on U . The Hermitian metric H is then a matrix valued function H = ( h j ¯ k ) on U . We define f ∧ H ¯ g locally on U by f t ∧ H ¯ g = X j,k f j ∧ h j ¯ k ¯ g k ∈ C ∞ ( M, Λ p + b,q + a T ∗ M ) . In this manner, we can define anti-linear isomorphisms ♯ H : C ∞ ( M, Λ p,q T ∗ M ⊗ E ) → C ∞ ( M, Λ q,p T ∗ M ⊗ E ∗ )by ♯ H u = H ¯ u , and(13) ¯ ∗ H = ♯ H ◦ ∗ : C ∞ ( M, Λ p,q T ∗ M ⊗ E ) → C ∞ ( M, Λ n − p,n − q T ∗ M ⊗ E ∗ )by ¯ ∗ H f = H ∗ f . There is a point-wise inner product on C ∞ ( M, Λ p,q T ∗ M ⊗ E ) definedby h f, g i = f ∧ ¯ ∗ H g and | u | H = h f, f i . It induces an inner product and a norm on C ∞ ( M, Λ p,q T ∗ M ⊗ E ) defined by(14) ( f, g ) H = ˆ M h f, g i , k f k H = ˆ M | f | H . If the metric ω is complete we get a Hilbert space L ( M, Λ p,q T ∗ M ⊗ E ) , containingthe space C ∞ ( M, Λ p,q T ∗ M ⊗ E ) as a dense subspace. Denote by D = D ′ + D ′′ theChern connection with D ′′ = d ′′ . The Chern curvature form(15) Θ H = id ′′ ( H − d ′ H ) = id ′ ( d ′′ H − ) H )is smooth Hermitian matrix from. The Hermitian vector bundle ( E, H ) is said to be
Nakano semi-positive (resp.
Nakano positive ), if the End ( E )-valued real (1 , i Θ H is positive semi-definite (resp. positive definite) quadratic form on each fiberof the vector bundle T M ⊗ E .We define D ′′∗ : C ∞ ( M, Λ p,q T ∗ M ⊗ E ) → C ∞ ( M, Λ p,q − T ∗ M ⊗ E ) by D ′′∗ = − ∗ D ′ ∗ = − ¯ ∗ H ∗ D ′′ ¯ ∗ H , which is the formal adjoint operator of D ′′ : C ∞ ( M, Λ p,q T ∗ M ⊗ ) → C ∞ ( M, Λ p,q +1 T ∗ M ⊗ E ) with respect to the inner product ( , ). In the sameway we denote D ′∗ the formal adjoint operator of D ′ with respect to the inner prod-uct ( , ). We denote by e ( α ) the left exterior product acting on C ∞ ( M, Λ p,q T ∗ M ⊗ E )by a form α ∈ C ∞ ( M, Λ a,b T ∗ M ). Then the adjoint operator e ( α ) ∗ with respect tothe inner product ( , ) H is defined by e ( α ) ∗ = ( − ( p + q )( a + b +1) ∗ e (¯ α ) ∗ . For instancewe set Λ = e ( ω ) ∗ . In the following write e ( α ) and e ( α ) ∗ simply as α and α ∗ forbrevity if without special explanations. Proposition 3.1.
If we change the metric H by multiplying a positive function η, the new adjoint operators of ∂ and ¯ ∂ with respect to the new metric Hη, denotedby D ′∗ η and ¯ ∂ ∗ η , are related to D ′∗ and D ′′∗ in the following way: D ′∗ η = D ′∗ − η ( d ′ η ) ∗ , D ′′∗ η = D ′′∗ − η ( d ′′ η ) ∗ . Proof.
For f ∈ C ∞ ( M, Λ p,q T ∗ M ⊗ E ) ,D ′∗ η f = − ¯ ∗ Hη ∗ D ′ (¯ ∗ Hη f ) = − ( 1 η ¯ ∗ H ∗ ) D ′ ( η ¯ ∗ H f ) = ¯ ∗ H ∗ D ′ ¯ ∗ H f − η ¯ ∗ H ∗ d ′ η ¯ ∗ H , hence we get the first equality and the second follows the same way. (cid:3) Twisted Bochner-Kodaira-Nakano identity.
The following Bochner-Kodaira-Nakano identity is basic in studying vanishing theorem and L -estimates in K¨ahlergeometry, we may find its proof in [Siu01],[Siu11],[DBook]. k D ′′ f k H + k D ′′∗ f k H = k D ′ f k H + k D ′∗ f k H + ([Θ H , Λ] f, f ) H . (16)Ohsawa and Takegoshi obtained a twisted form of it in [OT], where it plays a criticalrole in establishing the extension theorem with their name. In the following we willgive a short proof of the twisted Bochner-Kodaira-Nakano identity from the pointof view of deforming the Hermitian metric. Proposition 3.2.
Let M be a K¨ahler manifold with a complete K¨ahler metric ω and E be a Hermitian holomorphic vector bundle equipped with smooth Hermitian metric H. Let η, λ be smooth positive function on M. Then for any f ∈ L ( M, Λ n,q T ∗ M ⊗ E ) , k ( η + λ − ) D ′′∗ f k H + k η D ′′ f k H ≥ ([ η Θ H − id ′ d ′′ η − iλd ′ η ∧ d ′′ η, Λ] f, f ) H . (17) Proof.
Since C ∞ ( M, Λ n,q T ∗ M ⊗ E ) is dense in L ( M, Λ n,q T ∗ M ⊗ E ) , it suffices to prove(17) for u ∈ C ∞ ( M, Λ n,q T ∗ M ⊗ E ) . Change the metric by Hη and note D ′ f = 0 when f ∈ C ∞ ( M, Λ n,q T ∗ M ⊗ E ) we get from the Bochner-Kodaira identity the following k D ′′ f k Hη + k ( D ′′∗ η f k Hη = k ( D ′ η ) ∗ f k Hη + ([Θ Hη , Λ] f, f ) Hη . (18)By Proposition 3.1, k D ′′∗ η f k Hη = k η D ′′∗ f k H + k η − ( d ′′ η ) ∗ f k H − Re ( D ′′∗ f, ( d ′′ η ) ∗ f ) H . (19)Since i [ d ′′ η, Λ] = ( d ′ η ) ∗ and i [ d ′ η, Λ] = ( d ′′ η ) ∗ , we have ( d ′′ η )( d ′′ η ) ∗ − ( d ′ η ) ∗ ( d ′ η ) = i ( d ′′ η )[ d ′ η, Λ]+ i [ d ′′ η, Λ]( d ′ η ) = [ id ′ η ∧ d ′′ η, Λ] and hence for u ∈ L ( M, Λ n,q T ∗ M ⊗ E ) , k ( d ′′ η ) ∗ f k H = ([ id ′ η ∧ d ′′ η, Λ] f, f ) H . (20)Since Θ Hη = d ′′ ( η − H − d ′ ( ηH )) = Θ H + iη − d ′ η ∧ d ′′ η − iη − d ′ d ′′ η we have −k η − ( d ′′ η ) ∗ f k H + ([Θ Hη , Λ] f, f ) Hη = ([ η Θ H − id ′ d ′′ η, Λ] f, f ) H . (21)By (18)-(21) we have η D ′′ f k H + k η D ′′∗ f k H = ([ η Θ H − id ′ d ′′ η, Λ] f, f ) H + k D ′∗ f k Hη − Re ( D ′′∗ f, ( d ′′ η ) ∗ f ) H . (22)By Cauchy Schwartz inequality we have − Re ( D ′′∗ f, ( d ′′ η ) ∗ f ) H ≥ −k λ − D ′′∗ f k H − k λ ( d ′′ η ) ∗ f k H . (23)Note k η D ′′∗ f k H + k λ − D ′′∗ f k H = k ( η + λ − ) D ′′∗ f k H and by (20) we have k λ ( d ′′ η ) ∗ f k H = ([ iλd ′ η ∧ d ′′ η, Λ] f, f ) H . From these together with (22) and (23)we conclude (17). (cid:3)
Existence thorem for ¯ ∂ -equation. We will use the L existence theo-rem for solving ¯ ∂ -equations. Ohsawa and Takegoshi [OT] established their ex-tension theorem via solving the twisted form of ¯ ∂ -equations, and siu [Siu11] gavean elegant explanation of by a commutating diagram (page 1780 of [Siu11]). Let T f = D ′′ (( η + λ − ) f ) and Sf = η ( D ′′ f ) be the twisted ¯ ∂ operators acting on E -valued differential form f. Assume for some specific choice of the function λ and η we have(24) ( η Θ H − id ′ d ′′ η − iλd ′ η ∧ d ′′ η, Λ] f, f ) H ≥ k√ γ ( d ′′ ζ ) ∗ f k H − δ k f k H on M, where ζ is a non zero smooth function, γ is a positive function and δ ≥ L ( M, Λ n,k T ∗ M ⊗ E ) for 0 ≤ k ≤ m are denoted by the same signal || · || H . Then by (17) we have || T ∗ f || H + || Sf || H + δ || f || H ≥ k√ γ ( d ′′ ζ ) ∗ f k H . Assume g, g ∈ W := L ( M, Λ n,q T ∗ M ⊗ E ) are ( n, q ) form with Sg = 0 , and g = d ′′ ζ ∧ g + g with g ∈ W := L ( M, Λ n,q − T ∗ M ⊗ E ) . We want solve theequation
T f = g. In this paper we need to solve the equation in following twocases.3.3.1.
Case 1.
In this case g = 0 and δ = 0 . For any u ∈ W , write u = u + u with D ′′ u = 0 and u is in the orthogonal complement space of the kernel of D ′′ . Then ( g, u ) H = ( g, u ) H and we get( g, u ) H = (( d ′′ ζ ) ∗ u, g ) H ≤ || T ∗ u || H · || √ γ g || H , which means the functional T ∗ u → h g, u i defined on the Hilbert space W is con-tinuous, and hence by Riez Representation Theorem the equation D ′′ ˜ f = g with˜ f = ( η + λ − ) f has a solution with estimate ˆ M ( η + λ − ) − | ˜ f | H ≤ ˆ M √ γ | g | H . Case 2.
The second case we will need is to solve the approximate equationby taking δ → . So we assume δ > . Furthermore g is not necessary assumed tobe zero. In this case ( g, u ) H = (( d ′′ ζ ) ∗ u, g ) H + ( g , u ) H , by Cauchy-Schwartz inequality we have12 | ( g, u ) H | ≤ ||√ γ ( d ′′ ζ ) ∗ u || H · || √ γ g || H + δ || u || H · δ || g || H . gain we use the trick to write u = u + u ∈ W with D ′′ u = 0 and u is in theorthogonal complement space of the kernel of D ′′ . Then we get12 | ( g, u ) H | ≤ C ( g, γ, δ )( || T ∗ u || H + δ || u || H )with C ( g, γ, δ ) = || √ γ g || H + δ || g || H . Define a function on the Hilbert space sum W ⊕ W by ( T ∗ u, √ δu ) → h g, u i , then the inequality we got means it is a continuous functional, hence using RiezRepresentation Theorem we can solve the approximate ¯ ∂ equation D ′′ ˜ f + √ δh = ¯ ∂ζ ∧ g + g together with estimate ˆ M ( η + λ − ) − | ˜ f | H + ˆ M | h | H ≤ (cid:16) ˆ M √ γ | g | H + 1 δ ˆ M | g | H (cid:17) . As a summery we have the following existence theorem:
Proposition 3.3.
Let M as in Proposition 3.2, assume (24) holds for any f ∈ L ( M, Λ n,q T ∗ M ⊗ E ) . Then (i) . If δ = 0 , then for g = d ′′ ζ ∧ g ∈ L ( M, Λ n,q T ∗ M ⊗ E ) such that D ′′ g = 0 and ´ M √ γ | g | H < + ∞ , there exits f ∈ L ( M, V n,q − T ∗ M ⊗ E ) such that D ′′ f = g and ˆ M ( η + λ − ) − | f | H ≤ ˆ M √ γ | g | H . (ii) . If δ > , then for any g = d ′′ ζ ∧ g + g ∈ L ( M, Λ n,q T ∗ M ⊗ E ) such that D ′′ g = 0 and ´ M √ γ | g | H + δ ´ M | g | H < + ∞ , there exits f ∈ L ( M, V n,q − T ∗ M ⊗ E ) and h ∈ L ( M, V n,q T ∗ M ⊗ E ) such that D ′′ f + √ δh = ¯ ∂ζ ∧ g + g and ˆ M ( η + λ − ) − | f | H + ˆ M | h | H ≤ (cid:16) ˆ M √ γ | g | H + 1 δ ˆ M | g | H (cid:17) . Singular Hermitian metric on holomorphic vector bundle.
Let M bea complex manifold and E a holomorphic vector bundle of rank r on M. For any U ∈ D ′ kp,q ( M, End ( E )) and any continuous and differentiable of k -order matrixfunction V ∈ C k ( M, End ( E )) , we define the product V U : C k ( M, E ) × C k ( M, E ) → D ′ p,q ( M ) by ( V U )( ξ, η ) = U ( ξ, V η ) . In the same way we define (
U V )( ξ, η ) = U ( V ξ, η ) . Let U k , U ∈ D ′ kp,q ( M, End ( E )) be matrix currents. The sequence U k is called weak*-convergent to U if for any ξ, η ∈ C k ( M, E ) , the ( p, q )-currents U k ( ξ, η ) is weak*-convergent to U ( ξ, η ) . In this paper, we restrict to study the following type of singular Hermitian metricon the holomorphic vector bundles:
Definition 3.4.
A singular Hermitian metric of type I on E we means a measurableHermitian metric H such that d ′ H ∈ D ′ , ( M, End ( E )) and H − ∈ C ( M, End ( E )) is a metric with continuous coefficients. A singular Hermitian metric of type II on E we means a continuous Hermitian metric H such that H − is measurable and d ′′ H − ∈ D ′ , ( M, End ( E )) . By formula (15), if H and H − are smooth then id ′′ ( H − d ′ H ) = id ′ ( d ′′ H − ) H ) . For singular Hermitian metrics, the Chern connection is not necessary smooth andChern curvature forms id ′′ ( H − d ′ H ) and id ′ ( d ′′ H − ) H ) are only meaningful in thesense of distribution. For singular Hermitian metric of type I since H − d ′ H is a well efined matrix current, and its weak derivative d ′′ ( H − d ′ H ) is also well defined.For singular Hermitian metric of type II the curvature is also well defined in thesense of distribution. We call Θ H = id ′′ ( H − d ′ H ) (reap. Θ H = id ′ ( d ′′ H − ) H ))the curvature currents of the singular Hermitian metric of type I (reap. type II). Proposition 3.5.
The curvature current Θ H is a closed Hermitian matrix current.Proof. Here we give proof for the singular Hermitian metric of type I. The typeII case is prove in the same way. This is a local property, it suffices to provethe proposition on any small open set U over which E is a trivial bundle. For any ξ ∈ C ∞ ( U, Λ f,g T ∗ M | U ⊗ C n ) and η ∈ C ∞ ( U, Λ s,t T ∗ M | U ⊗ C n ) with f + s = g + t = n − . For simplicity we assume f = g = 0 and the other cases are checked in the sameway. Then we haveΘ H ( ξ, η ) = − iH − d ′ H ( d ′′ ξ, η ) + iH − d ′ H ( ξ, d ′ η ) . (25)Using the definition of weak differential of current we rewrite the R.H.S. of (25) as H − d ′ H ( d ′′ ¯ ξ, η ) = − ( d ′ d ′′ ξ, η ) − H ( d ′′ ξ, d ′′ ( H − η )) , (26) H − d ′ H ( ¯ ξ, d ′ η ) = − ( d ′ ξ, d ′ η ) − H ( ξ, d ′′ ( H − d ′ η )) , (27)where ( d ′ d ′′ ξ, η ) = ´ M ( d ′ d ′′ ξ ) t ∧ ¯ η. The second term of R.H.S. of (26) is H ( d ′′ ξ, d ( H − η )) = H ( d ′ ξ, ( d ′ ¯ H − ) η ) + ( d ′′ ξ, d ′′ η ) . (28)The second term of R.H.S. of (27) is H ( ξ, d ′′ ( H − d ′ η )) = H ( ξ, ( d ′ H − ) d ′′ η ) + ( ξ, d ′′ d ′ η ) . (29)By (25) and (26)-(29) we getΘ H ( ξ, η ) = i ¯ H ( d ′ ξ, ( d ′ H − ) η ) − i ¯ H ( ¯ ξ, ( d ′ H − ) d ′′ η )(30)Since H is a Hermitian matrix we haveΘ H ( ξ, η ) = iH (( d ′ H − ) η, d ′ ξ ) − iH (( d ′ H − ) d ′′ η, ξ ) , (31)hence we have Θ H ( ξ, η ) = id ′′ ( H ( d ′ H − ))( η, ξ ) . (32)In the same way we may check that(Θ H ) t ( ξ, η ) = id ′′ ( H ( d ′ H − ))( η, ξ ) . (33)By (32) and (33)) we know Θ H is closed and Θ H = (Θ H ) t , hence Θ H is a closedHermitian matrix current. (cid:3) Example 3.6.
Assume that s , · · · , s N are non zero holomorphic sections of therank r holomorphic vector bundle E and denote S = ( s , · · · , s N ) with s j =( s j , · · · , s rj ) t ∈ E for j = 1 , · · · , N. Let A = S ¯ S t = P Nj =1 s j ¯ s tj , then A = P Nj =1 | s j | P Nj =1 s j ¯ s j · · · P Nj =1 s j ¯ s rj P Nj =1 s j ¯ s j P Nj =1 | s j | · · · P Nj =1 s j ¯ s rj ... ... . . . ... P Nj =1 s rj ¯ s j P Nj =1 s rj ¯ s j · · · P Nj =1 | s rj | (34)is a r × r Hermitian matrix. We denote the Moore-Penrose pseudoinverse of A by H. Note that when H is a non-degenerate square matrix we have A = H − . Thereis a natural (possible singular) Hermitian metric on E defined by || ξ || = ξ t Hξ, for ξ ∈ E z . t is easy to check that this definition does not depend on the local trivializationof E and is well-defined. Note A is a nonnegative definite matrix and the set ofsingularity points of the metric are the points where A is degenerate, so it is the set Z = { z ∈ M | rank S ( z ) < r } , which are exactly the points where s , · · · , s N don’tgenerate the stalk E z . Clearly Z is an analytic subset of M and hence we say thatthe Hermitian metric has analytic singularity.If ( s , · · · , s N ) generated the stalk E z for all z ∈ M, then Z = ∅ . If ( s , · · · , s N )generated the stalk E z at least for one point z ∈ M (then ( s , · · · , s N ) generated thestalks over an open subset of M ), we call ( s , · · · , s N ) is generically generated. Inthis case we must have N ≥ r. Let Gr ( N, r ) be the complex Grassmannian manifoldof r -planes in complex N -dimensional vector space. Then there is a holomorphicembedding F : M → Gr ( N, r ) , p [ s ( p ) , · · · , s N ( p )] . Then the metric definedabove is smooth and it is exactly the pull back by F of the Fubiny-Study metric onthe universal vector bundle of Gr ( N, r ) . If moreover r = 1 , then || ξ || = | ξ | U e − ϕ . Here the metric weight ϕ = log( | s | + · · · + | s N | ) is a plurisubharmonic functionon U. Proposition 3.7. If ( s , · · · , s N ) is generically generated then the (possible) sin-gular Hermitian metric in Example 3.6 has positive curvature current.Proof. On the Grassmannian manifold Gr ( N, r ) , the universal bundle Q , which isthe quotient bundle of the trivial bundle C N → Gr ( N, r ) , and the quotient metricis given by h ( f ) = ( f ¯ f t ) − , where f ∈ Gr ( N, r ) is a r × N matrix whose rows givethe r -plane f. Choose holomorphic transformation such that f = ( I r , Z ) . In thisholomorphic local coordinate we have h = ( I + Z ¯ Z t ) − and h (0) = I, dh (0) = 0 . Hence at the origin, the Chern curvature of the universal bundle is i Θ h = id ′′ d ′ h = idZ ∧ dZ t . Since Gr ( N, r ) is homogeneous under the unitary group and h is aninvariant Hermitian metric, we know i Θ h is semi-positive (positive only when r = 1)at origin and hence is semi-positive everywhere.If ( s , · · · , s N ) globally generate the stalks of E, then F is smooth and hence i Θ H = iF ∗ (Θ h ) is semi-positive and smooth matrix of differential form. If ( s , · · · , s N )doesn’t generate the stalks of E then H defines a singular Hermitian metric on E. If( s , · · · , s N ) is merely generically generated we still can define the map F as above,which is a meromorphic map and not necessarily holomorphic. The curvature cur-rent i Θ H is the pull back of the curvature i Θ h of the Fubini-Study metric h onthe universal vector bundle Q of Gr ( N, r ) . We know i Θ H = iF ∗ (Θ h ) is a positivecurvature current. (cid:3) Remark 3.8.
The line bundle ∧ r Q → Gr ( N, r ) has an induced metric h r ( f ) =det( f ¯ f t ) with positive curvature which gives an invariant K¨ahler metric ω Gr = id ′′ d ′ log det( I + Z ¯ Z t ) on Gr ( N, r ) , which is exactly the pull-back of the FubiniStudy metric via Pl¨ucker embedding Gr ( N, r ) → P ( ∧ r C N ) . Using that d ′ log det( I + Z ¯ Z t ) = Tr(( I + Z ¯ Z t ) − ( dZ ) ¯ Z t ) we know ω Gr = i Tr { ¯ Z t ( I + Z ¯ Z t ) − ZdZ t ( I + Z ¯ Z t ) − dZ − ( I + Z ¯ Z t ) − dZ ∧ dZ t } . Note that ¯ Z t ( I + Z ¯ Z t ) − Z = I − ( I + ¯ Z t Z ) − , we get the expression of ω Gr asalready given in [Wong]: ω Gr = − i Tr { ( I + ¯ Z t Z ) − dZ t ( I + Z ¯ Z t ) − dZ } = i Tr { ( I + Z ¯ Z t ) − dZ ( I + ¯ Z t Z ) − dZ t } . (cid:3) Given a Hermitian metric H with measurable coefficients, we may smooth it viausing convolution construction. To illustrate it we take the Euclid flat case as anexample. Let H = ( h α ¯ β ) be a Hermitian matrix of measurable functions, we couldget a smooth matrix function H ǫ = H ⋆ ρ ǫ def = ( h α ¯ β ⋆ ρ ǫ ) , the convolution of its each ntry with a smooth kernel function ρ ǫ . Here ρ ǫ = ǫ n ρ ( zǫ ) and ρ ( z ) = χ {| z | < } · e − −| z | is the standard positive mollifier. Proposition 3.9.
Let H ∈ L loc ( C r , Her( C r )) . If H is semi positive definite then H ǫ is also semi positive definite; if H is strictly positive definite then H ǫ is strictlypositive definite too.Proof. Note for any a ∈ C n , the translation transformation i a : C ∞ ( C n ) → C ∞ ( C n ) , ξ ( x ) ξ ( x + a )is an algebra isomorphism of complex functions. For any ξ ∈ C ∞ ( C n ) , we know( ξ t H ǫ ¯ ξ )( x ) = P α,β (cid:16) ´ C n h α ¯ β ( x − y ) ρ ǫ ( y ) dy (cid:17) ξ α ( x ) ¯ ξ β ( x )= ´ C n P α,β h α ¯ β ( x − y )( i y ξ α )( x − y )( i y ξ β )( x − y ) ρ ǫ ( y ) dy. Hence if H is semi positive definite, then H ǫ is also semi positive definite; if H isstrictly positive definite then H ǫ is strictly positive definite too. (cid:3) Using Proposition 3.9, we may approximate a measurable Hermitian metric H by smooth Hermitian metric H ǫ . However we could not assure that the curvatureΘ H ǫ converges to the curvature Θ H , in some worse cases Θ H ǫ may have no anyconvergent sub sequences, in the topology that we usually use. Moreover in appli-cation the curvature of H usually has some positivity, but the curvature Θ H ǫ ofthe approximate metric may have no any positivity related to that of Θ H . In thenext section we will introduce some better singular Hermitian metric to avoid thesedifficulties.3.5.
Nef holomorphic vector bundle.
Let E be a holomorphic vector bundleon a Hermitian manifold M. Recall that E is called a nef vector bundle (in usualsense) [DPS] if the tautological line bundle O P ( E ∗ ) (1) is a nef line bundle over P ( E ∗ ) . Paper [DPS] gave a beautiful metric description of nefness when E is a line bundle.The metric description of nefness is very useful in application and generalizationof algebraic geometric results. However we only know that a Hermitian metric onthe tautological line bundle O P ( E ∗ ) (1) naturally induces a Finsler metric on E, andat present we don’t know how to define better Hermitian metrics on E via usingnefness metrics of the line bundle O P ( E ∗ ) (1) . In this paragraph we will define somestronger nefness concepts, directly using the Hermitian metric of E. Let H be a smooth Hermitian metric on E and G a smooth Hermitian metric M. Let( z , · · · , z n ) be local holomorphic coordinates of M and { e , · · · , e r } a localorthogonal frame of E with dual frame { e , · · · , e r } . The Chern curvature form ofa given Chern connection has the following formΘ H = i X j,k,α,β R βj ¯ kα dz j ∧ d ¯ z k ⊗ e α ⊗ e β ∈ Γ( M, ∧ T ∗ M ⊗ End ( E )) . where R γj ¯ kα = h γ ¯ β R j ¯ kα ¯ β and R j ¯ kα ¯ β = − ∂ h α ¯ β ∂z j ∂ ¯ z k + h γ ¯ δ ∂h α ¯ δ ∂z j ∂h γ ¯ β ∂ ¯ z k . Recall that E iscalled Griffiths semipositive if for any u = P j u j ∂∂z j and v = P α v α e α , we haveΘ H ( u ⊗ v, u ⊗ v ) = X j,k,α,β R j ¯ kα ¯ β u j ¯ u k v α ¯ v β ≥ E is called Nakano semipositive if for any u = P j,α u jα ∂∂z j ⊗ e α , we haveΘ H ( u, u ) = X j,k,α,β R j ¯ kα ¯ β u jα ¯ u kβ ≥ . e will use the following fact many times in this paper. Let u be a E -valued( n, q )-from on M. In holomorphic local coordinate write u = P α P | K | = q u αK dz ∧· · · ∧ dz n ∧ d ¯ z K ⊗ e α with K = { j , · · · , j q } and d ¯ z K = d ¯ z j ∧ · · · ∧ d ¯ z j q . Then,(35) h [Θ H , Λ] u, u i = X | S | = q − X j,k,α,β R j ¯ kα ¯ β u αjS ¯ u βkS . In particular, if E is Nakano semi positive then for any u ∈ K M ⊗ E we have h [Θ H , Λ] u, u i ≥ . Definition 3.10.
A holomorphic vector bundle E is said to be Griffiths nef, if forany ǫ > there exists a smooth Hermitian metric H ǫ on E such that its curvaturesatisfying Θ H ǫ ( u ⊗ v, u ⊗ v ) ≥ − ǫ || u || G || v || H ǫ for u ∈ T M and v ∈ E. It is said to beNakano nef, if for any ǫ > there exist a smooth Hermitian metric H ǫ on E suchthat its curvature satisfying Θ H ǫ ( u, u ) ≥ − ǫ || u || G ⊗ H ǫ for any u ∈ T M ⊗ E. Example 3.11. If E = L ⊕ · · · ⊕ L r is direct sum of nef holomorphic line bundles(in usual sense). Let ϕ j be the locally PSH function which gives the singular Her-mitian metric e − ϕ j on L j . Then E with diagonal metric H = diag( e − ϕ , · · · , e − ϕ r ) is a Griffiths nef vector bundle, and also a Nakano nef vector bundle. Proposition 3.12. If E is Griffiths nef vector bundle, then it is nef vector bundle(in usual sense).Proof. Let e , · · · , e r be the local holomorphic frame of E and e , · · · , e r the dualframe on E ∗ . The corresponding holomorphic coordinate of E ∗ is denoted by w =( W , · · · , W r ) and the homogeneous coordinate on fibers of P ( E ∗ ) is denoted by[ W : · · · : W r ] . We use ( w , · · · , w r − ) to denote the holomorphic coordinate onfibers of P ( E ∗ ) and z = ( z , · · · , z n ) to denote the holomorphic coordinate on X. Denote W = ( W , · · · , W r ) t as a column vector. The metrics H ǫ = ( h j ¯ k ) naturallyinduce Hermitian metrics H Lǫ on L := O P ( E ∗ ) (1) defined by H Lǫ ( z, w ) = h j ¯ k W j ¯ W k for ( z, w ) ∈ P ( E ∗ ) . Fix a ǫ, choose normal coordinates z , · · · , z n such that H ǫ (0) = I and dH ǫ (0) = 0 . Then Θ H Lǫ = id ′ d ′′ log H ( z, w ) andΘ H Lǫ = − P j,k ( ∂ ¯ ∂h j ¯ k ) W j ¯ W k P j,k h j ¯ k W j ¯ W k + P j,k h j ¯ k dW j ∧ dW k P j,k h j ¯ k W j ¯ W k − P j,k h j ¯ k dW j ¯ W k ∧ P j,k h j ¯ k W j dW k P j,k ( h j ¯ k W j ¯ W k ) . In matrix form, the curvatureΘ H Lǫ = (cid:0) Θ Hǫ ( W,W ) | W | Hǫ (cid:1) n × n (cid:12)(cid:12)(cid:12) −−−−−− −−−−−−−−−−−−− (cid:12)(cid:12)(cid:12) (cid:16) (1+ | w | ) δ jk − w k ¯ w j (1+ | w | ) (cid:17) ( r − × ( r − , where w = ( W W j , · · · , W j − W j , W j +1 W j , · · · , W r W j ) is the local coordinate of the open subsetΩ j = { W j = 0 } of the fiber P ( E ∗ | z ) . Note that (cid:16) (1+ | w | ) δ ij − w j ¯ w i (1+ | w | ) (cid:17) ( r − × ( r − is apositive definite matrix with eigenvalues 1 / (1 + | w | ) of order r − / (1+ | w | ) of order 1 . Since E is a Griffiths nef vector bundle we have Θ Hǫ ( W,W ) | W | Hǫ ≥− ǫI n × n . So E is a nef vector bundle (in usual sense). (cid:3) Proposition 3.13. If E is Griffiths nef, then E ⊗ det E Nakano nef. roof. We can prove this proposition by using discrete Fourier transformation, amethod used by Demailly and Skoda [De80]. Choose local coordinate on M andlocal orthogonal frame { e , · · · , e r } of E ⊗ det E such that G = ( δ jk ) and H ǫ = δ αβ . Then the curvature of E ⊗ det( E ) is expressed by R E ⊗ det( E ) j ¯ kα ¯ β = R j ¯ kα ¯ β + δ αβ P γ R j ¯ kγ ¯ γ and any u = P j,α u jα ∂∂z j ⊗ e α we have X j,k,α,β R E ⊗ det( E ) j ¯ kα ¯ β u jα ¯ u kβ = X j,k,α,β ( R j ¯ kα ¯ β u jα ¯ u kβ + R j ¯ kα ¯ α u jβ ¯ u kβ ) . (36)Fix a sufficiently large positive integer N and let S be the set of maps σ : { , · · · , r } →{ , e N Q i , · · · , e N − N Q i } . ˆ u σ = n X j =1 r X α =1 u jα σ ( α ) ∂∂z j ∈ T M , ˆ v σ = r X α =1 σ ( α ) e α ∈ E ⊗ det E. Then N r P σ ∈ S Θ H ǫ (ˆ u α ⊗ ˆ v α , ˆ u α ⊗ ˆ v α )= N r P σ ∈ S P j,k,α,β R j ¯ kα ¯ β ( P rγ =1 u jγ σ ( γ )( P rδ =1 u kδ σ ( δ )) σ ( α ) σ ( β )= N r P σ ∈ S P j,k,α,β R j ¯ kα ¯ β P N − p =1 P rγ,δ =1 u jγ u kδ σ ( γ ) σ ( δ ) σ ( α ) σ ( β )Note P σ ∈ S P rγ,δ =1 u jγ u kδ σ ( γ ) σ ( δ ) σ ( α ) σ ( β ) N r = ( P rγ =1 u jγ u kγ , if α = β,u jα u kβ , if α = β. (37)It together with (36) imply that X j,k,α,β R E ⊗ det( E ) j ¯ kα ¯ β u jα ¯ u kβ = 1 N r X σ ∈ S Θ H ǫ (ˆ u α ⊗ ˆ v α , ˆ u α ⊗ ˆ v α ) . (38)By the definition Griffiths nef we have X j,k,α,β R E ⊗ det( E ) j ¯ kα ¯ β u jα ¯ u kβ ≥ − ǫ N r X σ ∈ S || ˆ u α || G || ˆ v α || H ǫ . (39)Since || ˆ u α || G || ˆ v α || H ǫ = X σ ∈ S X j,γ,δ,α u jγ u jδ σ ( γ ) σ ( δ ) σ ( α ) σ ( α ) . (40)Using (37) again, we get from (38) to (40) that R E ⊗ det( E ) j ¯ kα ¯ β u jα ¯ u kβ ≥ − rǫ || u || G ⊗ H ǫ , hence E ⊗ det( E ) is Nakano nef. (cid:3) Definition 3.14.
The curvature current Θ E ∈ D ′ , ( M, Her(E)) of a singularHermitian metric is called Griffith pseudoeffective if Θ E is a positive Hermitiancurrent in the sense of Definition 2.6, i.e., if for any smooth local section u of E with compact support, Θ E ( u, u ) is a positive (1 , -current. It is called Nakanopseudoeffective if Θ E ( v, v ) is a positive distribution for any smooth local section v = P j u j ∂∂z j of E ⊗ T X with compact support, where u j are local sections of E. ote if E is Nakano pseudoeffective then it is Griffith pseudoeffective. In factwrite Θ H = P j,k T Ej ¯ k dz j ∧ d ¯ z k , where T Ej ¯ k are generalized matrix function. thenΘ E ( u, u ) = X j,k iT Ej ¯ k ( u, u ) dz j ∧ d ¯ z k , (41)take v = u ⊗ w j ∂∂z j with u a smooth local section of E. ThenΘ E ( v, v ) = X j,k iT Ej ¯ k ( u, u ) w j ¯ w k . (42)Clearly if the R.H.S. of (42) is positive distribution for any u ∈ E and P w j ∂∂z j ∈ T X then the R.H.S. of (41) is a positive (1 , u ∈ E .For a Hermitian metric H on a holomorphic vector bundle E of rank r over acompact complex manifold M with measurable entries h α ¯ β , we have the followingnorms for 1 ≤ p < ∞ : for any contractible coordinate open subset Ω ⊂ M, chooselocal trivialization of E on Ω and write the entries h α ¯ β as measurable function, andset || H || L p (Ω) = max ≤ α,β ≤ r || h α ¯ β || L p (Ω) . We call the metric H is in L p ( M, Her + (E)) if || H || L p ( M ) = P α || ϕ α H || L p (Ω α ) < + ∞ , where { ϕ α } is a smooth partition of unity of a coordinate open covering { Ω α } of M. We say the metric H is in W k,p ( M, Her + (E)) if H ∈ L p ( M, Her + (E)) and ∂ γ H ∈ L p ( M, Her + (E)) for 0 ≤ | γ | ≤ k. Here ∂ γ H = ( ∂ γ h α ¯ β ) and ∂ γ h α ¯ β are weakderivatives of h α ¯ β . Proposition 3.15.
Let M be a K¨ahler manifold with a complete K¨ahler metric ω and E be a Hermitian holomorphic vector bundle equipped with singular Hermitianmetric H of type I. Suppose there is a family of smooth Hermitian metric H ǫ suchthat lim ǫ → || H − ǫ − H − || C = 0 and lim ǫ → || H ǫ − H || W , = 0 . Let η, λ be smoothpositive function on M. Then for any f ∈ W , ( M, Λ n,q T ∗ M ⊗ E ) , k ( η + λ − ) D ′′∗ f k H + k η D ′′ f k H ≥ ([ η Θ H − id ′ d ′′ η − iλd ′ η ∧ d ′′ η, Λ] f, f ) H . (43) Proof.
We may assume f ∈ C ∞ ( M, Λ n,q T ∗ M ⊗ E ) since C ∞ ( M, Λ n,q T ∗ M ⊗ E ) isdense in W , ( M, Λ n,q T ∗ M ⊗ E ) , Using Proposition 3.2 for the smooth metric H ǫ weget k ( η + λ − ) D ′′∗ f k H ǫ + k η D ′′ f k H ǫ ≥ ([ η Θ H ǫ − id ′ d ′′ η − iλd ′ η ∧ d ′′ η, Λ] f, f ) H ǫ . Clearly for any f ∈ W , ( M, Λ n,q T ∗ M ⊗ E ) , lim ǫ → k ( η + λ − ) D ′′∗ f k H ǫ = k ( η + λ − ) D ′′∗ f k H , lim ǫ → k η D ′′ f k H ǫ = k η D ′′ f k H . Since η and λ are smooth, we also havelim ǫ → ([ − id ′ d ′′ η − iλd ′ η ∧ d ′′ η, Λ] f, f ) H ǫ = ([ − id ′ d ′′ η − iλd ′ η ∧ d ′′ η, Λ] f, f ) H . Hence to obtain (43) it suffices to prove that(44) lim ǫ → ([ η Θ H ǫ , Λ] f, f ) H ǫ = ([ η Θ H , Λ] f, f ) H . Since f is E -valued ( n, q )-form we have [Θ H ǫ , Λ] f = e (Θ H ǫ ) e ( ω ) ∗ f, this togetherthe definition of the inner product in (14) imply that([ η Θ H ǫ , Λ] f, f ) H ǫ = ˆ M ηe (Θ H ǫ ) e ( ω ) ∗ f ∧ H ǫ ∗ f . n the same way we rewrite R.H.S. of (44), and taking difference of both sides of(44), we get | ([ η Θ H ǫ , Λ] f, f ) H ǫ − ([ η Θ H , Λ] f, f ) H | = (cid:12)(cid:12)(cid:12) ˆ M ηe (Θ H ǫ ) e ( ω ) ∗ f ∧ ( H ǫ − H ) ∗ f (cid:12)(cid:12)(cid:12)| {z } = def I ( ǫ ) + (cid:12)(cid:12)(cid:12) ˆ M ηe (Θ H ǫ − Θ H ) e ( ω ) ∗ f ∧ H ∗ f (cid:12)(cid:12)(cid:12)| {z } = def I ( ǫ ) . In the following we will prove the limits of both terms I ( ǫ ) and I ( ǫ ) are zeroswhen ǫ → . In the term I ( ǫ ) , after taking out maximum of η, then integrating by part, weknow it is up bounded by the following three terms: I ( ǫ ) ≤ max M | η | n (cid:12)(cid:12)(cid:12) ˆ M H − ǫ e ( dH ǫ ) d ( e ( ω ) ∗ f ) ∧ ( H ǫ − H ) ∗ f (cid:12)(cid:12)(cid:12)| {z } = def ˆ I ( ǫ ) + (cid:12)(cid:12)(cid:12) ˆ M H − ǫ e ( dH ǫ ) e ( ω ) ∗ f ∧ ( d ( H ǫ − H )) ∗ f (cid:12)(cid:12)(cid:12)| {z } = def ˆ I ( ǫ ) + (cid:12)(cid:12)(cid:12) ˆ M H − ǫ e ( dH ǫ ) e ( ω ) ∗ f ∧ ( H ǫ − H ) d ( ∗ f ) (cid:12)(cid:12)(cid:12)| {z } = def ˆ I ( ǫ ) o . We claim that ˆ I ( ǫ ) ≤ M ω sup ǫ || H − ǫ || C ( || H ǫ || W , + || f || W , ) || H ǫ − H || L , (45) ˆ I ( ǫ ) ≤ M ω sup ǫ || H − ǫ || C ( || H ǫ || W , + || f || L ) || H ǫ − H || W , , (46) ˆ I ( ǫ ) ≤ M ω sup ǫ || H − ǫ || C ( || H ǫ || W , + || f || W , ) || H ǫ − H || L , (47)where M ω is a positive constant depending only on the K¨ahler metric ω. Here weonly give a proof of (45) since (46) and (47) follow in the same way. Write f = X | J | = q r X α =1 ( f Jα dz ∧ · · · ∧ dz n ∧ d ¯ z J ) e α By the definition of Hodge star operator in (13) we have ∗ f = X | J | = q, | I ∪ J | = n r X α =1 a J ( f Jα d ¯ z I ) e α , where a J is a function depending only on the K¨ahler metric ω of the K¨ahler manifold M. e ( dH ǫ ) d ( e ( ω ) ∗ f )= P | L | = q P rα =1 (cid:16) P nj,k =1 ( b jkL ∂H ǫ ∂z j ∂f Lα ∂ ¯ z k + c jkL ∂H ǫ ∂ ¯ z j ∂f Lα ∂z k ) dz ∧ · · · ∧ dz n ∧ d ¯ z L (cid:17) e α , where b jkL and c jkL are functions depending only ω.H − ǫ e ( dH ǫ ) d ( e ( ω ) ∗ f ) ∧ ( H ǫ − H ) ∗ f = P | J | = q P j ∈ J P α,β,γ,δ ( d jkJ ∂ ( H ǫ ) β ¯ γ ∂z j ∂f Jγ ∂ ¯ z k + e jkJ ∂ ( H ǫ ) β ¯ γ ∂ ¯ z j ∂f Jγ ∂z k )( H ǫ − H ) γ ¯ δ f Jδ dV here d jkJ and e jkJ are functions depending only ω and dV = Q nj =1 ( i dz j ∧ d ¯ z j ) . Now using the Cauchy-Schwartz inequality we have I ( ǫ ) ≤ M ω || H − ǫ || C P | J | = q,j ∈ J P α,β,γ (cid:16) ´ M P γ,δ | ( H ǫ − H ) γ ¯ δ | ) dV (cid:1) · (cid:16) ´ M ( | ∂ ( H ǫ ) β ¯ γ ∂ ¯ z j | + | ∂f Jγ ∂ ¯ z j | + | f Jδ | ) dV (cid:17) , from this inequality it is easy to conclude (45).Now form (45) to (47) we know I ( ǫ ) ≤ M ω || H − ǫ || C ( || H ǫ || W , + || f || W , ) || H ǫ − H || W , , hence we have lim ǫ → I ( ǫ ) = 0 . In the term I ( ǫ ) writeΘ H ǫ − Θ H = id ′′ ( H − ǫ d ′ ( H ǫ − H )) + id ′′ (( H − ǫ − H − ) d ′ H ) , then follow the same way as we done for I ( ǫ ) , we may get I ( ǫ ) ≤ f M ω [ || H − ǫ || C || H ǫ − H || W , + || H − ǫ − H || C || H || W , )( || H || W , + || f || W , )]hence we also have lim ǫ → I ( ǫ ) = 0 . (cid:3) The following version of Proposition 3.3 is a direct conclusion of Proposition3.15:
Proposition 3.16.
Let M and E as in Proposition 3.15,, assume (24) is truefor any f ∈ W , ( M, Λ n,q T ∗ M ⊗ E ) . Then (i) . If δ = 0 , then for g = d ′′ ζ ∧ g ∈ W , ( M, Λ n,q T ∗ M ⊗ E ) such that D ′′ g = 0 and ´ M √ γ | g | H < + ∞ , there exits f ∈ W , ( M, Λ n,q − T ∗ M ⊗ E ) such that D ′′ f = g and ˆ M ( η + λ − ) − | f | H ≤ ˆ M √ γ | g | H . (ii) . If δ > , then for any g = d ′′ ζ ∧ g + g ∈ W , ( M, Λ n,q T ∗ M ⊗ E ) such that D ′′ g =0 and ´ M √ γ | g | H + δ ´ M | g | H < + ∞ , there exits f ∈ W , ( M, Λ n,q − T ∗ M ⊗ E ) and h ∈ W , ( M, Λ n,q T ∗ M ⊗ E ) such that D ′′ f + √ δh = ¯ ∂ζ ∧ g + g and ˆ M ( η + λ − ) − | f | H + ˆ M | h | H ≤ (cid:16) ˆ M √ γ | g | H + 1 δ ˆ M | g | H (cid:17) . The Proposition 3.16 give the solution of ¯ ∂ -equation for holomorphic vectorbundle with singular Hermitian metric of type I (we also may have a correspondingversion for singular Hermitian metric of type II, for brevity we omit it here), howeverthe assumption imposed on the Hermitian metric in Proposition 3.15 is still verystrong. In application we need more singular Hermitian metrics. Before introducesuch singular metric we will use we firstly introduce some notions. We say that aHermitian metric H is equicontinuous and uniformly bounded if each entry h α ¯ β isequicontinuous and uniformly bounded on any open set Ω ⊂ M. Definition 3.17.
A holomorphic vector bundle E is said to be strong Griffiths( resp. Nakano) nef, if it is Griffiths nef and the metrics H ǫ in Definition 2.6satisfying the following two conditions: (i)There exists a p > r ( resp. p > ) suchthat || H − ǫ || L p ( M ) is uniformly bounded with respect to ǫ ; (ii) H ǫ ∈ C ( M, Her + (E)) are equicontinuous and uniformly bounded with respect to ǫ. Proposition 3.18. If E is a strong Griffiths nef vector bundle on a compact com-plex manifold M , then E is Griffiths pseudoeffective, E ⊗ det E is strong Nakanonef and Nakano pseudoeffective. roof. Since H ǫ are equicontinuous, H ǫ and || H − ǫ || L p ( M ) are uniformly bounded,by selecting subsequences one after another,one by one, on the finite entry positionsof H ǫ and H − ǫ and on a given finite covering of contractible coordinate open subsetsof M respectively, we at last get a subsequence such that(i). ( H ǫ k ) α ¯ β are uniformly convergent to g α ¯ β for all 1 ≤ α, β ≤ r, (ii). ( H − ǫ k ) α ¯ β are weakly convergent to h α ¯ β in L p ( M ) for all 1 ≤ α, β ≤ r. Let G = ( g α ¯ β ) r × r and H = ( h α ¯ β ) r × r . Firstly we will prove H = G − almost every-where. Note det( H ǫ k ) is uniformly convergent to det( G ) and ( H − ǫ k ) α ¯ β is pointwiseconvergent to ( G − ) α ¯ β = M α ¯ β det( g α ¯ β ) on det( g α ¯ β ) = 0 . Here M α ¯ β = ( − α + β det( f M α ¯ β )and f M α ¯ β is the submatrix of ( g α ¯ β ) by deleting its α -th row and β -th column. So M α ¯ β is a homogeneous polynomial of { g α ¯ β | ≤ α, β ≤ r } . For any relative com-pact open subset Ω ⊂ { z ∈ M | det( g α ¯ β ( z )) = 0 } , we have lim k →∞ || ( H − ǫ k ) α ¯ β − M α ¯ β det( g α ¯ β ) || L p (Ω) = 0 . i.e., ( H − ǫ k ) α ¯ β is strongly convergent to ( G − ) α ¯ β in L p (Ω) , so H = G − on { z ∈ M | det( g α ¯ β ( z )) = 0 } . Note { z ∈ M | h α ¯ β ( z ) = ∞} ⊆ ∪ ∞ k =1 { z ∈ M | ( H − ǫ k ) α ¯ β ( z ) = ∞} . The set { z ∈ M | ( H − ǫ k ) α ¯ β ( z ) = ∞} is of measure zero since H ǫ k ∈ L p ( M, Her + (E)) , hence { z ∈ M | h α ¯ β ( z ) = ∞} is of measure zero too.Let A = { z ∈ M |∃ α, β such that h α ¯ β ( z ) = ∞} and B = { z ∈ M | det( g α ¯ β ( z )) =0 } . We will prove that A = B up to a subset of measure zero. For any z suchdet( g α ¯ β ( z )) = 0 there is a small neighborhood Ω δ of z such that | det( g α ¯ β ( z )) | ≥ ǫ > δ , hence lim k →∞ || ( H − ǫ k ) α ¯ β − M α ¯ β det( g α ¯ β ) || L p (Ω δ ) = 0 . Note M α ¯ β det( g α ¯ β ) is bounded on Ω δ , hence h α ¯ β is bounded almost everywhere in the neighborhoodΩ δ of z . Therefore A ⊆ B. Assume A ( B and the measure of B is positive, thenthere exist small open subset Ω δ and a subset C ⊂ B ∩ Ω δ of positive measure, suchthat on the whole subset C, h α ¯ β ( z ) is almost everywhere bounded for all α, β butdet( g α ¯ β ( z )) = 0 . Since ( H − ǫ k ) α ¯ β is weakly convergent to h α ¯ β in L p ( M ) , we have ´ M (( H − ǫ k ) α ¯ β ( z ) − h α ¯ β ( z )) χ Ω δ ( z ) dµ → , which means ´ Ω δ (( H − ǫ k ) α ¯ β − h α ¯ β ) dµ → α, β as k → ∞ . But ( H − ǫ k ) α ¯ β are uniformly convergent to g α ¯ β on Ω δ for all1 ≤ α, β ≤ r. Since det( g α ¯ β )( z ) = 0 on C, hence det(( H − ǫ k ) α ¯ β ) is divergent to ∞ on C as k → ∞ . Therefore there exist α , β such that (( H − ǫ k ) α ¯ β ) is divergent to ∞ on a positive measure subset of C ⊂ Ω δ , and it is a contradiction to ´ Ω δ (( H − ǫ k ) α ¯ β − h α ¯ β ) dµ → α, β as k → ∞ . Hence { z ∈ M | det( g α ¯ β ( z )) = 0 } is of measurezero and H = G − almost everywhere.Now we claim that the curvatures Θ H ǫk = id ′ (( d ′′ H − ǫ k ) H ǫ k ) are weak ∗ -convergentto the curvature current Θ H = id ′ (( d ′′ H − ) H ) . For any u, v ∈ C ∞ ( M, E ) , by for-mula (25) we haveΘ H ǫk ( u, v ) = − id ′′ H − ǫ k ( H ǫ k d ′ u, v ) + id ′ H − ǫ k ( H ǫ k u, d ′′ v ) . (48)Since d ′′ H − ǫ k are weakly convergent to d ′′ H − and H ǫ k v and H ǫ k d ′ v are uniformlyconvergent to Hv and Hd ′ v respectively. Hencelim k →∞ Θ H ǫk ( u, v ) = − id ′′ H − ( Hd ′ u, v ) + id ′ H − ( Hu, d ′′ v ) . (49)for any u, v ∈ C ∞ ( M, C n ) . So Θ H ǫk are weak ∗ -convergent to Θ H . It suffices toprove Θ H is positive Hermitian current, but it is clear since Θ H ǫk ( u ⊗ v, u ⊗ v ) ≥− ǫ k || u || || v || . Let f H ǫ = H ǫ det( H ǫ ) be the Hermitian metrics on E ⊗ det E. Then H ǫ det( H ǫ )are equicontinuous and uniformly bounded with respect to ǫ. For any 1 < q < pr nd any contractible coordinate open subset Ω ⊂ M, by H¨older’s inequality ´ Ω | ( H − ǫ ) α ¯ β ( H − ǫ ) α ¯ β · · · ( H − ǫ ) α r ¯ β r | q dµ ≤ Vol(Ω) p − qrp ( ´ Ω | ( H − ǫ ) α ¯ β | p dµ ) qp ( ´ Ω | ( H − ǫ ) α ¯ β | p dµ ) qp · · · ( ´ Ω | ( H − ǫ ) α r ¯ β r | p dµ ) qp Expand det( H − ǫ ) as a sum of r ! monomials, we get || ( g H − ǫ ) α ¯ β || L q (Ω) ≤ r !Vol(Ω) p − qrpq || H − ǫ || r +1 L p ( M ) ≤ r !Vol( M ) p − qrpq || H − ǫ || r +1 L p ( M ) , hence || g H − ǫ || L q ( M ) are uniformly bounded with respect to ǫ. Hence e H ǫ are uni-formly convergent to e H = H det H and g H − ǫ are weakly convergent to e H − = H − (det H − ) in L q ( M ) . Hence E ⊗ det E is strong Nakano nef. Furthermorethe curvature current Θ e H of Hermitian metric e H is Nakano pseudoeffective since E ⊗ det E is Nakano nef by Proposition 3.13. (cid:3) Let E be a strong Grffiths (Nakano) nef holomorphic vector bundle, then byProposition 3.18 there is a family smooth Hermitian metric H ǫ on E which conver-gent to a Grifiths (Nakano) pseudo effective singular Hermitian metric of type II,denoted by H. We call H the limit Hermitian metric of the strong Grffiths (Nakano)nef vector bundle. 4. L -Extension theorems Using the twisted Bochner-Kodaira-Nakano inequality, Ohsawa-Takegoshi estab-lished a L -extension theorem for weighted L -integrable holomorphic functions ona bounded Stein domain in the fundamental paper [OT]. Manivel get a more gen-eral L -extension theorem in the framework of vector bundles using a more geomet-ric approach in [Man]. Siu’s work in algebraic geometry in [Siu98],[Siu01],[Siu02]gave very important applications of the Ohsawa-Takegoshi theorem. In the paper[De00], Demailly emphasized other applications Ohsawa-Takegoshi theorem in hisown work and gave many inspiring comments on how to deal with small negativecurvature term when constructing L -extension theorems via solving approximating¯ ∂ -equations, which Yi used in [Yi] in studying L -extension theorems on compactK¨ahler manifolds. The lecture paper [Var] gave a good introduction to Ohsawa-Takegoshi extension theorem and other L -techniques in complex geometry.In this section, we will consider the extension of holomorphic jet sections forhigh rank bundles whose supports are discrete points. Popovici [Pop] firstly got a L -extension theorem for the holomorphic jet sections of line bundles whose sup-ports are connected sub manifolds without codimension restriction. In our case thesupport may not be connected, however, its dimension is zero. We only considerextending holomorphic jet sections whose supports are sub schemes of dimensionzero since we only need such a case in proving our main theorem in the next section.With some more strength, it would be possible to get the extension of holomorphicjet sections whose supports are arbitrary subscheme whose underling reduced irre-ducible components are smooth complex sub manifolds. Recently the paper [GZ]obtained improvements of the upper bounds of the extension theorems founded be-fore, using undetermined function via solving ordinary differential equations. Weget our extension theorem via optimizing these techniques. Since we don’t needan effective upper bound constant in the next section, the techniques developedin [GZ] are used for a different purpose, rather for an optimal upper bound. Ourextension theorem 4.2 is sharper than main theorem of [Pop] when the support isan isolated point.In Subsection 4.2, we establish the L -extension theorem for jet section for strongNakano nef holomorphic vector bundles on bounded Stein domain by inductivemethod; in Subsection 4.3 we generalize our L -extension theorem further to anycompact K¨ahler manifold. We cut the compact K¨ahler manifold M into small ounded Stein open subsets and to extend a given jet section to each Stein opensubset in a compatible way, then glue these extended sections to a global holomor-phic section on M, via solving approximate ¯ ∂ -equation.4.1. Intrinsic norm on jet sections.
Let p = { p , · · · , p l } ⊂ M be a finiteset of distinct points, and E a holomorphic vector bundle on a Stein manifold M with (maybe singular) Hermitian metric H. A holomorphic section f ∈ H ( M, E ⊗ O M / J k +1 p · · · J k l +1 p l ) is a k -jet section of E (passing through p , · · · , p l with k =( k , · · · , k l )). We use J kp ( E ) to denote the set of all k -jet sections of E with support p . For any holomorphic section F of E, it defines a unique k -jet via quotient map E → E ⊗ O M / J k +1 p · · · J k l +1 p l , we denote it by J kp ( F ) . Conversely, for any jetsection f ∈ J kp ( E ) , if there exits F ∈ H ( M, E ) such that J kp ( F ) = f, we call F aholomorphic lifting section of f. Choose holomorphic local frame { e , · · · , e r } of E on a contractible neighborhood U j of p j for j = 1 , · · · , l. For any holomorphic section F of E, write F = r X j =1 F j e j , Denote ∂ α F ( p j ) = r X α =1 ∂ α F j ( p j ) e α the ∂ -partial derivative of α -order of F, where α = ( α , · · · , α r ) . And define(50) | ∂ α F | H ( p j ) = X j ¯ k H j ¯ k ∂ α j F j ( p j ) ∂ α k F k ( p j ) , where H j ¯ k = H ( e j , e k ) . If we change to another holomorphic local frame { ˜ e , · · · , ˜ e r } of E with transformation e j = φ jk ˜ e k , then F = P rj =1 ˜ F j ˜ e j with ˜ F j = P k F k φ kj . Hence under frame { ˜ e , · · · , ˜ e r } , the number | ∂ α F | H ( p j ) may take a different value(51) ^ | ∂ α F | H ( p j ) = X s,t,j,k ( H s ¯ t φ sj φ tk ) ∂ α j ( X n F n φ nj ) ∂ α k ( X n F n φ nj ) (cid:12)(cid:12)(cid:12) p j , where ( φ jk ) r × r is the inverse matrix of ( φ jk ) r × r . But the difference will not affectour definition of jet norm up to a positive constant in the following Proposition 4.1.Since M is Stein there is a surjective map H ( M, K M ⊗ E ) → H ( M, K M ⊗ E/ J k +1 p · · · J k l +1 p l ) . For any k -jet f ∈ J kp ( E ) , let F be an arbitrary holomorphiclifting to M. Now for f ∈ J kp ( E ) define k f k k = l X j =1 X | α |≤ k j α !) | ∂ α F | H ( p j ) , ] k f k k = l X j =1 X | α |≤ k j α !) ^ | ∂ α F | H ( p j ) , where | α | = max { α , · · · , α n } and α ! = α ! · · · α l ! . By (51) it is easy to check that
Proposition 4.1.
Fix the points p , · · · , p l . Suppose the Hermitian metric H isnon degenerate and bounded at p , · · · , p l , in particular if the metric is continuous,then k f k k is equal to ] k f k k up to a finite positive constant which is decided by thecoordinate transformation (and their derivatives) of the local frame. Hence up to apositive finite constant || f || k is a well defined norm. roof. Clearly k f k k = 0 if and only if ∂ k F j ( p s ) = 0 for all 0 ≤ k ≤ k j and j =1 , · · · , r and s = 1 , · · · , l. And ] k f k k = 0 if and only if P j [ φ sj ∂ k ( P n F n φ nj )]( p t ) = 0for all 0 ≤ k ≤ k t and s, t = 1 , · · · , r. Since { φ jk } are non degenerate coordinatetransformation we have(52) ∂ k ( X n F n φ nj )( p t ) = 0for all 0 ≤ k ≤ k t and t = 1 , · · · , l and j = 1 , · · · , r. Take k = 0 we get P n ( F n φ nj )( p t ) = 0 for t = 1 , · · · , l and j = 1 , · · · , r. Hence we have F n ( p t ) = 0 for n = 1 , · · · , r and t = 0 , · · · , l. Then take these into (52) then let k = 2 in (52) weget ∂ F n ( p t ) = 0 for n = 1 , · · · , r and t = 1 , · · · , r. Go on this way we finally have ∂ k F j ( p s ) = 0 for all 0 ≤ k ≤ k j and j = 1 , · · · , r and s = 1 , · · · , l. Hence k f k k = 0if and only if ] k f k k = 0 . If the metric H is bounded, then | ∂ α F | H ( p j ) for all α j ≤ k j are bounded if andonly if ^ | ∂ α F | H ( p j ) are bounded for all α j ≤ k j , since { φ jk } are smooth functions.Hence k f k k is bounded if and only if ] k f k k is bounded.It is also very easy to check the norms || f || k doesn’t depend on its lifting F, forbrevity we omit the details here. Hence up to a positive finite constant || f || k is awell defined norm. (cid:3) Here we define norm on jet sections not using intrinsic derivatives because theirsupport is a very simple set consisting of isolated points, another benefit is thatwe could see that || f || k doesn’t depend on the Chern connection of the Hermitianholomorphic vector bundle E, which is bad behaved if the Hermitian metric H isonly continuous.4.2. Extension theorems for jet sections on bounded Stein domains.
Throughout this subsection we assume M is a Stein domain in C n with finite diameter.Choose holomorphic local coordinate charts of M, and denote the coordinate of p , · · · , p l by w , · · · , w l . Let D denote the diameter of M. And s j ( z ) = z − w j D for j = 1 , · · · , l if without special explanation, and here log(log | s j | ) > . Extension theorems for Nakano semi positive vector bundles.Theorem 4.2.
Let M be a bounded Stein domain of diameter D, and E a holo-morphic vector bundle on M, equipped with a smooth Hermitian metric H such that Θ H is Nakano semi-positive. Then for any holomorphic jet section f ∈ J kp ( K M ⊗ E ) satisfying k f k k < ∞ , there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and ˆ M | F | H Q lj =1 (cid:16) | s j | n (cid:16) log | s j | (cid:17) δ j (cid:17) ≤ C k f k k , (A k ) where δ j > are any positive constants and C is a finite positive constant dependingonly on δ j and the minimal distance between any two points among p , · · · , p l . Proof.
Firstly note that it suffices to prove
Lemma 4.3.
Under the assumptions in Theorem 4.2, then for any holomorphicjet section f ∈ J kp ( K M ⊗ E ) satisfying k f k k < ∞ , here exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and forany ≤ j ≤ l ˆ M | F | H (cid:16) | s | · · · | s l | ) n (cid:16) log | s j | (cid:17) δ j ≤ C k f k k , (B k , j ) where δ j > are any positive constants and C is a finite positive constant dependingonly on δ j and the minimal distance between any two points among p , · · · , p l . Assume that Lemma 4.3 is true, then by the inequality of arithmetic and geo-metric means we have ˆ M | F | H Q lj =1 (cid:16) | s j | n (cid:16) log | s j | (cid:17) δjn (cid:17) ≤ l l X j =1 ˆ M | F | H (cid:16) | s | · · · | s l | ) n (cid:16) log | s j | (cid:17) δ j , hence Theorem 4.2 follows. Note for the inequalities ( A k ) and ( B k ,j ) it suffices toprove the case where 0 < δ j < δ j ≥ | sj | ) a ≤ | sj | ) b if a ≥ b > . Due to the symmetry, clearly it suffices to establish the inequalities ( B k ,j ) inLemma 4.3 for j = 1 . We will establish ( A k ) and ( B k , ) altogether by by inductionon | k | = k + · · · + k l . The case | k | = 0 is essentially a sharpening of a special case (the support ofthe section to be extended is of dimension zero) of the Demailly’s version [De00]of the Ohsawa-Takegoshi-Manivel extension theorem [OT],[Oh88],[Man]. Since M is Stein, the restriction map H ( M, K M ⊗ E ) → H ( M, K M ⊗ E/ J p · · · J p l ) issurjective, for any 0-jet section f ∈ H ( M, K M ⊗ E/ J p · · · J p l ) , we may write f = f + · · · + f n , with f j ∈ H ( M, K M ⊗ E/ J p · · · J p l ) satisfying(53) f j ( p j ) = f ( p j ) , and f j ( p k ) = 0 for k = j. It suffices to extend each f j to a section ˆ f j with the required properties and at thesame time satisfying ˆ f j ( p k ) = 0 for k = j. In the following we extend f to ˆ f andthe extension of f j follows in the same way.Let ρ ǫ be the smooth kernel function such that its support is contained in ( − ǫ, ǫ )and ´ ǫ − ǫ ρ ǫ ( t ) dt = 1 . Then(54) a τ,ǫ ( t ) = 11 − ǫ χ { τ − (1 − ǫ ) /
Without loss of generality we assume the coordinate of p isorigin, i.e. w = 0 . Take local trivialization of K M ⊗ E on the small neighborhoodof p . Write e F − F k − = ( ˜ f − f k − ) dz ∧ · · · ∧ dz n , where ˜ f and f k − are E -valued holomorphic function. Since e F is a lifting of f and J k − p ( F k − ) is the image of f in H ( M, K M ⊗ E ⊗ O / ( J k p J k +1 p · · · J k l +1 p l )) , we know ∂ α ˜ f ( p ) = ∂ α f k − ( p ) , ∀ | α | ≤ k − . Let dV = ∧ nj =1 ( idz j ∧ d ¯ z j ) be the volume element. Take coordinate transformation z j = e τ/ ˜ z j and ˜ z = (˜ z , · · · , ˜ z n ) . Then dV = e nτ ∧ nj =1 ( id ˜ z j ∧ d ¯˜ z j ) = e nτ dσ with dσ = ∧ nj =1 ( id ˜ z j ∧ d ¯˜ z j ) . Under the coordinate transformation we have | e F − F k − | H | s | k +2 n = h (27 D ) k +2 n | ˜ f − f k − | ( e τ/ ˜ z , · · · , e τ/ ˜ z n ) e k τ | ˜ z | k dσ, where h is a function depending only on metric. Notelim τ →−∞ ˆ M a τ ( φ ) | e F − F k − | H Q lj =1 | s j | n + k j ) = lim τ →−∞ ˆ e τ − / < | z |
Nowassume E is a strong Nakano nef vector bundle on a bounded Stein domain M, which means for any positive number ρ > , there exist a smooth Hermitian metric H ρ such that its curvature satisfying Θ H ρ ( u, u ) ≥ − ρ || u || for any u ∈ T M ⊗ E, and { H ρ } have a subsequence convergent to a Nakano pseudo effective Hermitianmetric H, called limit Hermitian metric. Theorem 4.5.
Let M be a bounded Stein domain of diameter D, and E a strongNakano nef holomorphic vector bundle on M, with a Nakano pseudo effective limitHermitian metric H. Then for any holomorphic jet section f ∈ J kp ( K M ⊗ E ) satis-fying k f k k < ∞ , there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and ˆ M | F | H Q lj =1 (cid:16) | s j | n (cid:16) log | s j | (cid:17) δ j (cid:17) ≤ C k f k k , where δ j > are any positive constants and C is a finite positive constant dependingonly on δ j and the minimal distance between any two points among p , · · · , p l . Proof.
We may prove this Theorem using the same inductive process as the proof ofTheorem 4.2, the only difference is that we will now solve an approximate ¯ ∂ -equationevery time instead when we solve a ¯ ∂ -equations during the proof of Theorem 4.2.After taking limit we will get the same estimates as in Theorem 4.2. For brevityhere we give a proof of the first step where | k | = 0 , since the the rest procedure isa repeat modification process as the case | k | = 0 . Write a given 0-jet section f ∈ H ( M, K M ⊗ E/ J p · · · J p l ) as a sum f = f + · · · + f n , with f j ∈ H ( M, K M ⊗ E/ J p · · · J p l ) satisfying (53). By symmetrypositions of f , · · · , f n , it suffices to extend f and the others are extended in thesame way. Let ˜ f be a section of H ( M, K M ⊗ E ) which is holomorphic liftingof f and let a τ,ǫ , b τ,ǫ , v τ,ǫ , φ, η ǫ be the functions as defined in (54),(55) and (56)respectively, here in (56) we take u ( x ) = x + δ x δ with 0 < δ < . Let e H = H ρ e − n P lj =1 log | s j | be a smooth Hermitian metric on K M ⊗ E. Note here H ρ are a family smoothHermitian metric on the strong Nakano nef bundle E, and h [Θ H ρ , Λ] ϑ, ϑ i ≥ − ρ | ϑ | H ρ or any E -valued ( n, ϑ by (35). Hence the operator B ǫ = [ η ǫ Θ e H − id ′ d ′′ η ǫ − iλd ′ η ǫ ∧ d ′′ η ǫ , Λ] satisfy h B ǫ ϑ, ϑ i ≥ h [ a τ,ǫ ( φ ) id ′ φ ∧ d ′′ φ ) , Λ] ϑ, ϑ i ≥ a τ,ǫ ( φ ) | ( d ′′ φ ) ∗ ϑ | H ρ − ρη ǫ | ϑ | H ρ . Note b τ,ǫ ( t ) = 0 for t < τ − − ǫ and b τ,ǫ ( t ) = 1 for t > τ + − ǫ . Hence if τ < − − ǫ, then v τ,ǫ ( t ) = t for t ≥ v τ,ǫ ( t ) ≤ − ( τ − − ǫ ) for t < . Hence for τ ≪ η ǫ = u ( − v τ,ǫ ( φ )) ≤ [ − ( τ − − ǫ − δ [ − ( τ − − ǫ δ ≤ τ . Take ρ = 1 τ . Then for τ ≪ ρη ǫ ≤ τ . Let L ,τ,ǫ = D ′′ ((1 − b τ,ǫ ( φ )) ˜ f ) = ( − a τ,ǫ ( φ ) d ′′ φ ) ˜ f . If τ ≪ { p , · · · , p l } ⊂ M \ φ − ([ τ − − ǫ , τ + − ǫ ]) , then L ,τ,ǫ ( p ) = · · · = L ,τ,ǫ ( p l ) = 0 . By (ii) of Proposition 3.3, there exist K ,τ,ǫ ∈ L ( M, Λ n, T ∗ M ⊗ E ) and h ,τ,ǫ ∈ L ( M, Λ n, T ∗ M ⊗ E ) such that D ′′ K ,τ,ǫ + 1 τ h ,τ,ǫ = L ,τ,ǫ , and ˆ M ( η ǫ + λ − ǫ ) − | K ,τ,ǫ | H ρ ( | s | · · · | s l | ) n + ˆ M | h ,τ,ǫ | H ρ ( | s | · · · | s l | ) n ≤ ˆ M a τ,ǫ ( φ ) | ˜ f | H ρ ( | s | · · · | s l | ) n . By Lemma 4.4 (using the similar proof) just like the estimate (58), the limitlim τ →−∞ lim ǫ → ´ M a τ,ǫ ( φ ) | ˜ f | Hρ ( | s |···| s l | ) n is bounded, hence the two integrals of L.H.S. ofthe above inequality are uniformly bounded. Since D ′′ h ,τ,ǫ = τ · D ′′ ( L ,τ,ǫ − D ′′ K ,τ,ǫ ) = 0 , and h ,τ,ǫ is square integrable with respect to the metric H ρ e − n P lj =1 log | s j | from the estimate in the above inequality. Hence by the existence theorem of ¯ ∂ -equation on the Stein domain, there exists E -valued function g ,τ,ǫ such that D ′′ g ,τ,ǫ = h ,τ,ǫ with the estimate ˆ M | g ,τ,ǫ | H ρ ( | s | · · · | s l | ) n ≤ C ˆ M | h ,τ,ǫ | H ρ ( | s | · · · | s l | ) n by H¨ormander L -estimate. Then D ′′ ((1 − b τ,ǫ ( φ ) ˜ f − K ,τ,ǫ − τ g ,τ,ǫ ) = 0 . Let f ,τ,ǫ = (1 − b τ,ǫ ( φ )) ˜ f − K ,τ,ǫ − τ g ,τ,ǫ . hen D ′′ f ,τ,ǫ = 0 and f ,τ,ǫ is holomorphic section. It in particular means that K ,τ,ǫ + τ g ,τ,ǫ is a smooth section. ´ M ( η ǫ + λ − ǫ ) − ( | s |···| s l | ) n | f ,τ,ǫ − (1 − b τ,ǫ ( φ )) ˜ f − τ g ,τ,ǫ | H ρ ≤ ´ M a τ,ǫ ( φ )( | s |···| s l | ) n | ˜ f | H ρ . So as ǫ → , f ,τ,ǫ , K ,τ,ǫ , g ,τ,ǫ and h ,τ,ǫ have subsequences which have weak limitsin L , denoted by ˆ f ,τ , ˆ K ,τ , ˆ g ,τ and ˆ h ,τ respectively, note ˆ f ,τ = (1 − b τ ( φ )) ˜ f − ˆ K ,τ − τ g ,τ is a holomorphic section.Let b τ = lim ǫ → b τ,ǫ and v τ = lim ǫ → v τ,ǫ , then(66) lim ǫ → ( η ǫ + λ − ǫ ) − = 1 δ (1 − δ ) (log 1 | s | ) δ for τ ≪ . Taking weak limits of the above two inequalities after passing to subse-quences, just like in the proof of Theorem 4.2, we have(67) ˆ M | ˆ K ,τ | H ρ (cid:16) ( | s | · · · | s l | ) n (cid:17) (log | s | ) δ ≤ δ (1 − δ ) ˆ M a τ ( φ ) | ˜ f | H ρ ( | s | · · · | s l | ) n and ˆ M | g ,τ | H ρ ( | s | · · · | s l | ) n ≤ C ˆ M | h ,τ | H ρ ( | s | · · · | s l | ) n ≤ C ˆ M a τ ( φ ) | ˜ f | H ρ ( | s | · · · | s l | ) n . Since log | s | > L -integrals ˆ M | ˆ K ,τ,ǫ + τ g ,τ,ǫ | H ρ ( | s | · · · | s l | ) n with resect to the weight function | s |···| s l | ) n are uniformly bounded. by countingthe orders of the singularity at p , · · · , p l of the denominator we have( ˆ K ,τ,ǫ + 1 τ g ,τ,ǫ )( p ) = · · · = ( ˆ K ,τ,ǫ + 1 τ g ,τ,ǫ )( p l ) = 0 . Note ρ = τ . By definition of strong Nakano nef vector bundle { H ρ } have asubsequence convergent to a limit Hermitian metric H with Nakano pseudo effectivecurvature current. After further taking subsequences we may assume ˆ f ,τ is weaklyconvergent to ˆ f . Note ˆ f ,τ = (1 − b τ ( φ )) ˜ f − ˆ K ,τ − τ g ,τ and any subseqences of(1 − b τ ( φ )) ˜ f and τ g ,τ are always weakly convergent to zeros. Hence take weaklimit as τ → −∞ of the inequality (67) we obtain ˆ M | ˆ f | H ( | s | · · · | s l | ) n (cid:16) log | s | (cid:17) δ ≤ C | ˜ f | H ( p ) . where C is a constant depending only on δ , the diameter of M, and the minimaldistance among the points in p , · · · , p l . Without loss of generality, in the following we assume f ,τ,ǫ , K ,τ,ǫ , g ,τ,ǫ and h ,τ,ǫ themselves (not passing to subsequences) are weakly convergent as ǫ → τ → −∞ . Since ˜ f ( p j ) = K ,τ,ǫ ( p j ) + τ g ,τ,ǫ ( p j ) = 0 for j ≥ f ,τ,ǫ ( p j ) =((1 − b τ,ǫ ( φ )) ˜ f ( p j ) − K ,τ,ǫ ( p j ) − τ g ,τ,ǫ ( p j ) = 0 for j ≥ f ( p j ) =lim τ →−∞ lim ǫ → f ,τ,ǫ ( p j ) = 0 . Note b τ,ǫ ( −∞ ) = 0 , therefore f ,τ,ǫ ( p ) = ((1 − τ,ǫ ( φ )) ˜ f ( p ) − K ,τ,ǫ ( p ) − τ g ,τ,ǫ ( p ) = ˜ f ( p ) = f ( p ) , and hence ˆ f ( p ) =lim τ →−∞ lim ǫ → f ,τ,ǫ ( p j ) = f ( p ) . Now let F = ˆ f + · · · + ˆ f l . Then F ( p j ) = f ( p j ) for j = 1 , · · · , l and | F | H ≤ l ( | ˆ f | H + · · · + | ˆ f l | H ) . By the inequality of arithmetic and geometric means we have ˆ M | F | H (cid:16)Q lj =1 ( | s j | n log | s j | ) δ j (cid:17) ≤ C X | f | H ( p j ) = C k f k ,C are finite positive constants depending only on δ j and the minimal distancebetween any two points among p , · · · , p l . This finishes the proof of the case | k | = 0 . (cid:3) Corollary 4.6.
Let M be a bounded Stein domain of diameter D, and E a strongNakano nef holomorphic vector bundle on M, with a Nakano pseudo effective limitHermitian metric H. Let s j ( z ) = z − w j for j = 1 , · · · , l. Then for any holomorphicjet section f ∈ J kp ( K M ⊗ E ) satisfying k f k k < ∞ , there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and ˆ M | F | H Q lj =1 (1 + | s j | ) n ≤ C k f k k where C is a finite positive constant depending only on the diameter of M and theminimal distance between any two points among p , · · · , p l . Corollary 4.7.
Let M be a bounded Stein domain of diameter D, and E a strongNakano nef holomorphic vector bundle on M, with a Nakano pseudo effective limitHermitian metric H. Then for any holomorphic jet section f ∈ J kp ( K M ⊗ E ) satis-fying k f k k < ∞ , there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and ˆ M | F | H ≤ C k f k k where C is a finite positive constant depending only on the diameter of M and theminimal distance between any two points among p , · · · , p l . Proof of Corollary 4.6 and 4.7 . If s j ( z ) = z − w j and D is diameter of M, then | s j D | < | sj | D ≥ . Set h ( x ) = (1+ x ) n − x n (cid:16) log | x D | (cid:17) δ j with 0 < x < D. Since lim x → + h ( x ) = 1 , there exist κ > x ≤ κ then h ( x ) > / δ j ≥ . Now assume x ≥ κ, then Dx − ≤ Dκ − h ( x ) ≥ x n (cid:16) (1 + x ) n − ( Dκ − δ j (cid:17) . Take δ j = inf D>x ≥ κ m log(1 + x )log( Dκ −
1) = m log(1 + D )log( Dκ − h ( x ) ≥ ≤ x ≤ D. Now for each j we take a fixed δ j as definedabove for j = 1 , · · · , l in Theorem 4.2, then we have | F | H Q lj =1 ( D + | s j D | ) n ≤ | F | H Q lj =1 (cid:16) | s j D | n (cid:16) log | sj D | (cid:17) δ j (cid:17) , rom it Corollary 4.6 follows immediately from Theorem 4.2, and Corollary 4.7follows directly from Corollary 4.6. (cid:3) Extension theorems on compact K¨ahler manifolds.
In this subsectionwe denote M a compact K¨ahler manifold and E a strong Nakano nef vector bundleon M. The family of a smooth Hermitian metrics on E is denoted by { H ρ | ρ > } and its limit Hermitian metric is denoted by H. We cut the compact K¨ahler manifold M into small bounded Stein open subsetsand to extend a given jet section f to each Stein open subset which insect the sup-port of f in a compatible way via using the Theorem 4.5 and Lemma 4.3 establishedin Section 4.2, then glue these extended sections to a smooth global section on M. After some small modification of this globally defined smooth section via solving¯ ∂ -equation, we will obtain the expected holomorphic section defined on M whichis an extension of f. Suppose the support of the jet section f is the set { p , · · · , p l } contained in aStein open ball U of M whose diameter is D. Choose holomorphic local coordinatecharts of M, and denote the coordinate of p , · · · , p l by w , · · · , w l . Let ( z , · · · , z n )be the holomorphic coordinate on U. Denote s j ( z ) = z − w j D for j = 1 , · · · , l as atthe beginning of Section 4.2. Theorem 4.8.
Let M be a compact K¨ahler manifold and E a strong Nakano nefholomorphic vector bundle on M, with a Nakano pseudo effective limit Hermitianmetric H. Let f ∈ J kp ( K M ⊗ E ) be a holomorphic jet section whose support is a setof finite distinct points locate in a stein open ball U of M, and satisfying k f k k < ∞ . Then there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and there is a finite positive constant C with ˆ M | F | H Q lj =1 (cid:16) | s j | n (cid:16) log | s j | (cid:17) δ j (cid:17) ≤ C k f k k , (C k , j ) where δ j > are any positive constants.Proof. Like the proof of Theorem 4.2, it suffices to prove the following
Lemma 4.9.
Under the assumptions in Theorem 4.8, then for any holomorphicjet section f ∈ J kp ( K M ⊗ E ) satisfying k f k k < ∞ , there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f and forany ≤ j ≤ l , there is a finite positive constant C such that ˆ M | F | H (cid:16) | s | · · · | s l | ) n (cid:16) log | s j | (cid:17) δ j ≤ C k f k k , (D k , j ) where δ j > are any positive constants. We will use almost the same induction procedure as the proof of Theorem 4.2,but solve different approximate ¯ ∂ -equations from that in the proof of Theorem 4.5,then take limits. Since in the proof of Theorem 4.5 we have done the inductive step | k | = 0 a second time, here we skip the step | k | = 0 . The essential technical ideasfor | k | = 0 and the proof we give here for the inductive step from | k | − | k | is thesame.Assume Lemma 4.9 have been proved for | k | = k + · · · + k l varies from 0 to | k | − . Without loss of generality we assume k ≥ . Let g be the image of f n H ( M, K M ⊗ E J k p J k p ··· J kl +1 pl ) . By the induction hypothesis, there exists a liftingsection F k − ∈ H ( M, K M ⊗ E ) such that J k − p ( F k − ) = g and(68) ˆ M | F k − | H (cid:16) | s | · · · | s l | ) n (cid:16) log | s j | (cid:17) δ j ≤ C k f k k − . Here we denote k − k − , k , · · · , k l ) for brevity. We will construct holomorphiclifting section of f − J k − p ( F k − ) via solving ¯ ∂ -equation by using H¨omander’s L -theory. Without loss generality we assume f − J k − p ( F k − ) = 0 throughout thissection, in particular || f || k = 0 . Let ˜ F be a holomorphic lifting of f on the Stein open subset U. We choose acovering of M by a finite number of bounded Stein open subsets { U α | α = 1 , · · · , N } such that U is a member of { U α } , and the maximal diameter of { U α } is less than D. Let f α ∈ H ( U α , K M ⊗ E ) be the extended holomorphic section of the jet section f − J kp ( F ) ∈ J k p ( K M ⊗ E ) on U α , obtained by using Lemma 4.3 (note Lemma 4.3is still true if the bundle E is only strong Nakano nef by the proof of Theorem 4.5).Then we have(69) ˆ U α | f α | H (cid:16) | s | · · · | s l | ) n (cid:16) log | s j | (cid:17) δ j ≤ C k f − J k − p ( F k − ) k k . We will glue theses holomorphic sections { f α } together to get a global extensionof ˜ F − F k − on M. Let { θ α } be a smooth partition of unity corresponding to thecovering { U α } , and ˜ f = X α θ α f α . is a smooth patching of { f α } . By definition of jet section we have ∂ γ ( ˜ F − F k − ) | p = 0 , for any | γ | ≤ k − . For any index β we have(70) D ′′ ˜ f | U β = X α f α ¯ ∂θ α | U β = X α ( f α − f β ) ¯ ∂θ α | U β ;here we use P α θ α = 1 and hence P α ¯ ∂θ α = 0 . Since both f α and f β are extensionsof the jet section f − J k − p ( F k − ), we have(71) ∂ γ f α | p = ∂ γ f β | p = 0 , for any | γ | ≤ k − . and(72) ∂ α f γ | p = ∂ γ f β | p = ∂ γ ˜ F | p , for any | γ | = k . Let a τ,ǫ , b τ,ǫ , v τ,ǫ , φ, η ǫ be the functions as defined in (54),(55) and (56) respec-tively, here in (56) we take u ( x ) = x + δ x δ with 0 < δ < . Let e H = H ρ e − P lj =1 ( n + k j ) log | s j | be a smooth Hermitian metric on K M ⊗ E. Note here H ρ are a family smoothHermitian metric on the strong Nakano nef bundle E, and h [Θ H ρ , Λ] ϑ, ϑ i ≥ − ρ | ϑ | H ρ for any E -valued ( n, ϑ by (35). Hence h [ η ǫ Θ e H − id ′ d ′′ η ǫ − iλd ′ η ǫ ∧ d ′′ η ǫ , Λ] ϑ, ϑ i ≥ a τ,ǫ ( φ ) | ( d ′′ φ ) ∗ ϑ | H ρ − ρη ǫ | ϑ | H ρ . or τ ≪ η ǫ ≤ τ as in the equality (65). Take ρ = τ then ρη ǫ ≤ τ . Let g τ,ǫ = D ′′ ((1 − b τ,ǫ ( φ )) ˜ f ) = ( − a τ,ǫ ( φ ) d ′′ φ ) ˜ f + (1 − b τ,ǫ ( φ )) D ′′ ˜ f . By (ii) of Proposition 3.3, there exist f τ,ǫ ∈ L ( M, Λ n, T ∗ M ⊗ E ) and h τ,ǫ ∈ L ( M, Λ n, T ∗ M ⊗ E ) such that(73) D ′′ f τ,ǫ + 1 τ h τ,ǫ = g τ,ǫ , and ´ M ( η ǫ + λ − ǫ ) − | f τ,ǫ | Hρ Q lj =1 | s j | n + kj ) + ´ M | h τ,ǫ | Hρ Q lj =1 | s j | n + kj ) ≤ (cid:16) ´ M a τ,ǫ ( φ ) | ˜ f | Hρ Q lj =1 | s j | n + kj ) + τ ´ M (1 − b τ,ǫ ( φ )) Q lj =1 | s j | n + kj ) | D ′′ ˜ f | H ρ (cid:17) . (74) Lemma 4.10.
For τ ≪ there is a bounded positive constant depending only on D and | ∂ γ ˜ F ( p ) | H with | γ | = k + 1 such that lim ǫ → ˆ M (1 − b τ,ǫ ( φ )) Q lj =1 | s j | n + k j ) | D ′′ ˜ f | H ρ ≤ Ce τ . Proof.
Take a small contractible open neighborhood U of p and trivialized E over U then we may write f α − f β = ( ˆ f α − ˆ f β ) dz ∧ · · · ∧ dz n , where ˆ f α , ˆ f β are local vector valued holomorphic functions. Let z = ( z , · · · , z m )be the local holomorphic coordinate of U such that p lies at the origin, i.e., w ( p ) = 0 . Then s = D z. Since the support of 1 − b τ,ǫ ( φ ) is contained in thesubset φ − (( −∞ , τ + − ǫ )) ⊂ M, for τ ≪ − b τ,ǫ ( φ ) is containedin U α ∩ U β . Note here φ = log | s | . We havelim ǫ → (1 − b τ,ǫ ( φ )) Q lj =1 | s j | n + k j ) ≤ (cid:16) Dd (cid:17) nl + | k |− k − n ) | s | k + n ) . By (71) and (72), on U we have Taylor expansionˆ f α − ˆ f β = X | γ |≥ k +1 ∂ γ ˆ f α − ∂ γ ˆ f β γ ! z γ , where γ = ( γ , · · · , γ n ) and | γ | = γ + · · · + γ n . Let dσ = ∧ nj =1 ( idz j ∧ d ¯ z j ) =2 n r n − drdS be the volume element and dS is area element of the unit sphere in R n . Note for 0 < ǫ < , P | γ | = k +1 ´ M (1 − b τ,ǫ ( φ )) | z | γ | s | k n ) dσ = P | γ | = k +1 ´ | s |≤ e τ + 1 − ǫ (1 − b τ,ǫ ( φ )) | z | γ | s | k n ) dσ ≤ D ) | γ | P | γ | = k +1 ´ | z |≤ e τ + 1 − ǫ | z | γ | z | k n ) dσ = 2 ( n +2) (27 D ) | γ | ( ´ e τ + 1 − ǫ rdr ) P | γ | = k +1 ¸ | z | =1 | z | γ | z | k n ) dS = C D e τ , here C D is a constant depending only on D. Hencelim ǫ → ´ M (1 − b τ,ǫ ( φ )) Q lj =1 | s j | n + kj ) | D ′′ ˜ f | H ρ ≤ γ !) (cid:16) Dd (cid:17) nl + | k |− k − n ) P | γ | = k +1 ´ M | z | γ ( | ∂ γ ˆ f α | Hρ + | ∂ γ ˆ f β | Hρ ) | ¯ ∂θ | | z | k n ) dσ ≤ max ≤ α ≤ Np ∈ M ( | ¯ ∂θ α | ( p )) max | γ | = k +1 p ∈ M ( | ∂ γ ˆ f α | H ρ ( p ) + | ∂ γ ˆ f β | H ρ ( p )) C D e τ + O ( e τ ) ≤ Ce τ . Since ∂ γ ˆ f α ( z ) → ∂ γ ˆ F ( p ) and ∂ γ ˆ f β ( z ) → ∂ γ ˆ F ( p ) as τ → −∞ , hence for τ ≪ C is a bounded positive constant depending only on D and | ∂ γ ˜ F ( p ) | H with | γ | = k + 1 . (cid:3) Now we continue to estimate the R.H.S. of (74). We may decompose the firstterm of the R.H.S. of (74) as sum of integral of each bounded Stein open set U α , ˆ M a τ ( φ ) | ˜ f | H ρ Q lj =1 | s j | n + k j ) ≤ X α ˆ U α a τ ( φ ) | ˜ f | H ρ Q lj =1 | s j | n + k j ) . Recall that each f α is a extension of f − J k − p F k − on U α . If U α ∩ U = ∅ then ´ U α a τ ( φ ) | ˜ f | Hρ Q lj =1 | s j | n + kj ) = 0 when τ ≪
0; if U α ∩ U = ∅ then by Lemma 4.4 (in fact (64)in the proof of Lemma 4.4 is enough),lim τ →−∞ ˆ U α a τ ( φ ) | ˜ f | H ρ Q lj =1 | s j | n + k j ) ≤ C k f − J k − p ( F k − ) k k ≤ C k f k k , here the length k f − J k − p ( F k − ) k k and k f k k are firstly with respect to H ρ , since(each entry of) H ρ is continuous and uniformly convergent to H, so finally thelength k f − J k − p ( F k − ) k k and k f k k in the above inequality are with respect to H after taking the limit of the metric when τ → −∞ . As a resultlim τ →−∞ ˆ M a τ ( φ ) | ˜ f | H ρ Q lj =1 | s j | n + k j ) ≤ C k f k k for a sufficiently large and positive number C < + ∞ . Hence the limit of the R.H.S.of (74)(75) lim τ →−∞ (cid:16) ˆ M a τ ( φ ) | ˜ f | H ρ Q lj =1 | s j | n + k j ) + τ ˆ M (1 − b τ ( φ )) Q lj =1 | s j | n + k j ) | D ′′ ˜ f | H ρ (cid:17) ≤ C k f k k , since τ e τ → τ → −∞ . So the integrals of the L.H.S. of (74) are uniformlybounded.By equation (73), D ′′ h τ,ǫ = τ · D ′′ ( g τ,ǫ − D ′′ f τ,ǫ ) = 0 , and h τ,ǫ is square integrablewith respect to the metric H ρ e − P lj =1 ( n + k j ) log | s j | from the estimate in (74) and(75). Hence by the existence theorem of ¯ ∂ -equation on Stein manifold, there exists E -valued function l ατ,ǫ defined on U α such that D ′′ l ατ,ǫ = h τ,ǫ ith the estimate ˆ U α | l ατ,ǫ | H ρ Q lj =1 | s j | n + k j ) ≤ C ′′ ˆ U α | h τ,ǫ | H ρ Q lj =1 | s j | n + k j ) by H¨ormander L -estimate. Let I τ,ǫ = (1 − b τ,ǫ ( φ )) ˜ f − f τ,ǫ + 1 τ l ατ,ǫ . Then D ′′ I τ,ǫ = 0 and I τ,ǫ is holomorphic section. It in particular means that f τ,ǫ − τ l ατ,ǫ ∈ C ∞ ( U α ) is a smooth section. Moreover(76) lim inf τ →−∞ lim inf ǫ → ˆ U α ( η ǫ + λ − ǫ ) − Q lj =1 | s j | n + k j ) | f τ,ǫ | H ρ ≤ C k f k k . So as ǫ → , I τ,ǫ , f τ,ǫ , h τ,ǫ and h ατ,ǫ have subsequences which have weak limits inweighted L -space on each U α , denoted by I τ , f τ , h τ and l ατ respectively. Note I τ is a holomorphic section and I τ = (1 − b τ ( φ )) ˜ f − f τ + τ l ατ . Taking weak limit as ǫ → ˆ U α | f τ | H ρ Q lj =1 | s j | n + k j ) (log | s | ) δ < + ∞ . In the same way we know ´ U α | l ατ | Hρ Q lj =1 | s j | n + kj ) < + ∞ . Since log | s | > U α ˆ U α | τ l ατ − f τ | H ρ Q lj =1 | s j | n + k j ) < + ∞ . Hence the integral ˆ U α | τ l ατ,ǫ − f τ,ǫ | H ρ Q lj =1 | s j | n + k j ) is uniformly bounded for each U α , since f τ,ǫ − τ l ατ,ǫ ∈ C ∞ ( U α ) is a smooth we musthave(78) ∂ γ ( f τ,ǫ − τ l ατ,ǫ )( p j ) = 0 , for | γ | ≤ k j , with j = 1 , · · · , l. Note ρ = τ . By definition of strong Nakano nef vector bundle { H ρ } have asubsequence convergent to a limit Hermitian metric H with Nakano pseudo effectivecurvature current. Hence take limit as τ → −∞ of the inequality (74) and use (75)we obtain(79) lim inf τ →−∞ lim inf ǫ → ˆ M ( η ǫ + λ − ǫ ) − Q lj =1 | s j | n + k j ) | f τ,ǫ | H ρ ≤ C k f k k . Since f τ and l ατ are weighted L -bounded, hence I τ are weighted L -bounded.After further taking subsequences we may assume the holomorphic sections I τ =(1 − b τ ( φ )) ˜ f − f τ + τ l ατ is weakly convergent to a holomorphic section, and thelimit we denote by F k . Note any subseqence of (1 − b τ ( φ )) ˜ f and τ h ατ are always eakly convergent to zeros on each open set U α . Hence by (66) and noting | s j | < j = 1 , · · · , l we get(80) ˆ M | F k | H ( | s | · · · | s l | ) n (cid:16) log | s | (cid:17) δ ≤ C k f k k , where C is a bounded positive constant.Now put F = F k + F k − . Note that I τ,ǫ = (1 − b τ,ǫ ( φ )) ˜ f − f τ,ǫ + τ l ατ,ǫ → F k when ǫ → τ → −∞ . Note by definition, (1 − b τ,ǫ ( φ )) is a globally defined smoothsection on M. since f τ,ǫ is obtained via solving ¯ ∂ -equations on M it is globallydefined on the whole M too. Though l ατ,ǫ is obtained by solving ¯ ∂ -equations on U α , they may not be able to glue up to a global section, but τ l ατ,ǫ is weakly convergentto zero as ǫ → τ → −∞ . Hence the weak limit F k of I τ,ǫ is well defined aglobal holomorphic section on M. Without loss of generality, in the following we assume I τ,ǫ , f τ,ǫ , h τ,ǫ and h ατ,ǫ themselves (not passing to subsequences) are weakly convergent as ǫ → τ → −∞ . Now we check F is the expected holomorphic extension. By definition of b τ,ǫ , when p is inside a sufficiently small neighborhood of p we have b τ,ǫ ( φ ( p )) = 0 andhence in a small neighborhood of p we have I τ,ǫ ( p ) = ˜ f − f τ,ǫ + τ l ατ,ǫ . By (78) wehave ∂ γ F ( p ) = ∂ γ ˜ f ( p )for any γ with | γ | ≤ k . For any p = p we havelim τ →−∞ lim ǫ → (1 − b τ,ǫ ( φ ( p ))) = 0 , In fact if τ ≪ b τ,ǫ ( φ ( p )) = 1 for any fixed p = p . In particular forany j ≥ ∂ γ F ( p j ) = − lim τ →∞ lim ǫ → ∂ γ ( 1 τ l ατ,ǫ − f τ,ǫ )( p j ) + ∂ α F k − ( p j ) = ∂ α ˜ F ( p j )for any γ with | γ | ≤ k j . Hence F is the expected extension of jet section f ∈ J kp ( K M ⊗ E ) . Note that | F | H ≤ | F k | H + | F k − | H ) , By the induction assumption(68) and (80), ˆ M | F | H ( | s | · · · | s l | ) n (cid:16) log | s | (cid:17) δ ≤ C k f k k + C k f k k − = C k f k k , where C is a finite positive constant. This establishes ( D k , ) and hence ( C k ) by usingarithmetic-geometric inequality as what we have done in the proof of Theorem 4.2. (cid:3) The following Corollaries are proved in the same way with Corollary 4.6 and 4.7,as they are concluded from Theorem 4.2.
Corollary 4.11.
Let M be a compact K¨ahler manifold and E a strong Nakano nefholomorphic vector bundle on M, with a Nakano pseudo effective limit Hermitianmetric H. Let { p , · · · , p l } be a finite set locate in a stein open ball U of M. Let z be holomorphic coordinate of U with z ( p j ) = w j and s j ( z ) = z − w j for j = 1 , · · · , l. Then for any holomorphic jet section f ∈ J kp ( K M ⊗ E ) satisfying k f k k < ∞ , here exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f andthere is a finite positive constant C with ˆ M | F | H Q lj =1 (1 + | s j | ) n ≤ C k f k k , where δ j > are any positive constants. Corollary 4.12.
Let M be a compact K¨ahler manifold and E a strong Nakano nefholomorphic vector bundle on M, with a Nakano pseudo effective limit Hermitianmetric H. Then for any holomorphic jet section f ∈ J kp ( K M ⊗ E ) whose supportlocate is a finite subset contained a Stein open ball U in M, satisfying k f k k < ∞ , there exists holomorphic section F ∈ H ( M, K M ⊗ E ) such that J kp ( F ) = f andthere is a finite positive constant C with ˆ M | F | H ≤ C k f k k , where δ j > are any positive constants. Proof of the main theorem
To prove the main theorem we will show M has many global holomorphic vec-tor fields if its tangential bundle is strong Griffiths positive. Geometrically, if thefollows of the holomorphic vector fields could connect with one another such that apoint start one place and go along the flows it could arrive everywhere of M then M is homogeneous. Unfortunately the point may stop at some places, which areexactly the zero points of the vector fields. If all extended holomorphic vector fieldshave no zero points, on the Lie group level which means the follows defined by theholomorphic vector fields has no fixed points, then the holomorphic automorphicgroup Aut(M) will act almost freely on M. If moreover the extended holomorphicvector fields could span the tangent space of M everywhere, which means the ac-tion of Aut(M) has open and dense orbit, then M is holomorphic homogeneous.Unfortunately, we could not guarantee the extended holomorphic vector fields haveno zero points outside the points where they already have the given nonzero values.To overcome this difficulty, we consider extending holomorphic jet sections. If wetake enough many holomorphic jet vector fields defined on a Stein neighborhood,and arrange their jet values such that they linearly spanned the tangent space in aneighborhood, and if these jet vector fields are all extendable on the whole manifold M, then it implies that Aut(M) acts transitively in the neighborhood. If we couldrepeat this process everywhere on M, then we may prove M is homogenous.Though as a Fano manifold, M is projective and we may cut a hyper surface L outof M such that the rest M \ L is a Stein manifold. Unfortunately it is very difficultto construct a complete K¨ahler metric on M \ L such that the diameter of M \ L isbounded. Since in Theorem 4.2, Corollary 4.6 and 4.7, the upper bounds of the L -estimate of the extended sections depend on the diameter of the Stein manifold andis not necessary uniformly bounded, even we could extend a holomorphic sectionto defined on the whole of M \ L, we could not extend it further to M. So we needusing the extension theorem established on the K¨ahler manifolds.
Theorem 5.1. If M is a compact K¨ahler manifold with strong Griffiths nef tangentbundle, then M is a homogeneous complex manifold under action of its holomorphicautomorphism group.Proof. Let Aut(M) denote the holomorphic automorphism group of M. The naturalholomorphic action Aut(M) × M → M is denoted by mapping ( g, p ) g · p. Firstlywe will prove the Aut(M) action is locally homogeneous. That means, for any point ∈ M and for an open ball B p ( r ) of sufficient small radius r with center p, for any q ∈ B p ( r ) there exist g ∈ Aut(M) such that g · q = p. Since the quotient map O M → O M J p J p J p ∼ = ⊕ j =1 ( C ⊕ T ∗ M | p j )is defined by f ( f ( p ) + df ( p ) , f ( p ) + df ( p ) , f ( p ) + df ( p )) is surjective,hence we have a surjective quotient map T M → T M J p J p J p ∼ = ⊕ j =1 ( T M | p j ⊕ End ( T M | p j )) . Take a jet X ∈ H ( M, T M ⊗ O M / J p J p J p ) with X ( p j ) = X ( p k ) and dX ( p j ) = dX ( p k ) for 1 ≤ j = k ≤ . Hence without loss generality we assume X ( p ) ∈ T M is not a zero vector and at the same time dX ( p ) = 0 . Now we claim that we could X to a global holomorphic vector field defined everywhere on M. Since the tangentbundle T M of M is strong Griffiths nef, by Proposition 3.18, E := K ∗ M ⊗ T M = det( T M ) ⊗ T M is strong Nakano nef. Hence there is a limit Hermitian metric H on E whose curva-ture current is Nakano pseudo effective. Moreover every entry of H is continuous on M by Proposition 3.18, hence || X || k < + ∞ with k = (2 , , . Note T M = K M ⊗ E. By Corollary 4.12, X could be extended to a global holomorphic vector field definedeverywhere on M and still denoted by X ∈ H ( M, T M ) . Take a smooth curve C pq ⊂ B p ( r ) connected p and q. For any p ′ ∈ C pq , wecould take jet vectors X ( p ′ ) , · · · , X m ( p ′ ) such that they span the tangent space T M | p ′ , at the same time we may assume dX ( p ′ ) , · · · , dX n ( p ′ ) are all nonzero. Nowextend all of them to global holomorphic vector fields X , · · · , X m on M, just inthe same way we extend X in the above paragraph. Then X , · · · , X m will spanthe tangent space T M in a neighborhood U p ′ of p ′ . So the flows of X , · · · , X n sweep out the open neighborhood U p ′ by the Frobenius theorem. Now for each p ′ ∈ C pq we repeat the process, we get an open cover { U p ′ | p ′ ∈ C p,q } of C pq . ByHeine-Borel’s theorem, we get a finite subcover. Hence there are finite number ofglobal holomorphic vector fields whose flow may sweep out an open neighborhoodof C pq . It in particular means that there exist g ∈ Aut(M) such that g · q = p. Since for each p ∈ M there is a open neighborhood B p ( r ) which is homogeneousunder the action of Aut(M) and M is compact, using Heine-Borel’s theorem onceagain, we know M is homogeneous under the holomorphic action of Aut(M) . Hence M is a homogeneous complex manifold. (cid:3) Corollary 5.2. If M is a compact K¨ahler manifold with Griffith semipoistive tan-gent bundle, then M is a homogeneous complex manifold under action of its holo-morphic automorphism group.Proof of the main theorem . By Theorem 5.1, M is a homogeneous complex manifold.Since M is meanwhile a projective K¨ahler manifold, by Borel-Remmert’s theorem[BR], M is s the direct product of a complex torus and a projective-rational man-ifold. Note any Fano manifold is simply connected, hence M is a homogeneousprojective-rational manifold. (cid:3) Acknowledgments
I’m grateful to professor Yum-Tong Siu for his enlighteningand stimulating discussions and for explaining to me many of his important worksand their backgrounds, in particular the works related to the extensions theorem nd positive closed currents; during the past year he not only taught me mathemat-ics but also guided me to think mathematics in a simple way. I would like to thankprofessor Shing-Tung Yau for encouragements and the friends in Yau’s studentssiminar for helping me in many ways. I would take this chance to thank Profes-sors Alan, Huckleberry, Peter, Heinzner and Takeo, Ohsawa, I studied under theirguidance before and their works on complex Lie group actions and L -techniquesin several complex variable inspired me to think the problem studied in this paperfor a long time. Finally I would like to thank the department of mathematics ofHarvard university for its hospitality. My visit to Harvard university is supportedby the Lingnan foundation of Sun Yat-Sen university. References [BT] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions.
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E-mail address : [email protected] Department of Mathematics, Sun Yat-Sen University, 510275, Guangzhou, P. R. CHINA.
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