∂ ¯ -equation on (p,q) -forms on conic neighbourhoods of 1 -convex manifolds
aa r X i v : . [ m a t h . C V ] J a n ∂ -EQUATION ON ( p, q ) -FORMS ON CONIC NEIGHBOURHOODS OF -CONVEX MANIFOLDS JASNA PREZELJ
Abstract.
Let X be a 1-convex manifold with the exceptional set S , which is also amanifold, Z a complex manifold, Z → X a holomorphic submersion, a : X → Z aholomorphic section and S ⊂ U ⋐ X an open relatively compact 1-convex set. We constructa metric on a vector bundle E → Z restricted to a neighbourhood V of a ( U ) , conic along a ( S ) with at most polynomial poles at a ( S ) and positive Nakano curvature tensor in bidegree( p, q ) . Introduction and the main theorem
Let π : Z → X be a submersion from a complex manifold Z to a 1-convex manifold X withthe exceptional set S, which is also a manifold. The motivation for this work was to constructsolutions of the ∂ -equation with at most polynomial poles at π − ( S ) in a particular geometricsituation (see Fig. 1), namely on a conic neighbourhood V of sections of π : Z → X. Theresults of this paper may provide one step in the proof of such a claim. It turned outthat it is possible to construct solutions of the ∂ -equation such that their L -norms on V δ := { z ∈ V, d ( z, π − ( S )) > δ } with respect to some ambient Hermitian metric h Z on Z and h E on E grow at most polynomially as δ → . To this end weights that give rise topositive Nakano curvature and have polynomial behaviour on π − ( S ) are constructed. Sup-norm estimates are given in the case q = 1 . The main theorem of the present paper is thefollowing:
Theorem 1.1 (Nakano positive curvature tensor in bidegree ( p, q )) . Let Z be a n -dimensionalcomplex manifold, X a -convex manifold, S ⊂ X its exceptional set, which is also a mani-fold, π : Z → X a holomorphic submersion with r -dimensional fibres, σ : E → Z a holomor-phic vector bundle and a : X → Z a holomorphic section. Let ϕ : X → [0 , ∞ ) be a plurisub-harmonic exhaustion function, strictly plurisubharmonic on X \ S and ϕ − (0) = S. Let U = ϕ − ([0 , c ]) for some c > be a given holomorphically convex set and let s ∈ { , . . . , n } . Then there exist an open neighbourhood V of a ( U \ S ) conic along a ( S ) , a Nakano posi-tive Hermitian metric h on E | V with at most polynomial poles on π − ( S ) and such that theChern curvature tensor i Θ( E ⊗ Λ s T Z ) restricted to V is Nakano positive and has at mostpolynomial poles and zeroes on π − ( S ) . For standard techniques for solving the ∂ -equation we refer to Demailly’s book Complexanalytic and algebraic geometry [Dem]. Our main tool is Mathematics Subject Classification.
Key words and phrases. heorem 1.2 (Theorem VIII-4.5, [Dem]) . If ( W, ω ) is complete and A E,ω > in bidegree ( p, q ) , then for any ∂ -closed form u ∈ L p,q ( W, E ) with Z W h A − E,ω u, u i dV < ∞ there exists v ∈ L p,q − ( W, E ) such that ∂v = u and k v k ≤ Z W h A − E,ω u, u i dV. It seems difficult to handle the commutator A E,ω (see Sect. VII-7, [Dem] for computations)in the case of ( p, q )-forms because it has terms of mixed signs for p < n.
Therefore we viewthe ( p, q )-form u as a ( n, q )-form u with coefficients in E = E ⊗ Λ n − p T Z by invoking theisomorphism Λ p T ∗ Z ≃ Λ n − p T Z ⊗ Λ n T ∗ Z. If u is closed so is u . In addition, in bidegree ( n, q ) , there is an analog of Theorem 1.2 for a noncomplete K¨ahler metric provided that the manifoldpossesses a complete one (Theorem VIII-6.1, [Dem]). Moreover, the positivity of A E ,ω follows from the positivity of the Chern curvature tensor i Θ( E ) = i Θ( E ) + i Θ(Λ n − p T Z ) . a(U)a(X)Z a(S) V Figure 1.
Conic neighbourhoods of a ( U \ S ) in Z To use both theorems we first need a Nakano positive Hermitian metric on E | V and aK¨ahler metric on V, both with polynomial zeroes or poles on π − ( S ). The K¨ahler metric ω = i∂∂ Φ constructed in Sect. 2 and the Nakano positive Hermitian metric given by Theorem1.1 in [Pre2] have the desired properties. The space (
V, ω ) is not complete but if we take asmaller neighbourhood conic along a ( N ( g )) , N ( g ) := g − (0) , for some holomorphic function g : X → C with g ( S ) = 0 , it contains a conic Stein neighbourhood V ′ which is completeK¨ahler (Fig. 2). With the above notation we have Corollary 1.3 ( ∂ -equation in bidegree ( p, q )) . Let g : X → C be holomorphic with N ( g ) = g − (0) ⊃ S. Let V ′ be an open Stein neighbourhood of a ( U \ N ( g )) , conic along a ( N ( g )) , let u be a closed ( p, q ) -form on V ′ , u the corresponding ( n, q ) -form with coefficients in E = E ⊗ Λ n − p T Z and let h be the metric on E from Theorem 1.1 with s = n − p and h the induced metric on E . Denote by A = A E ⊗ Λ n − p T Z,ω the commutator and assume that Z V ′ h A − u , u i h dV ω < ∞ hen there exist an ( n, q − -form v with k v k = Z V ′ h v , v i h dV ω ≤ Z V ′ h A − u , u i h dV ω . Corollary 1.4. If u is smooth and v is the minimal norm solution then v and the associated ( p, q − -form v are smooth. The L -norms of v on the sets V ′ δ := { z ∈ V ′ , d ( z, π − ( S )) > δ } with respect to h Z and h E grow at most polynomially with respect to δ as δ → and thesame holds for the L -norms of the corresponding ( p, q − -form v . If q = 1 we can use Lemma 4.5 in [Pre2] which is an adaptation of Lemma 3.2 in [ ? ] to getsup-norm estimates from the Bochner-Martinelli-Koppelman formula if we take a slightlysmaller Stein neighbourgood V ′′ ⊂ V ′ , conic along a ( N ( g )) (Fig. 2) Corollary 1.5. If q = 0 and the initial form u is smooth on a neighbourhood of a ( U \ S ) , the form v has at most polynomial poles on π − ( S ) . V'a(U)a(X) V"Z a(N(g))
Figure 2.
Conic neighbourhoods of a ( U \ N ( g )) in Z Notation.
The notation from Theorem 1.1 is fixed throughout the paper. Let h Z be aHermitian metric defined on the manifold Z and let σ : E → Z be a holomorphic vectorbundle of rank r equipped with a Hermitian metric h E . The local coordinate system in aneighbourhood V z ⊂ Z of a point z ∈ a ( U ) is ( z, w ) , where z denotes the horizontal and w the vertical (or fibre) direction and z = (0 , . More precisely, every point in a ( U ) has w = 0 and points in the same fibre have the same first coordinate. If the point z is in a ( S )we write the z -coordinate as z = ( z , z ) , where a ( S ) ∩ V z = { z = 0 , w = 0 } ∩ V z . Themanifold Z is n -dimensional and the dimension of the fibres Z z is constant, r = dim Z z . The notation ζ , . . . , ζ n is sometimes used for local coordinates in Z. Construction of the K¨ahler metric ω The K¨ahler metric ω will be obtained from the K¨ahler potential Φ = ϕ + ϕ and is similarto the one constructed in subsection 2.1 in [Pre2]. The only difference is that we choose aspecific plurisubharmonic function ϕ instead of the given ϕ in order to be able to study thecurvature properties of ω. The construction is explained below.Since the Remmert reduction p : X → ˆ X of X is a Stein space with finitely many isolatedsingular points it has a proper holomorphic embedding ˆ f : ˆ X → C M for some large M. onsequently, the holomorphic functions ˆ f ◦ p, . . . , ˆ f M ◦ p generate the cotangent space T ∗ X \ S and the function ˆ ϕ := P | ˆ f i ◦ p | is a plurisubharmonic exhaustion function of X , strictly plurisubharmonic on X \ S. The functions f i := ˆ f i ◦ p ◦ π are defined on Z andthey generate the horizontal cotangent space on a ( X \ S ) . Define ϕ := ˆ ϕ ◦ π = P | f i | anddenote by k the maximal order of degeneracy of i∂∂ϕ at π − ( S ) . For given l > ϕ replaced by ϕ yields almost holomorphic functions f M +1 , . . . f N , holomorphic to a degree l with zeroesof order at least k on π − ( S ) . To be precise, for every sufficiently large k by Theorem Athere exist sections F M +1 , . . . , F N ∈ Γ( U ′ , J ( a ( S )) k ( J ( a ( U ′ )) / J l +1 ( a ( U ′ )))), U ⋐ U ′ whichlocally generate the sheaf on U .
We further assume that k > k . The functions f M +1 , . . . f N , are obtained by patching together particular local lifts of these sections using the partitionof unity { χ j , U j } which we (can) choose to depend on the horizontal variables only and so f M +1 , . . . f N , are holomorphic in vertical directions. In local coordinates ( z, w ) near a ( S ) wethus have f i ( z, w ) = X | α | = k, < | β |≤ l c αβ z α w β + X i,j,l χ j ( z ) f ijl ( z, w )with f ijl ∈ O ( k z k k k w k l +1 ) holomorphic on open sets U j . The functions f i satisfy the es-timate ∂f i ( z, w ) ≈ k z k k k w k l +1 . Moreover, we can express z α w j = P g αij ( z ) f i ( z, w ) + O ( k z k k k w k l +1 ) with g αij holomorphic and from this we infer that ∂ w j f i , ≤ j ≤ r , M +1 ≤ i ≤ N generate the vertical cotangent bundle on a neighbourhood V T of a ( U ) in Z excepton π − ( S ) . Consequently, the matrix corresponding to ∂ w ∂ w P | f i | is of the form k z k k G, with G invertible. Define Φ := X f i f i and ω := i∂∂ Φ . We claim that the function Φ is a K¨ahler potential on a conic neighbourhood of a ( U \ S )and ω a K¨ahler metric.Write local coordinates as ( ζ , . . . ζ n ) = ( z, w ) and represent the Levi form i∂∂ Φ = i X h jk dζ j ∧ dζ k by a matrix H = { h jk } . In local coordinates ( z, w ) the nonnegative part of ω, ω + = i P ∂f i ∧ ∂f i , represented in the matrix form as H + , can be estimated from below by(2.1) H + ( z, w ) & (cid:20) k z k k + k w k k z k k − k w kk z k k − k w kk z k k − k z k k (cid:21) , where we have estimated the decay of ϕ by k z k k from below. The possibly negative part ω − = i P ∂∂f i f i + f i ∂∂f i + ∂f i ∧ ∂f i degenerates at least as k w k l k z k k − . It is clear that for k z k > δ and small k w k or k z k ≤ δ and k w k ≤ k z k the matrix H is strictly positive definiteand thus ω is a K¨ahler metric on a neighbourhood of a ( U \ S ) , conic along a ( S ) . Let ( z, w )be local coordinates near a ( S ) and define H ( z ) := H ( z,
0) = H + ( z, , H = H − H . Itfollows that H = O ( k w kk z k k − ) and that H decreases polynomially (in some directions)as we approach π − ( S ) and its degeneracy is bounded from below by k z k k ,H ( z ) & (cid:20) k z k k k z k k (cid:21) . otice that H is strictly positive on a neighbourhood of a ( U ) except on π − ( S ) (and there-fore invertible) and k H − k degenerates in the worst case as k z k − κ for some κ ≥ . Be-cause S is compact, there exists one κ for all points in a ( S ) . Write H = H ( I + H − H ) ,H − = ( I + H − H ) − H − , then k H − H k ≈ k z k k − − κ k w k in the worst case and this term is small, k H − H k < k z k κ + k on conic neighbourhoods ofthe form k w k ≤ k z k κ + k , and so(2.2) H − = H − + H − H ∞ X ( H − H ) n H − = H − + N, k N k ≤ k z k κ + k . Inside this cone the the degeneracy of the inverse H − is governed by H − . Theorems on curvatures
Basic theorems on curvatures.
Before proceeding to the proof we recall some for-mulae from Demailly’s Complex analytic and algebraic geometry [Dem].Let (
X, ω ) be a K¨ahler manifold, E → X a rank r vector bundle equipped with aHermitian metric h. The matrix H that corresponds to h in local coordinates is given by h u, v i h = P h λµ u λ v µ = u T Hv.
Let i Θ( E ) be the Chern curvature form of the metric and Λthe adjoint of the operator u → u ∧ ω, defined on ( p, q )-forms. Denote by L p,q ( X, E ) thespace of ( p, q )-forms with bounded L -norms with respect to the h and let A E,ω = [ i Θ( E ) , Λ]be the commutator.In bidegree ( n, q ) the positivity of A E,ω is equivalent to Nakano positivity of E. Let e , . . . , e r be a local frame of E. If the metric is locally represented by a matrix H then(3.1) i Θ( E ) = i∂ ( H − ∂H ) = i X c jkλµ dz j ∧ dz k ⊗ e ∗ λ ⊗ e µ , If e , . . . , e r is an orthonormal frame then the Hermitian form θ E defined on T X ⊗ E, whichis associated to i Θ( E ) , takes the form(3.2) θ E = X c jkλµ ( dz j ⊗ e ∗ λ ) ⊗ ( dz k ⊗ e ∗ µ ) . The curvature tensor (3.1) is
Griffiths positive if the form (3.2) is positive on decomposabletensors τ = ξ ⊗ v, ξ ∈ T X, v ∈ E, θ E ( τ, τ ) = P c jkλµ ξ j ξ k v λ v µ and Nakano positive if it ispositive on τ = P τ jλ ( ∂/∂z j ) ⊗ e λ , θ E ( τ, τ ) = P c jkλµ τ jλ τ kµ . In a nonotrhonormal frame wehave θ E ( τ, τ ) = P c jkλµ τ jλ τ kν h µν . Proposition VII-9.1,[Dem] states that(3.3) if θ ( E ) > Grif r Tr E ( θ ( E )) ⊗ h − θ ( E ) > Nak . The metric h on E induces the metric h s on Λ s E. Let L be an s -tuple of (not necessarilyordered) indices L = ( λ , . . . , λ s ) and denote e L := e λ ∧ . . . ∧ e λ s . If σ is a permutation then e σ ( L ) = sign( σ ) e L . Let
L, M ∈ { ( λ , . . . λ s ) , ≤ λ < . . . < λ s ≤ r } =: L , L = ( λ , . . . λ s ) ,M = ( µ , . . . , µ s ) . The coefficient H LM = h e L , e M i h s in the matrix H s representing theinduced metric h s is H sLM = det H ( λ ,...λ s ) , ( µ ,...,µ s ) , where H ( λ ,...λ s ) , ( µ ,...,µ s ) is a submatrix of H generated by rows λ , . . . λ s and columns µ , . . . , µ s of the matrix H. If e , . . . e r are orthonormal at z so are their wedge products { e L , L ∈ L} . he induced Chern curvature tensor on Λ s E, i
Θ(Λ s ( E )) = P j,k i Θ(Λ s ( E )) jk dz j ∧ dz k isdefined by formula V-(4.5’), [Dem],(3.4) i Θ(Λ s ( E )) jk ( e L ) = i X ≤ l ≤ s e λ ∧ . . . ∧ Θ( E ) jk e λ l ∧ . . . ∧ e λ s . It is known that E ≥ Nak s E ≥ Nak . The following lemma gives an explicit formulafor the curvature i Θ(Λ s ( E )) in terms of the curvature i Θ( E ) and shows that if the associateHermitian form θ ( E ) has at most polynomial poles on π − ( S ) , so does θ (Λ s ( E )) . Lemma 3.1. If i Θ( E ) is Nakano nonpositive (nonnegative) then i Θ(Λ s ( E )) , ≤ s ≤ r, isalso Nakano nonpositive (nonnegative).Proof. By formula (3.4) we have i Θ(Λ s ( E ))( e L ) jk = i X l,µ ( − l − c jkλ l µ e µ ∧ e L ′ l , where L ′ l is obtained from L by removing the l -th index. Let L ( λ, µ ) denote the (not ordered)multiindex obtained by replacing the index λ in the multiindex L by µ. We define that e L ( λ,µ ) = 0 if and only if λ / ∈ L or µ ∈ L \ { λ } . If λ l = λ then e µL ′ l = ( − l − e L ( λ,µ ) and i Θ(Λ s ( E )) = i X j,k,L,M c sjkLM dz j ∧ dz k ⊗ e ∗ L ⊗ e M = i X j,k,L X λ ∈ L,µ c jkλµ dz j ∧ dz k ⊗ e ∗ L ⊗ e L ( λ,µ ) = i X j,k | L ′ | = s − X λ,µ L ′ c jkλµ dz j ∧ dz k ⊗ e ∗ λL ′ ⊗ e µL ′ . Here the bijection between the sets { L : | L | = s } and { λL ′ : | L ′ | = s − , λ / ∈ L ′ } isused, where we compare multiindices as sets. Let τ = P j,L τ j,L ( ∂/∂z j ) ⊗ e L with additionalproperties τ j,σ ( L ) := sign( σ ) τ j,L for any permutation σ and τ j,L = 0 if there are at least twoequal indices in L. The bundle Λ s ( E ) is Nakano positive if the bilinear form θ Λ s ( E ) ( τ, τ ) = X c sjkLM τ j,L τ k,S h e M , e S i h s = X j,k, | S | = s | L ′ | = s − X λ,µ L ′ c jkλµ τ j,λL ′ τ k,S h e µL ′ , e S i h s is positive. Assume that the local frame e , . . . , e r is orthonormal. Then { e L , L ∈ L} areorthonormal and(3.5) θ Λ s ( E ) ( τ, τ ) = X j,k,L ′ X λ,µ c jkλµ τ j,λL ′ τ k,µL ′ = X L ′ θ E ( τ L ′ , τ L ′ ) , where the form τ L ′ is defined by τ L ′ = P j,λ τ j,λL ′ ( ∂/∂z j ) ⊗ e λ for any multiindex L ′ of length s − . Hence if θ E is Nakano nonpositive (nonnegative), so is θ Λ s E . (cid:3) .2. Almost nonpositivity of ω . In this section we study properties of the form ω con-structed in Sect. 2. Let V ⊂ Z be a neighbourhood of a ( U \ S ) , conic along a ( S ) . The metric ω on a vector bundle E → V is almost Nakano nonpositive if the curvarure tensor i Θ( E ) has adecomposition i Θ( E ) = i Θ ( E ) + i Θ ( E ) , where i Θ ( E ) is nonpositive and i Θ ( E ) is locallyof the form i Θ ( E )( z, w ) = O ( k w k l k z k k ) near points in a ( S ) and i Θ ( E )( z, w ) = O ( k w k l )near points in a ( U \ S ), for some l ∈ N , k ∈ Z .The main theorem in this subsection is Theorem 3.2 (Almost Nakano nonpositive K¨ahler metric) . Let
Z, X, S, a, U and ω be asin Theorem 1.1. There exist a neighbourhood V of a ( U \ S ) conic along a ( S ) such that themetric ω on T Z | V is almost Nakano nonpositive. Corollary 3.3.
Let h ω be the metric on T Z | V induced by ω. Then h ω e Φ is Nakano negativeon a smaller neighbourhood - which we again denote by V - of a ( U \ S ) , conic along a ( S ) . Proof of Theorem 3.2 . Write f = ( f , . . . f N ) T and P | f i | = f T f and let H denote thematrix corresponding to the metric. If D denotes the holomorphic and D the antiholomorphicderivative with respect to ( z, w ) ,Df = f ,z f ,z f ,z . . . f ,w r f ,z f ,z f ,z . . . f w r ... ... ... . . . ... f N,z f N,z f N,z . . . f N,w r then H = DD ( f T f ) = ( Df ) T Df + D ¯ f T Df + f T Lf + Lf T f The Levi form Lf of the vector is calculated as Lf = ( Lf , . . . , Lf N ) . With the notationdefined prior to (2.1) we have H + = ( Df ) T Df and H − = H − H + = D ¯ f T Df + f T Lf + Lf T f . Since f i are holomorphic or almost holomorphic to the degree l , we have ∂f = O ( k w k l +1 ) , terms ∂Df, Lf, H − are of the form O ( k w k l ) and because of the holomorphicity of f i in w -directions the terms ∂H − , ∂H − , ∂∂Df, ∂∂H − are of the form O ( k w k l − ) . The term H + gives ∂∂H + = ∂∂ (( Df ) ∗ Df ) = − ∂∂ (( Df ) ∗ Df ) = − ∂ ( ∂ ( Df ) ∗ Df + ( Df ) ∗ ( ∂Df ))= − ( ∂Df ) ∗ ∧ ( ∂Df ) + ∂ ( Df ) ∗ ∧ ∂Df + ( ∂∂ ( Df ) ∗ ) Df + ( Df ) ∗ ∂∂Df and so(3.6) ∂∂H = − ( ∂Df ) ∗ ∧ ( ∂Df ) + O ( k w k l − ) . By the above estimates we conclude that(3.7) ∂H = ∂ ( H + + H − ) = ( Df ) ∗ ( ∂Df ) + O ( k w k l − )and similarly(3.8) ∂H = ( ∂Df ) ∗ Df + O ( k w k l − ) . ince the metric is K¨ahler, we may assume that the coordinates ζ , . . . , ζ n near the point z ∈ a ( U \ S ) are such that H ( z ) = I to the second order. The curvature form equals i Θ( T Z ) = i∂ ( H − ∂H )= − iH − ∂HH − ∧ ∂H + iH − ∂∂H = − i∂∂ H = i ( ∂Df ) ∗ ∧ ( ∂Df ) + O ( k w k l − )by the above assumptions. We claim that i ( ∂Df ) ∗ ∧ ( ∂Df ) is Nakano nonpositive. Denotethe dual tangent vectors by e λ := ∂/∂ζ λ and let i Θ = i ( ∂Df ) ∗ ∧ ( ∂Df ) = i X j,k,λ,µ c jkλµ dζ j ∧ dζ k ⊗ e ∗ λ ⊗ e µ . Then we have c jkλµ = − N X i =1 ∂ f i ∂ζ j ∂ζ λ ∂ f i ∂ζ k ∂ζ µ and so θ ( τ, τ ) = − X i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X jλ ∂ f i ∂ζ j ∂ζ λ τ jλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . Remark 3.4.
If the functions f i were holomorphic, then the Nakano nonpositivity of i Θ( V T )on V could be inferred from the fact that the metric on T Z | V is the metric induced on thesubbundle F ≤ V × C N by the standard metric on C N via the holomorphic vector bundleisomorphism T Z V → F, T Z ∋ v ( df ( v ) , . . . , df N ( v )) ∈ C N and it is known that theNakano curvature decreases in subbundles by VII-(6.10), [Dem]. Because f i are almostholomorphic, we get an ‘error’ term, denoted by i Θ in the sequel and we will show that itdecreases arbitrarily fast on conic neighbourhoods. On the section a ( U \ S ) the ‘holomorphic’part of the curvature tensor is exactly i Θ . The curvature form i Θ( T Z ) is then almost Nakano nonpositive on an open neighbourhoodof a ( U \ S ) . If we want to prove that the neighbourhood is conic we have to show that thepart of the curvature which contains w -variables decreases polynomially sufficiently fast insome conic neighbourhood. Let ( z, w ) be local coordinates at ( z, ∈ a ( S ) . We can notassume that H = I to the second order at ( z,
0) because H ( z,
0) is degenerate on a ( S ) . We have to split the curvature tensor into the part depending only on z -variables - wehave just proved that it is nonpositive - and the rest, which we want to be small on conicneighbourhoods. By estimates (2.2), (3.6), (3.7), (3.8) we have for k w k ≤ k z k κ + k H − ∂∂H = H − (( ∂Df ) ∗ ∧ ( ∂Df ) + O ( k w k l − ))= H − ( ∂Df ) ∗ ∧ ( ∂Df ) + N ( ∂Df ) ∗ ∧ ( ∂Df ) + O ( k w k l − k z k − κ )= H − ( ∂Df ) ∗ ∧ ( ∂Df ) + O ( k z k κ + k ) + O ( k w k l − k z k − κ ) , − ∂HH − ∧ ∂H = ( H − + N )(( ∂Df ) ∗ Df + O ( k w k l − ))( H − + N ) ∧ (( Df ) ∗ ( ∂Df ) + O ( k w k l − ))= H − ( ∂Df ) ∗ Df H − ∧ ( Df ) ∗ ( ∂Df ) + H − ( ∂Df ) ∗ Df N ∧ ( Df ) ∗ ( ∂Df )+ N ( ∂Df ) ∗ Df H − ∧ ( Df ) ∗ ( ∂Df ) + N ( ∂Df ) ∗ Df N ∧ ( Df ) ∗ ( ∂Df )+ O ( k z k − κ ) O ( k w k l − )= H − ( ∂Df ) ∗ Df H − ∧ ( Df ) ∗ ( ∂Df ) + O ( k z k κ + k ) O ( k z k − κ )+ O ( k z k κ + k ) ) + O ( k z k − κ ) O ( k w k l − ) . Let Θ ( T Z ) := H − ( ∂Df ) ∗ ∧ ( ∂Df ) − H − ( ∂Df ) ∗ Df H − ∧ ( Df ) ∗ ( ∂Df ) . By Taylor seriesexpansion we see that i Θ ( T Z )( z, w ) = iH − ( ∂Df ( z,
0) + O ( k w k )) ∗ ∧ ( ∂Df ( z,
0) + O ( k w k )) − iH − ( ∂Df ( z,
0) + O ( k w k )) ∗ ( Df ( z,
0) + O ( k w k )) · H − ∧ ( Df ( z,
0) + O ( k w k )) ∗ ( ∂Df ( z,
0) + O ( k w k ))= i Θ ( T Z )( z,
0) + O ( k w k ) O ( k z k − κ + k z k − κ ) . Let i Θ ( T Z )( z, w ) = i Θ( T Z )( z, w ) − i Θ ( T Z )( z, . The form i Θ ( z,
0) is nonpositive on aneighbourhood of a ( U \ S ) , conic along a ( S ) . The ‘error term’ i Θ ( T Z ) is in the worst case O ( k w k ) O ( k z k − κ + k z k − κ ) + O ( k z k κ + k ) + O ( k w k l − k z k − κ )+ O ( k z k κ + k ) + O ( k z k κ + k ) ) + O ( k z k − κ ) O ( k w k l − )and it decreases at least as k z k k on conic neighbourhoods k w k ≤ k z k κ + k . (cid:3) Proof of Corollary 3.3 . The part H of the Levi form of Φ near ( z, ∈ a ( S ) that dominatesin the matrix H on a cone is bounded from below by k z k k I. The potentially positive partof the Nakano curvature, i Θ , is of the form O ( k z k k )on k w k ≤ k z k κ + k and can be compensated by − ∂∂ Φ / k > k thus making thecurvature tensor ( i Θ ( z, − i∂∂ Φ( z, w )) + ( i Θ ( z, w ) − i∂∂ Φ( z, w )))strictly Nakano negative. (cid:3) Proof of the main theorem
By theorem 1.1 in [Pre2] the bundle E can be endowed with a Nakano positive Hermitianmetric h on a conic neighbourhood of a ( U \ S ) with polynomial poles on π − ( S ) . Let ω = i∂∂ Φ be the given metric. In order to solve the ∂ -equation in bidegree ( p, q ) we have toshow that the curvature tensor i Θ( E ⊗ Λ s T Z ) + iL Ψ = i Θ( E ) + i Θ(Λ s T Z ) + iL Ψis positive (or at least nonnegative) for some strictly plurisubharmonic weight Ψ and s = n − p. he K¨ahler metric h induced by the K¨ahler form ω has almost nonpositive Nakano cur-vature. Let h s be the metric on Λ s T Z induced by h and h the metric induced by the form ω = ωe Φ . The latter has strictly Nakano negative curvature tensor i Θ( T Z ) ω = i Θ( T Z ) ω − i∂∂ Φby Corollary 3.3. If the original metric on
T Z is represented by H then the new one is H = He Φ and the induced metric h s on Λ s T Z is represented by the matrix H s = H s e s Φ . Since the Chern curvature tensor of the Hermitian metric ω on T Z is Nakano negative theinduced curvature tensor on Λ s T Z ω is also Nakano negative by Lemma 3.1 and equals i Θ(Λ s T Z ) h s = i Θ(Λ s T Z ) h s − is∂∂ Φ < Nak . Write F := Λ s T Z and let θ F be the bilinear form on T Z ⊗ F associated to i Θ( F ) . Since therank of the bundle F is (cid:0) ns (cid:1) , formula (3.3) gives θ = − (cid:18) ns (cid:19) Tr F ( θ F ) h s ⊗ h s + θ F hs > Nak . We observe that θ is the curvature form associated to the Chern curvature tensor of thebundle F ⊗ (det F ∗ )( ns ) with the metric induced by h . The induced metric on det F ∗ equals(4.1) h det F ∗ = (det H ) − ( n − s − ) = (det H ) − ( n − s − ) e − ( n − s − ) Φ by the identity(4.2) det F = det Λ s T Z = (det
T Z )( n − s − )and so i Θ(det F ∗ ) h = i Θ(det F ∗ ) h + (cid:18) n − s − (cid:19) i∂∂ Φ . Then θ = θ ((det F ∗ )( ns )) h ⊗ h s + e s Φ θ ( F ) h + (cid:18)(cid:18) ns (cid:19)(cid:18) n − s − (cid:19) − s (cid:19) ∂∂ Φ ⊗ h s > Nak θ = θ ((det F ∗ )( ns )) h ⊗ h s + θ ( F ) h + (cid:18)(cid:18) ns (cid:19)(cid:18) n − s − (cid:19) − s (cid:19) ∂∂ Φ ⊗ h s > Nak . To complete the proof we view this expression as part of the curvature of the metric e − Ψ h s on Λ s T Z.
Let Φ := ( (cid:0) ns (cid:1)(cid:0) n − s − (cid:1) − s )Φ . Then the weight e − Φ gives the last term of θ. Observethat i Θ(det F ∗ ) h = i∂∂ log(det H ) − ( n − s − ) = i∂∂ log(det H )( n − s − ) . Write F = F ⊗ (det F ∗ )( ns ) ⊗ (det F )( ns ) = F ⊗ (det F ∗ )( ns ) ⊗ (det T Z )( n − s − )( ns )by invoking (4.2) and define the Nakano positive metric on (det T Z )( n − s − )( ns ) in the followingway. Let v i be smooth sections of det T Z, given by Proposition 2.1 in [Pre2], which areholomorphic to the degree l > k on π − ( S ) and such that they enerate the bundle det T Z on a neighbourhood of a ( U \ S ) , conic along a ( S ) . Let v be alocal holomorphic section, v i = α i v and defineΦ := log X h v i , v i i H = log h v, v i H + log X | α i | . Then the metric e − Ψ h s for Ψ = Φ + Φ + (cid:0) ns (cid:1)(cid:0) n − s − (cid:1) Φ has i (cid:18) ns (cid:19)(cid:18) n − s − (cid:19) ∂∂ log det H + i Θ(Λ s T Z ) ω + i (cid:18)(cid:18) ns (cid:19)(cid:18) n − s − (cid:19) − s (cid:19) ∂∂ Φ+ i∂∂ Φ + i (cid:18) ns (cid:19)(cid:18) n − s − (cid:19) ∂∂ log X | α i | as a curvature tensor. The first three terms give a Nakano positive curvature by (4.3) andthe last is also Nakano positive in a suitable conic neighbourhood because the negative partof ∂∂ log P | α i | is of the form C k w k l k z k + C k w k l + C k w k l + C k w k l − and its modulus decreases at least as k z k k on conic neighbourhoods k w k ≤ k z k k +2 (see[Pre2], p.14 for details). Because i∂∂ Φ is strictly plurisubharmonic with the rate of degener-acy at most k z k k (independent of the shape of the cone if it is sharp enough) it compensatesthe negativity of ∂∂ log P | α i | provided k > k. (cid:3) References [Dem] Demailly, J.- P., Complex analytic and algebraic geometry, version June 21, 2016.[Pre1] Prezelj, J., A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces,Trans. Amer. Math. Soc. 362 (2010), 4213-4228.[Pre2] Prezelj, J., Positivity of metrics over 1-convex manifolds, Internat. J. Math., 27, No.4 (2016)
Jasna Prezelj, Faculty of Mathematics and Physics, Department of Mathematics, Uni-versity of Ljubljana, Jadranska 21, SI-1000 Ljubljana, SloveniaFaculty of Mathematics, Natural Sciences and Information Technologies, University ofPrimorska, Glagoljaˇska 8, SI-6000 Koper, Slovenia
E-mail address : [email protected]@fmf.uni-lj.si