Curvature formula for direct images of twisted relative canonical bundles endowed with a singular metric
aa r X i v : . [ m a t h . C V ] F e b CURVATURE FORMULA FOR DIRECT IMAGES OF TWISTEDRELATIVE CANONICAL BUNDLES ENDOWED WITH A SINGULARMETRIC by Junyan Cao, Henri Guenancia & Mihai P˘aun
Dedicated to Ahmed Zeriahi, on the occasion of his retirement
Abstract . —
In this note, we obtain various formulas for the curvature of the L metricon the direct image of the relative canonical bundle twisted by a holomorphic line bundleendowed with a positively curved metric with analytic singularities, generalizing some ofBerndtsson’s seminal results in the smooth case. When the twist is assumed to be relativelybig, we further provide a very explicit lower bound for the curvature of the L metric. Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Set-up and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. A few technicalities about Poincaré type metrics . . . . . . . . . . . . . . . . . . . 64. Curvature formulas and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215. A lower bound for the curvature in case of a -relatively- big twist . . . 31References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1. Introduction
Let p : X → D be a smooth, proper fibration from a ( n + ) -dimensional Kählermanifold X onto the unit disk D ⊂ C , and let ( L , h L ) be a holomorphic line bundleendowed with a possibly singular hermitian metric h L assumed to be positively curved(i.e. when i Θ h L ( L ) > F : = p ⋆ (( K X / D + L ) ⊗ I ( h L )) endowed with the L metric h F are well-known, cf e.g. [ Ber09, PT18, P ˘au18, DNWZ20 ]among many others. Moreover, when h L is smooth , we have at hand explicit formulasobtained by Berndtsson [ Ber09, Ber11 ] that compute the curvature of the L metric onthe direct image sheaf above.In this article we are aiming at the generalisation of Berndtsson’s curvature formulasin case where the metric h L has relatively simple singularities, e.g. analytic singularities .This is partly motivated by the need to have an interpretation of the "flat directions" JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN in the curvature of ( F , h F ) in this context. Our main result in this direction states asfollows. Theorem A . —
Let p : X →
D and ( L , h L ) → X as above, and let u ∈ H ( D , F ) . Weassume that • The metric h L has analytic singularities and i Θ h L ( L ) > in the sense of currents. • The section u is flat with respect to h F .Set E : = { h L = ∞ } . Then, there exists a continuous L -integrable representative u of u definedon the restriction X ⋆ \ E of the family p to some punctured disk D ⋆ such that ¯ ∂ u dt (cid:12)(cid:12)(cid:12) X t \ E = for any t ∈ D ⋆ and (1.1) D ′ u = Θ h L ( L ) ∧ u = on X ⋆ \ E. Here X ⋆ : = p − ( D ⋆ ) and u is L with respect to h L and a Poincaré type metric cf.Section 3. By "punctured disk" in the previous statement we mean that D \ D ⋆ is a discrete set,possibly empty. By L , we mean locally L with respect to the base D ⋆ .The result we are next mentioning concerns the case of a twisting line bundle L whichis p -big. It is then expected that the strict positivity of ( L , h L ) is inducing strongerpositivity properties of the curvature of the direct image than in the general case ofa semi-positively curved L . This is confirmed by the following statement, which is aversion of [ Ber11 , Thm 1.2].
Theorem B . —
Let p : X →
D be a smooth projective fibration and let ( L , h L ) → X be a linebundle such that • h L has analytic singularities and i Θ h L ( L ) > in the sense of currents. • For any t ∈ D, the absolutely continuous part ω L : = ( i Θ h L ( L )) ac satisfies R X t ω nL > .Then there exists a punctured disk D ⋆ ⊂ D such that for any u ∈ H ( D , F ) we have thefollowing inequality (1.2) h Θ h F ( F ) u , u i t > c n Z X t c ( ω L ) u ∧ ue − φ L for any t ∈ D ⋆ . In the statement above we identify Θ h F ( F ) with an endomorphism of F by "dividing"with idt ∧ dt . Moreover, c n = ( − ) n is the usual unimodular constant. We denoteby c ( ω L ) : = ω n + L ω nL ∧ idt ∧ dt the geodesic curvature associated to ω L , cf. Definition 2.3 for aprecise definition in the degenerate case.Actually we can provide some details about the punctured disk D ⋆ in Theorem B.Under the hypothesis of this result, it turns out that the L metric h F is smooth in acomplement of a discrete subset of D . We will show that the formula (1.2) is valid forpoints t ∈ D in the neighborhood of which the metric h F is smooth, and such that F t = H ( X t , ( K X t + L ) ⊗ I ( h L | X t )) , cf Remark 5.5. URVATURE FORMULA IN A SINGULAR SETTING • Strategy of the proof
Roughly speaking, the idea of the proof of Theorem A and B respectively is as follows:we endow the complement
X \ E with a complete metric of Poincaré type and proceedby taking advantage of what is known in the compact case, combined with the exis-tence of families of cut-off functions specific to the complete setting. There are howeverquite a few difficulties along the way. Probably the most severe stems from the Hodgedecomposition in the complete case: the image of the usual operators ¯ ∂ and ¯ ∂ ⋆ maynot be closed. We show in Section 3.2 that at least in bi-degree ( n , 1 ) this is the case,cf. Theorem 3.6, as consequence of the fact that the background metric has Poincarésingularities.In order to construct the form u in Theorem A, we start with a representative of u givenby the contraction with the lifting V of ∂∂ t with respect to a Poincaré metric ω E . It turnsout that this specific representative has all the desired properties needed to fit into the L -theory. Then we "correct" it: this is possible by the flatness hypothesis, and it boilsdown to solving a fiberwise ¯ ∂ ⋆ -equation. It is both in the resolution of this equation aswell as in the study of the regularity of the resulting solution that Theorem 3.6 is used.Another important ingredient of the proof is Proposition 4.1, which gives a generalcurvature formula for ( F , h F ) when h L has e.g. analytic singularities. It provides arather wide generalisation of a result due to Berndtsson.As for Theorem B, the starting point is the fact that the positivity properties of ( L , h L ) allow us to construct a family of Poincaré metric ( ω ε ) ε > on X \ E . Then, the repre-sentatives u ε of u - obtained as above as the contraction with the lifting V ε of ∂∂ t withrespect to ω ε - enjoy a special property that allows us to extract the desired inequalityfrom the general curvature formula from Proposition 4.1 and a limiting argument when ε approaches zero. Although the use of this special representative goes back to Berndts-son, several new analytic inputs are required to deal with the present singular situation. • Organization of the paper ◦ In Section 2, we introduce our set-up, notation and main objects of study (the L metric on F , the geodesic curvature). ◦ In Section 3, we review two aspects of Poincaré metrics: first, the integrabilityproperties of representatives u of sections u of F constructed via such metrics(Lemma 3.3) and then, we investigate the closedness of the image of the operators¯ ∂ , ¯ ∂ ∗ on a hermitian line bundle with analytic singularities (Theorem 3.6). ◦ In Section 4, we establish a general curvature formula (Proposition 4.1). This al-lows us to find very special representatives of flat sections of F (Theorem 4.6),leading to the proof of Theorem A. ◦ In Section 5, we analyze the relatively big case in the "snc situation" (Theorem 5.1),from which we then deduce Theorem B . • Acknowledgements
H.G. has benefited from State aid managed by the ANR un-der the "PIA" program bearing the reference ANR-11-LABX-0040, in connection withthe research project HERMETIC. J.C. thanks the excellent working conditions provided
JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN by the IHES during the main part of the preparation of the article. M.P. gratefully ac-knowledge the support of DFG.
It is our privilege to dedicate this article to our friend and colleague Ahmed Zeriahi, with ouradmiration for his outstanding mathematical achievements and wishing him a very happy andactive retirement !
2. Set-up and notation
The set of assumptions we need for our results to hold is the following.
Set-up 2.1 . —
Let p : X → D be a smooth, proper fibration from a ( n + ) -dimensionalKähler manifold X onto the unit disk D ⊂ C , and let ( L , h L ) be a holomorphic linebundle endowed with a possibly singular hermitian metric h L .We assume that there exists a divisor E = E + · · · + E N whose support is contained inthe total space X of p . such that the following requirements are fulfilled.(A.1) For every t ∈ D the divisor E + X t has simple normal crossings. Let Ω ⊂ X be a coordinate subset on X . We take ( z , . . . . z n , t = z n + ) a coordinate systemon Ω such that the last one z n + corresponds to the map p itself and such that z . . . z p = E ∩ Ω .(A.2) The metric h L has generalised analytic singularities along E ; i.e. its local weights ϕ L on Ω can be written as ϕ L ≡ p ∑ i = a i log | z i | − ∑ I b I log φ I ( z ) − log (cid:0) ∏ i ∈ I | z i | k i (cid:1)! modulo C ∞ functions, where a i , b I are positive real numbers, k i are positive inte-gers and ( φ I ) I are smooth functions on Ω . The set of indexes in the second sumcoincides with the non-empty subsets of {
1, . . . , p } .(A.3) The Chern curvature of ( L , h L ) satisfies i Θ h L ( L ) > X .We then set F : = p ∗ (( K X / D + L ) ⊗ I ( h L )) and assume that that this vector bundle on D has positive rank. As a consequence ofthe previous requirements (A.1)-(A.3), we have the following statement. Lemma 2.2 . —
Under the assumptions (A.1)-(A.3), we have F t = H ( X t , ( K X t + L ) ⊗ I ( h L | X t )) for every t ∈ D. Moreover, the canonical L metric (cf. Notation 2.4) on F is non-singular.Proof . — We first remark that F is indeed locally free given that it is torsion-free and D ⊂ C is a disk. URVATURE FORMULA IN A SINGULAR SETTING The fibers of F are indeed identified with H ( X t , ( K X t + L ) ⊗ I ( h L | X t )) because ofthe transversality hypothesis (A.1), combined with the type of singularities we are al-lowing for h L in (A.2). The point is that a holomorphic function f defined on the co-ordinate subset Ω belongs to I ( h L ) exactly when the restriction f | Ω ∩ X t belongs to theideal I ( h L | X t ) . On the other hand the Kähler version of Ohsawa-Takegoshi theorem[ Cao17 ] implies that any element of H ( X t , ( K X t + L ) ⊗ I ( h L | X t )) extends to X –it is atthis point that the hypothesis (A.3) plays a crucial role.Concerning the smoothness of the L -metric on F , we can use partitions of unityto reduce to checking that integrals of the form R Ω ∩ X t | f t | e − ϕ L vary smoothly with t ,where f t = f | X t for some f ∈ I ( h L ) | Ω and ϕ L is given by the expression in (A.2). Now itis clear there that all derivatives in the t , ¯ t variables of ϕ L are bounded, so that the resultfollows from general smoothness results for integrals depending on a parameter. • A few comments about the conditions (A.1)-(A.2).
The point we want to make here is that the transversality requirements in (A.1)-(A.2)can be obtained starting from a quite general context.We consider p : X → D a proper fibration from a ( n + ) -dimensional Kähler mani-fold X onto the unit disk D ⊂ C , and let ( L , h L ) be a holomorphic line bundle endowedwith a possibly singular hermitian metric h L . We assume that (A.3) holds true, and thatthe singularities of h L are of the form(2.1) ϕ L ≡ p ∑ i = a i log | f i | − p ∑ i = b i log (cid:0) τ i − log | g i | (cid:1) modulo C ∞ functions, where a i , b i are positive real numbers, f i , g i are holomorphic, and τ i are smooth.If (A.1)-(A.2) are not satisfied for p : X → D , then one can consider a log resolution π : X ′ → X of ( X , I Z ) where Z is the singular set of h L . Set p ′ : = p ◦ π : X ′ → D , ( L ′ , h L ′ ) : = ( π ∗ L , π ∗ h L ) , E is the reduced divisor induced by π − ( Z ) . It is immediatethat π ∗ (( K X ′ / D + L ′ ) ⊗ I ( h L ′ )) = ( K X / D + L ) ⊗ I ( h L ) so that, in particular, π ∗ (( K X ′ / D + L ′ ) ⊗ I ( h L ′ )) = F .The map p ′ may not be smooth anymore (e.g. some components of E may be irre-ducible components of fibers of p ′ ). Set D reg to be the Zariski open set of regular valuesof p ′ , D : = D reg ∩ D , X : = p ′− ( D ) , p : = p ′ | X , ( L , h L ) := ( L ′ , h L ′ ) | X . Then, thetriplet ( p , L , h L ) satisfies the assumptions (A.1)-(A.3).In conclusion, starting with a map p as above and singular metric h L as in (2.1), we canuse our results on the family p restricted to some punctured disk D ⋆ ⊂ D . • The geodesic curvature in a degenerate setting.
Let p : X → D be a smooth, proper fibration where X is a Kähler manifold of dimen-sion n +
1. Let ω be a closed positive (
1, 1 ) -current on X such that ω is smooth on anon-empty Zariski open subset X ◦ ⊂ X . Let ω X be a Kähler metric on X . JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
Definition 2.3 . —
The geodesic curvature c ( ω ) of ω on X ◦ is defined byc ( ω ) : = lim ε → c ( ω + εω X ) = lim ε → ( ω + εω X ) n + ( ω + εω X ) n ∧ idt ∧ d ¯ t .A few explanations are in order.First, if ω is relatively Kähler on X , one recovers the usual definition.Next, it is easy to observe that c ( ω + εω X ) = k dt k ω + εω X . In particular, that non-negative quantity is non-increasing when ε decreases towards 0, hence it admits a limit.By the same token, one can see that the limit is independent of the choice of the Kählermetric ω X .Finally, if t ∈ D is such that X ◦ t : = X t ∩ X ◦ is dense in X t , then one defines c ( ω ) on the whole X t by extending it by zero across X t \ X ◦ t . Note that if the absolutelycontinuous part of ω satisfies ω ac C ω X on X t for some constant C >
0, then c ( ω ) is a bounded function on X t (this follows e.g. from the inequality c ( ω ) k ∂∂ t k ω for aset of coordinates ( z , . . . , z n , z n + = t ) such that p ( z ) = t ). In particular, the integral R X t c ( ω ) ω n X is finite. Notation 2.4 . —
In the Set-up 2.1 above: · We set X ◦ : = X \ E , X ◦ t : = X t ∩ X ◦ , L t : = L | X t , h L t : = h L | X t . · We use interchangeably h L and e − φ L ; when working in a trivializing chart of L , wewill denote by ϕ L the local weight of h L . The (
1, 0 ) -part of the Chern connection of ( L , h L ) over X \ E is denoted by D ′ . · Under assumption (A.1), we will write E : = ∑ Ni = E i for the decomposition of E intoits (smooth) irreducible components. Next, let s i be a section of O X ( E i ) that cuts out E i , and let h i be a smooth hermitian metric on O X ( E i ) . In the following, | s i | stands for | s i | h i , and we assume that | s i | < e − . · We will interchangeably denote by k · k or h F the L metric on F ; i.e. if u ∈ F t = H ( X t , ( K X t + L t ) ⊗ I ( h L t )) , then k u k : = c n R X t u ∧ ¯ ue − φ Lt with c n = ( − ) n . Lemma2.2 ensures that the L metric is smooth on D . We denote by ∇ the (
1, 0 ) part of theChern connection of ( F , k · k ) on D .
3. A few technicalities about Poincaré type metrics
Throughout this section we adopt Set-up 2.1. Let ω be a fixed Kähler metric on X , andlet ω E : = ω + dd c h − N ∑ i = log log 1 | s i | i on X ◦ be a metric with Poincaré singularities along E . Thanks to (A.1) we infer that ω E | X ◦ t is acomplete Kähler metric on X ◦ t with Poincaré singularities along E ∩ X t for each t ∈ D .In the next subsections we will be concerned with the following two main themes.Let V be the horizontal lift of ∂∂ t with respect to the Poincaré-type metric ω E . Weestimate the size of its coefficients near the singularity divisor E , and then show that therepresentatives of direct images constructed by using V have the expected L properties URVATURE FORMULA IN A SINGULAR SETTING allowing us to use them in the computation of the curvature of F . This is the contentof Subsection 3.1.In Subsection 3.2 we establish a few important properties of the L -Hodge decompo-sition for ( n , 1 ) -forms with values in ( L , h L ) , where the background metric is ( X , ω E ) .The main result here is that the image of ¯ ∂ ⋆ is closed, cf Theorem 3.6, a very useful result per se and for the next sections of this paper as well. We choose local coordinates ( z , . . . , z n , z n + = t ) on X such that p ( z , t ) = t –as in (A.1)– and in which ω E is locallygiven by ω E = g t ¯ t idt ∧ d ¯ t + ∑ α g α ¯ t idz α ∧ d ¯ t + ∑ α g t ¯ α idt ∧ d ¯ z α + ∑ α , β g α ¯ β idz α ∧ d ¯ z β By the estimates in [
Gue14 , §4.2] the coefficients of ω E are can be written as follows(3.1) g j ¯ k = g j ¯ k + δ j · δ jk | z j | log | z j | + δ j A j z j log | z j | + δ k B k ¯ z k log | z k | + p ∑ ℓ = C ℓ log | z ℓ | where A j , B k , C ℓ , g j ¯ k are smooth functions on Ω . We use the notation δ j = δ j ∈{ p } and δ ij is the usual Kronecker symbol.In order to present the computations to follow in a reasonably simple way, we introducefor i =
1, . . . , n the functions(3.2) f i ( z ) : = ( w i : = z i log | z i | if i ∈ {
1, . . . , p } f i ( z ) = δ i w i + − δ i more concisely. The coefficients of the metric ω E can be writtenas(3.3) g α ¯ β = f α ¯ f β Ψ α ¯ β ( z ′′ , w , ρ ) .The notations we are using in (3.3) are:(a) Ψ α ¯ β is a smooth function defined in the neighborhood of 0 ∈ C n − p × C p × C p .(b) w : = ( w , . . . , w p ) (cf. (3.2)) and z ′′ : = ( z p + , . . . , z n ) .(c) For i =
1, . . . , p we introduce ρ i : = | z i | .Given that ω E is a metric, the functions Ψ α ¯ β are not arbitrary (since the matrix ( Ψ α ¯ β ) α ¯ β is definite positive at each point). We simply want to emphasize in (3.3) the generalshape of the coefficients, which will be useful in the statement that follows. Lemma 3.1 . —
The following estimates hold true:(i) det ( g ) = n ∏ α = | f α ( z ) | − (cid:0) + Ψ ( z ′′ , w , ρ ) (cid:1) , where the "1" inside the parentheses means astrictly positive constant, and Ψ is smooth such that Ψ ( ) = . JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN (ii) For each pair of indexes α , β we haveg ¯ βα = f α ( z ) ¯ f β ( z ) Ψ α ¯ β ( z ′′ , w , ρ ) , where g βα are the coefficients of the inverse of ( g α ¯ β ) .Proof . — Both statements above are obtained by a direct calculation, using the expres-sion (3.3) of the coefficients. We skip the straightforward details.Thanks to Lemma 3.1 it is easy to infer the following useful estimates: for each set ofindexes α , β , q , r we have ∂ g βα ∂ z q ( z ) = (cid:18) δ α q δ α ( + log | z α | ) ¯ f β ( z ) + δ β q δ β f α ( z ) z q z q (cid:19) Ψ α ¯ β ( z ′′ , w , ρ ) (3.4) + ( + δ q ( log | z q | − )) f α ( z ) ¯ f β ( z ) Ψ α ¯ β ( z ′′ , w , ρ ) + δ q z q log | z q | f α ( z ) ¯ f β ( z ) Ψ α ¯ β ( z ′′ , w , ρ ) as well as ∂ g βα ∂ z q ∂ z r ( z ) = O ( ) ∂∂ z r (cid:18) δ α q δ α ( + log | z α | ) ¯ f β ( z ) + δ β q δ β f α ( z ) z q z q (cid:19) (3.5) + O ( ) (cid:18) + δ r z r log | z r | (cid:19) (cid:18) δ α q δ α ( + log | z α | ) ¯ f β ( z ) + δ β q δ β f α ( z ) z q z q (cid:19) + O ( ) δ q δ rq | z q | log | z q | f α ( z ) ¯ f β ( z )+ O ( ) (cid:18) + δ q z q log | z q | (cid:19) (cid:18) + δ r z r log | z r | (cid:19) f α ( z ) ¯ f β ( z )+ O ( ) (cid:18) + δ q z q log | z q | (cid:19) (cid:18) δ β r δ β ( + log | z β | ) f α ( z ) + δ α r δ α ¯ f β ( z ) z r z r (cid:19) Again, in the relations (3.4)–(3.5) we are using Ψ αβ and O ( ) as generic notation, thesefunctions are allowed to change from one line to another, the point is that they are ofthe same type. The verification of formula (3.5) is immediate, one simply takes thederivative in (3.4). Horizontal lift, µ and η . — One can define the lift V of ∂∂ t with respect to thePoincaré type metric ω E , cf. [ Siu86, Ber11, Sch12 ]. It is a vector field of type (
1, 0 ) on X ◦ such that dp maps it to ∂∂ t pointwise on X ◦ and which is orthogonal to T X t forany t . In local coordinates, one has the following formula(3.6) V = ∂∂ t − ∑ α , β g ¯ βα g t ¯ β ∂∂ z α . URVATURE FORMULA IN A SINGULAR SETTING Let u be a holomorphic section of p ∗ ( O ( K X / D + L ) ⊗ I ( h L )) . One can choose an arbi-trary representative U of u , this is an L -valued ( n , 0 ) -form on X which coincides with u t on X t . Now, let(3.7) u : = V y ( dt ∧ U ) .One can write locally (using the previous system of coordinates): U ∧ dt = a ( z , t ) dt ∧ dz ∧ . . . ∧ dz n where a ( z , t ) is holomorphic with values in L . We have an explicit formula:(3.8) u = a ( z , t ) (cid:0) dz ∧ . . . ∧ dz n − ∑ α , β ( − ) α g ¯ βα g t ¯ β dt ∧ dz ∧ . . . ∧ c dz α ∧ . . . ∧ dz n (cid:1) .By construction, we have(3.9) dt ∧ u = dt ∧ U .Therefore, although u is only well defined on X ◦ , u ∧ dt can be extended as a smoothform on X . Remark 3.2 . —
The representative u in (3.8) has the following interesting property(3.10) u ∧ ω E | Ω = a ( z , t ) g t ¯ t dt ∧ dt ∧ dz ∧ . . . ∧ dz n .In particular, we have u ∧ ω E dt (cid:12)(cid:12)(cid:12) X t = t .Moreover, as U is a smooth representative of u on X and u is a holomorphic form, wehave ¯ ∂ ( U ∧ dt ) = X . Combining with (3.9), we know that ¯ ∂ u ∧ dt = X ◦ . Asa consequence, we can find a smooth ( n −
1, 1 ) -form η on X ◦ such that(3.11) ¯ ∂ u = dt ∧ η on X ◦ and one has D ′ u = loc ∂ u − ∂ϕ L ∧ u (3.12) = dt ∧ µ for some ( n , 0 ) -form µ on X ◦ . Estimates for η and µ . — In this section our goal is to establish the following state-ment.
Lemma 3.3 . —
We consider the form η | X t , µ | X t induced by the representative u constructedin the previous section in (3.11) and (3.12) respectively. Then η | X t , µ | X t as well as ¯ ∂µ | X t are L with respect to ( ω E , e − φ L ) .Moreover, u , η , µ and ¯ ∂µ are also in L ( X ◦ ) with respect to ( ω E , e − φ L ) , say up to shrinkingD.Proof . — This is routine: we can easily obtain the explicit expression of η and µ , andthen we simply evaluate their respective L norms by using the estimates (3.4)–(3.5).We detail to some extent the calculations next. JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
First of all, we have η | X t = a ( z , t ) ∑ α , β , r ( − ) α + ( g ¯ βα ,¯ r g t ¯ β + g ¯ βα g t ¯ β ,¯ r ) dz ∧ . . . ∧ c dz α ∧ . . . ∧ dz n ∧ d ¯ z r .where we use the notation g ¯ βα ,¯ r : = ∂ g ¯ βα ∂ z r .We consider first the quantity(3.13) (cid:12)(cid:12)(cid:12) g ¯ βα ,¯ r g t ¯ β (cid:12)(cid:12)(cid:12) .Thanks to the equality (3.4), up to a constant it is smaller than(3.14) 1 | f β | δ α r δ α log | z α | | f β | + δ β r δ β | f α | + (cid:16) − δ r + δ r | z r | log | z r | (cid:17) | f α f β | ! which simplifies to | f α | + δ r h δ α r log | z r | + δ β r | f α | | f r | + (cid:16) | z r | log | z r | − (cid:17) | f α | i .Since (cid:12)(cid:12)(cid:12) dz ∧ . . . ∧ c dz α ∧ . . . ∧ dz n ∧ d ¯ z r (cid:12)(cid:12)(cid:12) ω E dV ω E . | f r | | f α | dV ω we eventually find(3.15) (cid:12)(cid:12)(cid:12) g ¯ βα ,¯ r g t ¯ β (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) dz ∧ . . . ∧ c dz α ∧ . . . ∧ dz n ∧ d ¯ z r (cid:12)(cid:12)(cid:12) ω E dV ω E C ( + δ r δ α r log | z r | ) dV ω .The term(3.16) (cid:12)(cid:12)(cid:12) g ¯ βα g t ¯ β ,¯ r (cid:12)(cid:12)(cid:12) is bounded by(3.17) δ β r δ β ( + log | z β | ) | f α | | f β | + (cid:18) − δ r + δ r | w r | (cid:19) | f α | and we see that the same thing as before occurs, i.e.(3.18) (cid:12)(cid:12)(cid:12) g ¯ βα g t ¯ β ,¯ r (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) dz ∧ . . . ∧ c dz α ∧ . . . ∧ dz n ∧ d ¯ z r (cid:12)(cid:12)(cid:12) ω E dV ω E C ( + δ r log | z r | ) dV ω .Thus, the restriction of η to any fiber of the family p : X → D is L with respect to ( ω E , e − φ L ) since the holomorphic function a ( z , t ) belongs to the multiplier ideal sheafdefined by h L . Indeed, setting ν : = | a | dV ω , one has e − ϕ L ∈ L + ε ( ν ) for some ε > ϕ L has analytic singularities and e − ϕ L ∈ L ( ν ) ) while log | z r | ∈ L p ( ν ) for all p >
0, so that Hölder inequality shows the claim.
URVATURE FORMULA IN A SINGULAR SETTING The local expression of the form µ | X t is obtained by restricting D ′ u dt to the fiber X t ; itreads as µ | X t dz = a , t − a ∑ α , β ( g ¯ βα , α g t ¯ β + g ¯ βα g t ¯ β , α ) − ∑ α , β a , α g ¯ βα g t ¯ β (3.19) − a ( z , t ) ϕ L , t + a ( z , t ) ∑ α , β ϕ L , α g ¯ βα g t ¯ β where dz : = dz ∧ · · · ∧ dz n .By our transversality conditions, the function a , t is still L with respect to h L . The term g ¯ βα , α g t ¯ β + g ¯ βα g t ¯ β , α is treated as we did for (3.13) and (3.16), with the exception that theindexes r and β coincide (and the type of the form is different). We have up to someconstant (cid:12)(cid:12)(cid:12) g ¯ βα , α g t ¯ β (cid:12)(cid:12)(cid:12) ( − log | z α | ) | f β | + δ α β δ α | f α | + h ( − δ α ) + δ α | z α | log | z α | i · | f α f β | ! · | f β | (3.20) . − log | z α | and we can bound this term as before. The second term satisfies g ¯ βα g t ¯ β , α = g ¯ βα g α ¯ β , t since ω E is Kähler, hence it is bounded.Next, we have (cid:12)(cid:12)(cid:12) g ¯ βα g t ¯ β (cid:12)(cid:12)(cid:12) C | z α | log | z α | , and also that(3.21) | a , α | | z α | log | z α | e − ϕ L = O ( | a | log | z α | e − ϕ L ) ∈ L for any α =
1, . . . , p again by the transversality/ L conditions we impose to a ( z , t ) , sothe third term in (3.19) is in L . A similar argument applies to the second line of (3.19).The last part of the proof of our lemma concerns ¯ ∂µ ; the computations are using (3.5).We will only discuss the term(3.22) ∂∂ z r (cid:16) g ¯ βα , α g t ¯ β + g ¯ βα g t ¯ β , α (cid:17) dz ∧ dz r since for (3.22) the computations are the most involved. The reason why we are able toconserve the L property is that the partial derivative with respect to z r will induce anew term of order O ( | z r | ) if r p , and its square will be compensated by | dz r | ω E . Asfor the computations: the singularities induced by g ¯ βα , α ¯ r g t ¯ β are bounded by the followingquantity (cid:12)(cid:12)(cid:12) g ¯ βα , α ¯ r g t ¯ β (cid:12)(cid:12)(cid:12) δ α δ r α | z r | + δ α δ β r | f β | log 1 | z α | log 1 | z β | + δ α δ r αβ | f β || z α | (3.23) + δ r | w r | (cid:18) δ α log 1 | z α | + δ α δ αβ (cid:19) + δ α δ r | w α w r | | f α | + δ α | w α | (cid:18) δ β δ β r log 1 | z β | | f α || f β | + δ α δ α r (cid:19) JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN from which we see that the first part of (3.22) is L . The remaining terms are(3.24) (cid:16) g ¯ βα , α g t ¯ β ,¯ r + g ¯ βα ,¯ r g t ¯ β , α + g ¯ βα g t ¯ β , α ¯ r (cid:17) dz ∧ dz r for which one could use the fact that the metric ω E is Kähler and so we have(3.25) g t ¯ β , α = g α ¯ β , t , g t ¯ β , α ¯ r = g α ¯ β , t ¯ r .The equalities (3.25) are simplifying a bit the calculations, since the derivative withrespect to t does not increase at all the order of the singularity.For the first term of (3.24), we have, up to a multiplicative constant (cid:12)(cid:12)(cid:12) g ¯ βα , α g t ¯ β ,¯ r (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ α ( − log | z α | ) ¯ f β + δ α δ αβ f α + ( − δ α ) + δ α z α log | z α | ! f α ¯ f β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ r δ r β | z r | log | z r | + f β + δ r ¯ z r log | z r | !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ( − δ α log | z α | ) · (cid:18) + δ r | z r | ( − log | z r | ) ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) In particular, we get (cid:12)(cid:12)(cid:12) g ¯ βα , α g t ¯ β ,¯ r (cid:12)(cid:12)(cid:12) · | d ¯ z r | ω E . − δ α log | z α | and we are done with this term as before.For the second term of (3.24), we have, using (3.25) (cid:12)(cid:12)(cid:12) g ¯ βα ,¯ r g α ¯ β , t (cid:12)(cid:12)(cid:12) | f α f β | · δ α r δ α ( − log | z α | ) | f β | + δ β r δ β | f α | + h ( − δ r ) + δ r ( − log | z r | + | z r | log | z r | i | f α f β | ! + δ r − log | z r | | z r | ( − log | z r | ) . In particular, we get (cid:12)(cid:12)(cid:12) g ¯ βα ,¯ r g α ¯ β , t (cid:12)(cid:12)(cid:12) · | d ¯ z r | ω E . − δ r log | z r | and we are done.As for the last term of (3.24), we use (3.25) and (3.5) to see that the expansion of g ¯ βα g t ¯ β , α ¯ r will only involve terms like ψ α ¯ β ,¯ r , ∂ ¯ r f α f α , ∂ ¯ r ¯ f β ¯ f β which are respectively of order δ r z r log | z r | , δ r δ α r z r log | z r | , δ r δ β r ¯ z r . URVATURE FORMULA IN A SINGULAR SETTING All in all, we find (cid:12)(cid:12)(cid:12) g ¯ βα g t ¯ β , α ¯ r (cid:12)(cid:12)(cid:12) · | d ¯ z r | ω E . − δ r log | z r | and this is the end of the main part of the proof.The integrability of u , η , µ on X ◦ follows directly from the estimates we have obtainedabove. Concerning ¯ ∂µ there is one additional term given by ∂∂ t of the expression in(3.19). This is however harmless: given the shape of the coefficients ( g αβ ) (i.e. thetransversality conditions), the additional anti-holomorphic derivative with respect to t induces no further singularity and the estimates e.g. for the term ∂∂ t (cid:16) g ¯ βα , α g t ¯ β (cid:17) will be completely identical to those already obtained g ¯ βα , α g t ¯ β . We leave the details tothe interested reader. Remark 3.4 . —
Using quasi-coordinates adapted to the Poincaré metric ω E (cf. e.g.[ CY75, Kob84, TY87 ]), we can prove easily that η and its derivatives are in L . However,that argument cannot be applied to µ because of the singularity in the Chern connectionof ( L , h L ) . L Hodge theory. —
We recall briefly a few results of L -Hodge theory for a complete manifold endowed with a Poincaré type metric, followingclosely [ CP20 ]. We are in the following setting.Let X be a n -dimensional compact Kähler manifold, and let ( L , h L ) be a line bundleendowed with a (singular) metric h L = e − φ L such that • h L has analytic singularities; • Its Chern curvature satisfies i Θ h L ( L ) > π : b X → X of X such that the support of the singularitiesof ϕ L ◦ π is a simple normal crossing divisor E . As usual, we can construct π such thatits restriction to b X \ E is an biholomorphism. Then(3.26) ϕ L ◦ π | Ω ≡ p ∑ α = e α log | z α | modulo a smooth function. Here Ω ⊂ b X is a coordinate chart, and ( z α ) α = n arecoordinates such that E ∩ Ω = ( z . . . z p = ) .Let b ω E be a complete Kähler metric on b X \ E , with Poincaré singularities along E , andlet(3.27) ω E : = π ⋆ ( b ω E ) be the direct image metric. We note that in this way ( X ◦ , ω E ) becomes a completeKähler manifold, where X ◦ : = X \ ( h L = ∞ ) . JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
Remark 3.5 . — If u is a L -valued ( p , 0 ) -form on X ◦ which is L with respect to ω E ,then it is also L with respect to an euclidean metric on ˆ X (or X , too).Therefore, if u is holomorphic, then it extends holomorphically to X and more generally any smoothcompactification of X ◦ .The main goal of this section is to establish the following decomposition theorem,which is a slight generalization of the corresponding result in [ CP20 ]. Theorem 3.6 . —
Consider a line bundle ( L , h L ) → X endowed with a metric h L with analyticsingularities, as well as the corresponding complete Kähler manifold ( X ◦ , ω E ) , cf. (3.27) . Ifi Θ h L ( L ) > on X, we have the following Hodge decompositionL n ,1 ( X ◦ , L ) = H n ,1 ( X ◦ , L ) ⊕ Im ¯ ∂ ⊕ Im ¯ ∂ ⋆ . Here H n ,1 ( X ◦ , L ) is the space of L ∆ ′′ -harmonic ( n , 1 ) -forms. The proof follows closely the aforementioned reference, in which the case Θ h L ( L ) = Lemma 3.7 . —
There exists a family of smooth functions ( µ ε ) ε > with the following properties. (a) For each ε > , the function µ ε has compact support in X ◦ , and µ ε . (b) The sets ( µ ε = ) are providing an exhaustion of X ◦ . (c) There exists a positive constant C > independent of ε such that we have sup X ◦ (cid:0) | ∂µ ε | ω E + | ∂ ¯ ∂µ ε | ω E (cid:1) C .We have also the Poincaré type inequality for the ¯ ∂ -operator acting on ( p , 0 ) -forms. Proposition 3.8 . — [ CP20 ] Let ( Ω j ) j = N be a finite union of coordinate sets of b X coveringE, and let b U be any open subset contained in their union and U : = π ( b U ) . Let τ be a ( p , 0 ) -formwith compact support in a set U \ π ( E ) ⊂ X and values in ( L , h L ) . Then we have (3.28) 1 C Z U | τ | ω E e − φ L dV ω E Z U | ¯ ∂τ | ω E e − φ L dV ω E where C is a positive numerical constant. We emphasize that the constant C in (3.28) only depends on the distortion between themodel Poincaré metric on Ω j with singularities on E and the global metric b ω E restrictedto Ω j . Another important observation is that by using the cut-off function µ ε in Lemma3.7, we infer that (3.28) holds in fact for any L -bounded form with compact support in U . • Quick recap around the Bochner-Kodaira-Nakano formula.
We recall the following formula, which is central in complex differential geometry(3.29) ∆ ′′ = ∆ ′ + [ i Θ h L ( L ) , Λ ω E ] URVATURE FORMULA IN A SINGULAR SETTING where ∆ ′′ = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ and ∆ ′ = D ′ D ′∗ + D ′∗ D ′ where D ′ is the (
1, 0 ) -part of the Chernconnection on ( L , h L ) . Let us also recall the well-known fact that the self-adjoint opera-tor A : = [ i Θ h L ( L ) , Λ ω E ] is semi-positive when acting on ( n , q ) forms, for any 0 q n as long as i Θ h L ( L ) > L -integrable form u with values in L of any type in the domains of ∆ ′ and ∆ ′′ , we have(3.30) k ¯ ∂ u k L + k ¯ ∂ ∗ u k L = k D ′ u k L + k D ′∗ u k L + Z X ◦ h Au , u i dV ω E where k · k L (resp. h , i ) denotes the L -norm (resp. pointwise hermitian product) takenwith respect to ( h L , ω E ) . Let ⋆ : Λ p , q T ∗ X ◦ → Λ n − q , n − p T ∗ X ◦ be the Hodge star with respectto ω E ; we introduce for any integer 0 p n the space(3.31) H ( p ) : = { F ∈ H ( X ◦ , Ω pX ◦ ⊗ L ) ∩ L ; Z X ◦ h A ⋆ F , ⋆ F i dV ω E = } and we can observe by Bochner formula that for a L integrable, L -valued ( p , 0 ) -form F , one has(3.32) ∆ ′′ ( ⋆ F ) = ⇐⇒ ∆ ′ ( ⋆ F ) = Z X ◦ h A ⋆ F , ⋆ F i dV ω E = ⇐⇒ F ∈ H ( p ) .The proof of Theorem 3.6, which we give below, makes use of the following propositionwhich is the ¯ ∂ -version of the Poincaré inequality established in [ Auv17 ]. Proposition 3.9 . —
Let p n be an integer. There exists a positive constant C > such thatthe following inequality holds (3.33) Z X ◦ | u | ω E e − φ dV ω E C (cid:18) Z X ◦ | ¯ ∂ u | ω E e − φ dV ω E + Z X ◦ h A ⋆ u , ⋆ u i dV ω E (cid:19) for any L-valued form u of type ( p , 0 ) which belongs to the domain of ¯ ∂ and which is orthogonalto the space H ( p ) defined by (3.31) . Here ⋆ is the Hodge star operator with respect to the metric ω E .Proof of Proposition 3.9 . — If a positive constant as in (3.33) does not exists, then weobtain a sequence u j of L -valued forms of type ( p , 0 ) orthogonal to H ( p ) such that(3.34) Z X ◦ | u j | ω E e − φ dV ω E =
1, lim j Z X ◦ | ¯ ∂ u j | ω E e − φ L dV ω E = j Z X ◦ h A ⋆ u j , ⋆ u j i dV ω E = u ∞ of ( u j ) is holomorphic and belongs to H ( p ) . On theother hand, each u j is perpendicular to H ( p ) , so it follows that u ∞ is equal to zero.Let us first show that the weak convergence u i ⇀ u ∞ also takes places in L ( X ◦ ) . Tothat purpose, let us pick a small Stein open subset U ⋐ X ◦ . By solving the ¯ ∂ -equation U , we can find w j such that ¯ ∂ w j = ¯ ∂ u j on U and R U | w j | →
0. Therefore u j − w j JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN is holomorphic on U and converges weakly, hence strongly to u ∞ | U . In particular u j converges to u ∞ in L on U . As u ∞ =
0, we have(3.35) u j | K → L for any compact subset K ⊂ X ◦ .The last step in the proof is to notice that the considerations above contradict the factthat the L norm of each u j is equal to one. This is not quite immediate, but is preciselyas the end of the proof of Lemma 1.10 in [ Auv17 ], so we will not reproduce it here.The idea is however very clear: in the notation of Proposition 3.8, we choose V smallenough so that it admits a cut-off function χ with small gradient with respect to ω E .Then, we decompose each u j as u j = χ u j + ( − χ ) u j . Then the L norm of χ u j is smallby (3.35). The L norm of ( − χ ) u j is equally small by (3.28), and this is how we reacha contradiction.We have the following direct consequences of Proposition 3.9. Corollary 3.10 . —
There exists a positive constant C > such that the following inequalityholds (3.36) Z X ◦ | u | ω E e − φ dV ω E C (cid:18) Z X ◦ | ¯ ∂ u | ω E e − φ dV ω E (cid:19) for any L-valued form u of type ( n , 0 ) which belongs to the domain of ¯ ∂ and which is orthogonalto the kernel of ¯ ∂ .Proof . — This follows immediately from Proposition 3.9 combined with the observa-tion that the curvature operator A is equal to zero in bi-degree ( n , 0 ) .The next statement shows that in bi-degree ( n , 2 ) the image of the operator ¯ ∂ ⋆ is closed. Corollary 3.11 . —
There exists a positive constant C > such that the following holds true.Let v be a L-valued form of type ( n , 2 ) . We assume that v is L , in the domain of ¯ ∂ and orthog-onal to the kernel of the operator ¯ ∂ ⋆ . Then we have (3.37) Z X ◦ | v | ω E e − φ L dV ω E C Z X ◦ | ¯ ∂ ⋆ v | ω E e − φ L dV ω E . Proof . — Let us first observe that the Hodge star u : = ⋆ v , of type ( n −
2, 0 ) , is orthogo-nal to H ( n − ) . This can be seen as follows. Let us pick F ∈ H ( n − ) ; it follows from (3.32)that we have ¯ ∂ ⋆ ( ⋆ F ) =
0. In other words, ⋆ F ∈ Ker ¯ ∂ ⋆ . We thus have Z X ◦ h u , F i dV ω E = Z X ◦ h v , ⋆ F i dV ω E = v and using the facts that ¯ ∂ v = v is orthog-onal to Ker ¯ ∂ ∗ ) and that ¯ ∂ ∗ u = k ¯ ∂ ∗ v k L = k ¯ ∂ u k L + Z X ◦ h A ⋆ u , ⋆ u i dV ω E This proves the corollary by applying Proposition 3.9.
URVATURE FORMULA IN A SINGULAR SETTING We discuss next the relative version of the previous estimates. Let p : X → D and ( L , h L ) be the family of manifolds and the line bundle, respectively fixed in the previoussection. We assume that(3.39) D ∋ t dim (cid:0) Ker ( ∆ ′′ t ) (cid:1) is constantwhere the Laplace operator ∆ ′′ t is the one acting on L ( n , 1 ) -forms with respect to ( ω E , h L ) .The next result is a consequence of the proof of Proposition 3.9. Corollary 3.12 . —
Under the additional assumptions (3.39) and (A.1) , there exists a constantC > independent of t such that (3.40) Z X ◦ t | u | ω E e − φ dV ω E C (cid:18) Z X ◦ t | ¯ ∂ u | ω E e − φ dV ω E + Z X ◦ t h A ⋆ u , ⋆ u i dV ω E (cid:19) for all L forms u orthogonal to the space H ( p ) t defined in (3.31) on the fiber X t . Here the constant C is uniform in the sense that for any subset U of compact supportin D , we can find a constant C depending on U such that (3.40) is satisfied for any t ∈ UProof . — We first show that every form F on the central fiber which is in the space H ( p ) can be written as limit of F t i ∈ H ( p ) t i . This is of course well-known in the compact case,but we include a proof here since we could not find a reference fitting in our context.Let ( F t ) t ∈ D ⋆ any family of L -valued holomorphic p -forms on the fibers above thepointed disk D ⋆ such that(3.41) Z X ◦ t | F t | ω E e − φ L dV ω E = F ∞ on the central fiber X , but in principle itcould happen that F ∞ ≡ Ω for X (3.42) F t | Ω = ∑ f I dz I ⊗ e L ,where the coefficients f I are holomorphic, and of course depending on t . We can as-sume that the multiplier ideal sheaf of h L is trivial, given the transversality conditionsthat we have imposed (we can simply divide F t with the corresponding sections). Ifthe weak limit of F t is zero, we can certainly extract a limit in strong sense, because the L norm with respect to a smooth metric is smaller than the L norm with respect toPoincaré metric, cf. also Remark 3.5.In this case, the sup norm of the coefficients f I above converges to zero as t → t ≪ ( F t , j ) of the space H ( p ) t (this is obtained by the ⋆ t ofan orthonormal basis for the Ker ( ∆ ′′ t ) , for example). The previous considerations willallow us to construct by extraction an orthonormal family ( F ∞ , j ) in H ( p ) ; this will be abasis because of dimension considerations. JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
We argue by contradiction and assume that the smallest constant C t for which (3.33)holds true to for the fiber X t tends to infinity when t →
0. Then we get u i on X t i suchthat u i is orthogonal to the space H ( p ) t i and such that(3.43) Z X ti | u i | ω E e − φ L dV ω E =
1, lim i Z X ti | ¯ ∂ u i | ω E e − φ L dV ω E = u : = lim i → u i is still orthogonal to H ( p ) : this is exactly where the previous considerations are needed.The rest of the proof of the corollary follows the arguments already given for Proposi-tion 3.9, so we simply skip it.Now we can prove Theorem 3.6. Proof of Theorem 3.6 . — This statement is almost contained in [
Dem12 , chapter VIII,pages 367-370]. Indeed, in the context of complete manifolds one has the followingdecomposition(3.44) L n ,1 ( X ◦ , L ) = H n ,1 ( X ◦ , L ) ⊕ Im ¯ ∂ ⊕ Im ¯ ∂ ⋆ .We also know (see loc. cit. ) that the adjoints ¯ ∂ ⋆ and D ′ ⋆ in the sense of von Neumanncoincide with the formal adjoints of ¯ ∂ and D ′ respectively.It remains to show that the range of the ¯ ∂ and ¯ ∂ ⋆ -operators are closed with respect tothe L topology. In our set-up, this is a consequence of the particular shape of the metric ω E at infinity (i.e. near the support of π ( E ) ): we are simply using the inequalities (3.36)and (3.37). The former shows that the image of ¯ ∂ is closed, and the latter does the samefor ¯ ∂ ⋆ .We finish this section with the following result (relying of the decomposition theoremobtained above), identifying the L -integrable ∆ ′′ -harmonic forms of bi-degree ( n , 1 ) on ( X ◦ , ω E , h L ) with the vector space H ( X , K X ⊗ L ⊗ I ( h L )) - which is independent of ω E . Proposition 3.13 . —
In the setting of Theorem 3.6, we have a natural isomorphism H n ,1 ( X ◦ , L ) ≃ −→ H ( X , K X ⊗ L ⊗ I ( h L )) where H n ,1 ( X ◦ , L ) is the space of L integrable, ∆ ′′ -harmonic ( n , 1 ) -forms on X ◦ .Proof . — We proceed in several steps. • Step 1. Reduction to the snc case.
The first observation is that the statement is invariant by blow-up whose centers lie on X \ X ◦ . It is obvious for the LHS while it follows from the usual formula π ∗ ( K X ′ ⊗ π ∗ L ⊗ I ( π ∗ h L )) ≃ K X ⊗ L ⊗ I ( h L ) as well as Grauert-Riemenschneider vanishing R π ∗ ( K X ′ ⊗ π ∗ L ⊗ I ( π ∗ h L )) = Mat16 , Cor. 1.5]) valid for any modification π : X ′ → X . So from now on, we assume that the singular locus of h L is an snc di-visor. In the following, we pick a finite Stein covering ( U i ) i ∈ I of X . • Step 2. Statement of the claim to solve the ¯ ∂ -equation. URVATURE FORMULA IN A SINGULAR SETTING Our main tool in the proof will be the following estimate
Claim 3.14 . —
Let v be a ( n , 1 ) -from on X ◦ with values in ( L , h L ) , and such that ∆ ′′ v = Z X | v | ω E e − ϕ L dV ω E < ∞ . Then for each coordinate set Ω ⊂ X there exists an ( n , 0 ) -form u on Ω such that (3.45) ¯ ∂ u = v , Z Ω | u | ω E e − ϕ L dV ω E < ∞ ,For bi-degree reasons, the ( n , 0 ) -form u in (3.45) is L with respect to h L (indepen-dently of any background metric). We postpone the proof of the claim for the momentand we will use it in order to prove Proposition 3.13. • Step 3. The map "harmonic to cohomology".
We first construct an application(3.46) Φ : H n ,1 ( X ◦ , L ) −→ ˇ H ( X , K X ⊗ L ⊗ I ( h L )) as follows. Let f ∈ H n ,1 ( X ◦ , L ) ; by definition we have ∆ ′′ f =
0. Therefore, on can solveon each U ◦ i : = U i ∩ X ◦ the equation ¯ ∂ u i = f where u i is an L -valued ( n , 0 ) -form on U ◦ i satisfies the condition (3.45). In particular, the form u ij : = u i − u j is a holomorphic L -valued n -form on U ◦ ij such that Z U ◦ ij | u ij | e − ϕ L Z U ◦ ij | f | ω E e − ϕ L dV ω E It follows that u ij extends holomorphically across E as a section of K X ⊗ L ⊗ I ( h L ) on U ij and therefore it defines a 1-cocycle of the latter sheaf. It is straighforward to checkthat the class Φ ( f ) : = { ( u ij ) i , j ∈ I } ∈ ˇ H ( X , K X ⊗ L ⊗ I ( h L )) is independent of the choice of the L -integrable form u i solving ¯ ∂ u i = f . • Step 4. The map "cohomology to harmonic".
Next, we have a natural morphism(3.47) Ψ : ˇ H ( X , K X ⊗ L ⊗ I ( h L )) −→ H n ,1 ( X ◦ , L ) Indeed, given a cocycle v : = ( v ij ) i , j ∈ I and a partition of unity ( θ k ) k ∈ K we use the Lerayisomorphism and consider as usual the L -valued ( n , 0 ) -form(3.48) τ k : = ∑ i ∈ I θ i v ki on U k and then the local L -valued forms of type ( n , 1 ) ¯ ∂τ k are glueing on overlapping sets. Let β v be the resulting form. We have(3.49) ¯ ∂β v = β v ∈ L , JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN where the second property in (3.49) is due to the fact that ω E > ω . Under the canonicaldecomposition Ker ¯ ∂ = H n ,1 ⊥ ⊕ Im ¯ ∂ from Theorem 3.6, we define Ψ ( v ) to be the orthogonal projection of β v onto H n ,1 . It isclear that Ψ above is well-defined: if v ij = v i − v j , then τ k − v k is a global, L form andour β v is exact and therefore its projection onto the kernel of ∆ ′′ is zero. • Step 5. Compatibility of the maps.
We are left to showing that the maps Φ and Ψ in (3.46) and (3.47) are inverse to eachother. Let f ∈ H n ,1 , u i ∈ L such that ¯ ∂ u i = f on U ◦ i and u = ( u ij ) . Then on U ◦ k , one has β u − f = ¯ ∂ ( ∑ i θ i u ki − u k ) = ¯ ∂ − ∑ i ∈ I θ i u i ! and that last form is globally exact in X ◦ and L , hence Ψ ( Φ ( f )) = f .In the other direction, let v : = ( v ij ) i , j ∈ I be a cocycle and let us write β v = Ψ ( v ) + ¯ ∂ w for some L -integrable ( n , 0 ) -form w . On U k , one has Ψ ( v ) = ¯ ∂ ( τ k − w ) so that Φ ( Ψ ( v )) is represented by the cocycle ( τ i − τ j ) i , j ∈ I = v . • Step 6. Proof of Claim 3.14.
In order to complete the proof of Proposition 3.13, we need to prove the Claim 3.14 thatwe used in the course of the proof.By (3.32), the form ⋆ v is holomorphic and its restriction to a coordinate subset Ω canbe written as(3.50) ⋆ v | Ω = ∑ ( − ) i − α i b dz i ⊗ e L , ∑ j Z Ω | α j | | f j | e − ϕ L d λ < ∞ where the α i are holomorphic on Ω and f j is as in (3.2).Then we have(3.51) v | Ω = ( − ) n ∑ i , k α i g ik dz ∧ dz k ⊗ e L where g ik are the coefficients of the metric ω E . The construction of the metric at thebeginning shows that(3.52) g ik = ∂ ∂ z i ∂ z k φ − ∑ j log log 1 | s j | ! where φ is a local potential for the smooth metric ω . Therefore we can get a primitivefor v | Ω by defining(3.53) u = ∑ i α i ∂∂ z i φ − ∑ j log log 1 | s j | ! dz ⊗ e L .By equality (3.52) it verifies ¯ ∂ u = v and in is also in L as one can see by an directexplicit computation combined with the second inequality in (3.50).The proof of Proposition 3.13 is now complete. URVATURE FORMULA IN A SINGULAR SETTING
4. Curvature formulas and applications
In this section, we use the Set-up 2.1. We also borrow the Notation 2.4 for the L met-ric denoted by h F on the direct image bundle F = p ⋆ ( O ( K X / D + L ) ⊗ I ( h L )) inducedby e − φ L .Let u ∈ H ( D , F ) and let u be a ( n , 0 ) -form on X ◦ representing u . Thanks to (3.9),for any smooth function f ( t ) with compact support in D , we have(4.1) Z D k u k h F · dd c f ( t ) = c n Z X ◦ u ∧ ¯ u e − φ L ∧ dd c f ( t ) .Recall that h F is smooth by Lemma 2.2.The aim of this section is to generalize formulas [ Ber09 , (4.4), (4.8)] to our singularsetting, cf Proposition 4.1 and Proposition 4.5.
In this context we establish the following generalformula, which generalise the corresponding result in [
Ber09 , (4.4)].
Proposition 4.1 . —
Let u be a continuous representative of u such that: (i) u , D ′ u and ¯ ∂ ( D ′ u ) are L on X ◦ with respect to ω E , h L , (ii) ¯ ∂ u ∧ ¯ ∂ u is L on X ◦ with respect to ω E , h L .Then the following formula holds true ∂ ¯ ∂ k u k h F = c n h − p ⋆ (( Θ h L ( L )) ac ∧ u ∧ ¯ u e − φ L ) + ( − ) n p ⋆ ( D ′ u ∧ D ′ u e − φ L ) (4.2) + ( − ) n p ⋆ ( ¯ ∂ u ∧ ¯ ∂ u e − φ L ) i Here ( Θ h L ( L )) ac is the absolutely continuous part of the current Θ h L ( L ) . Remark 4.2 . —
Here we merely require the L integrability of ¯ ∂ u ∧ ¯ ∂ u and not the L -integrability of ¯ ∂ u . The reason is that for later application in Theorem 4.6, we couldonly obtain the former condition. It is not clear whether the term ¯ ∂ u in Theorem 4.6 is L .The proof of Proposition 4.1 will require a few preliminary computations and will begiven on page 23 below. First, we start with the following result legitimizing integrationby parts. Lemma 4.3 . — If u and D ′ u are L on X ◦ , we have Z X ◦ u ∧ ¯ u e − φ L i ∂ ¯ ∂ f ( t ) = − Z X ◦ D ′ u ∧ ¯ u e − φ L i ¯ ∂ f ( t ) . Proof . — Let ψ ε be the cut-off fonction in Lemma 3.7. Since u is L bounded with re-spect to φ L and ω E , we have Z X ◦ u ∧ ¯ u e − φ L i ∂ ¯ ∂ f ( t ) = lim ε → Z X ◦ ψ ε u ∧ ¯ u e − φ L i ∂ ¯ ∂ f ( t ) . JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
An integration by parts yields Z X ◦ ψ ε u ∧ ¯ u e − φ L i ∂ ¯ ∂ f ( t ) = − Z X ◦ i ∂ψ ε ∧ u ∧ ¯ u e − φ L ¯ ∂ f ( t ) (4.3) − Z X ◦ ψ ε ∧ D ′ u ∧ ¯ u e − φ L i ¯ ∂ f ( t ) − ( − ) n Z X ◦ ψ ε ∧ u ∧ ¯ ∂ u e − φ L i ¯ ∂ f ( t ) . Since u is a representative of a holomorphic section u , we know by (3.11) that ¯ ∂ u = dt ∧ η , hence(4.4) ¯ ∂ u ∧ dt = u is assumed to be globally L -integrable. Similarly, we see that the second term of RHS of (4.3) tends to − Z X ◦ D ′ u ∧ ¯ u e − φ L i ¯ ∂ f ( t ) .The lemma is thus proved.As a corollary of Lemma 4.3 above, we can compute the Chern connection of ( F , h F ) as follows. Corollary 4.4 . —
Let u ∈ H ( D , F ) and let u be a smooth representative of u. We have ∇ u = P ( µ ) dtwhere- ∇ is the (
1, 0 ) -part of the Chern connection on ( F , h F ) .- µ is defined by D ′ u = dt ∧ µ , cf. (3.12) , and µ | X t only depends on u.- P ( µ ) is the fiberwise projection onto H ( X t , ( K X t + L t ) ⊗ I ( h L t )) with respect to theL -norm.Proof . — Let ∇ be the (
1, 0 ) -part of the Chern connection of ( F , h F ) . Then we have ∇ u = σ ⊗ dt ,where σ = ∇ udt ∈ C ∞ ( D , F ) . Let u , v be two holomorphic sections of F and let f be asmooth function with compact support in D . Since v is a holomorphic, we have Z D h u , v i i ∂ ¯ ∂ f ( t ) = Z D h∇ u , v i ∧ i ¯ ∂ f ( t ) .Let u and v be the representatives of u and v respectively given by (3.7). The argumentalready used in Lemma 4.3 shows that we have Z X ◦ u ∧ ¯ v e − φ L i ∂ ¯ ∂ f ( t ) = Z X ◦ D ′ u ∧ ¯ v e − φ L i ¯ ∂ f ( t ) .Here D ′ is, as before, the Chern connection on ( L → X ◦ , h L ) . As a consequence, wehave Z D h∇ u , v i ∧ i ¯ ∂ f ( t ) = Z X ◦ D ′ u ∧ ¯ v e − φ L i ¯ ∂ f ( t ) . URVATURE FORMULA IN A SINGULAR SETTING Since we can choose f on the base D arbitrarily, we infer Z X ◦ t h σ t , v t i = def Z X ◦ t h ∇ udt , v i t = Z X ◦ t D ′ u dt ∧ ¯ v e − φ L = Z X ◦ t µ ∧ ¯ v t e − φ L = Z X ◦ t P ( µ | X t ) ∧ ¯ v t e − φ L for each t ∈ D .As the above holds for any holomorphic section v , we obtain thus ∇ u = P ( µ ) dt on D .We can now complete the proof of Proposition 4.1. Proof of Proposition 4.1 . — Let f ∈ C ∞ c ( D ) . By (4.4), we have Z X ◦ ψ ε u ∧ ¯ ∂ u e − φ L ¯ ∂ f ( t ) = ε . By integration by parts, we obtain(4.5) Z X ◦ ¯ ∂ψ ε ∧ u ∧ ¯ ∂ u e − φ L f ( t ) + Z X ◦ ψ ε ¯ ∂ u ∧ ¯ ∂ u e − φ L f ( t ) + ( − ) n Z X ◦ ψ ε u ∧ D ′ ¯ ∂ u e − φ L f ( t ) = ( − ) n Z X ◦ ¯ ∂ψ ε ∧ u ∧ ¯ ∂ u e − φ L f ( t ) = − Z X ◦ ¯ ∂ψ ε ∧ D ′ u ∧ u e − φ L f ( t ) + Z X ◦ ∂ ¯ ∂ψ ε ∧ u ∧ u e − φ L f ( t ) − Z X ◦ ¯ ∂ψ ε ∧ u ∧ u e − φ L ∧ ∂ f ( t ) . Recall that d ψ ε and dd c ψ ε are uniformly bounded with respect to ω E and converge tozero pointwise. Since u and D ′ u are L by assumption, we see from Lebesgue dom-inated convergence theorem that the RHS tends to 0. Therefore the first term of (4.5)tends to 0.Since ¯ ∂ u ∧ ¯ ∂ u is L , the second term of (4.5) tends to R X ◦ ¯ ∂ u ∧ ¯ ∂ u e − φ L f ( t ) . We obtainthus(4.6) Z X ◦ ¯ ∂ u ∧ ¯ ∂ u e − φ L f ( t ) = ( − ) n − lim ε → Z X ◦ ψ ε u ∧ D ′ ¯ ∂ u e − φ L f ( t ) .We complete in what follows the proof of the proposition. We have Z X ◦ u ∧ ¯ u e − φ L ∧ ¯ ∂∂ f ( t ) = lim ε → Z X ◦ ψ ε u ∧ ¯ u e − φ L ∧ ¯ ∂∂ f ( t )= − lim ε → h Z X ◦ ¯ ∂ψ ε ∧ u ∧ ¯ u e − φ L ∧ ∂ f ( t ) + Z X ◦ ψ ε ∧ ¯ ∂ u ∧ ¯ u e − φ L ∧ ∂ f ( t )+ ( − ) n Z X ◦ ψ ε ∧ u ∧ D ′ u e − φ L ∧ ∂ f ( t ) i JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
Note that the first term tends to 0 since u is L . The second term vanishes because of(4.4). Then we have Z X ◦ u ∧ ¯ u e − φ L ∧ ¯ ∂∂ f ( t ) = ( − ) n − lim ε → Z X ◦ ψ ε ∧ u ∧ D ′ u e − φ L ∧ ∂ f ( t ) .Applying again integration by part, the RHS above becomes (4.7)lim ε → ( − ) n − Z X ◦ ∂ψ ε ∧ u ∧ D ′ u e − φ L f ( t ) + ( − ) n − Z X ◦ ψ ε D ′ u ∧ D ′ u e − φ L f ( t ) − Z X ◦ ψ ε u ∧ ¯ ∂ D ′ u e − φ L f ( t ) . As u and D ′ u are L , the first term of (4.7) tends to 0, and the second term of (4.7)tends to R X ◦ D ′ u ∧ D ′ u e − φ L f ( t ) . For the third term, as ¯ ∂ D ′ u = Θ h L ( L ) − D ′ ¯ ∂ u , we have(4.8) Z X ◦ ψ ε u ∧ ¯ ∂ D ′ u e − φ L f ( t ) = − Z X ◦ ψ ε u ∧ D ′ ¯ ∂ u e − φ L f ( t ) − Z X ◦ ψ ε Θ h L ( L ) u ∧ u e − φ L f ( t ) .Combining with (4.6), we obtainlim ε → Z X ◦ ψ ε u ∧ ¯ ∂ D ′ u e − φ L f ( t ) = ( − ) n Z X ◦ ¯ ∂ u ∧ ¯ ∂ u e − φ L f ( t ) − Z X ◦ Θ h L ( L ) ∧ u ∧ u e − φ L f ( t ) .All three terms of the RHS of (4.7) have now been calculated and the sum is just ( − ) n − Z X ◦ D ′ u ∧ D ′ u e − φ L f ( t ) + ( − ) n − Z X ◦ ¯ ∂ u ∧ ¯ ∂ u e − φ L f ( t ) + Z X ◦ Θ h L ( L ) ∧ u ∧ u e − φ L f ( t ) . The proposition is thus proved.
Now for applications, we need to general-ize [
Ber09 , Prop 4.2] and formula [
Ber09 , (4.8)] to our singular setting. Following theargument of [
Ber09 , Prop 4.2], we have the following.
Proposition 4.5 . —
We assume that the coefficients b I in the Set-up condition (A.2) are equalto zero. Let u be a holomorphic section of F on D such that ∇ u ( ) = . Then u can berepresented by a smooth ( n , 0 ) -form u on X ◦ , L with respect to h L , ω E , such that ¯ ∂ u = dt ∧ η for some L -form η which is primitive (with respect to ω E | X ◦ ) on X ◦ , andD ′ u = dt ∧ µ for some µ satisfying µ | X ◦ = . Here X ◦ : = X ∩ X ◦ is on the central fiber.Proof . — Let u be the representative constructed in (3.7). We have D ′ u = dt ∧ µ ¯ ∂ u = dt ∧ η .Then µ | X ◦ is orthogonal to the space of L -holomorphic section by Corollary 4.4.By Remark 3.2 our representative u has the following property(4.9) u ∧ ω E = dt ∧ dt ∧ u for some ( n , 0 ) -form u on X ◦ . It follows that we have(4.10) η ∧ ω E = X t . URVATURE FORMULA IN A SINGULAR SETTING Moreover, as µ | X ◦ is orthogonal to Ker ¯ ∂ , Theorem 3.6 shows that µ | X ◦ is ¯ ∂ ⋆ -exact,i.e., there exists a ¯ ∂ -closed L -form β on X ◦ such that¯ ∂ ⋆ β = µ | X ◦ .Let e β be an arbitrary (globally L ) extension of ⋆ β . Then u − dt ∧ e β is the represen-tative we are looking for.The result above produces a representative enjoying nice properties in restriction tothe central fiber. In order to generalize that to each fiber, we consider the case where u ∈ H ( D , F ) is a flat section with respect to h F . For that purpose, we introduce anadditional cohomological assumption.In the Set-up 2.1, assume that the coefficients b I appearing in (A.2) vanish. That is tosay, h L has analytic singularities in the usual sense. We let ∆ ′′ t be the Laplace operatoron L -integrable ( n , 1 ) -forms with values in L on X ◦ t , taken with respect to ω E , h L . Letus consider the following assumption.(A.4) The dimension dim Ker ∆ ′′ t is independent of t ∈ D .Note that by Bochner formula, we already know that dim Ker ∆ ′′ t < + ∞ . Indeed, if α is a ∆ ′′ t -harmonic L ( n , 1 ) -form on X ◦ t , then (3.30) shows that α is ∆ ′ t -harmonic and ( D ′ ) ∗ α =
0. In particular, ⋆ α is a L -holomorphic section. Thanks to Remark 3.5, weget an injection Ker ∆ ′′ t ֒ → H ( X t , Ω n − X t ⊗ L t ) . In particular, the former space is finite-dimensional.The main result of this subsection states as follows. Theorem 4.6 . —
In the Set-up 2.1, assume that h L has analytic singularities and that thecondition (A.4) above is satisfied.Let u ∈ H ( D , F ) be a flat section with respect to h F . Then, we can find a continuous ( n , 0 ) -form u on X \
E representing u such that ( i ) u is L and D ′ u = , ( ii ) η | X ◦ t = for any t ∈ D, and ¯ ∂ u ∧ ¯ ∂ u = , where η is –as usual– given by ¯ ∂ u = dt ∧ η .Moreover, the equality (4.11) Θ h L ( L ) ∧ u = holds true point-wise on X \ E. Remark 4.7 . —
Let us collect a few remarks about the theorem.(a) The content of Theorem 4.6 is clear: it "converts" the abstract data ¯ ∂ u = ∇ u = i Θ h L ( L ) on Λ n T ∗X ◦ has u in its kernel. Proof . — As in the proof of Proposition 4.5, we start with a representative u given by(3.7) (i.e. constructed via the contraction with the canonical lifting of ∂∂ t with respect ω E ). JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
Since u is flat on D , we have D ′ u = dt ∧ µ where µ | X ◦ t is L and ¯ ∂ ⋆ t -exact for every t ∈ D . Therefore we can solve the ¯ ∂ ∗ -equation fiberwise, namely there exists a unique L -form β t on X ◦ t such that β t is orthogonal to the Ker ¯ ∂ ⋆ t and such that¯ ∂ ⋆ t β t = µ | X ◦ t .By taking the ¯ ∂ in both side and taking into account the fact that β t is orthogonal to theKer ¯ ∂ ⋆ t , we obtain(4.12) ∆ ′′ t β t = ¯ ∂ ( µ | X ◦ t ) on X ◦ t and β t ⊥ Ker ∆ ′′ t .By analogy to the compact case it is expected that the minimal solution of a ∆ ′′ t equationvaries smoothly provided that dim Ker ∆ ′′ t is constant. We partly confirm this expecta-tion in Proposition 4.8 by showing that it is continuous; for the moment, we will admitthis fact and finish the proof of the theorem.We set u : = u − dt ∧ ( ⋆ t β t ) .It is a continuous, fiberwise smooth form on X ◦ and we show now that u is a repre-sentative for which the points (i)-(ii) above are satisfied.By construction, D ′ u = X ◦ . By (4.18) of Proposition 4.8 below, the L -norm of β t is smaller than the L -norm of ¯ ∂µ | X ◦ t , i.e., k β t k L C k ¯ ∂µ | X ◦ t k L for some constant C independent of t . Moreover, we recall the estimates in Lemma 3.3:¯ ∂µ | X ◦ t is uniformly L -bounded. Therefore dt ∧ ( ⋆ t β t ) is L and so our representative u is L .We have ¯ ∂ u = dt ∧ (cid:0) η + ¯ ∂ (cid:0) ⋆ t β t (cid:1)(cid:1) and since ¯ ∂ (cid:0) ⋆ t β t (cid:1) ∧ ω E = ¯ ∂β t =
0, it follows that(4.13) ¯ ∂ u dt (cid:12)(cid:12)(cid:12) X t ∧ ω E = ∂ u ∧ ¯ ∂ u ∈ L . To thisend, we write¯ ∂ u ∧ ¯ ∂ u = dt ∧ η ∧ dt ∧ η + dt ∧ ¯ ∂ ( ⋆ t β t ) ∧ dt ∧ η (4.14) + dt ∧ η ∧ dt ∧ ¯ ∂ ( ⋆ t β t ) + dt ∧ ¯ ∂ ( ⋆ t β t ) ∧ dt ∧ ¯ ∂ ( ⋆ t β t ) .(4.15)By the estimates in Lemma 3.3, η is L . Then the first term of RHS of (4.14) is L . Degreeconsiderations show that we have(4.16) dt ∧ ¯ ∂ ( ⋆ t β t ) ∧ dt ∧ η = dt ∧ ¯ ∂ t ( ⋆ t β t ) ∧ dt ∧ η ,where ¯ ∂ t is the ¯ ∂ -operator on X t . Since ∆ ′′ t β t = ¯ ∂µ and β t is of degree ( n , 1 ) , Bochnerformula shows that the L -norm of ¯ ∂ t ( ⋆ t β t ) is equal to the L norm of the form ( D ′ ) ⋆ t β t (this is due to the fact that β t is ¯ ∂ -closed), which in turn is bounded by the L -norm of¯ ∂µ . Once again, the estimates provided by Lemma 3.3 show that the L -norm of ¯ ∂µ | X t is bounded uniformly with respect to t . It follows that dt ∧ ¯ ∂ t ( ⋆ t β t ) is L . URVATURE FORMULA IN A SINGULAR SETTING Therefore dt ∧ ¯ ∂ ( ⋆ t β t ) ∧ dt ∧ η is L -bounded by using (4.16). The same type of argu-ments show that the two terms in (4.15) are also L .We apply Proposition 4.1 for the representative u of u . The flatness of u imply that c n Z X ◦ (( − ) n ¯ ∂ u ∧ ¯ ∂ u + i Θ h L ( L ) ∧ u ∧ u ) e − φ L = i Θ h L ( L ) > ( β t ) our result isproved.The proof of the following result is very similar to the familiar situation in which thecouple of metrics ( ω E , h L ) are non-singular. We provide a complete argument becausewe were unable to find a reference. Proposition 4.8 . —
The minimal solution β t in (4.12) varies continuously with respect to t.Proof . — We have divided our proof in a few steps. Step 1 . — Let ( u t ) be a family of L -valued, L forms of ( n , 1 ) –type on the fibersof p , such that we have(4.17) ∆ ′′ t v t = u t on the fiber X t . If moreover we assume that each v t is perpendicular to Ker ∆ ′′ t , then weclaim that(4.18) Z X t | v t | ω E e − φ L dV ω E C Z X t | u t | ω E e − φ L dV ω E for some constant C uniform with respect to t .Indeed, our claim follows instantly from Corollary 3.12 and (3.32) applied to u : = ⋆ t v t . Step 2 . — Let λ ∈ C such that 0 < | λ | ≪ A λ , t : = λ − ∆ ′′ t is invertible, which we show by proving that the equation A λ , t v = u admits a solution v , as soon as u is in L . This can be seen via the usual Riesz representation theorem, asfollows.We define on L n ,1 the functional I ( φ ) = Z X t h u , φ i e − φ L dV ω E We write u = u + u and φ = φ + φ according to the decomposition L n ,1 = Ker ∆ ′′ t ⊕ ( Ker ∆ ′′ t ) ⊥ . Then we have I ( φ ) = Z X t h u , φ i e − φ L dV ω E + Z X t h u , φ i e − φ L dV ω E and then the squared absolute value of the second integral is smaller than(4.19) Z X t | ∆ ′′ t φ | e − φ L dV ω E JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN up to a uniform constant, by Step 1. Therefore, we get(4.20) | I ( φ ) | . Z X t | φ | e − φ L dV ω E + Z X t | ∆ ′′ t φ | e − φ L dV ω E .Since (cid:12)(cid:12)(cid:12) Z X t h φ , ∆ ′′ t φ i dV ω E (cid:12)(cid:12)(cid:12) Z X t | φ | e − φ L dV ω E · Z X t | ∆ ′′ t φ | e − φ L dV ω E we see that from (4.20) that(4.21) | I ( φ ) | C u , λ Z X t | λφ − ∆ ′′ t φ | e − φ L dV ω E as we see from the previous step, provided that | λ | C , where C is the constant in(4.18). Moreover, C u , λ is of the form C λ · k u k L .Taking φ = u in the identity above, we see that A λ , t : = λ − ∆ ′′ t is injective. Moreover,the functional J : Im A λ , t −→ C A λ , t φ I ( φ ) is well-defined and continuous by (4.21). In particular, it extends to F = Im A λ , t andRiesz theorem provides us with an element v ∈ F satisfying(4.22) ∀ ψ ∈ F , J ( ψ ) = Z X t h v , ψ i e − φ L dV ω E and k v k L C λ k u k L .The equality J ( A λ , t φ ) = I ( φ ) for any φ in L shows that A λ , t v = u . This concludes thisstep. Step 3 . — Let λ ∈ C as in the previous step, and let u t be a continuous L -family.We show that v t : = ( λ − ∆ ′′ t ) − u t is continuous with respect to t (with respect to the L -norm).It would be sufficient to check the continuity at one point 0 ∈ D . For any ε > v ε : = µ ε v with compact support in X \ E . We then have k v ε − v k L ε , k ( λ − ∆ ′′ ) v ε − ( λ − ∆ ′′ ) v k L ε by the properties of ( µ ε ) ε > .We next construct a smooth extension v ε of v ε as follows. Let ( Ω i ) i ∈ I be a finite cov-ering of p − ( D ) by coordinate charts, and let ( θ i ) i ∈ I be a partition of unity subordinateto this covering. The L -valued form v ε : = µ ε ∑ i θ i ( z , t ) v i extends v ε and it is compactly supported in X \ E . Here we denote by v i is the localexpression of v ε | X ∩ Ω i , extended trivially to Ω i (note that this is still L ).Since the metrics ω E , h L are smooth in X \ E , u ε , t : = ( λ − ∆ ′′ t )( v ε | X t ) is a smooth L -family. URVATURE FORMULA IN A SINGULAR SETTING By the second part in (4.22) we have k v t − v ε | X t k L C k u t − u ε , t k L .As u t and u ε , t are continuous with respect to t , we infer that we have k v t − v ε | X t k L C ε O ( | t | ) + C k u − u ε ,0 k L . It follows that we have k v t − v ε | X t k L o ( ) + C ε as | t | →
0. The -small- quantity o ( ) here depends on ε , but since by construction thefamily v ε | t is continuous with respect to t and its continuity modulus is independent of ε we infer that v t is continuous at 0.Now we define the operator Pr t : = R λ ∈ Γ ( λ − ∆ ′′ t ) − d λ where Γ is a small circle centeredat 0. We have proved above that Pr t is continuous with respect to t . Moreover, Pr t coincides with the orthogonal projection onto Ker ∆ ′′ t : we postpone the proof of thisclaim for the moment, see the Remark 4.9 below. Step 4 . — This is the main step in the proof of the proposition. Let β t be the¯ ∂ ∗ -solution on X t in question. Then ∆ ′′ t β t = ¯ ∂ ( µ | X ◦ t ) .By the estimates in Proposition 4.5, the RHS is L and continuous with respect to t . Let s t be a continuous L -family (continuous with respect to t ) such that s = β . Then wehave λ s t − ∆ ′′ t β t = λ s t − ¯ ∂ ( µ | X ◦ t ) for every t , where 0 < | λ | ≪ γ t such that λ s t − ∆ ′′ t β t = λ γ t − ∆ ′′ t γ t for every t . Then ∆ ′′ t ( β t − γ ⊥ t ) = λ ( s t − γ t ) , where γ ⊥ t : = γ t − Pr t γ t is the projectiononto ( Ker ∆ ′′ t ) ⊥ .Now β t is orthogonal to Ker ∆ ′′ t by construction. Then β t − γ ⊥ t is orthogonal to Ker ∆ ′′ t .By Step 1 we thus have k β t − γ ⊥ t k L C k s t − γ t k L .Note that s = γ and s t and γ t are continuous, then k β t − γ ⊥ t k L = o ( ) . By Step 4, γ ⊥ t is continuous, therefore β t is continuous at 0.Summing up, the continuity with respect to the L norm of ( β t ) t ∈ D is established. Step 5 . — We show here that the form(4.23) dt ∧ ⋆ t β t induced by the family ( β t ) t ∈ D in (4.12) is continuous on X \ E . This is a consequenceof the fact that the family of operators ( ∆ ′′ t ) t ∈ D is smooth and it has a smooth variationwhen restricted to a compact subset K ⊂ X \ E , combined with the continuity propertyestablished in Proposition 4.8. JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
Let Ω ⋐ X \ E be a small coordinate chart. We can interpret the ( ∆ ′′ t ) t ∈ D as family ofoperators on the forms defined on Ω , since p is locally trivial. Then we have(4.24) ∆ ′′ β t = ¯ ∂µ t + ( ∆ ′′ − ∆ ′′ t )( β t ) from which it follows that(4.25) ∆ ′′ ( β t − β ) = ¯ ∂µ t − ¯ ∂µ + ( ∆ ′′ − ∆ ′′ t )( β t ) .The equality (4.25) combined with the usual a-priori estimates for the elliptic operatorsimply that(4.26) k β t − β k W C (cid:0) k ¯ ∂µ t − ¯ ∂µ k L + k β t − β k L (cid:1) + δ t k β t k W where δ t → t →
0. We infer that(4.27) k β t − β k W δ t for some (other) function δ t tending to zero.The usual boot-strapping method implies that lim t → β t = β smoothly on any compactsubset in X \ E . In global terms this translates as dt ∧ ⋆ t β t is a continuous ( n , 0 ) -form on X \ E , so our lemma is proved. Remark 4.9 . —
For the sake of completeness, we provide the details for the fact thatthe linear operator Pr t : = R λ ∈ Γ ( λ − ∆ ′′ t ) − d λ is the orthogonal projection onto Ker ∆ ′′ t .Let H t be the Hilbert space ( Ker ∆ ′′ t ) ⊥ . We need to prove two points: ( i ) Pr t u = u ∈ H t ; ( ii ) Pr t u = u for any u ∈ Ker ∆ ′′ t .For the first point, we have the following equality(4.28) ( λ − ∆ ′′ t ) − = − ∑ k > λ k G k + t on H t where G t is the inverse of the operator ∆ ′′ t restricted to H t . The sum in (4.28) is in-deed convergent (for the operator norm), given the estimates (4.18) and the fact that λ belongs to the circle Γ of small enough radius.It follows that we can exchange integration/sum and then we have(4.29) Z λ ∈ Γ ( λ − ∆ ′′ t ) − ud λ = − ∑ k > G k + t ( u ) Z λ ∈ Γ λ k d λ and this shows that Pr t ( u ) = u ∈ H t .For the second point, let u ∈ Ker ∆ ′′ t . For any α ∈ Ker ∆ ′′ t , we have h u λ , α i = h ( λ − ∆ ′′ t )( λ − ∆ ′′ t ) − u λ , α i = h ( λ − ∆ ′′ t ) − u , α i .Therefore h u , α i = Z λ ∈ Γ h u λ , α i d λ = Z λ ∈ Γ h ( λ − ∆ ′′ t ) − u , α i d λ . URVATURE FORMULA IN A SINGULAR SETTING Then u − Pr t u is orthogonal to Ker ∆ ′′ . On the other hand, thanks to the equality ∆ ′′ t ◦ ( λ − ∆ ′′ t ) − ( u ) = ( λ − ∆ ′′ t ) − ◦ ∆ ′′ t u = t u ∈ Ker ∆ ′′ t . Therefore Pr t u = u . Remark 4.10 . —
Actually the form β t can be obtained as usually via an integral for-mula,(4.30) β t = − Z λ ∈ Γ λ ( λ − ∆ ′′ t ) − ( ¯ ∂µ | X ◦ t ) which gives the hope that its variation with respect to t is actually smooth. This canprobably be obtained along the same lines as in [ Kod86 , Thm 7.5] modulo the fact thatin the present situation, we have to deal with the additional difficulty induced by thefact that we are working with singular metrics ω E and h L .We can now end this section by providing a proof of Theorem A. Proof of Theorem A . — Up to shrinking D to a punctured disk D ⊂ D , one may assumethat the assumptions (A.1)-(A.3) are satisfied (with b I = I ), cf. Section 2.Next, there exists another punctured disk D ⊂ D such that the coherent sheaf R p ∗ ( K X / D ⊗ L ⊗ I ( h L )) is locally free and commutes with base change; i.e. its fiberat t ∈ D is given by H ( X t , K X t ⊗ L | X t ⊗ I ( h L | X t )) and the dimension of the latteris independent of t ∈ D . Thanks to Proposition 3.13, the dimension of the space ofharmonic ( n , 1 ) -forms, i.e. dim Ker ( ∆ ′′ t ) , is independent of t ∈ D . In other words, thecondition (A.4) is satisfied over D .Theorem A is now a direct consequence of Theorem 4.6.
5. A lower bound for the curvature in case of a -relatively- big twist
Let p : X → D be a smooth, projective family, and let L → X be a line bundleendowed with a metric h L = e − φ L satisfying the following requirements.(B.1) There exist a smooth, semi-positive real (1,1)-form ω L as well as an effective R -divisor E on X such that i Θ h L ( L ) = ω L + [ E ] where we denote by [ E ] the current of integration associated to the R -divisor E .(B.2) ω L is relatively Kähler, i.e., ω L | X t > t .(B.3) The support of the divisor E : = Supp ( E ) is snc, and transverse to the fibers of p .Let c ( φ L ) : = ω n + L ω nL ∧ idt ∧ dt be the so-called geodesic curvature associated to ω L .Our goal here is to establish the following result. JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN
Theorem 5.1 . —
Under the assumptions (B.1)-(B.3) above, let ( F , h F ) be the direct imagebundle p ⋆ (cid:0) ( K X / D + L ) ⊗ I ( h L ) (cid:1) endowed with the L metric. Then for every u ∈ H ( D , F ) and every t ∈ D, the following inequality holds (5.1) h Θ h F ( F ) u , u i t > c n Z X t c ( φ L ) u ∧ ue − ϕ L where we identify Θ h F ( F ) with an endomorphism of F by "dividing" with idt ∧ dt. Prior to providing the arguments for Theorem 5.1 we propose here the following prob-lem.
Question 5.2 . —
We assume that Y is a foliation on a Kähler manifold Z. In which casesthe bundle K Y + L admits a positively curved metric? That is to say, is there some analogueof the Bergman metric on twisted relative canonical bundles in the more general contexts of afoliation? If yes, can we equally obtain a lower bound of the curvature form?Proof of Theorem 5.1 . — The idea of our proof is to construct an approximation of themetric h L so that the resulting absolutely continuous part of the associated curvatureform has Poincaré singularities along the support of E . Then we can use the curvatureformulas we have obtained in the previous sections, and finally conclude by a limitargument. Approximation of the metric . — Let ε > ω ε : = ω L − ε ∑ i ∈ I dd c log log 1 | s i | where I is the set of irreducible components of E , and s i cut outs exactly one of thesefor a given i ∈ I , following notation in Section 2. We note that ω ε is positive and hasPoincaré singularities along E as soon as the metrics h i used to measure the norm of s i are suitably scaled, which is what we assume from now on.Next, we introduce the following weight on L (5.3) φ ε : = φ L − ε ∑ i ∈ I log log 1 | s i | .Clearly, φ ε has generalized analytic singularities in the sense of (A.2) in Set-up 2.1 andit satisfies dd c φ ε = ω ε .The properties of ( φ ε ) ε are collected in the following statement. Lemma 5.3 . —
Let ( Ω , ( z , . . . , z n , t = z n + ) be a coordinate system on X adapted to the pair ( X , E ) as in Set-up 2.1. Then the following hold.(i) The geodesic curvature c ( φ ε ) is uniformly bounded from above.(ii) We have lim ε → c ( φ ε ) = c ( φ L ) point-wise on X \
E.(iii) For every ε small enough, the multiplier ideal sheaf of h ε : = e − φ ε coincides with I ( h L ) .Moreover, the induced L metric, say H ε on the direct image is smooth, and it convergesto h F as ε → . URVATURE FORMULA IN A SINGULAR SETTING Proof . — For the point ( i ) , we write c ( ω ε ) = k dt k ω ε and the result follows from thetransversality conditions (B.3) and e.g. the estimates for the coefficients provided in(3.1).The point ( ii ) follows easily from the local smooth convergence φ ε → φ L on X \ E combined with the positivity requirement (B.2).As for the third point ( iii ) , the smoothness of H ε and its convergence to h F is a conse-quence of the transversality assumption (B.3) by the same arguments as for Lemma 2.2.As for the statement about multiplier ideal sheaves, one has clearly I ( φ ε ) ⊂ I ( φ L ) while the reverse inclusion is an easy consequence of (B.1)-(B.3). Application of the curvature formula . — We consider u a local holomorphic sectionof the bundle F , and let u ε be the representative of u constructed in (3.7), by using thecontraction with the vector field V ε associated to the metric ω ε .Let(5.4) ¯ ∂ u ε = dt ∧ η ε , D ′ u ε = dt ∧ µ ε where D ′ = D ′ ε is the Chern connection corresponding to ( L , h ε ) . Moreover we have ω ε ∧ u ε ∧ u ε = c ( φ ε ) u ε ∧ u ε ∧ p ⋆ ( dt ∧ dt ) on X \ E by [ Ber11 , Lem 4.2]. Proposition 4.1 then gives(5.5) − ∂ ∂ t ∂ t ( k u k H ε ) = c n Z X t c ( φ ε ) u ε ∧ u ε e − ϕ ε + Z X t | η ε | e − φ ε dV ω ε − Z X t | µ ε | e − φ ε dV ω ε ,since η ε is primitive on fibers of p . We discuss next the terms which occur in (5.5). • The LHS of (5.5) is equal to(5.6) h Θ H ε ( F ) u , u i − k P ( µ ε ) k by the usual formula of the Hessian of the norm of a holomorphic section of a vectorbundle. Then (5.5) becomes(5.7) h Θ H ε ( F ) u , u i t = c n Z X t c ( φ ε ) u ∧ u e − ϕ ε + Z X t | η ε | ω ε e − φ ε dV ω ε − Z X t | µ ⊥ ε | ω ε e − φ ε dV ω ε ,where µ ε = P ( µ ε ) + µ ⊥ ε is the L decomposition of µ ε according to the Ker ¯ ∂ and itsorthogonal. • As observed in [
Ber11 , Lem 4.4], we have(5.8) ¯ ∂µ ε = D ′ η ε –even if the curvature is not zero!–, and actually µ ⊥ ε is the solution of (5.8) whose L normis minimal . By [ CP20 , Thm 1.6], we have the precise estimate(5.9) Z X t | µ ⊥ ε | ω ε e − φ ε dV ω ε Z X t | η ε | ω ε e − φ ε dV ω ε and then we get(5.10) h Θ H ε ( F ) u , u i t > c n Z X t c ( φ ε ) u ∧ ue − φ ε .as consequence of (5.7). JUNYAN CAO, HENRI GUENANCIA & MIHAI P ˘AUN • The last step in our proof is to notice that as the parameter ε approaches zero, theinequality (5.10) implies(5.11) h Θ h F ( F ) u , u i t > c n Z X t c ( φ L ) u ∧ ue − φ L .Indeed, we are using Lemma 5.3 for the LHS of (5.10) and Lemma 5.3 (i) combined withdominated convergence theorem for the RHS. Theorem 5.1 is proved.In the last lines, we now explain how to deduce Theorem B from Theorem 5.1 above. Proof of Theorem B . — We start by making the observation that if π : X ′ → X is aproper birational morphism inducing birational morphisms X ′ t → X t , then one has R X ′ t c ( φ ′ L ) u ′ ∧ ¯ u ′ e − φ ′ L = R X t c ( φ L ) u ∧ ¯ ue − φ L , with the self-explanatory notation.Therefore, by blowing up X and restricting the family to a punctured disk D ⊂ D ,one can from now on assume that the conditions (B.1) and (B.3) are satisfied.Now, one has to show that one can further assume that condition (B.2) is satisfied.This is a bit more involved and can be shown as follows.Since R X t ω nL > ω L is smooth, it follows from e.g. [ Bou02 ] that [ ω L ] is p -big. Inparticular, there exists a punctured disk D ⊂ D , an effective, horizontal R -divisor F and an ample R -line bundle A on X such that(5.12) [ ω L ] = A + F in H ( X , R ) .After blowing-up once again and restricting to a smaller punctured disk D ⊂ D , onecan assume without loss of generality that E + F is snc and transverse to the fiber. Ofcourse, the pull-back of A is not ample anymore, but there exists an effective divisor G contained in the exceptional locus of the blow-up such that A − G is ample. All in all,one will assume from now on that one has a decomposition (5.12) where A is ampleand E + F is snc and transverse to the fibers.We pick a strictly psh smooth weight φ A on A and set φ E (resp. φ F ) for the singularpsh weight on the corresponding R -divisor.For δ >
0, we introduce the psh weight φ δ on L defined by φ δ = ( − δ ) φ L + δ ( φ A + φ F + φ E ) .Clearly, φ δ has analytic singularities along the divisor E + F and ( dd c φ δ ) ac is a relativeKähler metric for any δ >
0. That is, the metric h L , δ : = e − φ δ satisfies (B.2).Thanks to Theorem 5.1, the proof of Theorem B will be complete once we show thefollowing Claim 5.4 . —
With the notation above, one has(i) I ( φ δ ) = I ( φ L ) for δ small enough.(ii) The L metric H δ induced by h L , δ on F is smooth and converges smoothly to h F when δ → .(iii) For any t ∈ D and u ∈ F t , one has lim δ → Z X t c ( φ δ ) u ∧ ¯ ue − φ δ = Z X t c ( φ L ) u ∧ ¯ ue − φ L . URVATURE FORMULA IN A SINGULAR SETTING Proof of Claim 5.4 . — Since φ L − φ E it is smooth (its curvature is nothing but ω L ), wehave I ( φ L ) = I ( φ E ) and I ( φ δ ) = I ( φ E + δφ F ) , which coincides with I ( φ E ) when δ issmall enough. This shows ( i ) .The item ( ii ) can be proved along the same lines as Lemma 2.2, using the fact that E + F is snc and transverse to the fibers.As for item ( iii ) , we have pointwise convergence c ( φ δ ) → c ( φ L ) on a Zariski openset of each X t , t ∈ D , cf. Definition 2.3. Moreover, the Kähler metric ( dd c φ δ ) ac on X isuniformly bounded above by a fixed Kähler metric on X (for instance, ω L + dd c φ A ). Inparticular c ( φ δ ) is uniformly bounded above (say over compact subsets of D ) and onecan apply Lebesgue dominated convergence theorem to conclude.The proof of Theorem B is now complete. Remark 5.5 . —
The following limit argument shows that we can take D ⋆ ⊂ D to be theset of t ∈ D such that the following hold: • the metric h F is smooth locally near t ; • the fiber F t coincides with H ( X t , ( K X t + L ) ⊗ I ( h L | X t )) .Let 0 ∈ D be a point which satisfies these requirements. Let U ⋐ X \ ( h L = ∞ ) be anyopen subset of X whose closure does not meet the singular locus of the metric h L . Thenwe have(5.13) Z U ∩ X c ( φ L ) u ∧ ue − φ L = lim t → Z U ∩ X t c ( φ L ) u ∧ ue − φ L since all the objects involved are non-singular.The next observation is that since h F is smooth near 0 -by assumption-, the function t
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