Regularization of closed positive currents and intersection theory

Abstract We prove the existence of a closed regularization of the integration current associated to an effective analytic cycle, with a bounded negative part. By means of the King formula, we are reduced to regularize a closed differential form with L1loc coefficients, which by extension has a test value on any positive current with the same support as the cycle. As a consequence, the restriction of a closed positive current to a closed analytic submanifold is well defined as a closed positive current. Lastly, given a closed smooth differential (qʹ, qʹ)-form on a closed analytic submanifold, we prove the existence of a closed (qʹ, qʹ)-current having a restriction equal to that differential form. After blowing up we deal with the case of a hypersurface and then the extension current is obtained as a solution of a linear differential equation of order 1.


Introduction
Let .X; !/ be a compact Kähler manifold of dimension n and let T be a closed positive current in X of bidegree .q; q/. We prove (see Proposition 9.2) the following result.
Proposition. There are a constant C 0 and a sequence T l of C 1 closed differential .q; q/-forms in X , weakly converging in X to T and satisfying T l C ! q in X for any l.
This theorem is a version in any codimension q of Demailly's regularization theorem (see [11]) known only for q D 1. Recall the statement of Demailly's theorem in its most precise form. Let u be a C 1 positive differential .1; 1/-form on X such that i 2 ‚ TX C u˝id TX 0 in the sense of Griffiths in X , where ‚ TX 2 C 1 1;1 .X; T X˝TX / is the Chern curvature form. For T of bidegree .1; 1/, we can write T D lim l T l weakly in X with .T l / a sequence of C 1 closed differential .1; 1/-forms in X such that T l l u where . l / is a decreasing sequence of continuous functions in X satisfying l .x/ ! .T; x/ for every x 2 X , with .T; x/ the Lelong number of T at x. In particular, l u converges to 0 weakly in X .
Recall also the regularization theorem of Dinh-Sibony (see [13]) which applies to any q but only claims the existence of a closed regularization with a negative part that is bounded in the L 1 loc sense. This theorem asserts precisely the existence of a constant C dependent of ! such that all T in X can be written T D lim l .T C l T l / weakly in X , where T C l ; T l are some closed positive differential .q; q/-forms of class C 1 on X satisfying R X kT C l k Ä C R X kT k and R X kT l k Ä C R X kT k. Here, we denote by kT k the mass measure of T . A canonical decomposition T D T C T with T C ; T closed positive is then obtained.
We prove by a direct method (see Propositions 2.2 and 5.1) the following result, similar to that of Dinh-Sibony.
Proposition. There is a sequence .T l / of C 1 differential .q; q/-forms closed in X weakly converging in X to T and such that T l u l with .u l / a sequence of C 1 differential .q; q/-forms in X converging to 0 for the L 1 loc topology in X .
The existence of the sequence T l in the first above Proposition is obtained here by a biduality argument and T l has no explicit construction.
On the other hand we give an explicit construction of a closed regularization in the sense of Dinh-Sibony of the integration current associated with an effective algebraic cycle P j m j Z j , using a Green form of this cycle (see [22,24]). The Green form is obtained from a locally free projective resolution of the ideal sheaves, following Bismut-Bost-Gillet-Soulé (see [4,6]), but here without any assumption of smoothness and without any hypothesis of compatibility between the Hermitian metrics.
Specifically we assume X projective and we consider Z an analytic subset of codimension q of X which is the locus of zeroes of a holomorphic section s of a Hermitian holomorphic vector bundle E above X . To calculate a Green form of the codimension q cycle P j m j Z j associated to s, we denote by I O X the ideal sheaves generated by the components of s and we suppose given a projective resolution 0 ! O.F n / g n ! O.F n 1 / g n 1 ! : : : with holomorphic vector bundles F i defined in X , equipped with Hermitian metrics h 0;i of class C 1 in X . We get explicitly a differential form with coefficients of class C 1 in X X Z and L 1 loc in X such that in the sense of currents on X , we have with c q .‚ E / the q th Chern form of the curvature form ‚ E of .E; h/ and with c.‚ F i / D P k c k .‚ F i / the total Chern form of ‚ F i , for Hermitian metrics h and h i naturally derived from that of E and h 0;i .
We proceed here directly from the formula of King X j m j OEZ j D .dd c log ksk/ q 1l XXZ .dd c log ksk/ q : Applying the Bott-Chern calculations of transgression forms for the Chern classes involved in the exact sequence 0 ! Cs ! E ! E=Cs ! 0 in X X Z, we first make explicit a differential form 1 with coefficients of class C 1 in X X Z and L 1 loc in X, such that in X in the sense of currents, we have X A generalization of the Poincaré-Lelong formula was also obtained by Andersson (see [1,2]). To express c q .‚ E=Cs / we again argue on X X Z by breaking the resolution in short exact sequences of holomorphic vector bundles. The local integrability of is obtained as a result of the construction of MacPherson of the graph (see [3]), which gives the existence on the blow up of X along Z of an exact complex of holomorphic vector bundles G i , extending the one given on X X Z.
In conclusion and as an application, we obtain (see Proposition 8.3) the following result.
Proposition. There is an explicit sequence of differential forms Q W l of class C 1 , closed in X , converging weakly in X to P j m j OEZ j and satisfying Q W l Q U l where Q U l is of class C 1 in X and converges to 0 for the L 1 loc topology.
When Z is smooth, it is further shown that every differential .q 0 ; q 0 /-form of class C 1 that is closed in Z extends as a closed .q 0 ; q 0 /-current in X . This allows for any manifold X to extend formula (1) by setting with Â q a closed .q; q/-current satisfying fÂ qjZ g D c q .N X Z/ and with 1 a differential .q 1; q 1/-form explicitly calculated (see Proposition 7.1).
2 Closed regularization with negative part converging weakly towards 0 Let .X; !/ be a compact Kähler manifold of dimension n and let T be a closed positive current in X of bidegree .q; q/. We give here a closed regularization of T obtained using the Green operator of .X; !/ (see [5,16,23]).
There is a linear operator N W fcurrent of bidegree .q; q/ in Xg ! fcurrent of bidegree .q; q/ in X g; that is continuous for the weak topology and such that Â N.Â / is closed for every current Â of bidegree .q; q/ in where G is the Green operator. With the @-Laplacian associated with !, then I G is the orthogonal projection on the space of harmonic differential .q; q/-forms on X . Moreover @G D G@ and @G D G@. See [16,23] for the construction of G and for the continuity of G for the weak topology of currents and the L 1 loc topology. We set˛D A.Â /, then we can write˛D h 0 C @A 0 .˛/ C A 0 .@˛/ with harmonic h 0 , Since @ commutes with @ , then @ commutes with A 0 and we have Â D h C @@A 0 .˛/ C N.Â / with N.Â/ D A 0 .@@˛/ C A.@Â /. Then Â N.Â / D h C @@A 0 .A.Â // is closed in X . More we have @@˛D @@A.Â / D @Â @A.@Â / D @Â A.@@Â / since @ commutes with A. Thus N.Â/ D A 0 .A.@@Â // A 0 .@Â / C A.@Â / satisfies N.Â / D 0 if Â is closed.
There is a sequence .T l / of closed C 1 differential .q; q/-forms on X that weakly converge to T in X and satisfy T l u l , where .u l / is a sequence of C 1 differential .q; q/-forms on X that weakly converge to 0.
Proof. .U˛/ is a finite covering of X with open sets of coordinate maps and . ˛/ is a C 1 partition of the unit subordinate to .U˛/. With 0 of class C 1 approximating the Dirac mass at 0 in C n , we set which is a positive differential form of class C 1 in X weakly converging in X to T . So Â N.Â / is of class C 1 closed in X , weakly converges in X to T N.T / D T and is N.Â / with N.Â / weakly converging in X to N.T / D 0.
Note that N.Â / is smooth but not necessarily positive nor closed.

Closed regularization constructed from the formula of King
We will give a closed regularization of the current of integration built using an approximation formula of [18]. Let Z be an analytical subset of codimension q of X , let I O X be coherent ideal subsheaves such that supp .O X =I/ D Z, let .Z j / be the family of irreducible analytic components of Z of codimension exactly q and let m j 2 N be the generic multiplicity of I along Z j . So with a function C 1 in X always 0 such that 1 .0/ D Z built using a method of [9] to be reminded now. Let .U˛/ be an open finite covering of X and let f˛W U˛! C N˛b e a holomorphic map satisfying: for every x 2 U˛, the germs in x of the components of f˛generate the ideal I x and therefore f 1 .0/ D Z \ U˛. We denote by H˛.x/ a positive definite Hermitian N˛ N˛matrix dependent in the C 1 manner of x 2 U˛. We set there is a constant C 0 such that dd c log. C / C C ! 0 for all > 0.
Proposition 3.1. We have the approximation formula of the integration current Proof. Using the binomial theorem. Just look for k with 1 Ä k Ä q, we have weakly in X when ! 0 C . By the extension theorem of Skoda-El Mir, the Monge-Ampère operator .dd c log / q k is a differential form with L 1 loc coefficients in X. So C .dd c log. C // q k weakly converges in X to .dd c log / q k when ! 0 C .
We apply the Proposition 2.1 taking Â D C V . We set U D N.Â / and W D Â N.Â /, so W is a differential form of class C 1 , closed in X . When ! 0 C , Â weakly converges in X to P j m j OEZ j which is closed and therefore U weakly converges in X to 0. Then W weakly converges in X to with harmonic h convergent when ! 0 C in the space of differential forms of class C 1 of bidegree .q; q/ in X . As C V is positive and bounded by mass, by extracting a sequence, we can assume C V weakly converging to a positive current, thus a current with measure coefficients. Because of the orders of singularity in G and A D @ G, then G. C V / converges for the L 1 loc topology and A. C V / too. This implies that A 0 .A. C V // converges for the L 1 loc topology. Finally since V is closed and therefore N.V / D 0. But C V converges for the L 1 loc topology since . 1 2 dd c log / q D . P j m j OEZ j C . 1 2 dd c log / q jXnZ / and jZ D 0.
. C / 2 g^V converges for the L 1 loc topology. In effect @ C ^V converges for the L 1 loc topology since @ ^.
loc coefficients in X , as being the direct image by the blow up of X of center Z of a differential form with L 1 loc coefficients in Q X . Also converges for the L 1 loc topology. In the same manner @@.
for the orthogonal projection of Â D C V on the space of harmonic differential .q; q/-forms and note that cohomologically P j m j fZ j g D lim !0 fh g.

Restriction of a closed positive current to a closed complex submanifold
Suppose Z is a closed complex submanifold of codimension q in X and T is a closed positive current in X of bidimension .p 0 ; p 0 /. To define the restriction T jZ , we have to define the intersection T^OEZ under the formula i .T jZ / D T^OEZ where i W Z ! X is the canonical injection.
Proposition 4.1. With the notation of Proposition 3.1, there is a sequence l ! 0 C such that the positive not necessarily closed currents converge towards a closed positive current which is T^OEZ.
Proof. We use that Also a restriction T jZ can be expressed by means of T which is closed when T is closed.
A priori the limit T^OEZ in the Proposition 4.1 depends on the sequence l ! 0 C . 5 Closed regularization with negative part converging to 0 for the L 1 loc topology We now use the writing with ‚ a C 1 differential .q; q/-form closed in X and S a .q 1; q 1/-current in X . This is obtained from Proposition 2.1 which gives that T @@A 0 .A.T // is of class C 1 since T is closed.
There is a sequence .T l / of C 1 differential .q; q/-forms closed in X weakly converging in X to T and such that T l u l with .u l / a sequence of C 1 differential .q; q/-forms in X converging to 0 for the L 1 loc topology in X.
Proof. Since T is positive thus with measure coefficients, As a consequence @S is with L 1 loc coefficients and since S is real, @S too. In other words, one can choose S with L 1 loc coefficients, with @S and @S with L 1 loc coefficients. We set We have Q T Q u with Q u which converges for the L 1 loc topology necessarily to 0.
Note that Q u is smooth but not necessarily positive nor closed. Since Q u ! 0 for the L 1 loc topology when ! 0 C , there are l ! 0 C and g 0 and L 1 loc in X such that k Q u l k Ä g for all l (see the theorem of Fischer-Riesz in [12], Proposition 13.11.4 (ii)). The regularization obtained is therefore of the same type of the Dinh-Sibony regularization (see [13]).

Closed extension of a closed current defined on a closed complex submanifold
Suppose Z is a closed complex submanifold of codimension q in X and Â is a C 1 differential form closed in Z of bidegree .q 0 ; q 0 /. Let W Q X ! X be the blow up of center Z and let T be a closed current of bidegree .q 0 ; q 0 / in X . We define the restriction of T to Z by by denoting by H Q X the exceptional divisor. We are looking for T such that T jZ D Â and it will be so when . T / jH D˛with˛D . jH / Â which is closed in H . We will set T D S with S a current closed in Q X satisfying S jH D˛. This equality returns to Proof. Locally D z n e where e is a local frame of O.H / and OEH D C ı.z n /dz n^d z n : But D D dz n˝e C z n De and D D dz n˝e C z n De and therefore dz n^d z n D .dz n˝e /^.dz n˝e / D .D z n De/^.D z n De / D D ^D modulo terms containing z n or z n . For the other formula,˛is C 1 closed in H and there isˇa differential form The aim is therefore of finding S closed in Q X satisfying j ˛D S^OEH ,ˇ ^D ^D D S ^D ^D : with ‚ the .1; 1/-form of curvature of the connection D. Thus u; v; w; t must satisfy the linear differential equation of order 1 Then we can assume that U D V and W D T and a solution S will be real if u D v and w D t . The differential of ˝.
is the adjoint of ˝.Du C ‚w C U / C D ˝.u Dw C W / and its differential is the adjoint of ˝.DU C ‚W / C D ˝.U DW / which is The existence of a solution to the equation (3) therefore causes the necessary condition But we know that ˝U C V˝ C D ˝W C T˝D is exact thus closed. Thus it is known that i.e. that equation (4) is satisfied. S is first constructed in Q X n H ie in 6 D 0 and then extends as a closed current in Q X .
Proposition 6.2. There is a current T closed in X of bidegree .q 0 ; q 0 / such that T jZ D Â .

Formula of King
The restriction W H D P .N X Z/ ! Z of the blow up in the exceptional divisor H is a fibration and jH D i ı where i W Z ! X is the canonical injection. For y 2 Q X the differential d .y/ W T y Q X ! T .y/ X satisfies d .y/.T y H / T .y/ Z if y 2 H . Hence there is an induced map We will now express the term .dd c log k k/ q jXnZ considering the exact sequence For 0 Ä k Ä q 1, using Proposition 6.2, we can write where Â q k is a closed current in X of bidegree .q k; q k/. Since i ı D ı j we have with ' a closed current in X of bidegree .q; q/ which satisfies f' jZ g D 0. Replacing Â q by Â q ', we can assume Let m be an integer such that .k k m w/ is C 1 in X and 0 on Z, so Proposition 7.1. With Â q k the currents closed in X defined by (5), we can write King's formula (see [17,20]) expresses here that the differential form .log ksk/ dd c log ksk q 1 which is C 1 in X X Z is with locally integrable coefficients in X and that where .Z j / still refers to the family of irreducible analytic components of Z of dimension p exactly and m j 2 N is the generic multiplicity of I along Z j . The current dd c ..log ksk/ dd c log ksk q 1 / is also denoted by .dd c log ksk/ q and can be obtained from a Monge-Ampère operator in the sense of [10,14]. Now we will express the term dd c log ksk q jXXZ following the usual method of Bott-Chern (see [7,8]) that we will remember.
Consider an exact sequence 0 ! L ! E ! Q ! 0 of holomorphic vector bundles on X with rk L D 1. The C 1 Hermitian metric on E induces metrics on L and Q. Denote by ‚ E ; ‚ L and ‚ Q the .1; 1/-forms of curvature of the Chern connections respectively on E; L and Q.
Denoting by c.‚ E / D P k c k .‚ E / the total Chern form associated to ‚ E , we have to make explicit a solution ' of class C 1 in X of the equation where we did not include here the sign^. To define ' we use the following notations: Hom.E; E/ D E˝E injects itself into the exterior algebra V .E˚E / and the total Chern of ‚ E is then written as c.
With v a holomorphic local frame of L, v 2 E the adjoint, we set D vv kvk 2 and˛D DvDv kvk 2 where D means the Chern connection on E. We can then take We have also c.‚ Q / D c.‚ L / 1 c.‚ E / C dd c .c.‚ L / 1 '/ and since c.‚ L / D 1 C c 1 .‚ L /, we have . c 1 .‚ L // k^' q k 1 with ' q k 1 the component of bidegree .q k 1, q k 1/ of '.
Let W Q X ! X be the blow up of X along I and let H be the exceptional divisor. We take L the line sub-bundle of E such that I D O.L / (see [21]). Let us apply the above to the exact sequence on Q X and take the direct images by . Since where D P 0ÄkÄq 1 dd c log ksk k^' q k 1 and ' is given by (6) with D ss ksk 2 and˛D DsDs ksk 2 . Being the direct image by the blow up of a differential form C 1 in Q X , as c q .‚ E=Cs / is with L 1 loc coefficients in X . Then, since in X X Z log ksk.dd c log ksk/ k 1^c q k .‚ E / C : Then the following result generalizes the Poincaré-Lelong formula, a generalization is also due to Andersson (see [1,2]). The differential forms c q .‚ E=Cs / and 0 are C 1 in X X Z, are with L 1 loc coefficients in X and we have in X the equality between currents with each F i a holomorphic vector bundle over X, with a Hermitian metric h 0;i of class C 1 in X. Denote by h 0;0 the Hermitian metric induced on E by that of E.
In X X Z, we have an exact sequence of holomorphic vector bundles 0 ! F n g n ! F n 1 g n 1 ! : : : we can break into short exact sequences of holomorphic vector bundles x/ for x 2 X and we denote by h i the metric induced in F i . We will now still apply the Bott-Chern calculations in the case of an exact sequence but this time without any assumption on rk L. We can then take and W E ! E the orthogonal projection onto L. We can write in X X Z the equalities for 1 Ä i Ä n 2 and c.‚ K n 2 ;q n 2 / D c.‚ F n ;h n 1 / 1 c.‚ F n 1 / 1 C dd c c.‚ F n 1 / 1ˆn 1 noting q i the quotient metric on K i D F iC1 =K iC1 andˆi the differential form ' of (9) for the exact sequence with M t the Hermitian metric .1 t /q i C t h i on K i and with L K i ;t the endomorphism of K i for writing the Hermitian form h i q i with M t (see [7], page 83), we have with K n 1 D F n and c.‚ K n 1 ;q n 1 / D c.‚ F n /, the metric q n 1 being obtained from the exact sequence 0 ! F n ! K n 1 ! 0 and with c.‚ K 0 / D c.‚ K 0 ;h 0 /, the metric h 0 being obtained from the exact sequence 0 ! K 0 ! E ! C ! 0 which amounts to the exact sequence 0 ! Cs ! E ! E=Cs ! 0.
We equip E with the Hermitian metric h dual of the Hermitian metric h 0 on F 0 D E and E=Cs with the Hermitian metric induced by h. So c q .‚ K 0 / D . 1/ q c q .‚ E=Cs ; h/ and we get a differential .q 1; q 1/-form ‰ of class C 1 in X X Z such that in X X Z, we have To prove that ‰ is with L 1 loc coefficients in X and that the equality holds in X in the sense of currents, MacPherson's construction using the graph (see [3] page 120, [15]) gives the following result. There is an exact sequence 0 ! G n n ! G n 1 n 1 ! : : : Proof. Suppose X connected and let N i be the rank of F i . So for x 2 X X Z the graph of g i .x/ is a vector subspace of dimension N i of .F i˚Fi 1 / x . Consider the Grassmann bundles G.N i ; F i˚Fi 1 / ! X and the fiber product G D G.N n ; F n˚Fn 1 / X : : : X G.N 0 ; F 0˚F 1 / ! X: This gives a holomorphic map " W X X Z ! G. Note i the tautological vector bundle over G.N i ; F i˚Fi 1 / and pr i i its inverse image over G which is a holomorphic vector sub-bundle of rank N i of .F i˚Fi 1 /. Since for x 2 X X Z, .pr i i / ".x/ ' . F i / ".x/ and the complex of the F i is exact in X X Z so we have According to [3], page 122, with ".X X Z/ the closure of ".X X Z/ in G, then W ".X X Z/ ! X is the blow up of X along Z and in the K-theory of Q X , we still have Over Q X , we still have pr 2 .G i / G i 1 and therefore When Z is singular, Q X can be singular and we reduce to the case when Q X is smooth considering a modification of Q X which is a desingularization of Q X . The exact complex of holomorphic vector bundles on Q X remains exact when the inverse images over the desingularization are taken.
Let's break down the exact sequence 0 ! G n n ! G n 1 n 1 ! : : : with the Hermitian metric induced by the initial Hermitian metric of F i˚ F i 1 .

So in Q
X X H we have short exact sequences and the Hermitian metric on F i induced by h i and that obtained by the isomorphism with G i are the same. So for 0 Ä k Ä n we have the equality between Chern forms and given by the formula (9) written for the exact sequence 0 given as in formula (10).
We finally obtain the following result which explicitly gives a Green form of the algebraic cycle P j m j Z j .
Proposition 8.2. The differential forms ‰ and D . 1/ q 1 ‰ C 0 C .log ksk/.dd c log ksk/ q 1 are with coefficients of class C 1 in X X Z, L 1 loc in X and we have in X in the sense of currents E being equipped with the Hermitian metric h and each F i with the Hermitian metric h i . Since O X =I/C: : :, Proposition 8.2 gives effectively the formula of Grothendieck P j m j fZ j g D . 1/ q 1 c q .O X =I/ expressing the cohomology class of the algebraic cycle P j m j Z j using the q th Chern class of the coherent sheaf O X =I.
Note also that if one uses only the existence of an exact sequence 0 ! G n n ! G n 1 n 1 ! : : : of holomorphic vector bundles on Q X satisfying G ij Q X XH isomorphic to . F i / j Q X XH with the property (iii) of Proposition 8.1, then one has for any Hermitian metric on G i , instead of the previous equality c k .
where A i;k is a differential .k 1; k 1/-form with coefficients L 1 loc in Q X , j W H ! Q X is the canonical injection and R i;k is a .k 1; k 1/-current closed in H . The formula (13) becomes so that c.‚ F i / is replaced by c.‚ F i / C P k j R i;k and Proposition 8.2 does not generalize a priori because in order to calculate L ‰ from the formula generalizing the formula (11), we must make products of currents. Let as in the proof of Proposition 5.1 .U˛/ be a finite covering of X with open sets of coordinate maps and let . ˛/ be a C 1 partition of the unit subordinate to .U˛/. With 0 of class C 1 approximating the Dirac mass at 0 in C n , we set D X ˛ . jU˛/ which is a differential form of class C 1 in X , weakly converging in X to . Set In conclusion, we have the following result.
Proposition 8.3. The differential forms Q W are of class C 1 closed in X , weakly converge in X to P j m j OEZ j and satisfy Q W Q U where the Q U are of class C 1 in X and converge to 0 for the L 1 loc topology, when ! 0 C .
Proof. See the proof of Proposition 5.1.
In particular since Q U ! 0 for the L 1 loc topology when ! 0 C , there are a sequence l ! 0 C and a function g 0 and L 1 loc in X such that k Q U l k Ä g for all l.

Closed regularization with bounded negative part
Using the formula of King, we will show for the current P j m j OEZ j of integration the existence of a closed regularization with bounded negative part.
First log ksk is a quasi-plurisubharmonic function in X because dd c log ksk C f i 2 ‚ E .s/;sg ksk 2 0 where ‚ E 2 C 1 1;1 .X; E˝E / is the curvature of E, that satisfies i ‚ E Hermitian. We choose u a differential .1; 1/-form C 1 closed positive in X such that ksk 2 u f i 2 ‚ E .s/; sg. For > 0 we have by the Cauchy-Schwarz inequality

0.
King's formula is written as T with T D . i 2 @@ log.ksk 2 C / C u/ q R where R is real C 1 closed, weakly converging in X to .dd c log ksk C u/ q jX XZ .
Proof. If Â is a .n q; n q/-current in X , according to Poly, Â exists as a current in Q X , in the sense that Â is a current such that Â D Â (see [19]). But Â ! Â is not weakly continuous. By Proposition 4.1 if Â 0 is closed, then Â is 0 and closed. If Â D d with of class C 1 , then For Â not necessarily exact, Â will be replaced by Â j Â where Â satisfies .j Â / D 0 and Â D if Â D d . This will still be an inverse image of Â by , in the sense that its direct image by will be equal to Â .
We have the formula If there is R closed of class C 1 satisfying lim !0 R X R ^Â D L.Â / for every current Â 0 with supp Â [ j Z j and for every Â smooth, then in particular L.Â 0 Â 1 / D 0 when Â 0 0 and Â 1 0 are such that supp Â i [ j Z j and Â 0 Â 1 D d with a current. This necessary condition is satisfied.
Thus one concludes that there is R closed C 1 satisfying lim !0 R X R ^Â D R Q X . C u/ q^ Â 2 R for every current Â 0 with supp Â [ j Z j and weakly converging to .dd c log ksk C u/ q jX XZ . This does not necessarily mean the convergence of the integral R X .dd c log ksk C u/ q jXXZ^Â since the R ^Â do not necessarily converge in X to the product .dd c log ksk C u/ q jXXZ^Â of currents. We have and this last integral can be defined by means of the Hörmander-Lojasiewicz division theorem, when Â is a current. Thus by changing the first term in T , we can assume that lim !0 R X T ^Â exists in R for every current Â 0 in X with supp Â [ j Z j .
With ! a Kähler metric in X , set C.Â / D sup >0 . R X T ^Â R X ! q^Â /. If there is a sequence of smooth differential forms Â j 0 such that lim j C.Â j / D C1, replacing Â j by . R X ! q^Â j / 1 Â j which is bounded by mass and using an argument of double limit, it can be assumed Â j converging to a current Â 0 such that lim !0 R X T ^Â is Ä thus D 1 and such that supp Â [ j Z j by an application of the Lebesgue-Nikodym theorem. This is a contradiction and there exists a constant C 0 such that R X T ^Â Ä C R X ! q^Â for every smooth differential form Â 0. Finally T C ! q .
Proposition 9.2. There exists a sequence T l of closed smooth differential .q; q/-forms in X that weakly converge in X towards P j m j OEZ j and satisfy T l C ! q for all l, where C is a certain constant 0.
Remark. Since the restriction of OEZ in X X Z is 0, we can write OEZ D lim !0 T with T smooth closed in X such that (i) R X T ^Â converges for all positive current Â in X with supp Â Z, (ii) R X T ^Â converges to 0 for all current Â in X with compact support in X X Z. As a consequence, T j converges to 0 in the space of smooth differential .q; q/-forms in , for all relatively compact open subset X X Z.
Now for T a closed positive current of bidegree .q; q/ in X , using the formula T D pr 1 .OED X ^pr 2 T / with D X the diagonal in X X , we can write T D lim !0 T with T smooth closed in X satisfying T C ! q for all > 0, where C is a certain constant 0. Moreover if T is smooth on an open subset U X , then T j converges to T j in the space of smooth differential .q; q/-forms in , for all relatively compact open subset U . For the proof, we write OED X D lim !0 with smooth closed in X X such that C 0 .pr 1 ! C pr 2 !/ n for all > 0, with C 0 0. We set T D pr 1 . ^pr 2 T / which is smooth closed in X and weakly converges in X towards T . Moreover T C 0 pr 1 .pr 1 ! C pr 2 !/ n^p r 2 T and using the binomial theorem we have pr 1 .pr 1 ! C pr 2 !/ n^p r 2 T D n X kD0 C k n pr 1 .pr 1 ! k^p r 2 ! n k^p r 2 T / D n X kD0 C k n ! k^p r 1 pr 2 .! n k^T /: Since ! n k^T is a current on X , we can take n k C q Ä n by considering the bidegree. Since the fibers of pr 1 are of dimension n, we can take n Ä n k C q. In such a way, we are reduced to take k D q, therefore we conclude that T C 0 C q n ! q pr 1 pr 2 .! n q^T / where the constant pr 1 pr 2 .! n q^T / is the volume of T with respect to !. On the other hand, let be a smooth function on X with compact support in U , equal to 1 on an open neighbourhood of . We write T D pr 1 ^pr 2 . T / C pr 1 ^pr 2 .1 /T and we use the hypocontinuity of the convolution of a distribution by a smooth function with compact support. Since supp . T / U , T is smooth in X thus pr 1 . ^pr 2 . T // converges to T in the space of smooth differential .q; q/-forms in X and . T / j D T j . For the second term, we use that if x 2 and y 2 supp ..1 /T /, then y 6 2 f D 1g, thus .x; y/ 6 2 D X . Since converges to 0 in the space of smooth differential forms on every relatively compact open subset .X X / X D X , this second term converges to 0 in the space of smooth differential forms on .