DESINGULARIZATION OF QUASIPLURISUBHARMONIC FUNCTIONS

Let T be a positive closed current of bidegree (1,1) on a compact complex surface. We show that for all e > 0, one can find a finite composition of blow-ups π such that π*T decomposes as the sum of a divisorial part and a positive closed current whose Lelong numbers are all less than e.


Introduction
Let X be a compact complex surface (i.e.dim C X = 2).It is a classical result that if C is a complex (singular) curve in X, then one can find π : X → X a finite composition of blow-ups such that the strict transform C of C is smooth.
Let [C] denote the current of integration along the curve C. It has Lelong number 0 at points in X\C, 1 at regular points of C, and Lelong number ≥ 2 at every singular point.Thus the above result can be seen as an attenuation of singularities, in the sense that [ C] has all its Lelong numbers ≤ 1.More generally, given T a positive closed current of bidegree (1, 1) on X, one can wonder whether it is possible to attenuate its singularities.Denote by ν(T, p) the Lelong number of the current T at point p (we refer the reader to [4] for basic facts on positive currents and Lelong numbers).The aim of this note is to give an elementary proof of the following result: Theorem 1.1.Let T be a positive closed current of bidegree (1, 1) on a compact complex surface X.For any ε > 0, there exists π : X → X a finite composition of blow-ups such that where the Cj 's are smooth curves with normal crossings, c j ≥ 0 and T is a positive closed current of bidegree (1, 1) on X such that sup x∈ X ν( T , x) < ε.
A semi-local version of this result was first obtained by Mimouni [11].Theorem 1.1 has been also proved independently by Favre and Jonsson (see [7,Theorem 7.2]) by using an interesting (but difficult) analysis on the "valuative tree" [6].Their proof also works in a local context.The proof we present here is quite elementary and follows the approach of Mimouni.
When T = [C] is the current of integration along a complex curve, then c j = 1 automatically and T = 0 as soon as ε ≤ 1, so Theorem 1.1 is indeed a generalization of the desingularization of curves.
When the surface X is Kähler, there is an alternative way of stating this result.Let us recall that a quasiplurisubharmonic function (qpsh for short) is an uppersemi-continuous L 1 -function ϕ on X whose curvature is bounded from below by a smooth form, say where ω is a smooth closed (1, 1)-form on X and d = ∂ + ∂, d c = i π ( ∂ − ∂) are both real differential operators.Such functions are locally given as the difference of a plurisubharmonic function and a smooth function.Thus qpsh functions, once normalized, are in 1-to-1 correspondence with positive closed currents of bidegree (1, 1) on X: one can associate to ϕ the positive current for some qpsh function ϕ (uniquely determined up to an additive constant): this is the celebrated "dd c -lemma" of Kähler geometry (see e.g.[8, p. 149]).Our result can therefore be interpreted as an attenuation of singularities for qpsh functions.This is of practical importance in complex geometry, for example, where such functions arise as positive (singular) metrics of holomorphic line bundles.We refer the reader to [5,9] for more information on this point of view.

Proof of the Theorem
In the sequel we let T (X) denote the cone of positive closed currents of bidegree (1, 1) on X. Basic facts on positive currents can be found, e.g. in [4].Of crucial importance is the following decomposition result of Siu [12]: if T ∈ T (X), then T can be written where the C j 's are (singular) complex (closed) curves in X, the c j 's are non-negative constants, and T 0 ∈ T (X) does not charge any proper analytic subset of X. Equivalently, the set is at most countable.We shall say in the sequel that a current T 0 is diffuse when it does not charge curves.
We start by recalling a classical result linking the Lelong number of a current T at a point p and Lelong numbers of the total transform π * T under the blow-up π at point p.Lemma 2.1.Let π : X → X be the blow up of X at point p and let E = π −1 (p) denote the exceptional divisor.If T ∈ T (X), then where T ∈ T ( X) does not charge E.
Proof.It follows from Siu's decomposition result that π * T = c[E] + T where c ≥ 0 and T ∈ T ( X) does not charge E. Note that π is a local biholomorphism near each point q ∈ X \ E, hence ν( T , q) = ν(T, π(q)).
Let us emphasize that these numbers depend on the choice of local coordinates (actually on the choice of axes), however this ambiguity is fixed here by the choice of the direction q ∈ E P 1 .We refer the reader to [10,4] for basic facts on Kiselman numbers.
The next lemma is an elementary observation that is crucial for the proof of Theorem 1.1.

Lemma 2.2. Assume
where {T } 2 denotes the self-intersection of {T } ∈ H 2 (X, R), the cohomology class of T .
Proof.It follows from the work of Demailly [3] that there exists C > 0 such that for all ε > 0, one can find closed real currents T ε cohomologous to T such that We infer that the currents T ε have local potentials whose gradients are locally in L 2 .One can therefore define the pointwise wedge-product of T ε with itself (see [1,2]) which is bounded from below by −C 2 ε 2 ω 2 .On cohomology classes this yields Proof of Theorem 1.1.Fix T ∈ T (X) and ε > 0. It follows froms Siu's decomposition result that where the C j 's are (possibly singular) curves, c j ≥ 0 and T 0 is a diffuse current.
Step 1. Set It is another result of Siu [12] that E ε (T 0 ) is a closed proper analytic subset of X.Since T 0 is diffuse, this is a finite set of points.If this set is empty, this is the end of Step 1. Otherwise we label these points, where Consider now E ε (T 1 ).If this set is empty we stop here, otherwise E ε (T 1 ) = {p 1  1 , . . ., p 1 s1 } is finite.Let π 2 : X 2 → X 1 denote the blow-up of X 1 at these points.Then where E 2 j = π −1 2 p 1 j and T 2 ∈ T (X 2 ) is again a diffuse current.We infer from Lemma 2.2 Observe that T 2 can be written where the E 2 j 's denote, for 1 + s 1 ≤ j ≤ s 1 = s 0 + s 1 , the strict transform of the E 1 j 's under π 2 .Going on by induction, we end up with Thus the series on the left-hand side is convergent, so sup x∈X ν(T n , x) < ε for n large enough.This ends Step 1.
Step 2. Going back to our current T , what we have proved so far writes where sup x∈Xn ν(T n , x) < ε and the C j 's are either exceptional divisors or strict transforms of the curves C j .Observe now that E ε (π * T ) is a proper analytic subset of X n , so only finitely many C j 's may produce Lelong numbers ≥ ε.Thus we can rewrite (after possibly relabelling) where T n is not necessarily diffuse anymore but satisfies sup x∈Xn ν(T n , x) < ε.It only remains to blow-up a few more times in order to desingularize the C j 's.This may of course modify the current T n , but it will not increase the Lelong numbers thanks to Lemma 2.1.
Remark 2.3.In [11], Mimouni considers psh functions ϕ in the unit ball B of C 2 and obtains the same attenuation result under the hypothesis that ϕ is locally bounded near the boundary ∂B.Note that such functions can easily be extended as qpsh functions on the complex projective space X = P 2 , hence Mimouni's result can be seen as a particular case of Theorem 1.1.