Weak Convergence of Currents

Weak Convergence of Currents Yang Xing

Abstract. We give certain condictions to guarantee weak convergence uk Tk → uT , where uk , u are plurisubharmonic functions, and Tk , T are positive closed currents. As applications we obtain that convergence in capacity of plurisubharmonic functions uk implies weak convergence of the complex Monge-Amp`ere measures (ddc uk )n if all of the plurisubharmonic functions uk are bounded from below by one of some sorts of plurisubharmonic functions.

1. Introduction Denote by P SH(Ω) the set of plurisubharmonic (psh) functions in a domain Ω in Cn . Write d = ∂ + ∂ and dc = i(∂ − ∂). The standard K¨ahler form is β = ddc |z|2 and we also write βp = β p /p!. Let T be a positive closed current of bidimension (p, p) in Ω. It is well known that one cannot define the wedge product ddc u ∧ T without any problem for all of psh functions u in Ω, see an example of Kiselman in [K]. However, it is very useful in algebraic geometry and many other mathematical branches to study the product ddc u ∧ T for some kinds of psh functions u. Approximation theorems serve as an important tool in the theory. Fornaess and Sibony have obtained several results in this direction, see [F-S]. We also like to mention that one of main purpose in study of such currents is to study the complex Monge-Amp`ere operator (ddc )p . For a locally bounded psh function u in Ω, the positive closed current (ddc u)p is defined in [B-T2] inductively by the equality 8

Ω

c

p

(dd u) ∧ φ =

8

u(ddc u)p−1 ∧ ddc φ,

Ω

for every test form φ of bidegree (n − p, n − p) in Ω. Bedford and Taylor have proved that the operator (ddc )p is continuous under monotone limits for bounded psh functions. The monotonicity hypothesis of the sequence of functions in their result is essential , but not necessary. In [X1]-[X3] for some kinds of psh functions we use convergence of functions in some capacity to get weak convergence of Monge-Amp`ere measures of the functions. Capacity provides us an effective tool in study of the Monge-Amp`ere operator. It is known that for psh functions uk , u and positive closed currents Tk , T weak convergences uk → u and Tk → T can not imply weak convergence uk Tk −→ uT , see Cegrell’s example in [C1]. In this paper we give several sufficient condictions of weak convergence uk Tk −→ uT in 1

terms of capacity. Our approximation theorems strengthen some results from [F-S] and [D]. The assumptions given in our results are quite sharp. We also use Hausdorff s-content  s as a tool to obtain an approximation theorem. We know that weak convergence of psh H functions uk to u is very strong type of convergence, for instance, it implies convergence of the functions uk to u in Lploc for any p ≥ 1. But it is still not good enough to guarantee the weak convergence of (ddc uk )n to (ddc u)n . In our opinion the essential reason of this is that weak convergence of psh functions uk to u is only equivalent to convergence of  s for some s > 2n − 2, whereas one needs convergence of the uk to u the uk to u in H  2 to get (ddc uk )n −→ (ddc u)n . Finally as an application we also prove that at least in H convergence in capacity of psh functions uk implies weak convergence of Monge-Amp`ere measures (ddc uk )n if all of the functions uk are bounded from below by some psh function introduced by Bedford and Taylor in [B] or by a fixed psh function g, which is bounded near the boundary of Ω. This result generalizes an approximation theorem in [X3] to unbounded psh functions. 2. Weak Convergence of Currents uk Tk In this section we shall use capacities and Hausdorff contents as tools to study weak convergence of currents of type uk Tk . For a positive closed current T of bidimension (p, p) e e $ and a given subset E of Ω, we define the trace measure eT eE (E1 ) = E∩E1 T ∧ βp for any subset E1 in Ω. It is well known that the current uT cannot be defined for all of psh functions u without loss of some of the essential properties that currents should have, see [K]. We begin with a characterization of well defined currents uT , where u are certain unbounded psh functions. Theorem 1. Assume u ∈ P SH(Ω) and assume that T is a positive closed current of bidimension (p, p) in Ω. Then the following assertions are equivalent. ∞ e e  eT e (u < −j) < ∞ for each subset E ⊂⊂ Ω. (i) E j=0

(ii) The function u is locally integrable in Ω with respect to T ∧ βp . (iii) There exists E0 ⊂⊂ Ω such that the function u is locally integrable in Ω \ E0 with respect to T ∧ βp .

Proof. We assume without loss of generality that the e u e is negative in Ω. We first prove e (i) ⇒ (ii). The convergence assumption implies that T eE has zero mass on the pluripolar set {u = −∞}. On the other hand, it follows from Abel transformation in [Z, p.3] that for any E ⊂⊂ Ω and any positive integer J 8

(−u) T ∧ βp =

{−J≤u 0 such that Cs1 (E1 ) ≤ AE Cs2 (E1 ) for all E1 ⊂ E. Hence, by Theorem 3.5 in [B-T2] we obtain that any psh function u in Ω is quasicontinuous with respect to Cs for all s = 0, 1, . . . , n, that is, for any constant ε > 0 there exists a subset Eε ⊂ Ω with Cs (Eε ) < ε such that the restriction of u on Ω \ Eε is continuous. On the other hand, there exists a smooth, real closed curve in Cn with a positive capacity Cn , see 3

[M ]. Therefore, by Proposition 1 of this paper we have that the capacity Cn cannot be controlled by the capacity Cn−1 . Recall that positive measures μk are said to be uniformly absolutely continuous with respect to capacity Cs on a set E, or we write that μk Cs on E uniformly for all k, if for any constant ε > 0 there exists a constant δ > 0 such that for any E1 ⊂ E with Cs (E1 ) < δ the inequality μk (E1 ) < ε holds for all k. Recall also that a sequence of functions uk is said to be convergent to a function u in‚ capacity Cs on a set \ E if for each constant δ > 0 we have Cs z ∈ E; |uk (z) − u(z)| > δ −→ 0 as k → ∞. In [X3] we obtain that if locally uniformly bounded psh functions uk converge to a psh function u in capacity Cn−1 on each E ⊂⊂ Ω, then (ddc uk )n → (ddc u)n weakly in Ω. Generalizations of this result to some kinds of unbounded psh functions can be found i [X1] and [X2]. Now we prove the following approximation theorem for currents of type uT . Theorem 2. Suppose that T and Tk , where k = 1, 2, . . ., are positive currents of bidimension (p, p) in Ω and suppose that psh functions u and uk are locally integrable with respect to T ∧ βp and Tk ∧ βp in Ω respectively. If there exists an integer 1 ≤ s ≤ n such that (a) uk → u in Cs on each E ⊂⊂ Ω, (b) Tk −→ T as currents in Ω, (c) |u| T ∧ βp Cs on each E ⊂⊂ Ω and |uk | Tk ∧ βp Cs on each E ⊂⊂ Ω uniformly for all k, then uk Tk −→ u T and hence ddc uk ∧ Tk −→ ddc u ∧ T as currents in Ω. Proof. Without loss of generality we can assume that all the uk and u are negative in Ω. For each E ⊂⊂ Ω and each a > 0 we have 8 e8 e D i e e max(uk , −a) Tk + max(u, −a) T ∧ βp e e (uk Tk + u T ) ∧ βp − E

E

e e =e

8

(uk + a) Tk ∧ βp −

E∩{uk δ} inequality and the Chern-Levine-Nirenberg estimate in [B-T2] that there exists A > 0 such that 8

(ddc w)n−p ∧ βp ≤ A

w8

D

(u − uk )ddc (uk − u) ∧ ddc (uk + u)

Ω

{|uk −u|>δ}

in−p−1

∧ βp

W1/2n−p

for all k, δ and such functions w, see the proof of Theorem 2 in [X3]. It follows from quasicontinuity of psh functions and Hartog’s Lemma that for any ε > 0 there exists k0 > 0 and U ⊂ E with Cn−p (U ) < ε such that u(z) + ε ≥ uk (z) in Ω \ U for all k ≥ k0 . Since all the uk and u are locally uniformly bounded, then for k ≥ k0 the last integral does not exceed 8 p Q D in−p−1 (ε + u − uk )ddc (uk − u) ∧ ddc (uk + u) ∧ βp + O Cn−p (U ) + ε Ω\U

D i2 ≤ (n − p)!

8

E\U

(ε + u − uk )

n−p 3 l=0

(ddc uk )l ∧ (ddc u)n−p−l ∧ βp + O(ε)

D i2 8 D i (n − p)! = |z|2 (ddc u)n−p+1 − (ddc uk )n−p+1 ∧ βp−1 + O(ε), p Ω

5

where the integral in the last sum converges to zero as k → ∞ since (ddc uk )n−p+1 − (ddc u)n−p+1 tends weakly to zero and has a compact support in Ω. Hence, we have proved that uk → u in Cn−p on Ω and the proof of Theorem 3 is complete. Using the characterization of currents given Theorem 1, we can give another version of Theorem 2. Theorem 4. Let T and Tk be positive currents of bidimension (p, p) in Ω. Suppose that uk ∈ P SH(Ω) and that u ∈ P SH(Ω) is locally integrable with respect to T ∧ βp in Ω. If there exists an integer 1 ≤ s ≤ n such that (a) uk → u in Cs on each E ⊂⊂ Ω, (b) DTk −→ Ti as currents in Ω, Cs on each E ⊂⊂ Ω uniformly for all k, (c) T + Tk ∧ βp ∞ e e  eTk e (uk < −j) < ∞ are uniformly convergent for all k on each subset (d) E j=0

E ⊂⊂ Ω, then uk Tk −→ u T as currents in Ω.

Proof. It follows from Theorem 1 and (d) that each currents uk Tk has locally bounded mass in Ω. Since |u| T ∧ βp 0 we take balls B(zj , rj ) such that B(zj , rj ) ⊃ E1 and

 j

j

2(n−s) rj

 2(n−s) (E1 ) + ε. Assume that u ∈ P SH(Ω) satisfies 0 < u < 1. For 0, see [L]. The last inequality 2(n−s) B(z,r) r is trivial when rj ≥ δ. By the Chern-Levine-Nirenberg inequality in [B-T2] we have $ $  2(n−s) $ 1 (ddc u)s ∧βn−s ≤ (ddc u)s ∧βn−s ≤ δ2(n−s) rj (ddc u)s ∧βn−s ≤ E1 E1 ∩B(zj ,rj ) 2δ j j E D i  AE,δ H2(n−s) (E1 ) + ε with some constant AE,δ independing of u. Taking supremum over i D  2(n−s) (E1 ) + ε , which is what we need all of such functions u one gets Cs (E1 ) ≤ AE,δ H to prove.

Using Proposition 1 for s = n − 1 and Corollary 1 for q = n we get that if a se 2 then quence of functions uk in a Bedford’s and Taylor’s class converges locally to u in H c n c n (dd uk ) −→ (dd u) weakly. However, the following proposition shows that weak conver s for some s > 2n − 2, and thus is gence of psh functions is equivalent to convergence in H not good enough to quarantee weak convergence of (ddc uk )q for any q ≥ 2. Proposition 2. Suppose that uk , u ∈ P SH(Ω) and that the sequence {uk } is bounded in  s on each E ⊂⊂ Ω for some L1loc (Ω). Then uk → u weakly in Ω if and only if uk → u in H s > 2n − 2

Proof. The “if” part is clear since the set {uk } is bounded and hence relatively compact in L1 (Ω). The “only if” part follows essentially from the proof of Theorem 4.4.1 in [A]. We present now a similar result of Theorem 4 by means of the Hausdorff s-content. Theorem 5. Let T and Tk be positive currents of bidimension (p, p) in Ω. Suppose that uk ∈ P SH(Ω) and that u ∈ P SH(Ω) is locally integrable with respect to T ∧ βp in Ω. If we have 8

 2p on each E ⊂⊂ Ω, and u is quasicontinuous with respect to uk → u in H  2p in Ω, H (b) Tk −→ T as currents in Ω, ∞ e e  eTk e (uk < −j) < ∞ are uniformly convergent for all k on each subset (c) E (a)

j=0

E ⊂⊂ Ω, then uk Tk −→ u T as currents in Ω.

Observe that (c) in Theorem 5 holds for any bounded sequence of functions uk . It then follows from Hartog’s Lemma that Theorem 5 is a slightly stronger version of Theorem 3.9 in [F-S]. $ Proof. For any fixed point z ∈ Ω, since r12p B(z,r) (T + Tk ) ∧ βp are increasing functions of r, see [L, p.73], it turns out from (b) that the measures (T + Tk ) ∧ βp are uniformly  2p on each E ⊂⊂ Ω. Using a completely similar absolutely continuous with respect to H proof of Theorem 2 and Theorem 4 we get Theorem 5. Finally we will finish this section by giving another type of convergence theorem for currents. A slightly weak version of the following theorem is Proposition 3.2 in [F-S]. Theorem 6. Suppose that T and Tk , where k = 1, 2, . . ., are positive currents of bidimension (p, p) in Ω and suppose that negative psh functions u and uk are locally integrable with respect to T ∧ βp and Tk ∧ βp in Ω respectively. If we have (a) uk → u weakly in Ω, (b) Tk −→ T as currents in Ω, (c) there exist open subsets Ω1 ⊂ Ω2 ⊂ . . . ⊂ Ωj ⊂ . . . ⊂ Ω with Ωj = Ω such j $ $ that lim inf Ωj uk Tk ∧ βp ≥ Ωj u T∧ βp > −∞ for all j, k→∞

then uk Tk −→ u T as currents in Ω.

Proof. By Proposition 3.2 in [F-S] we know that every weak limit L of uk Tk satisfies L ≤ u T . On the other hand, let L = lim uki Tki as currents. By (c), for each j and any φ ∈ C0∞ i→∞ $ $ $ with 0 ≤ φ ≤ 1 we have Ωj φ L ∧ βp = lim Ωj φ uki Tki ∧ βp ≥ lim inf Ωj uk Tk ∧ βp ≥ i→∞ $ $ k→∞ $ u T β . Taking supremum over all of such functions φ, we get L ∧ βp ≥ Ωj u T∧ βp , ∧ p Ωj Ωj which implies L = u T on each Ωj and hence we have proved Theorem 6. 3. Weak Convergence of Currents (ddc uk )n In this section we give one more application of Theorem 2 to the complex MongeAmp`ere operator (ddc )n . Our purpose is to generalize the following convergence theorem in [X3] to psh functions with bounded values near the boundary of Ω. 9

Proposition 3. Suppose that uk are uniformly bounded psh functions in Ω for all k. If uk converge to a psh function u in capacity Cn on each E ⊂⊂ Ω, then (ddc uk )n → (ddc u)n weakly in Ω . We need two lemmas, which is a weaker version of ”integration by parts”. D i Lemma 1. If u, v ∈ P SH(Ω) satisfy lim inf u(z) − v(z) ≥ 0, then the inequality z→∂Ω

8

c

u