aa r X i v : . [ m a t h . D S ] S e p AN EQUIDISTRIBUTION THEOREM FOR BIRAITONAL MAPS OF P k TAEYONG AHNA
BSTRACT . We prove an equidistribution theorem of positive closed currents for a certainclass of birational maps f + : P k → P k of algebraic degree d ≥ satisfying S n ≥ f n − ( I + ) ∩ S n ≥ f n + ( I − ) = ∅ , where f − is the inverse of f + and I ± are the sets of indeterminacy for f ± , respectively.
1. I
NTRODUCTION
Let ω be a Fubini-Study form chosen so that R P k ω k = 1 . For an integer ≤ p ≤ k , C p denotes the space of positive closed ( p, p ) -currents of unit mass on P k where the mass isdefined by k S k := h S, ω k − p i for positive closed ( p, p ) current S on P k . For an open subset W ⊆ P k , C p ( W ) is the set of currents S ∈ C p with supp S ⋐ W .In [2], the following equidistribution theorem for regular polynomial automorphismsof C k was proved. Theorem 1.1 (Theorem 1.3 in [2], See also [1], [8]) . Let f : C k → C k be a regularpolynomial automorphism of C k of degree d ≥ and s > an integer such that dim I + = k − s − and dim I − = s − where I ± are the sets of indeterminacy of f, f − , respectively.Then, for an integer < p ≤ s , for S ∈ C p whose super-potential U S of mean is continuousnear I − , d − pn ( f n ) ∗ S converges to the Green ( p, p ) -current T p + for f in the sense of currentswhere T p + = lim n →∞ d − pn ( f n ) ∗ ω p . (For the notion of the local continuity and H¨older continuity of super-potentials, seeSection 3.) The motivation of this note is to futher study Theorem 1.1 in the case ofcertain birational maps of P k .Let f + : P k → P k be a birational map of algebraic degree d ≥ and f − its inverse.Let δ denote the algebraic degree of f − . Let I ± denote the indeterminacy sets of f ± and I ±∞ := S n ≥ f n ∓ ( I ± ) , respectively. Let s > be an integer such that dim I + = k − s − and dim I − = s − . The main theorem of this note is as follows: Theorem 1.2.
Let f + : P k → P k be a birational map of algebraic degree d ≥ such that I + ∞ ∩ I −∞ = ∅ and that there exists an open subset V ⊂ P k such that V ∩ I + ∞ = ∅ and I −∞ ⊂ f + ( V ) ⋐ V . Then, for an integer < p ≤ s , for every S ∈ C p whose super-potential isH¨older continuous in V , then we have d − pn ( f n + ) ∗ S converges to T p + in the sense of currentswhere T p + = lim n →∞ d − pn ( f n + ) ∗ ω p . If we assume I −∞ is attracting for f + , we obtain Date : September 22, 2020. Theorem 1.3.
Assume the hypotheses in Theorem 1.2 and further that I −∞ is attracting for f + . Then, for an integer < p ≤ s , for a generic analytic subset H of pure dimension k − p which means H ∩ I −∞ = ∅ , we have d − pn ( f n + ) ∗ [ H ] → cT p + in the sense of currents where c is the degree of H . Here, a compact subset A of P k is called an attracting set if it has an open neighborhood U , called a trapping neighborhood, such that f + ( U ) ⊆ U and A = S n ≥ f n + ( U ) where f n + := f + ◦ · · · ◦ f + , n -times.Among numerous studies on the birational maps on P k , listing some works related toequidistribution of inverse images of positive closed currents, in [4], Diller proved thatin P , the equidistribution is true for S ∈ C with supp S ∩ I −∞ = ∅ . In [6], Dinh-Sibonydefined notions of regular birational maps and P C p ( V ) -currents for an open subset V of P k , which is equivalent to its super-potential U S being continuous in P k \ V and provedthat if the initial current S ∈ C p satisfies a regularity condition in terms of P C p ( V ) , thenthe equidistribution is true in a certain open subset of P k where V is an open neigh-borhood of I + ∞ . In [10], De Th´elin-Vigny prove that for f + with I + ∞ ∩ I −∞ = ∅ , outsidea super-polar set of C s , equidistribution in Theorem 1.2 holds. They gave a sufficientcondition in terms of super-potentials for the equidistribution. The condition in [10] isnot stated in terms of the set I −∞ . In this note we focus on a sufficient condition in termsof the set I −∞ of critical values of f + as in [4] and equidistribution on the whole P k .The condition I + ∞ ∩ I −∞ = ∅ was introduced in [4] and [10]. From the dynamical viewpoint, that is, considering iteration of f ± , it seems reasonable to regard I −∞ for birationalmaps as a generalization of I − in Theorem 1.1 rather than I − alone. Along the samelines, the regularity condition in Theorem 1.1 may be translated into I + ∞ ∩ I −∞ = ∅ forbirational maps. If we compare the class of birational maps in this note, in [10] and in[6], ours contains the case of [6] and ours belongs to the case of [10].For the proof, we basically follow and refine the proof of Theorem 1.1 and 1.4 in[2]. The main difficulty is to get uniform estimates of R W S ∧ U Λ n ( R ) with respect to n ∈ N in a neighborhood W of I −∞ for smooth R ∈ C k − p +1 where U ( · ) denotes the Greenquasi-potential of a given current and Λ is a constant multiple of the operator ( f + ) ∗ . Forthis, we use the idea in the proof of Theorem 1.1 in [2], which essentially means thata Green quasi-potential can be approximated from below by a negative closed currentwith a small error. (See the proof of Proposition 2.3.6 in [8].) Also, there are subtledifferences between Theorem 1.2 and the case of regular polynomial automorphisms of C k . Firstly, we do not know whether for every U ⋐ P k \ I + ∞ , a super-potential of T p + iscontinuous in U . This is needed to bound the dynamical super-potentials from above.As a replacement for this, we will use a convergence appearing in a proof of Theorem3.2.4 in [10]. Next, in general, I − may not be invariant under f + and I −∞ may not be anattracting set. For the former part, we construct another invariant current for f + which isdenoted by R ∞ in Proposition 5.9 and for the latter part, the assumption of the existenceof the neighborhood V such that V ∩ I + ∞ = ∅ and I −∞ ⊂ f + ( V ) ⋐ V resolves the difficulty. In this note, the C α -norm k · k C α , the uniform norm k · k , the uniform norm k · k U on aset U are computed in terms of the sum of the coefficients of a given form with respectto a fixed finite atlas. 2. C URRENTS
In this note, we assume some familiarity of the reader to pluripotential theory andcurrents. For details, consult [3] and [11] for instance. In this section, we introducesome notions and notations that we will use in this note.Let ≤ q ≤ k and W an open subset of P k . The following spaces and norms areuseful in the study of currents. For instance, see [9], [5], [2]. Let D q be the realvector space spanned by C q and D q ( W ) the subspace of D q of currents R which arecohomologous to and satisfy supp R ⋐ W . We define k R k ∗ := inf {k R + k : R = R + − R − , R ± positive and closed } on D q ( W ) . Let e D q ( W ) := { R ∈ D q ( W ) : k R k ∗ ≤ } .The topology on e D q ( W ) is the subspace topology of the space of currents in W . Notethat the space e D q ( W ) is compact. The norm k · k ∗ bounds the mass norm. So, e D q ( W ) ismetrizable. More precisely, if γ > is a constant, we define for R ∈ e D q ( W ) k R k − γ := sup {|h R, φ i| , φ is a test form of bi-degree ( k − q, k − q ) with k φ k C γ ≤ } . In a similar fashion, we have
Definition 2.1 (See [7]) . Let φ : P k → P k be an L -function. We say that φ is aDSH function if outside a pluripolar set, φ can be written as a difference of two quasi-plurisubharmonic functions. Two DSH functions are identified if they are equal to eachother outside a pluripolar set.If φ is a DSH function on P k , we define the DSH-norm of φ by k φ k DSH := k φ k L + k dd c φ k ∗ . On P k , we have a good smooth approximation of positive closed currents. The follow-ing is from [8]. We will simply call it the standard regularization or the θ -regularizationof a current. Since Aut( P k ) ≃ PGL( k + 1 , C ) , we choose and fix a holomorphic chart suchthat | y | < and y = 0 at id ∈ Aut( P k ) . We denote by τ y the automorphism correspondingto y . We choose a norm | y | of y so that it is invariant under the involution τ → τ − .Fix a smooth probability measure ρ with compact support in { y : | y | < } such that ρ is radial and decreasing as | y | increases. Then, the involution τ → τ − preserves ρ . Let h θ ( y ) := θy denote the multiplication by θ ∈ C and for | θ | ≤ define ρ θ := ( h θ ) ∗ ρ . Then, ρ becomes the Dirac mass at id ∈ Aut( P k ) . We define for R ∈ C q , R θ := Z Aut( P k ) ( τ y ) ∗ Rdρ θ ( y ) = Z Aut( P k ) ( τ θy ) ∗ Rdρ ( y ) = Z Aut( P k ) ( τ θy ) ∗ Rdρ ( y ) . Note that R θ ∈ C q . Proposition 2.2 (Proposition 2.1.6 in [8]) . If θ = 0 , then R θ ∈ C q is a smooth form whichdepends continuously on R . Moreover, for every α ≥ there is a constant c α independent of R such that k R θ k C α ≤ c α k R k| θ | − k − k − α .
3. S
UPER - POTENTIALS
For the details of super-potentials on P k , we refer the reader to [8]. For the reader’sconvenience, we summarize some definitions and properties of super-potentials on P k . Definition 3.1.
Let < q ≤ k be an integer. For smooth S ∈ C q , the super-potential U S of S of mean is a function defined on C k − q +1 by U S ( R ) = h U S , R i where R ∈ C k − q +1 and U S is a quasi-potential of S of mean , which is a ( q − , q − -currentsuch that S − ω s = dd c U S and h U S , ω k − q +1 i = 0 .For a general current S ∈ C q , U S ( R ) = lim θ → U S θ ( R ) where S θ is the standard regularization of S as in Section 2 and U S θ is its super-potential of S θ of mean . Among various quasi-potentials, there is a good one for a computational purpose. It iscalled the Green quasi-potential and given by an integral formula.
Proposition 3.2 (Proposition 2.3.2 in [8]) . Let ∆ be the diagonal submanifold of P k × P k and Ω a closed real smooth ( k, k ) -form cohomologous to [∆] . Then, there is a negative ( k − , k − -form K on P k × P k smooth outside ∆ such that dd c K = [∆] − Ω which satisfiesthe following inequality near ∆ : k K ( · ) k ∞ . − dist( · , ∆) − k log dist( · , ∆) and k∇ K ( · ) k ∞ . dist( · , ∆) − k . Moreover, there is a negative dsh function η and a positive closed ( k − , k − -form Θ smooth outside ∆ such that K ≥ η Θ , k Θ( · ) k ∞ . dist( · , ∆) − k and η − log dist( · , ∆) isbounded near ∆ . Here, the inequalities are up to a constant multiple independent of the point in P k × P k \ ∆ . The norm k∇ K k ∞ is the sum P j |∇ K j | , where the K j ’s are the coefficients of K for a fixed atlas of P k × P k .We consider a fixed kernel K throughout the rest of the note. The Green quasi-potential U S of S is defined by U S ( z ) := Z ζ = z K ( z, ζ ) ∧ S ( ζ ) . Using the notion of super-potentials, we can define the operator f ∗ + on C q where f + isa birational map in Theorem 1.2. Definition 3.3 (Definition 5.1.4 in [8]) . We say that S ∈ C q is f ∗ + -admissible if there is acurrent R ∈ C k − q +1 which is smooth on a neighborhood of I + , such that the super-potentialof S are finite at Λ k − q +1 ( R ) . Proposition 3.4 (Proposition 5.1.8 in [8]) . Let S be an f ∗ + -admissible current in C p . Let U S and U L ( ω p ) be super-potentials of S and L p ( ω p ) . Then, we have λ p ( f + ) − λ p − ( f + ) U S ◦ Λ k − p +1 + U L p ( ω p ) is equal to a super-potential of L p ( S ) for R ∈ C k − p +1 , smooth in a neighborhood of I + . Lemma 3.5 (Lemma 3.1.5 in [10]) . Let S ∈ C q for < q ≤ k . Let n > be such that S is ( f n + ) ∗ -admissible then for all j with ≤ j ≤ n − , L j ( S ) is well defined, f ∗ + -admissible and L j +1 ( S ) = L j +1 ( S ) . In particluar, L n ( S ) = L n ( S ) .
4. L
OCALLY REGULARITY OF SUPER - POTENTIALS
In [2], the notions of locally bounded/continuous superpotentials were given as below.Similarly, we define local H¨older continuity. The notion of the H¨older continuity of super-potentials was given in [5]. We will write U S for the super-potential of a current S ∈ C q of mean . Definition 4.1.
Let ≤ q ≤ k . Let S ∈ C q and W an open subset of P k . The super-potential U S of S of mean m is said to be bounded in W if there exists a constant C S > such thatfor any smooth current R ∈ e D k − q +1 ( W ) , we have | U S ( R ) | ≤ C S . The super-potential U S of S of mean m is said to be continuous in W if U S continuouslyextends to e D k − q +1 ( W ) with respect to the subspace topology of the space of currents in W .The super-potential U S of S of mean m is said to be H¨older continuous in W if U S continuously extends to e D k − q +1 ( W ) and H¨older continuous with respect to one of the norms k · k − γ on e D k − q +1 ( W ) . Note that when W = P k , our notion coincides with the definition in [5]. Since e D k − q +1 ( W ) is compact, U S is continuous in an open subset W ⊂ P k , then it is boundedin an open subset W ⊂ P k . Remark 4.2.
By interpolation theory, for any γ ≥ γ ′ > , there is a constant c > suchthat k · k − γ ≤ k · k − γ ′ ≤ c ( k · k − γ ) γ ′ /γ . So, if U S is H¨older continuous for one k · k − γ , then itis H¨older continuous for all k · k − γ . Remark 4.3.
For S ∈ C q , its super-potential U S is continuous in an open subset W ⊂ P k ifand only if S is P C q ( P k \ W ) . The equivalence can be observed via the Green quasi-potentialkernel in Proposition 3.2. Proposition 4.4.
If a super-potential U S of S ∈ C q is bounded in an open subset W ⊂ P k and if R ∈ C k − q +1 is smooth outside a compact subset K ⋐ W , then U S ( R ) is finite.Proof. Let χ : P k → [0 , be a smooth cut-off function such that supp χ ⋐ W and χ ≡ on K . Then, dd c ((1 − χ ) U R ) is a smooth ( k − q + 1 , k − q + 1) -current and dd c ( χU R ) is acurrent in D k − q +1 ( W ) . Hence, U S ( R ) = U S ( dd c ((1 − χ ) U R )) + U S ( dd c ( χU R )) + U S ( ω k − p +1 ) and so, it is finite as desired. (cid:3)
5. B
IRATIONAL M APS
In this section, we summarize well-known properties of birational maps on P k . Fordetails, see [10] for instance.Let f + : P k → P k be a birational map of algebraic degree d ≥ and f − its inverse.Let δ denote the algebraic degree of f − . Let I ± denote the indeterminacy sets of f ± and I ±∞ := S n ≥ f n ∓ ( I ± ) , respectively. Let s > be an integer such that dim I + = k − s − and dim I − = s − . Let C ± be the critical sets for f ± : C + := f − ( I − ) and C − := f − − ( I + ) . Then, we have I + ⊂ C + , I − ⊂ C − and f + : P k \ C + → P k \ C − is a biholomorphism (p.42in [10]). We also have f − ◦ f + = id on P k \ C + and f + ◦ f − = id on P k \ C − ; f + ( P k \ I + ) ⊆ ( P k \ C − ) ∪ I − and f − ( P k \ I − ) ⊆ ( P k \ C + ) ∪ I + . For ≤ q ≤ k and n > , we define λ q ( f n + ) by λ q ( f n + ) := k ( f n + ) ∗ ( ω q ) k = k ( f n + ) ∗ ( ω k − q ) k . Proposition 5.1 (Proposition 3.1.2 in [10]) . We have λ q ( f + ) = d q for q ≤ s and λ q ( f + ) = δ k − q for q ≥ s . In particular, d s = δ k − s . Proposition 5.2 (Corollary 3.1.4 in [10]) . We have ( f ∗ + ) n = ( f n + ) ∗ for smooth currents in C q and λ q ( f n + ) = ( λ q ( f + )) n for all ≤ q ≤ k . We define two operators acting on C q : L q := ( λ q ( f + )) − f ∗ + and Λ q := ( λ k − q ( f + )) − ( f + ) ∗ . Note that the operators L q and Λ q are well-defined for currents in C q which are smoothnear I − and I + , respectively. Proposition 5.3.
Let < q ≤ k . Let R be a smooth current of bidegree ( q, q ) . Then, ( f + ) ∗ R = ( f − ) ∗ R as a current on P k and supp( f + ) ∗ R = ( f − ) ∗ R ⊆ ( f + ) − (supp R \ C − ) = f − (supp R \ C − ) .Proof. Notice that ( f + ) ∗ R and ( f − ) ∗ R are both forms with L -coefficients. So, they do notcharge any algebraic sets of dimension ≤ k − . Let ϕ be a smooth test form of bidegree ( k − q, k − q ) . Then, we have h ( f + ) ∗ R, ϕ i = h R, ( f + ) ∗ ϕ i = h R, ( f + ) ∗ ϕ i P k \ C + . Since f − : P k \ C − → P k \ C + is a biholomorphism, the change of coordinates by f − implies h R, ( f + ) ∗ ϕ i P k \ C + = h ( f − ) ∗ R, ( f − ) ∗ (( f + ) ∗ ϕ ) i P k \ C − = h ( f − ) ∗ R, ϕ i P k \ C − = h ( f − ) ∗ R, ϕ i . The second last inequality is from f + ◦ f − : P k \ C − → P k \ C − being identity on P k \ C − .The support property is from direct computations together with the fact that the cur-rents ( f + ) ∗ R and ( f − ) ∗ R have L -coefficients. (cid:3) Corollary 5.4.
For R ∈ C k − q +1 smooth outside I −∞ , Λ nk − q +1 ( R ) is smooth outside I −∞ . Together with Lemma 3.5, we obtain the following proposition:
Proposition 5.5. If S ∈ C q admits a super-potential bounded in a neighborhood of I −∞ , thenfor every m, n ≥ , L nq ( S ) is well defined and L m + nq ( S ) = L mq ( L nq ( S )) . Proof.
Let R ∈ C k − q +1 be a smooth current. Then, by Corollary 5.4 and Lemma 3.5, Λ nk − q +1 ( R ) is well defined and smooth outside I −∞ . So, by Proposition 4.4, the super-potential of S is finite at Λ nk − q +1 ( R ) for every n . Definition 3.3 and Lemma 3.5 finish theproof. (cid:3) Now, we consider the Green current of order q associated to f + . We further assumethe existence of an open subset V ⊂ P k such that V ∩ I + ∞ = ∅ and I −∞ ⊂ f + ( V ) ⋐ V asin Theorem 1.2. Then, there is a strictly positive distance between I + ∞ and I −∞ . Then, thissatisfies Hypothesis 3.1.6 in [10]. As a result of it, we have the existence of the Greencurrent of order q as below: Theorem 5.6 (Theorem 3.2.2 in [10]) . Let < q ≤ s . The sequence ( L mq ( ω q )) convergesin the Hartog’s sense to the Green current T q + of order q of f . Further, U T s + ([ I − ]) > −∞ . The following proposition can be obtained via a slight modification of Lemma 5.4.2and Lemma 5.4.3 in [8].
Proposition 5.7.
Suppose that there is an open subset V ⊂ P k such that V ∩ I + ∞ = ∅ and I −∞ ⊂ f + ( V ) ⋐ V as in Theorem 1.2. Let < q ≤ s . Then, the Green current T q + of order q is H¨older continuous on C k − q +1 ( V ) . Here, the H¨older continuity is with respect to the same k · k − γ -norm ( γ > ) as in Def-inition 4.1, but the difference is that we are taking a different set C k − q +1 ( V ) of currentsother than e D k − q +1 ( V ) .In the rest of this section, we construct a ( k − p + 1 , k − p + 1) -current R p ∞ such that Λ k − p +1 ( R p ∞ ) = R p ∞ and supp R p ∞ ⊂ I −∞ where Λ k − s +1 = d − ( p − ( f + ) ∗ for < p ≤ s .We will first construct such a current R s ∞ of bidegree ( k − s + 1 , k − s + 1) and thenconsider the case of bidegree ( k − p + 1 , k − p + 1) . The current [ I − ] denotes the current ofintegration on the regular part of I − . It is not difficult to prove the following proposition: Proposition 5.8.
Suppose that I + ∞ ∩ I −∞ = ∅ . For all i = 0 , , , · · · , the currents Λ ik − s +1 ([ I − ]) are well-defined positive closed currents of bidegree ( k − s + 1 , k − s + 1) and they have thesame mass as [ I − ] does. Also, we have suppΛ ik − s +1 ([ I − ]) ⊆ f i + ( I − ) . Consider the following sequence of currents: R n := ( n + 1) − n X i =0 Λ ik − s +1 ([ I − ]) Then, the sequence { R n } has bounded mass. So, there exists a convergent subsequence { R n j } in the sense of currents. Let R s ∞ denote one of its limit currents. Proposition 5.9. Λ k − s +1 ( R s ∞ ) = R s ∞ , supp R s ∞ ⊂ I −∞ . Proof.
The first part is clear from supp R n j ⊂ I −∞ for all j ∈ N . For a convergent subse-quence { R n j } , we have Λ k − s +1 ( R n j ) − R n j = ( n j + 1) − (Λ n j +1 k − s +1 ([ I − ]) − [ I − ]) as j → ∞ . Since Λ k − s +1 is continuous for currents in C k − s +1 smooth near I + and themass of Λ n j k − s +1 is bounded in j ∈ N , we see that Λ k − s +1 ( R s ∞ ) = R s ∞ by letting j → ∞ . (cid:3) For < p ≤ s , from Proposition 5.7, a super-potential U T s − p + of T s − p + is H¨older contin-uous in V . Since supp R s ∞ ⊂ I −∞ ⋐ V , the current R p ∞ := ( T s − p + ) ∧ R s ∞ is well-defined. Proposition 5.10.
Let < p ≤ s . Assume the existence of the neighborhood V of I −∞ inTheorem 1.2. Then, we have Λ k − p +1 ( R p ∞ ) = R p ∞ . Proof.
By use of the standard regularization and the Hartog’s convergence, we may as-sume that T p − s + is smooth. Note that, R s ∞ has support in V . Then, we have Λ k − p +1 ( R p ∞ ) = d − ( p − ( f + ) ∗ (( T s − p + ) ∧ R s ∞ ) = d − ( p − ( f + ) ∗ ( d − ( s − p ) ( f + ) ∗ ( T s − p + ) ∧ R s ∞ )= d − ( s − T s − p + ∧ ( f + ) ∗ R s ∞ = T s − p + ∧ R s ∞ = R p ∞ . (cid:3) We also obtain the following corollary from Proposition 5.7.
Corollary 5.11.
Let < p ≤ s . Assume the existence of the neighborhood V of I −∞ inTheorem 1.2, then the value U T p + ( R p ∞ ) is finite. The argument in p.53 of [10] works for any < p ≤ s . So, we obtain Proposition 5.12.
The sequence d − n U T p + ◦ Λ nk − p +1 goes to on smooth forms in C k − p +1 .
6. P
ROOF OF T HEOREM
AND T HEOREM S ∈ C p denotes the current in Theorem 1.2. Forthe proof of Theorem 1.2, we set up environment as in [8] and [2]. For simplicity, wewill write L and Λ for L p and Λ k − p +1 , respectively. Also, S n := L n ( S ) = d − pn ( f n + ) ∗ ( S ) and T p := T p + . Definition 6.1.
For S ∈ C p , we define the dynamical super-potential V S by V S := U S − U T p − c S , where c S := U S ( R ∞ ) − U T p ( R ∞ ) and the dynamical Green quasi-potential of S by V S := U S − U T s − ( m S − m T s + c S ) ω p − where U S , U T p are the Green quasi-potentials of S , T p , and m S , m T p are their mean, respec-tively. The lemma below can be proved in the same way as in Lemma 5.5.5 in [8].
Lemma 6.2 (See Lemma 5.5.5 in [8]) . (1) V S ( R p ∞ ) = 0 , (2) V S ( R ) = h V S , R i for smooth R ∈ C k − p +1 and (3) V L ( S ) = d − V S ◦ Λ for currents in C k − p +1 smooth near I + ∞ . Different from the case of Theorem 1.1, since we cannot say that a super-potential U T p of T p is continuous in an open subset W ⋐ P k \ I + ∞ , we cannot say that U S − V S isbounded from above on C k − p +1 ( W ) by a constant independent of S in general. However,we have Proposition 5.12 as an alternative for this.We further introduce some more notations. Let R ∈ C k − p +1 be a smooth current.We choose and fix a constant λ such that < λ < d throughout the proof. Let η n :=min { η, − λ n } + λ n where η is a DSH function in Proposition 3.2. Then, the DSH-norm of η n is bounded in dependent of n and K ≥ η Θ ≥ η n Θ − λ n Θ where η and Θ are a functionand a current in Proposition 3.2.For a positive or negative current S ′ and each n ∈ N , define U ′ n,S ′ := Z ζ = z λ n Θ( ζ , z ) ∧ S ′ ( ζ ) , U ′′ n,S ′ := Z ζ = z η n ( ζ , z )Θ( ζ , z ) ∧ S ′ ( ζ ) and U ′ Λ n ( R ) := Z ζ = z λ n Θ( ζ , z ) ∧ Λ n ( R )( ζ ) , U ′′ Λ n ( R ) := Z ζ = z η n ( ζ , z )Θ( ζ , z ) ∧ Λ n ( R )( ζ ) . Note that if S ′ is closed, then, U ′ n,S ′ is closed and its mass is c m λ n k S ′ k for a constant c m > which is independent of n and S ′ .For the proof of Theorem 1.2, we need the following estimate. Observe that Lemma2.3.9 in [8] does not need closedness of the current. Its proof consists of disintegrationand singularity estimate. so, we have Lemma 6.3 (Lemma 2.3.9 in [8]) . Let S ′ be a positive current of bidegree ( p, p ) withbounded mass. Then, we have (cid:12)(cid:12)(cid:12)(cid:12)Z U ′′ n,S ′ ∧ ω k − p +1 (cid:12)(cid:12)(cid:12)(cid:12) . e − λ n k S ′ k . The inequality is up to a constant multiple independent of n and S ′ . Now, we start to prove Theorem 1.2. It is a direct consequence of the following propo-sition.
Proposition 6.4.
Assume the hypotheses in Theorem 1.2. Let R ∈ C k − p +1 be a smoothcurrent. Then, we have V S n ( R ) = d − n V S (Λ n ( R )) → in the sense of currents. We begin with an estimate near I −∞ in Lemma 6.10. Lemma 6.5.
There exist open subsets W ⋐ W ⋐ W ⋐ W ⋐ V such that f + ( W i ) ⋐ W i .Proof. Note that f + is holomorphic outside I + . Since V ∩ I + = ∅ , f + ( V ) is compact in V .So, simply take f + ( V ) ⋐ W ⋐ W ⋐ W ⋐ W ⋐ V . (cid:3) Let χ : P k → [0 , be a cut-off function such that χ ≡ on W and supp χ ⋐ W . Let M > be a constant such that k Df − k P k \ W < M . Here, k · k P k \ W denotes the uniformnorm of coefficients on P k \ W with respect to a fixed finite atlas of P k . Lemma 6.6.
Let R ∈ C k − p +1 be smooth outside I −∞ . Then, there exists a constant c > independent of R and n such that k dd c ( χU Λ n ( R ) ) k ∗ ≤ cM kn k R k P k \ W where U ( · ) denotes the Green quasi-potential of a given current.Proof. We can write dd c ( χU Λ n ( R ) ) = dd c χ ∧ U Λ n ( R ) + dχ ∧ d c U Λ n ( R ) + dU Λ n ( R ) ∧ d c χ + χ (Λ n ( R ) − ω k − p +1 ) . The last term is bounded below by ω k − p +1 . Since the first three terms are all smoothsince Λ n ( R ) is smooth outside I −∞ . They are all bounded by the C -norm of U Λ n ( R ) on the support of dχ or d c χ which is compact outside W . Hence, by Proposition 5.3, k Λ n ( R ) k P k \ W ≤ c M kn k R k P k \ W for some c > . Then, due to Lemma 2.3.5 in [8], weget the desired estimate. (cid:3) Since the above estimate only depends on the uniform norm of Λ n ( R ) on the supportof dχ and d c χ , we see that there exists a constant δ > , which only depend on thedistance between W and P k \ W , such that the same estimate as in Lemma 6.6 holds forall δ > with δ < δ and for all n ∈ N : k dd c ( χU (Λ n ( R )) δ ) k ∗ ≤ cM kn k R k P k \ W . So, we have c − M − kn k R k − P k \ W dd c ( χU (Λ n ( R )) δ ) ∈ e D k − p +1 ( W ) for δ > with | δ | < δ . Lemma 6.7.
For < δ < δ , we have k dd c ( χU Λ n ( R ) ) − dd c ( χU (Λ n ( R )) δ ) k − . δ. The inequality is up to a constant multiple independent of n and δ .Proof. Let ϕ be a smooth test form with k ϕ k C ≤ . We have h dd c ( χU Λ n ( R ) ) − dd c ( χU (Λ n ( R )) δ ) , ϕ i = h U Λ n ( R ) − U (Λ n ( R )) δ , χdd c ϕ i = h Λ n ( R ) − (Λ n ( R )) δ , U χdd c ϕ i ≤ h Λ n ( R ) , U χdd c ϕ − ( U χdd c ϕ ) δ i Hence, by Lemma 2.3.5 in [8], we have |h dd c ( χU Λ n ( R ) ) − dd c ( χU (Λ n ( R )) δ ) , ϕ i| . k U χdd c ϕ k C δ . k χdd c ϕ k ∞ δ . δ. (cid:3) Lemma 6.8.
Let S θ be a standard regularization of S for sufficiently small < | θ | ≪ . For < δ < δ , we have (cid:12)(cid:12)(cid:12)(cid:12)Z U S θ ∧ dd c ( χU (Λ n ( R )) δ ) (cid:12)(cid:12)(cid:12)(cid:12) . δ − k − k − e − λ n + λ n . Here, the inequality is independent of θ , δ and n . Indeed, the θ is chosen so that supp( dd c χ ∧ U ′ n, (Λ n ( R )) δ ) θ ⋐ V . This condition is com-pletely determined by the function χ . Proof.
We have Z U S θ ∧ dd c ( χU (Λ n ( R )) δ ) = Z S θ ∧ χU (Λ n ( R )) δ − Z χω p ∧ U (Λ n ( R )) δ ≥ Z S θ ∧ χU ′ n, (Λ n ( R )) δ + Z S θ ∧ χU ′′ n, (Λ n ( R )) δ − Z χω p ∧ U (Λ n ( R )) δ We estimate the first integral R S θ ∧ χU ′ n, (Λ n ( R )) δ . Note that U ′ n, (Λ n ( R )) δ is closed and itsmass is a constant multiple of λ n . Z S θ ∧ χU ′ n, (Λ n ( R )) δ = Z U S θ ∧ dd c χ ∧ U ′ n, (Λ n ( R )) δ + Z ω s ∧ χU ′ n, (Λ n ( R )) δ & − λ n . Since supp dd c χ ∧ U ′ n, (Λ n ( R )) δ ⋐ W and k dd c χ ∧ U ′ n, (Λ n ( R )) δ k ∗ . λ n , the first integral isestimated from the boundedness of U S θ in W ⋐ V . The second integral is bounded bythe mass of U ′ n, (Λ n ( R )) δ . So, we get the last inequality.We estimate the second integral R S θ ∧ χU ′′ (Λ n ( R )) δ . From the negativity of η n and thepositivity of Θ , we have Z S θ ∧ χU ′′ (Λ n ( R )) δ = Z χ ( z ) S θ ( z ) ∧ η n ( z, ζ ) ∧ Θ( z, ζ ) ∧ (Λ n ( R )) δ ( ζ ) & δ − k − k − Z P k × P k χ ( z ) S θ ( z ) ∧ η n ( z, ζ ) ∧ Θ( z, ζ ) ∧ ω k − s +1 ( ζ )= δ − k − k − Z U ′′ n,χS θ ∧ ω k − p +1 & − δ − k − k − e − λ n . The last inequality is from Lemma 6.3. (cid:3)
From the hypothesis on S ∈ C p in Theorem 1.2, let α > and C α > be two constantssuch that for all θ ∈ C with sufficiently small | θ | as in Lemma 6.8, | U S θ ( R ) − U S θ ( R ′ ) | ≤ C α ( k R − R ′ k − ) α for R, R ′ ∈ e D k − p +1 ( W ) . Lemma 6.9.
Let R ∈ C k − p +1 be a current smooth outside I −∞ . Let S θ be a standard regular-ization of S for sufficiently small | θ | as in Lemma 6.8. We have Z W S θ ∧ U Λ n ( R ) & − λ n for all sufficiently large n . The inequality is independent of θ and n .Proof. From the negativity of the Green quasi-potential, we have Z W S θ ∧ U Λ n ( R ) ≥ Z χS θ ∧ U Λ n ( R ) = Z U S θ ∧ dd c ( χU Λ n ( R ) ) + Z χω p ∧ U Λ n ( R ) Let δ > be a small constant to be determined later. Then, the last quantity can bewritten as Z U S θ ∧ dd c ( χU (Λ n ( R )) δ ) + Z U S θ ∧ ( dd c ( χU Λ n ( R ) ) − dd c ( χU (Λ n ( R )) δ )) + Z χω p ∧ U Λ n ( R ) From the H¨older continuity of U S θ in W with Lemma 6.6 and Lemma 6.7, the secondintegral can be estimated as below: (cid:12)(cid:12)(cid:12)(cid:12)Z U S θ ∧ ( dd c ( χU Λ n ( R ) ) − dd c ( χU (Λ n ( R )) δ )) (cid:12)(cid:12)(cid:12)(cid:12) . C α cM kn k R k P k \ W (cid:18) δcM kn k R k P k \ W (cid:19) α . Since the mass of quasi-potential is uniformly bounded, the third integral is uniformlybounded. From Lemma 6.8, the first integral can be approximated by Z U S θ ∧ dd c ( χU (Λ n ( R )) δ ) & − δ − k − k − e − λ n − λ n . Altogether, if we choose δ = 1 / (2 M k ) n/α , we have Z W S θ ∧ U Λ n ( R ) & − λ n for all sufficiently larget n . (cid:3) Lemma 6.10.
Let R ∈ C k − p +1 be a current smooth outside I −∞ . Let S θ be a standardregularization of S for sufficiently small | θ | as in Lemma 6.8. We have Z W U S θ ∧ Λ n ( R ) & − λ n for all sufficiently larget n . The inequality is independent of θ and n .Proof. Z W U S θ ∧ Λ n ( R ) = Z z ∈ W Z ζ = z S θ ( ζ ) ∧ K ( z, ζ ) ∧ Λ n ( R )( z )= Z z ∈ W Z ζ ∈ W \{ z } S θ ( ζ ) ∧ K ( z, ζ ) ∧ Λ n ( R )( z )+ Z z ∈ W Z ζ ∈ P k \ W S θ ( ζ ) ∧ K ( z, ζ ) ∧ Λ n ( R )( z ) From the estimate of K in Proposition 3.2, the second integral is bounded by a con-stant independent of θ and n . From the negativity of the Green quasi-potential, the firstintegral is bounded by Z W S θ ∧ U Λ n ( R ) . Hence, by Lemma 6.9, we get the estimate. (cid:3)
Proof of Proposition 6.4.
Let R ∈ C k − p +1 be a smooth current. By Lemma 6.2, we canwrite V S n ( R ) = d − n V S (Λ n ( R )) . By the definition, we have V S (Λ n ( R )) = U S (Λ n ( R )) − U T s (Λ n ( R )) − c S k Λ n ( R ) k . Since thesuper-potentials on P k are upper semicontinuous on C k − s +1 which is compact, U S (Λ n ( R )) is bounded from above. So, Proposiiton 5.12 implies that lim sup n → d − n V S (Λ n ( R )) ≤ .So, we only consider the estimate of d − n V S (Λ n ( R )) from below.We consider U S (Λ n ( R )) . From the definition of the super-potential, we have U S (Λ n ( R )) = lim θ → U S θ (Λ n ( R )) . Hence, we estimate U S θ (Λ n ( R )) for θ ∈ C with sufficiently small | θ | .In the rest of the proof, the inequalities . , & are up to a constant multiple independentof θ and n .Let ε n > be a sufficiently small positive number to be determined later and U ( · ) denotes the Green quasi-potential of a given current with respect to a fixed Green quasi-potential kernel in Proposition 3.2. Since S θ is smooth and has the same mean and massas S does, we can write U S θ (Λ n ( R )) = Z U S θ ∧ Λ n ( R ) − m S k Λ n ( R ) k = Z U S θ ∧ (Λ n ( R ) − (Λ n ( R )) ε n ) + Z U S θ ∧ (Λ n ( R )) ε n − m S k Λ n ( R ) k From the negativity of the Green quasi-potential, we have U S θ (Λ n ( R )) ≥ Z W U S θ ∧ Λ n ( R ) + Z P k \ W U S θ ∧ (Λ n ( R ) − (Λ n ( R )) ε n )+ Z U S θ ∧ (Λ n ( R )) ε n − m S k Λ n ( R ) k . We estimate the first integral. From Lemma 6.10, we have Z W U S θ ∧ Λ n ( R ) & − λ n for all sufficiently large n .For the second integral, we will use the fact that the mass of the Green quasi-potentialis uniformly bounded. Proposition 5.3 implies that Λ n ( R ) is smooth in P k \ W , and from k Df − k P k \ W < M and f + ( W ) ⋐ W , we have k Λ n ( R ) k P k \ W . M kn . So, we have k Λ n ( R ) − (Λ n ( R )) ε n k P k \ W . M − kn ε n and since the mass of U S θ is uniformly bounded, the second integral is Z P k \ W U S θ ∧ (Λ n ( R ) − (Λ n ( R )) ε n ) & − M − kn ε n For the last integral, thanks to Lemma 3.2.10 in [8] and Proposition 2.1.6 in [8], we have Z U S θ ∧ (Λ n ( R )) ε n & log ε n . If we choose ε n := min { / , M / } kn , then we have U S (Λ n ( R )) & − λ n for all sufficientlylarge n . Since < λ < d , we have lim inf n → d − n U S (Λ n ( R )) ≥ . Together with Proposi-tion 5.12 again, we get lim inf n → d − n V S (Λ n ( R )) ≥ (cid:3) Proof of Theorem 1.2.
Let ϕ be a smooth test form of bidegree ( k − p, k − p ) . Since ϕ issmooth, there exists m ϕ > such that m ϕ ω k − s +1 + dd c ϕ ≥ . Then, we have h S n − T p , ϕ i = h dd c V S n , ϕ i = h V S n , dd c ϕ i = h V S n , m ϕ ω k − p +1 + dd c ϕ i − h V S n , m ϕ ω k − p +1 i If we apply Proposition 6.4 to both terms, we see the desired convergence. (cid:3) Proof of Theorem 1.3.
Let < p ≤ s . Let H be an analytic subset of pure dimension k − p and suppose that H ∩ I −∞ = ∅ . Let c H denote the degree of H . Let W be a subset ofthe trapping neiborhood U of I −∞ such that H ∩ W = ∅ . Then, since U is a trappingneighborhood, there exists an N such that I −∞ ⋐ f N + ( W ) ⋐ W . Indeed, there exists N such that I −∞ ⊂ f N + ( U ) ⋐ W . Note that I + ∞ , I −∞ and T p remain the same if we replace f by f N .In the proof of Proposition 6.4, by applying the same lemmas and propositions to ( f + ) N , λ N and Λ j ( R ) in place of f + , λ and R , we obtain U c − H [ H ] ((Λ N ) n (Λ j ( R ))) & − ( λ N ) n for each j = 0 , · · · , N − . By considering a subsequence, Proposition 5.12 impliesthat d − Nn U T p ((Λ N ) n (Λ j ( R ))) → as n → ∞ for each j = 0 , · · · , N − . Applyingthe same argument in Proposition 6.4, we see that for a given smooth R ∈ C k − p +1 , V L Nn ( c − H [ H ]) (Λ j ( R )) → as n → ∞ holds for each j = 0 , · · · , N − , which means d − p ( Nn + j ) ( f Nn + j + ) ∗ ( c − H [ H ]) converges to T p for each j = 0 , , · · · , N − . So, the onlylimit points of d − pn ( f n + ) ∗ ( c − H [ H ]) is T p , which completes the proof. (cid:3) Remark 6.11.
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