Weak normality of families of meromorphic mappings and bubbling in higher dimensions
aa r X i v : . [ m a t h . C V ] S e p WEAK NORMALITY OF FAMILIES OF MEROMORPHICMAPPINGS AND BUBBLING IN HIGHER DIMENSIONS
S. IVASHKOVICH, F. NEJI
Abstract.
Our primary goal in this paper is to understand wether the sets of normalityof families of meromorphic mappings between general complex manifolds are pseudocon-vex or not. It turns out that the answer crucially depends on the type of convergenceone is interested in. We examine three natural types of convergence introduced by oneof us earlier and prove pseudoconvexity of sets of normality for a large class of targetmanifolds for the so called weak and gamma convergencies. Furthermore we determinethe structure of the exceptional components of the limit of a weakly/gamma but notstrongly converging sequence, they turn to be rationally connected . This observationallows to determine effectively when a weakly/gamma converging sequence fails to con-verge strongly. An application to the Fatou sets of meromorphic self-maps of compactcomplex surfaces is given.
Contents
1. Introduction 12. Topologies on the space of meromorphic mappings 63. Pseudoconvexity of sets of normality 104. Convergence of mappings with values in projective space 135. Bloch-Montel type normality criterion 176. Behavior of volumes of graphs under weak and gamma convergence 227. Rational connectivity of the exceptional components of the limit 248. Fatou components 27References 32
1. Introduction
When one works with sequences ofmeromorphic functions and, more generally, mappings one finds himself bounded to con-sider several notions of their convergence. Some of these notions were introduced in [Fu]and [Iv2], we shall recall the essentials below. An important question is: what can besaid about the maximal open sets where the given sequence converge? It occurs thatpseudoconvexity or not of domains of convergence/normality in the case of meromorphicmappings crucially depends on the type of convergence one is looking for.Now let briefly describe the ways one can define what does it means that a sequence { f k } of meromorphic mappings between complex manifolds U and X converges. We startwith the most obvious one. A sequence { f k } of meromorphic mappings between complex Date : October 30, 2018.2010
Mathematics Subject Classification.
Primary - 32H04, Secondary - 32H50, 32Q45.
Key words and phrases.
Meromorphic mapping, convergence, normal family. manifolds U and X is said to converge strongly to a meromorphic map f if the graphs Γ f k converge over compacts in U to the graph Γ f in Hausdorff metric. Our first result showsthat Γ f k then converge to Γ f in a stronger topology of cycles . Theorem 1. If f k strongly converge to f then for every compact K ⋐ U the volumes Γ f k ∩ ( K × X ) are uniformly bounded and therefore Γ f k converge to Γ f in the topology ofcycles. This type of convergence is natural and has some nice features. For example the stronglimit of a sequence of holomorphic maps is holomorphic and vice versa, if the limit f is holomorphic then for every compact K ⋐ U all f k for k ≫ K and uniformly converge there to f . This statement was called theRouch´e Principle in [Iv2].But strong convergence has also some disadvantages. The first, crucial for us is thefact that the sets of strong convergence, i.e., maximal open subsets of U where a givensequence converges strongly on compacts, are not pseudoconvex in general. Moreover,the sets of strong normality (see later on) of families of meromorphic mappings can bejust arbitrary, see Example 2.1. Also if one takes X = P N the ”most immediate” notionof convergence doesn’t correspond to the strong one.Therefore in [Iv2] along with the notion of strong convergence we proposed two weakerones. We say that f k converge weakly to f if they converge strongly to f on compactsoutside of some analytic set A in U of codimension at least two. It turns out that this A can be taken to be the set I f of points of indeterminacy of the limit map f and then forevery compact K in U \ I f all weakly converging to f mappings f k will be holomorphicon K (for k big enough) and converge to f uniformly on K , see Remark 3.1.One more notion of convergence from [Iv2], which we need to recall here, is the gammaconvergence (Γ-convergence). We say that f k gamma -converge to f if they strongly con-verge to f outside of an analytic set (now it can be of codimension one) and for everydivisor H in X and every compact K ⋐ U the intersections f ∗ k H ∩ K have bounded volumecounted with multiplicities, see more about the last condition in section 3.2. Remark 1.
Strong convergence (or s - convergence) will be denoted by f k → f , the weakone (or w - convergence) as f k ⇀ f , and Γ - convergence as f k Γ −→ f . Note that in thesecond and third definitions we suppose that the limit f is defined and meromorphic onthe whole of U if, even, the convergence takes place only on some part of U . In the firstcase the limit exists on the whole of U automatically.For the better understanding of these notions let us give a description of the listedtypes of convergence in the case when X is projective, i.e., imbeds into P N for some N .In that special case the notions of convergence listed above permit an explicit analyticdescription as follows. Every meromorphic mapping f with values in P N can be locallyrepresented by an ( N + 1)-tuple of holomorphic functions f ( z ) = [ f ( z ) : ... : f N ( z )] , (1.1)where not all of f , ..., f N are identically zero, see section 4. More precisely, if f : U → P N is a meromorphic mapping then for every point x ∈ U there exists a neighborhood V ∋ x and holomorphic functions f , ..., f N in V satisfying (1.1). If the zero sets of f j contain acommon divisor then we can divide all f j by its equation and get a representation suchthat GCD ( f , ..., f N ) = 1 in every O x , x ∈ V . In that case the indeterminacy set of f is ntroduction 3 I f ∩ V = { z ∈ V : f ( z ) = ... = f N ( z ) = 0 } (1.2)and has codimension at least two. Representation (1.1) satisfying (1.2) is called reduced .We shall prove the following Theorem 2.
Let { f k } be a sequence of meromorphic mappings from a complex manifold U to P N . Then: i) f k Γ −→ f if and only if for any point x ∈ U there exists a neighborhood V ∋ x , reducedrepresentations f k = [ f k : ... : f Nk ] and not necessarily reduced representation f = [ f : ... : f N ] such that for every j N the sequence f jk converges to f j uniformly on V ; ii) f k ⇀ f if and only if f k Γ −→ f and the limit representation f = [ f : ... : f N ] is reduced ; iii) f k → f if and only if f k ⇀ f and corresponding non-pluripolar Monge-Amp`ere massesconverge, i.e., for every p n = dim U one has (cid:0) dd c k z k (cid:1) n − p ∧ (cid:0) dd c ln k f k k (cid:1) p → (cid:0) dd c k z k (cid:1) n − p ∧ (cid:0) dd c ln k f k (cid:1) p (1.3) weakly on compacts in U . Here in (1.3) we suppose that V = ∆ n , z , ..., z n are standard coordinates and k f k = | f | + ... + | f N | , i.e., dd c ln k f k is the pullback of the Fubini-Study form by f . Non-pluripolar MA mass of ln k f k of order p in V here means Z V \ I f (cid:0) dd c k z k (cid:1) n − p ∧ (cid:0) dd c ln k f k (cid:1) p , (1.4)where I f is given by (1.2), i.e., is the indeterminacy set of f . Remark 2. a)
Reducibility or not of the limit representation f = [ f : ... : f N ] in thistheorem doesn’t depend on the choice of converging representations f k = [ f k : ... : f Nk ],provided they are taken to be reduced (the last can be assumed always). Indeed, any otherreduced representation of f k has the form f k = [ g k f k : ... : g k f Nk ], where g k are holomorphicand nowhere zero. If the newly chosen representations converge to some representationof f then g k must converge, say to g , and this g is nowhere zero by Rouch´e’s theorem.Therefore the obtained representation of the limit is f = [ gf : ... : gf N ] and it is reducedif and only if f = [ f : ... : f N ] was reduced. b) The case when the representation f = [ f : ... : f N ] of the limit is not necessarilyreduced was studied for mappings with values in P N by H. Fujimoto in [Fu], who called it meromorphic , or m -convergence. According to the part ( i) of our theorem it turns out thatour Γ-convergence (in the case of X = P N ) is equivalent to m -convergence of Fujimoto. In this paper we considertwo classes of complex manifolds: projective and Gauduchon, the last is the class ofcomplex manifolds carrying a dd c -closed metric form - a Gauduchon form. Let F be afamily of meromorphic mappings between complex manifolds U and X . F is said to be strongly/weakly or gamma normal if from every sequence of elements of F one can extracta subsequence converging on compacts in U in the corresponding sense. The maximalopen subset N F ⊂ U on which F is normal is called the set of normality. As it wasalready told the sets of strong normality could be arbitrary. In subsection 3.1 we provethe following Section 1
Theorem 3.
Let U be a domain in a Stein manifold ˆ U such that ˆ U is an envelope ofholomorphy of U and let f k : ˆ U → X be a weakly converging on U sequence of meromorphicmappings with values in a disk-convex complex manifold X . Then:(a) If the weak limit f on f k meromorphically extends from U to ˆ U then f k weaklyconverge to f on the whole of ˆ U .(b) If, in addition, the manifold X carries a pluriclosed metric form then the weak limit f of f k meromorphically extends to ˆ U and then the part (a) applies. As a result the sets of weak normality are locally pseudoconvex provided the target isdisk-convex and Gauduchon. Recall that an open subset N of a complex manifold U iscalled locally pseudoconvex if for every point p ∈ ∂ N there exists a Stein neighborhood V of p in U such that V ∩ N is Stein. Corollary 1.
Let
F ⊂ M ( U, X ) be a family of meromorphic mappings from a complexmanifold U to a disk-convex Gauduchon manifold X . Then the set of weak normality N F of F is locally pseudoconvex. If F = { f k } is a sequence then the set of its weak convergenceis locally pseudoconvex. Remark 3.
This corollary clearly follows from Theorem 3. Sets of Γ-normality are alsolocally pseudoconvex under the same assumptions, see Proposition 3.1 in section 2.As one more supporting argument in favor of weak convergence we prove in section 5the following normality criterion.
Theorem 4.
Let { H i } di =0 , d > , be hypersurfaces in projective manifold X such that Y := X \ S di =0 H i is hyperbolically imbedded to X . Let F be a family of meromorphicmappings from a complex manifold U to X such that: i) for every i = 0 , ..., d and every compact K ⋐ U the volumes f ∗ H i ∩ K counted withmultiplicities are uniformly bounded for f ∈ F ; ii) F uniformly separates every pair H i , H j , i < j d .Then the family F is weakly normal on U . Conditions ( i) and ( ii) are explained in section 5, they are intuitively clear and moreor less necessary. The classically known case of a system of divisors with hyperbolicallyimbedded complement is 2 N + 1 hypersurfaces in P N in general position - Theorem ofBloch, see [Gr]. A criterion for m -normality ( i.e., Γ-normality in our sense) was given byFujimoto in [Fu].
Strongconvergence obviously implies the weak one and the latter implies the gamma-convergence,see Remark 3.4: s -convergence = ⇒ w -convergence = ⇒ Γ- convergence . (1.5)Our second principal task in this paper is to understand what obstructs a weakly/gammaconverging sequence to converge strongly. The problem is that by Theorem 1 the volumesof graphs of a strongly converging sequence are uniformly bounded over compacts in thesource. When dimension n of the source U is two and X is K¨ahler the volumes of thegraphs of a weakly converging sequence are still bounded, see [Iv2]. The same is true ifis X an arbitrary compact complex surface (and again dim U = 2), see [Ne]. We shall saymore about this in section 6. But this turns out not to be the case starting from dimension ntroduction 5 three, i.e., the volumes of graphs of a weakly converging sequence can diverge to infinityover compacts of U . Via (1.3) this turns out to be a geometric counterpart of a well knowndiscontinuity of Monge-Amp`ere masses, see Example 6.1 in section 4. Nevertheless for asequence Γ f k of Γ-converging meromorphic graphs we can consider the Hausdorff limit ˆΓ(its always exists after taking a subsequence). Set Γ := ˆΓ \ Γ f , where Γ f is the graph ofthe limit map f , and call Γ a bubble . For a ∈ γ := pr (Γ) set Γ a := pr ( pr − ( a ) ∩ Γ), here pr and pr are natural projections, see section 2. We prove the following statement. Theorem 5.
Let X be a disk-convex Gauduchon manifold and let f k : U → X be a weaklyconverging sequence of meromorphic mappings which doesn’t converge strongly. Then forevery point a ∈ γ the fiber Γ a is rationally connected. If X is, moreover, projective thenthe same is true also for a Γ -converging sequences. Here by saying that a closed subset Γ a of a complex manifold is rationally connected we mean that every two distinct points p, q ∈ Γ a can be connected by a chain of rationalcurves which is entirely contained in Γ a , see section 7 for more details. Families of a special interest are thefamilies of iterates f n := f ◦ ... ◦ f of some fixed meromorphic self-map of a compact complex manifold X . The maximal open subset X where { f n } is relatively compactis called the Fatou set of f . Depending on the sense of convergence that one wishes toconsider one gets different Fatou sets: strong, weak or gamma Fatou sets. We denote themas Φ s , Φ w and Φ Γ respectively, their dependance on f will be clear from the context. Corollary 2.
Let f be a meromorphic self-map of a compact complex surface. Then theweak Fatou set Φ w of f is locally pseudoconvex. If Φ s is different from Φ w then: a ) X is bimeromorphic to P ; b ) Φ w = X \ C , where C is a rational curve in X ; c ) the weak limit of any weakly converging subsequence { f n k } of iterates is a degeneratemap of X onto C . It should be pointed out that our Fatou sets are different from the Fatou sets as theywere considered in [FS]. In [FS] the Fatou set of f is the maximal open subset Φ of X \ S ∞ n =0 f − n ( I f ) where the family { f n } is equicontinuous (remark that on X \ S ∞ n =0 f − n ( I f )all iterates are holomorphic). If, for example, f : P → P is the Cremona transformation[ z : z : z ] → [ z z : z z : z z ] then Φ s = Φ w = Φ Γ = P but Φ = P \ { three lines } .In subsection 8.2 an example of higher degree and with an interesting dynamics on theindeterminacy set is given. This is one more instance which shows how crucially canchange a picture when the notion of convergence changes. Notes. 1.
Let us make a final note about the goals of this paper. On our opinion themost interesting information about a converging sequence of meromorphic mappings isconcentrated near the “limit“ of their indeterminacy sets. We describe the most reasonable(in our opinion) notions of convergence of meromorphic mappings and conclude that the weak one is the most appropriate. At the same time we detect that if a weakly/gamma converging sequence doesn’t converge strongly then this imposes very serious restrictionson the target manifold (it is forced to contain many rational curves). In some cases (ex.iterations) this puts strong constraints also on the sequence itself. Domains of convergence of holomorphic functions of several variables were, probably,for the first time considered by G. Julia in [J]. In [J] and then in [CT] it was proved that
Section 2 these domains are (in some sense) pseudoconvex. Domains of convergence of meromorphicfunctions of several variables were studied in [Sa] and then in [Ru]. In these early papersconvergence was understood as holomorphic ( i.e., uniform) convergence outside of theunion of indeterminacy sets of meromorphic mappings in question.
Acknowledgement.
We are grateful to Alexander Rashkovskii for explaining to us theExample 6.1 with unbounded Monge-Amp`ere masses.
2. Topologies on the space of meromorphic mappings
Our manifolds will be Haus-dorff and countable at infinity if the opposite is not explicitly stated. We shall alsoeverywhere suppose that they are disk-convex . Definition 2.1.
A complex manifold X is called disk-convex if for every compact K ⋐ X there exists a compact ˆ K such that for every h ∈ O (∆ , X ) ∩ C ( ¯∆ , X ) such that h ( ∂ ∆) ⊂ K one has h ( ¯∆) ⊂ ˆ K . The minimal such ˆ K is called the disk envelope of K . Let X be equipped with someHermitian metric h . By ω h denote the (1 , h . We saythat the metric h is d - closed or K¨ahler if dω h = 0. We say that h is pluriclosed or Gauduchon if dd c ω h = 0. In [Ga] it was proved that on a compact complex surface every Hermitianmetric is conformally equivalent to the unique dd c -closed one. Remark 2.1.
We shall need only the existence of such metric forms on compact complexsurfaces and this can be proved by duality: non existence of a positive dd c -closed (1 , ω on U . In the case of a polydisk U = ∆ n we willwork with the standard Euclidean metric e . The associated form will be denoted by ω e = dd c k z k = i P nj =1 dz j ∧ d ¯ z j . By pr : U × X −→ U and pr : U × X −→ X denotethe projections onto the first and second factors. On the product U × X we consider themetric form ω = pr ∗ ω + pr ∗ ω h .A meromorphic mapping f between complex manifolds U and X is defined by an irreducible analytic subset Γ f ⊂ U × X such that • the restriction pr | Γ f : Γ f → U of the natural projection to Γ f is a proper modifi-cation, i.e., is proper and generically one to one. Γ f is called the graph of f . Due to the irreducibility of Γ f and the Remmert propermapping theorem the set of points over which pr is not one to one is an analytic subsetof U of codimension at least two. This set is called the set of points of indeterminacyof f and is usually denoted as I f . Therefore an another way to define a meromorphicmapping f between complex manifolds U and X is by considering a holomorphic map f : U \ A → X , where A is an analytic subset of U of codimension at least two, suchthat the closure Γ f of its graph is an analytic subset of the product U × X satisfying thecondition above. Remark that the analyticity of the closure of the holomorphic graph isnot automatic. Think about the natural projection f : C \ { } → C \ { } /z ∼ z of C \ { } onto a Hopf surface. The properness of the restriction of the projection pr tothe closure is, unless X is disk convex, not automatic too.The volume of the graph Γ f of a meromorphic mapping f is given by opologies on the space of meromorphic mappings 7 n ! Vol (Γ f ) = Z Γ f ω n = Z Γ f ( pr ∗ ω + pr ∗ ω h ) n = Z U (cid:0) ω + f ∗ ω h (cid:1) n , (2.1)where n = dim U . Remark 2.2.
Let us make a few remarks concerning the notion of a meromorphic map-ping. a) If V is a subvariety of U such that V I f then the restriction f | V of f to V is definedby taking as its graph Γ f | V the irreducible component of the intersection Γ f ∩ ( V × X )which projects onto V generically one to one. Therefore Γ f | V ⊂ Γ f ∩ ( V × X ) and theinclusion here is proper in general. The full image of a set L ⊂ U under f is defined as f [ L ] := pr (Γ f ∩ [ L × X ]). b) It is probably worth to notice that x ∈ I f if and only if dim f [ x ] >
1. This follows fromthe obvious observation that I f = pr (cid:16) { ( x , x ) ∈ Γ f : dim ( x ,x ) pr | − f ( x ) > } (cid:17) . c) If dim V = 1 then the irreducible component of Γ f ∩ ( V × X ) which projects onto V is a curve. Since the projection is generically one to one it is on to one everywhere andtherefore the restriction f | V is necessarily holomorphic . d) Let us give the sense to f ∗ ω h in the formula (2.1). The first integral there has perfectlysense since we are integrating a smooth form over a complex variety. Denote by I εf the ε -neighborhood of the indeterminacy set I f of f . Then (2.1) shows that the limit lim ε → Z U \ ¯ I εf (cid:0) ω + f ∗ ω h (cid:1) n = lim ε → Z U \ ¯ I εf n X p =0 C pn ω n − p ∧ f ∗ ω ph (2.2)exists. Therefore all f ∗ ω ph are well defined on U as positive currents. Before turning to the notions of convergence ofmeromorphic mappings let us recall the natural topologies on the space of analytic subsetsof a complex manifold.Recall that an analytic cycle of dimension r in a complex manifold Y is a formal sum Z = P j n j Z j , where { Z j } is a locally finite sequence of reduced analytic subsets of puredimension r and n j are positive integers called multiplicities of Z j . The set | Z | := S j Z j iscalled the support of Z . In our applications Y will be U × X and r will be the dimension n = dim U . By a coordinate chart adapted to Z we shall understand a relatively compactopen set V in Y such that V ∩ | Z | 6 = ∅ together with a biholomorphism j of V onto aneighborhood ˜ V of ¯∆ r × ¯∆ q in C r + q , r + q = dim Y , such that j − ( ¯∆ r × ∂ ∆ q ) ∩ | Z | = ∅ .We shall denote such chart by ( V, j ). The image j ( Z ∩ V ) of the cycle Z ∩ V underbiholomorphism j is the image of the underlying analytic set together with multiplicities.Following Barlet and Fujiki, see [Ba] and [Fj], we call the quadruple E = ( V, j, ∆ r , ∆ q ) a scale adapted to Z .If pr : C r × C q → C r is the natural projection, then the restriction pr | j ( Z ∩ V ) : j ( Z ∩ V ) → ∆ r is a branched covering of degree say d . This branched covering defines in a naturalway a holomorphic mapping ϕ j,Z : ∆ r → Sym d (∆ q ) to the d -th symmetric power of ∆ q by setting ϕ j,Z ( z ′ ) = (cid:8) ( pr | j ( Z ∩ V ) ) − ( z ′ ) (cid:9) . The latter denotes the unordered set of allpreimages of z ′ under the projection in question. This construction, due to Barlet, allows Section 2 to represent a cycle Z ⊂ Y by a set of holomorphic maps ϕ j α ,Z : ∆ r → Sym d (∆ q ), where { ( V α , j α ) } is some open covering of | Z | by adapted coordinate charts. Definition 2.2.
One says that Z k converges to Z in the topology of cycles if for everycoordinate chart ( V, j ) adapted to Z there exists k such that ∀ k > k this chart will beadapted to Z k and the sequence of corresponding holomorphic mappings ϕ j,Z k converge to ϕ j,Z uniformly on ∆ r . This defines a metrizable topology on the space C r ( Y ) of r -cycles in Y . This topologyis equivalent to the topology of currents : Z k → Z if for any continuous ( r, r )-form χ withcompact support one has Z Z k χ → Z Z χ, see [Fj]. It is also equivalent to the Hausdorff topology under an additional condition of boundedness of volumes . Recall that the Hausdorff distance between two subsets A and B of a metric space ( Y, ρ ) is a number ρ ( A, B ) = inf { ε : A ε ⊃ B, B ε ⊃ A } . Here by A ε wedenote the ε -neighborhood of the set A , i.e. A ε = { y ∈ Y : ρ ( y, A ) < ε } .Now, Z k → Z if and only if for every compact K ⋐ Y there exists C K > Vol r ( Z k ∩ K ) C K and Z k ∩ K → Z ∩ K with respect to the Hausdorff distance. Thisstatement is the content of the Harvey-Shiffman’s generalization of Bishop’s compactnesstheorem. For the proof see [HS]. We denote the space of r -cycles on Y endowed with thetopology described as above by C loc r ( Y ). Let { f k } be a sequence ofmeromorphic mappings of a complex manifold U to a complex manifold X . Definition 2.3.
We say that f k converge strongly to a meromorphic map f : U → X ( s -converge) if the sequence of graphs Γ f k converge over compacts to Γ f in Hausdorff metric, i.e., for every compact K ⋐ U one has Γ f k ∩ ( K × X ) H −→ Γ f ∩ ( K × X ) . Now let us prove Theorem 1 from the Introduction, i.e., that Hausdorff convergence inthe case of graphs implies the boundedness of volumes (over compacts) and therefore theconvergence in the topology of cycles . Let us underline at this point that in this theoremone doesn’t need to suppose anything on the target manifold X . Proof of Theorem 1.
The reason why Hausdorff convergence of graphs implies their strongerconvergence in the topology of cycles is that, being the graphs, the analytic cycles Γ f k converge to Γ f with multiplicity one. Now let us give the details. Let a ∈ U \ I f bea regular point of f and set b = f ( a ). Then we can find neighborhoods D ∋ a and D ∋ b biholomorphic to ∆ n , n = dim U and ∆ p , p = dim X respectively such that Γ f ∩ (cid:0) ¯ D × ∂D (cid:1) = ∅ . In particular V = D × D is an adapted chart for Γ f , let ( V, j, ∆ n , ∆ p )be a corresponding scale. Here j : V → ∆ n × ∆ p is some biholomorphism. From Hausdorffconvergence of Γ f k to Γ f we see that for k ≫ f k ∩ (cid:0) ¯ D × ∂D (cid:1) = ∅ . ThereforeΓ f k ∩ ( D × D ) → D is a ramified covering ( i.e., is proper) of some degree d k . But Γ f k isone to one over a generic point of D . Therefore d k = 1 and Γ f k ∩ V converge to Γ f ∩ V asgraphs (in particular as cycles). We proved that f k converge to f on compacts of U \ I f as holomorphic mappings.Let now a ∈ I f and take some b ∈ f [ a ]. As above take a neighborhood V = D × D ∼ =∆ n × ∆ p of ( a, b ), where a = 0 and b = 0 in these coordinates. Denote by ( w ′ , w ′′ ) the opologies on the space of meromorphic mappings 9 coordinates in ∆ n × ∆ p . Perturbing the slope of coordinate w ′′ we can suppose that( { } × ∆ p ) ∩ Γ f has 0 as its isolated point. Remark 2.3.
After perturbation of the slope of w ′′ the decomposition j ( V ) = ∆ n × ∆ p will not correspond to the decomposition U × X .For sufficiently small ε > nε and ∆ pε in (perturbed) coordinates(actually only w ′′ needs to be perturbed). We get an adapted chart for Γ f which possedthe following property: i) ˜ V := j ( V ) writes as ˜ V = ∆ n × ∆ p , p = dim X . ii) Local coordinates ( w ′ , w ′′ ) of ∆ n × ∆ p enjoy the property that z ′ := w ′ ∈ ∆ n is alocalcoordinate in U (but w ′′ is not a local coordinate on X ). iii) j (Γ f ∩ V ) → ∆ n is a ramified covering of degree d > f k to Γ f we get that for all k ≫ j (Γ f k ∩ V ) is a ramified covering of ∆ n of degree d k . Obviously d k > d for k ≫
1. If forsome subsequence d k > d we shall get a contradiction as follows. In that case someirreducible component of Γ f ∩ V will be approached by Γ f k ∩ V at least doubly. Let Γstands for this irreducible component. Since dim [Γ f ∩ ( I f × X )] n − f ) we see that Γ f k multiply approach every compact of Γ \ ( I f × X ). Take a point c ∈ Γ \ ( I f × X ) having a relatively compact neighborhood W ⊂ Γ \ ( I f × X ) such that pr | W : W → W is biholomorphic, i.e., W is the graph of f over W ⋐ U \ I f . Now itis clear that Γ f k ∩ ( W × X ) cannot approach Γ f ∩ ( W × X ) = W with multiplicity morethan one, because Γ f k is a graph of a holomorphic map over W for k ≫ f ∩ V the graphs Γ f k approach with mul-tiplicity one. Therefore j (Γ f k ∩ V ) → ∆ n is a ramified covering of the same degree d as j (Γ f ∩ V ) → ∆ n for k ≫
1. This proves at a time that Γ f k converge to Γ f in the topologyof cycles and that their volumes are uniformly bounded. (cid:3) Strong convergence has some nice features, one was mentioned in the Introduction.Moreover, as it is explained in [Iv4], strong topology is natural in studying fix pointsof meromorphic self-mappings of compact complex manifolds. But domains of strongconvergence and strong normality are quite arbitrary. We shall explain this in moredetails. Let F be a family of meromorphic mappings from a complex manifold U to adisk convex complex manifold X . Definition 2.4.
The set of normality of F is the maximal open subset N F of U such that F is relatively compact on N F . If F = { f k } is a sequence then the set of convergence of F is the maximal open subset of U such that f k converge on compacts of this subset. To be relatively compact in this definition means that from every sequence of elementsof F one can extract a converging on compacts subsequence. The sense of convergence(strong, weak or other) should be each time specified. Example 2.1. 1.
Let X be a Hopf three-fold X := C \ { } /z ∼ z . Denote by π : C \ { } → X the canonical projection. Let D ⋐ C be any bounded domain. Take asequence { a n } ⊂ D accumulating to every point on ∂D . Let g n : C → C be defined as g n ( z ) = ( z − a n , /n ). Set f n := π ◦ g n . Then the set of normality of F = { f n } has D asone of its connected components. Remark 2.4.
For an analogous example with X projective see Example 4 from [Iv2]. The same example is instructive when understanding the notion of weak convergence.Take a converging to zero sequence a n . Then f n from this example will converge oncompacts of C \ { } but the limit will not extend to zero meromorphically. I.e., f n willnot converge weakly in any neighborhood of the origin.
3. Pseudoconvexity of sets of normality
In view of such examples aweaker notion of convergence for meromorphic mappings was introduced in [Iv2]. Let f ∈ M ( U, X ) be a meromorphic map from U to X and let { f k } ⊂ M ( U, X ) be a sequenceof meromorphic mappings.
Definition 3.1.
We say that f k converge weakly to f ( w -converge) if there exists ananalytic subset A in U of codimension at least two such that f k converge strongly to f on U \ A . Remark 3.1. f k converge weakly to f if and only if for every compact of U \ I f all f k are holomorphic in a neighborhood of this compact for k big enough and uniformlyconverge there to f as holomorphic mappings. Indeed, let A be the minimal analyticset of codimension > f k converge strongly to f on U \ A . Then A must becontained in I f because if there exists a point a ∈ A \ I f then f is holomorphic in someneighborhood V ∋ a and then, by Rouch´e Principle of [Iv2] f k for k ≫ V \ A and converge uniformly (on compacts) to f there. From here andthe fact that codim A > f k are holomorphic on compacts in V andconverge to f .Now let us turn to the sets of weak convergence/normality. Sets of strong normalityobviously are well defined, i.e., they do exist. The existence of sets of weak normalitywas proved in [Iv2], see Corollary 1.2.1a. Remark 3.2.
In the formulation of this Corollary the Author of [Iv2] speaks about “weakconvergence” but the proof is about “weak normality“.Domains of weak convergence of meromorphic mappings turn to be pseudoconvex fora large class of target manifolds. This follows from the ”mutual propagation principle”stated in Theorem 3 in the Introduction. Let us give a proof of it.
Proof of Theorem 3.
Let us prove the part (b) first.
Step 1. Extension of the limit.
First of all by the main result of [Iv3] every meromorphicmap f : U → X extends to a meromorphic map f : ˆ U \ A → X , where A is closed, complete( n − U of Hausdorff (2 n − a ∈ A there exists a coordinate neighborhood V ∼ = ∆ n − × B of a = 0such that A ∩ (∆ n − × ∂ B ) = ∅ and for every z ′ ∈ ∆ n − the intersection A z ′ := A ∩ B z ′ is a zero dimensional complete pluripolar subset of B z ′ := { z ′ } × B . Here B stands forthe unit ball in C . Moreover, if A = ∅ then f ( S z ′ ) is not homologous to zero in X . Here S z ′ = ∂ B z ′ is the standard three-dimensional sphere in C .Let U ′ be the maximal open subset of ˆ U \ ( I f ∪ A ) such that f k converge to f on compactson U ′ as holomorphic mappings. seudoconvexity of sets of normality 11 Step 2. U ′ is locally pseudoconvex in ˆ U \ ( I f ∪ A ) . If not then by Docquier-Grauert criterion,see [DG], there would exist a point b ∈ ∂U ′ \ ( I f ∪ A ) and a Hartogs figure h : H nε → U ′ imbedded to U ′ such that the image h (∆ n ) of the corresponding polydisk contains b . Allthis is local and therefore we can assume that h (∆ n ) is relatively compact in U \ ( I f ∪ A ).Pulling back f k and f to ∆ n we arrive to contradiction as follows. By the Theorem ofSiu, see [Si2], there exists a Stein neighborhood V of the graph of f ◦ h in ∆ n × X . Sincefor every compact K ⋐ H nε we have that the graph of f k | K is contained in V we concludethe same for every compact of ∆ n . Now f k ◦ h converge to f ◦ h on compacts in ∆ n asholomorphic mappings. But that mean that they converge also around the preimage of b .Contradiction. Since ˆ U was supposed to be the envelope of holomorphy of U we obtainthat U ′ = ˆ U \ ( I f ∪ A ). Step 3. Removing A . Suppose A is non-empty. Take a sphere S z ′ as described in Step1 for some point a ∈ A . Using the fact that I f is of codimension > S z ′ not to intersect I f as well. I.e., S z ′ ⊂ U ′ . f k ( S z ′ ) is homologous to zero in X , because f k meromorphically extends to the corresponding B z ′ . Moreover f k converge to f in aneighborhood of S z ′ . This implies that f ( S z ′ ) is also homologous to zero and therefore A should be empty. Contradiction. Part (b) is proved.The proof of (a) is a particular case of the Step 2 of the proof of part (b). (cid:3)
Remark 3.3.
We gave a proof of Theorem 3 here because the proof of an analogousstatement in [Iv2] uses a stronger extension claim from the subsequent paper [Iv3]. Namelythe Author claimed that A appearing in the Step 1 of the proof is analytic of codimensiontwo. This was not achieved in [Iv3] (and is not clear for us up to know). Therefore wefind necessary to remark that vanishing of (2 n − A togetherwith homological characterization of the obstructions for the meromorphic extension is,in fact, sufficient for our particular task here. Let again f k be a sequenceof meromorphic mappings between complex manifolds U and X , the last is supposed tobe disk-convex. Let f ∈ M ( U, X ) be a meromorphic map.
Definition 3.2.
We say that f k Γ -converge to f if: i) there exists an analytic subset A ⊂ U such that f k strongly converge to f on U \ A ; ii) for every divisor H in X , such that f ( U ) H and every compact K ⋐ U thevolumes of f ∗ k H ∩ K counted with multiplicities are uniformly bounded for k ≫ . Remark 3.4.
This notion is strictly weaker than the weak convergence because A canhave components of codimension one, and remark that the item ( ii) is automaticallysatisfied by a weakly converging sequence, because divisors f ∗ H extend from U \ A to U and if they have bounded volume on compacts of U \ A then the same is true oncompacts of U . All this obviously follows from the ingredients involved in the proof ofBishop’s compactness theorem, see [Bi] or [St]. It might be convenient to add to A theindeterminacy set of f and then, see Remark 3.1, f k will converge to f uniformly oncompacts of U \ A as holomorphic mappings. Example 3.1. a) Consider the following sequence of holomorphic mappings f k : ∆ → P : f k : z → (cid:20) z + ... + 1 z k k ! (cid:21) = (cid:20) z k : z k + z k − + ... + 1 k ! (cid:21) . (3.1) It is clear that f k converges on compacts of ∆ \ { } to f ( z ) = [1 : e z ] but, as it isclear from the second expression in (3.1) the preimage counting with multiplicities of thedivisor H = { Z = 0 } is k [0] (here [ Z : Z ] are homogeneous coordinates in P ), i.e., hasunbounded volume. And indeed, this sequence should not be considered as convergingone, because its limit is not holomorphic on ∆. b) Set f k ( z ) = [ z : z − k ] : ∆ → P . This sequence clearly converges to the constant map f ( z ) = [ z : z ] = [1 : 1] on compacts of ∆ \ { } . Moreover, the preimage of any divisor H = { P ( z , z ) = 0 } under f k is { z ∈ ∆ : P ( z, z − k ) = 0 } , i.e., is a set of points, uniformlybounded in number counting with multiplicities. Therefore this sequence Γ-converge (butdoesn’t converge weakly). Example 3.2.
Consider the following sequence of meromorphic functions on ∆ ( i.e., meromorphic mappings to P ): f k ( z , z ) = [ z : 2 − k z k ] . The limit map is constant f ( z ) = [1 : 0]. f k converge to f strongly (uniformly in fact)on compacts of ∆ \ { z = 0 } . If [ Z : Z ] are homogeneous coordinates in P then thepreimage of the divisor [ Z = 0] is k [ z = 0], i.e., this sequence doesn’t converge even inΓ-sense on ∆ . Remark 3.5.
Examples 3.1 (a) and 3.2 are examples of converging outside of an analyticset of codimension one sequences which are not Γ-converging. In the first case the limitdoesn’t extend to the whole source, in the second it does. Convergence of meromorphicmappings of this type was introduced and studied by Rutishauser in [Ru].If in Definition 2.4 the underlying convergence is Γ-convergence we get the correspond-ing notions of a convergence/normality set. Let us conclude this general discussion withthe following
Proposition 3.1.
Let X be a disk-convex Gauduchon manifold. Then the sets of Γ -convergence/normality of meromorphic mappings with values in X are locally pseudocon-vex. Proof.
We shall prove the statement for the sets of Γ-normality, the case of sets ofconvergence obviously follows. Let D be the maximal open subset of U where the family F is Γ-normal. Suppose that D is not pseudoconvex. Then by Docquier-Grauert criterion,see [DG], there exists an imbedding h : H nε → D of a Hartogs figure into D such that h extends to an immersion of the polydisk to U with h (∆ n ) ∩ ∂D = ∅ . Recall that Hartogsfigure is the following domain H nε := (cid:0) ∆ n − ε × ∆ (cid:1) ∪ (cid:0) ∆ n − × A − ε, (cid:1) , (3.2)where A − ε, := ∆ \ ¯∆ − ε is an annulus. Let us pull-back our family to ∆ n by h and there-fore without loss of generality we can suppose that U = ∆ n , F is a family of meromorphicmappings from ∆ n to X , H nε ⊂ D ⊂ ∆ n is the set of Γ-normality of F such that D = ∆ n .That means that there exists a sequence { f k } ⊂ F , which converges on D but doesn’tnot Γ-converge on compacts in ∆ n . Let us see that this is impossible. Let f : D → X be the Γ-limit of f k . Denote by A the analytic set in D such that f k converge to f oncompacts of D \ A . By [Iv3] f extends to ∆ n \ S , where S is closed ( n − n . Let A ′ be the pure ( n − A . By the theorem ofGrauert we have two cases. onvergence of mappings with values in projective space 13 Case 1. The envelope of holomorphy of D \ A ′ is ∆ n . In that case the Theorem 3 is applicablewith U = D \ A ′ and ˆ U = ∆ n and gives us the weak (and therefore Γ) convergence of f k on ∆ n . Case 2. A ′ extends to a hypersurface ˜ A in ∆ n and ∆ n \ ˜ A is the envelope of holomorphy of D \ A ′ . In that case again by Theorem 3 f k weakly converge to f on ∆ n \ ˜ A . S \ ˜ A isremovable for f , see the Step 3 in the proof of Theorem 3. Therefore f k strongly convergeto f outside of a proper analytic set A ∪ I f . We need now to prove that f is extendableto ∆ n , i.e., that S is empty. By Lemma 7.2 below the areas of disks f k (∆ z ′ ) are boundeduniformly on k and on z ′ ∈ ∆ n − (1 − ε ) for any fixed ε >
0, here ∆ z ′ := { z ′ }× ∆. Thereforethe areas of f (∆ z ′ ) are bounded to. Theorem 1.5 together with Proposition 1.9 from [Iv3]imply now that f meromorphically extends onto ∆ n − (1 − ε ) × ∆. Therefore it extends to∆ n . The condition ( i) of Definition 3.2 is fulfilled.Let H be a divisor in X . Then for every compact K ⋐ H nε the volumes of f ∗ k H ∩ K counted with multiplicities are bounded. By Oka-Riemenschneider theorem, see [Rm],the volumes of the extensions of these divisors are bounded on compacts of ∆ n to. Thisverifies the condition ( ii) of Definition 3.2. Proposition is proved. (cid:3)
4. Convergence of mappings with values in projective space
Now let us examine our notions of convergence on the example when the target manifoldis a complex projective space.
Let a meromorphicmapping f : U → P N be given. Without loss of generality we suppose that the image of f is not contained in a hyperplane. Then the (complete) inverse image f − ( H ) under f of ahyperplane H is a divisor in U . By f − ( P n \ H ) we shall understand U \ f − ( H ). Denoteby [ w : w : ... : w N ] the homogeneous coordinates of P N . Let U j = { w ∈ P N : w j = 0 } and let w w j , ..., w N w j be affine coordinates in U j . Set D j := f − ( U j ), i.e., D j = U \ f − ( H j ),where H j := { w j = 0 } . Since U is isomorphic to C N the restriction f | D : D −→ U is given by holomorphic functions w w = f ( z ) , ..., w N w = f N ( z ). The coordinate changein P N shows that f | D ∩ D j : D ∩ D j −→ P N is given by functions w w = f j ( z ) , ..., w N w = f N ( z ) f j ( z ) which are holomorphic in D j . Therefore functions f , ..., f N are meromorphic on D ∪ D j . This proves that f , ..., f N are meromorphic on S Nj =0 D j ⊂ U . We have that U \ S Nj =0 D j = T Nj =0 f − ( H j ), i.e., for every point from this set the image of every itsneighborhood intersects every H j . Such point can be only an indeterminacy point of f .I.e., U \ S Nj =0 D j ⊂ I f . I f is analytic of codimension > f j meromorphically extends to U .If f ≡ ... ≡ f n ≡ f ( U ) ≡ ∈ U . If not, let f
0. One finds holomorphicfunctions h j et g j ≤ j ≤ N in a polydisk neighborhood V of a given point x ∈ U , g j = 0such that f = h g , ..., f N = h N g N and therefore gets f := (cid:20) h g : ... : h N g N (cid:21) = " N Y j =1 g j : h N Y j =2 g j : ... : h N N − Y j =1 g j . This proves that f can be locally written in the form f ( z ) := [ f ( z ) : f ( z ) : ... : f N ( z )] (4.1)as claimed. Let usprove now the part ( ii) of Theorem 2 from the Introduction. I.e.,
Proposition 4.1.
A sequence of meromorphic mappings f k from a complex manifold U to P N converges weakly on compacts of U if an only if for every point z ∈ U there existsa neighborhood V ∋ z and reduced representations f k = [ f k : ... : f Nk ] , f = [ f : ... : f N ] in V such that for every j N f jk converge to f j uniformly on V . ⇒ Let f k ⇀ f , i.e., f k converge to f weakly. Shrinking U we suppose that all f k and f admit reduced representations f k = [ f k : ... : f Nk ] (4.2)and f = [ f : ... : f N ] (4.3)correspondingly. Up to making a linear coordinate change in P N we can suppose that f [ U ] is not contained in any of coordinate hyperplanes, i.e., that f j j N .Set Z j = { z ∈ U : f j ( z ) = 0 } , and note that T Nj =1 Z j = I f . Since f k converge on compacts in U j := U \ Z j to f , seeRemark 3.1, we see, taking j = 0, that f jk f k ⇒ f j f (4.4)for all j on compacts in U . Denote by Z k the zero divisors of f k and note that they leaveevery compact of U as n → ∞ . Lemma 4.1.
Divisors Z k converge to Z in cycle space topology. Let us prove this Lemma first. Fix a point a ∈ Z \ Z j (if Z \ Z j is not empty) andtake a relatively compact neighborhood V ∋ a such that ¯ V ∩ Z j = ∅ . We have that f k /f jk ⇒ f /f j on ¯ V . The Rouch´e’s theorem easily implies now that Z k ∩ V converge to Z ∩ V as currents. Remark 4.1.
In fact the cycle space topology on the space of divisors coincides with thetopology of uniform convergence of defining them holomorphic functions, see [Stl]. Andthis immediately gives the previous assertion. onvergence of mappings with values in projective space 15
We conclude from here that Z k converge to Z as cycles on compacts in U \ I f . Butthen by [Ni], Theorem II, we obtain that they converge on the whole of U . Lemma 4.1 isproved.We continue the proof of the Theorem. Shrinking U if necessary we can suppose that U is biholomorphic to ∆ n = ∆ n − × ∆ and Z k ∩ U regularly covers ∆ n − for k ≫
1. Noweach Z k can be written as the zero set of a uniquely defined unitary polynomial P k from O ∆ n − [ z n ] and these P k uniformly converge to P - the defining polynomial for Z . Aftermultiplying each [ f k : ... : f Nk ] by the unit P k /f k we get the reduced representations f k = [ P k : g k ... : g Nk ] . The same with f = [ P : g ... : g N ] . But now P k ⇒ P and therefore from (4.4), which reads now as g jk P k ⇒ g j P (4.5)on compacts in U , we get that for every 1 j N g jk ⇒ g j on compacts in U = U \ Z .But from the maximum principle it follows that g jk ⇒ g j on compacts in U . ⇐ For proving the inverse statement we start with converging reduced representations(4.2) to (4.3), i.e., f jk ⇒ f j on U . Then for every 0 j N on every U j = U \ Z j we geta convergence on compacts f k f jk , ..., f Nk f jk ! ⇒ (cid:18) f f j , ..., f N f j (cid:19) . And since the codimension of I f = T Z j is at least two we deduce the weak convergenceof f k to f . (cid:3) Strongconvergence of meromorphic maps into P N can be described in the following way. First,if f k → f then f k ⇀ f . Therefore [ f k : ... : f Nk ] ⇒ [ f : ... : f N ] for an appropriate reducedrepresentations. According to (2.2) the volume of the graph of f k is Z U \ I fk (cid:0) ω + f ∗ k ω F S (cid:1) n = Z U \ I f n X j =0 C jn ω j ∧ f ∗ k ω n − jF S . (4.6)Since f ∗ k ω F S = dd c ln k f k k this is nothing but the non-pluripolar Monge-Amp`ere mass ofln k f k k as appeared in (1.3). By Proposition 1 volumes of Γ f k converge to the volume ofΓ f , i.e., Z U \ I fk ω j ∧ (cid:0) dd c ln k f k k (cid:1) n − j → Z U \ I f ω j ∧ (cid:0) dd c ln k f k (cid:1) n − j (4.7)for 0 j < n . In the case U = ∆ n this gives (1.3). Vice versa, if one has convergence ofvolumes the appearance of an exceptional component is impossible and we conclude thepart ( iii) of Theorem 2: Proposition 4.2. f k converge to f strongly if and only if i) the appropriate reduced representations converge uniformly; ii) for every j n − one has (4.7). Now let us descend to the convergence of meromorphic functions. Meromorphic func-tions on a complex manifold U are exactly the meromorphic mappings from U to P . I.e.,all our previous results and notions are applicable to this case. Proposition 4.3.
If a sequence { f k } of meromorphic functions converge weakly then itconverge strongly. Proof.
Let f be the weak limit of f k . We shall see in a moment, see Corollary 5.2 thatvolumes of graphs in this case are uniformly bounded over compacts in U . Thereforeafter going to a subsequence we get that the Hausdorff limit ˆΓ := lim Γ f k is a purely n -dimensional analytic subset of U × P . We claim that lim Γ f k = Γ f in fact, i.e., thatthere are no exceptional components. If not take any irreducible component Γ of thislimit different from Γ f . Denote by γ its projection to U . γ is a proper analytic set ofcodimension at least two U . But then Γ should be contained in γ × P and the last analyticset is of dimension dim U −
1. This is impossible, because all components of lim Γ f k are ofpure dimension dim U . Therefore γ = ∅ and lim Γ f k = Γ f . (cid:3) In [Fu] and subsequent papers of Fu-jimoto the following type of convergence of meromorphic mappings with values in P N was considered, it was called the m -convergence (or meromorphic convergence): f k m -converge to f if there exist reduced (admissible in the terminology of [Fu]) representations f k = [ f k : ... : f Nk ] which converge uniformly on compacts to f = [ f : ... : f N ], but the last is not supposed to be reduced ( i.e., admissible), only not all f j are identically zero. Letus prove the item ( i) of Theorem 2. Proposition 4.4.
When the target manifold X is the complex projective space P N the Γ -convergence of meromorphic mappings is equivalent to m -convergence in the sense ofFujimoto. Proof. ⇒ Suppose that f k Γ −→ f . Let γ be the an analytic subset of U such that oursequence converge strongly on compacts of U \ γ . We add to γ also the indeterminaciesof the limit f and therefore f k will converge to f on U \ γ in compact open topology.Let f = [ f : ... : f N ] be some reduced representation of the limit map. Making linearchange of coordinates we can suppose, without loss of generality that f i.e., that f ( U ) H , where H = { Z = 0 } in homogeneous coordinates [ Z : ... : Z N ] of P N . Wehave that f ∗ k H converge on compacts in U in the cycle space topology (after taking asubsequence).Take some a ∈ γ and choose a chart ( V, j ) adapted both to γ and f ∗ H with coordinates z , ..., z n around a in such a way that a = 0 and ( γ ∪ f ∗ H ) ∩ (∆ n − × ∆) projects to ∆ n − properly. Then f ∗ k H ∩ V also projects to ∆ n − properly for k ≫
1. After going to asubsequence once more we can fix the degree d of ramified coverings f ∗ k H ∩ V → ∆ n − and write the corresponding polynomials P k ∈ O ∆ n − [ z n ] defining f ∗ k H ∩ V . P k convergeto some P on (compacts of) ∆ n − . Let f k = [ f k : ... : f Nk ] be some reduced representationsof f k on V . Notice that f ∗ k H ∩ V = { f k = 0 } . Divide each such representation by the loch-Montel type normality criterion 17 unit f k /P k and get representations f k = [ P k : g k : ... : g Nk ] with converging first terms P k .At the same time ( g k /P k , ..., g Nk /P k ) represents f k in nonhomogeneous coordinates of thechart Z = 0 on P N . Therefore g jk /P k converge to some f j on compacts of V \ ( γ ∪ f ∗ H ).Therefore g jk converge to g j := f j P on compacts of V \ ( γ ∪ f ∗ H ) to. By maximumprinciple they converge everywhere on V to the extension of g j . We get that reducedrepresentations f k = [ P k : g k : ... : g Nk ] converge term by term to a (may be non reduced)representation [ P : g : ... : g N ] and this can be only a representation of f . ⇐ Suppose now that f k m -converge to f . Again change coordinates in P N , if necessary,in such a way that f ( U ) H . Let V be a neighborhood of some point a ∈ U . If a ∈ f ∗ H then take ( V, j ) to be an adapted chart to this divisor. In any case take V to bebiholomorphic to ∆ n − × ∆. Let F k = ( f k , ..., f Nk ) be the lifts of f k to C N +1 in V such that F k converge to the lift F = ( f , ..., f N ) of f . From here one gets immediately that f jk /f k converge to f j /f uniformly on compacts of V \ { f = 0 } , i.e., that our maps convergestrongly outside of a divisor.Now let H = { P ( Z , ..., Z N ) = 0 } be a divisor such that f ( U ) H . Using convergenceof lifts F k = ( f k , ..., f Nk ) to F = ( f , ..., f N ) one gets that f k ( U ) H for k ≫
1. One hasalso that P ( f k , ..., f Nk ) uniformly converge to P ( f , ..., f N ) and this is equivalent to theconvergence of divisors. (cid:3) Remark 4.2.
The relation between weak/gamma convergence and m -convergence for thecase of X = P N was indicated without proof in [Iv2].
5. Bloch-Montel type normality criterion
The aim of this section is to test the notion of weak convergence on the Bloch-Monteltype normality statement, i.e., we are going to prove here the Theorem 4 from theIntroduction.
Before proceeding with the proof let us recall few basic facts. Westart with an extended version of Zalcman’s lemma, see [Me]:
Lemma 5.1.
A family F of holomorphic mappings from ∆ n to a compact Hermitianmanifold ( X, h ) is not normal at z ∈ ∆ n if and only if there exist sequences z k → z , r k ց , f k ∈ F such that f k ( z k + r k w ) converge uniformly on compacts in C n to a non-constant entire mapping f : C n → X such that k df ( w ) k h for all w ∈ C n . This f may well have rank one. We shall also need the following result from [IS1], whichis a precise version of Gromov compactness theorem (we shall need it in the integrablecase only): Proposition 5.1.
Let u k : ∆ → X be a sequence of holomorphic maps into a disk-convexHermitian manifold ( X, h ) with uniformly bounded areas, which uniformly converges onsome annulus A − ε, adjacent to the boundary ∂ ∆ . Then u k converge to stable complexcurve over X after a reparametrization. Moreover, the compact components of the limitare rational curves. For the notions of stable curve over X , convergence after a reparametrization, as wellas for the proof we refer to [IS1]. The obvious conclusion from this type of convergenceis the following: Corollary 5.1. If u k converge in stable sense to u and u (∆) intersects a divisor H in X ,but us not contained in H , then all u k (∆) intersect H for k ≫ . Proof.
It was proved in [IS2] (more details are given in [IS1]) that for any k ≫ u k with u by a holomorphic one parameter family of stable maps, see Proposition2.1.3 in [IS2] for the exact statement. For us it is sufficient to understand that there existsa normal complex surface Y π −→ ∆ foliated over the disk ∆ such that all fibers Y s := π − ( s )are disks and a holomorphic mapping U : Y → X such that U | Y = u and U | Y s = u k forsome s ∈ ∆ and some k . Remark 5.1.
The fact that this family can be contracted to a surface with normal pointsis proved in Lemma 2.2.6 in [IS2].Let h be a defining holomorphic function of the divisor H near the point of intersection u (∆) ∩ H . Then h ◦U is holomorphic on Y (for this one might need to take disks of smallerradii) and is equal to zero at 0 ∈ Y ⊂ Y . At the same time it cannot vanish on S s ∈ ∆ ∂Y s because U | Y s ( ∂Y s ) is close to u ( ∂ ∆) for all s ∈ ∆. Therefore the zero set of h ◦ U mustintersect every Y s . And that means that u k (∆) intersects H . (cid:3) Let us make one more remark. Let ω F S be the Fubini-Study form on P N . For aholomorphic map f : ¯∆ → C N (we always suppose f to be defined in a neighborhood ofthe closure ¯∆), the area of f (∆) with respect to the Fubini-Study form is area F S f (∆) = Z ∆ f ∗ ω F S . (5.1)Denote by Z = ( Z , ..., Z N ) coordinates in C N +1 and let π : C N +1 \ { } → P N be thestandard projection. Consider the following singular (1 , C N +1 ω = dd c ln k Z k . (5.2)The following statement is a simple case of King’s residue formula, but we shall give asimple proof for the sake of completeness. Lemma 5.2.
For a holomorphic lift F = ( f , ..., f N ) : ¯∆ → C N +1 of f : ¯∆ → P N ( i.e., f = π ◦ F ) such that F | ∂ ∆ doesn’t vanishes one has area F S f (∆) = Z ∂ ∆ d c ln k F k − N F . (5.3) Here N F is the number of zeroes of F counted with multiplicities. Proof.
By the very definition of the Fubini-Study form one has π ∗ ω F S = ω . Andtherefore it is immediate to check that in a neighborhood of a point a ∈ ∆ such that F ( a ) = 0 one has that f ∗ ω F S = F ∗ ω . As the result area F S f (∆) = Z ∆ f ∗ ω F S = Z ∆ \ Z F F ∗ ω , (5.4) loch-Montel type normality criterion 19 where Z F := { z , ..., z k } is the set of zeroes of F , i.e., such z l that f j ( z l ) = 0 for all j = 0 , ..., N . Let n l be the multiplicity of zero z l . Then F ( z ) = ( z − z i ) n l ( g ( z ) , ..., g N ( z )),where at least one of g j -s is not zero at z l . We have that dd c ln k F k = n l δ z l + dd c ln k G k , where G ( z ) = ( g ( z ) , ..., g N ( z )). Therefore dd c ln k G k is an extension of F ∗ ω to z l . Therest obviously follows from the Stokes formula. (cid:3) Let us observe the following immediate corollary from this lemma.
Corollary 5.2.
Let f k : U → P N be a Γ -converging sequence of meromorphic mappingsand let L be a divisor in U such that f k converge uniformly on compacts of U \ L . Let V ∼ = ∆ n − × ∆ be a scale adapted to L and to the limit M of f ∗ k H , where H = [ Z = 0] .Then the areas of the analytic disks f k (∆ z ′ ) are uniformly bounded in z ′ ∈ ∆ n − and k ∈ N . Proof.
Let ( z ′ , z n ) be coordinates in ∆ n − × ∆. Denote by F k = ( f k , ..., f Nk ) lifts of f k to C N +1 . Consider restrictions f k | ∆ z ′ . Due to the fact that our chart is adapted to M = lim f ∗ k H we have that f k doesn’t vanishes on ∂ ∆ z ′ for k ≫ L the lifts F k = ( f k , ..., f Nk ) converge in a neighborhood of ∂ ∆ z ′ . By (5.3) wehave area F S f k (∆ z ′ ) Z ∂ ∆ z ′ d c ln k F k k c, (5.5) i.e., the areas are uniformly bounded for z ′ ∈ ∆ n − and all k . (cid:3) Remark 5.2.
For a family F of meromorphic mappings from a manifold U to a projectivemanifold X to be normal an obvious necessary condition is that for any fixed hypersurface H ⊂ X and any fixed compact K ⋐ U the volumes counting with multiplicities ofintersections f ∗ H ∩ K should be uniformly bounded for f ∈ F . It was proved by Fujimotoin [Fu] that this condition (in the case X = P N and H i are hyperplanes) turns out tobe also sufficient, but only for the meromorphic ( i.e., Γ) normality. We in this paper areinterested in the normality in the weak convergence sense (which is, that’s to say, strongerthan meromorphic one). In that case there is one more necessary condition. Take twohypersurfaces H and H in X . Let { f ∗ H : f ∈ F } and { f ∗ H : f ∈ F } be the families oftheir preimages by elements of our family f ∈ F . By boundedness of volumes conditionfor every sequence f ∗ k H i , i = 0 ,
1, some subsequences f ∗ k j H i converge to divisors L and L . If there exist coinciding (without taking to account the multiplicities) components L ′ and L ′ of L and L respectively, then f k j cannot weakly converge in a neighborhood of L ′ = L ′ . Indeed, since the limit f is a holomorphic map outside of I f , the preimages f ∗ H and f ∗ H cannot have common components. But L ′ and L ′ are such components.Contradiction. This will be formalized in the following definition.Let F be a Γ-normal family in M ( U, X ). Fix a divisor H in X . Remark that for everyrelatively compact D ⋐ U the intersections f ∗ H ∩ ¯ D from a pre-compact family of setswhen f is running over F . Therefore one can find a finite collection of scales { E α } suchthat every f ∗ H ∩ ¯ D can be covered by some of corresponding V α and the members of this covering are adapted to f ∗ H . This collection { E α } of scales depends on D ⋐ U , butdoesn’t depend on f ∈ F and, moreover, doesn’t depend on H taken in some compactfamily of divisors, in our case this family is { H , ..., H d } , i.e., is finite. Definition 5.1.
We say that a family F of meromorphic mappings from a complex man-ifold U to a complex manifold X uniformly separates hypersurfaces H and H from X if for any f ∈ F and any adapted for both f ∗ H and f ∗ H scale E α = ( V α , j α , ∆ n − , ∆) as above, the Hausdorff distance between f ∗ H ∩ V α and f ∗ H ∩ V α for f ∈ F is boundedfrom below by a strictly positive constant. Hausdorff distance is taken here in the Euclidean metric of C n . A constant in questionmay well depend on divisors H , H and adapted chart V α , but it is supposed not todepend on f ∈ F . We are going to prove nowTheorem 4 from the Introduction. Recall that a relatively compact open subset Y of acomplex manifold X is said to be hyperbolically imbedded to X if for any two sequences { x n } and { y n } in Y converging to distinct points x ∈ ¯ Y and y ∈ ¯ Y one has lim sup n →∞ k Y ( x n , y n ) > , where k Y is the Kobayashi pseudodistance of Y . Y ⋐ X is said to be locally hyperbolicallycomplete (l.h.c) if for every y ∈ ¯ Y there exists a neighborhood V y ∋ y such that V y ∩ Y ishyperbolically complete. For example every Y ⋐ X of the form X \{ divisor } is obviouslyl.c.h. It was proved in [Ki] that if Y is hyperbolically imbedded into X and is l.h.c. then Y is complete hyperbolic.These notions are connected to complex lines in ¯ Y by Theorem of Zaidenberg, see [Za].By a complex line in Y (or in X ) one understands an image of a non-constant holomorphicmap u : C → Y (or X ). Sometimes one requires that (cid:13)(cid:13) d z u ( ∂x ) (cid:13)(cid:13) h z ∈ C , where h is some Hermitian metric on X . Complex line u : C → ¯ Y ⋐ X is called limiting for Y ifthere exists a sequence of holomorphic mappings u n : ∆( R ) → Y converging on compactsin C to u : C → ¯ Y . Theorem of Zaidenberg says now that: for a relatively compactl.c.h. domain Y in a complex manifold X to be complete hyperbolic and hyperbolicallyimbedded in X it is necessary and sufficient that Y doesn’t contain complex lines anddoesn’t admits limiting complex lines.Now we turn to the proof. Let { f k } be a sequence from F , where F satisfies theassumptions of Theorem 4 from the Introduction. { H i } di =0 our set of divisors. Step 1. Convergence outside of a divisor.
By Bishop’s compactness theorem for every i some subsequence from f ∗ k H i converges to a (may empty) hypersurface in U . Denote thislimit hypersurface as L i . Set L := d [ i =0 L i . In order not to complicate notations we will not introduce subindexes when extractingsubsequences.If L is empty then for every compact K ⋐ U all f k with k big enough send K to X \ S di =0 H i , the last is Stein. In particular they are holomorphic in a neighborhood of loch-Montel type normality criterion 21 K and we can use Zalcman’s Lemma 5.1 together with Zaidenberg’s characterization toextract a converging subsequence.Therefore from now on we suppose L is nonempty. Take a point z ∈ U \ L and takea relatively compact neighborhood V ∋ z biholomorphic to a ball such that ¯ V ∩ L = ∅ .Then for k big enough f k ( ¯ V ) ⊂ X \ S di =0 H i . This implies that they all are holomorphicon V and we again can find a converging subsequence on V as before. Therefore somesubsequence of { f k } (still denoted as { f k } ) converge on compacts of U \ L in the usualsense of holomorphic mappings. Denote by f its limit. f is a holomorphic map from U \ L to X . Step 2. Convergence across the divisor.
Take a point z ∈ L , if L is empty we can re-numerate L i -s. fix an imbedding i : X → P N and let H be the intersection of X ( i.e., of i ( X )) with hyperplane { Z = 0 } in the standard homogeneous coordinates [ Z : ... : Z N ]of P N . After going to a subsequence we have that f ∗ k H converge, denote by M thelimit. Let ( V, j ) be an adapted chart for L ∪ M (and therefore also for L ) at z with thescale E = ( V, j, ∆ n − , ∆). Let P k [ z n ] ∈ O ∆ n − [ z n ] be the defining unitary polynomial for f ∗ k H ∩ V . P k converges to the defining polynomial P of M ∩ V .Let f k = [ f k : ... : f Nk ] be reduced representations of f k on V (we write f k for i ◦ f k ).Then multiplying this representation by the unit P k /f k we obtain a reduced representation f k = [ P k : g k : ... : g Nk ]. We have that g jk /P k converge on compacts of V \ ( L ∪ M ). Therefore g jk converge there to, denote by g j its limit. We see that lifts F k = ( P k , g k , ..., g Nk ) convergeto F := ( P, g , ..., g N ) on compacts of V \ ( L ∪ M ). By maximum principle they convergeon V .In particular f extends to a meromorphic mapping from U to X . Remark 5.3.
It is worth of noticing that at this stage we proved the Γ-normality of ourfamily. For the case X = P N with H i hyperplanes this was proved in [Fu]. One more pointworth of noticing is that the extendibility of f also follows from usual complex hyperbolicgeometry, see [Ko]. Step 3. Convergence outside of codimension two.
Changing indices of H i , if necessary, wecan suppose that our family uniformly separates H and H . Take a point z ∈ L \ S i =0 L i such that L in addition is smooth at z . Take an adapted scale E = ( V, j, ∆ n − , ∆) for L near z which intersects L only by the smooth part of L and, moreover, such that j ( L ∩ V ) = d · [∆ n − × { } ] for some multiplicity d >
1. Fix coordinates ( z ′ , z n ) on∆ n − × ∆. By Corollary 5.2 the areas of analytic disks f k | ∆ z ′ are uniformly bounded. Fixsome z ′ ∈ ∆ n − and take a subsequence f k such that f k | ∆ z ′ converge in stable topology to f | ∆ z ′ plus a chain C z ′ of rational curves. By Corollary 5.1 if C z ′ intersects some H with i = 0 then f k | ∆ z ′ (∆ z ′ ) intersects H to. But then f ∗ k H ∩ V is nonempty and converge to L ∩ V . This can be only L ∩ V with some multiplicity, because V was chosen in sucha way that L ∩ V = L ∩ V . The last violates the assumed uniform separability of thepair H , H by F . Therefore C z ′ is empty. That means that (some subsequence of) f k | ∆ z ′ uniformly on ∆ z ′ converge to f | ∆ z ′ . This implies that the whole sequence f k restricted to∆ z ′ converge to f . Therefore f k converge to f on U \ Sing L in compact open topology asholomorphic mappings. This proves the Theorem. (cid:3) Remark 5.4.
Theorem of Bloch, see also [Gr], states that Y = P N \ S Nj =0 H i is hy-perbolically imbedded to P N , where H i are hyperplanes in general position. Therefore Y = P N \ S Ni =0 H i is an example for our Theorem 4.
6. Behavior of volumes of graphs under weak and gamma convergence
In this section we are concerned with the following question: let meromorphic mappings f k : U → X converge in some sense to a meromorphic map f , what can be said aboutthe behavior of volumes of graphs of f k over compacts in U ? If f k converge to f stronglythen, as it was proved in Theorem 1, for every relative compact V ⋐ U we have that Vol (Γ f k | V ) → Vol (Γ f | V ) . (6.1)When f k converges only weakly one cannot, of course expect anything like (6.1). At mostwhat one can expect is that volumes of Γ f k stay bounded over compacts in U and convergeto the volume of Γ f plus volumes of exceptional components. I.e., the question is if for aweakly converging sequence { f k } one has that for every relatively compact open V ⋐ U there exists a constant C V such that Vol (Γ f k | V ) C V for all k. (6.2)This turns to be wrong in general, the following example was communicated to us byA. Rashkovskii. There exists a sequence ε k ց holomorphic mappings f k : B → P defined as f k : ( z , z , z ) → [ z : z − ε k : z : z k ] (6.3)converge weakly to f ( z ) = [ z : z : z : 0] on compacts of the unit ball B ⊂ C , but thevolumes of graphs of f k over the ball B (1 /
2) of radius 1 / Vol (Γ f k ) ∩ ( B (1 / × P ) > k. (6.4)Consider the following family of plurisubharmonic functions on the unit ball B in C : u ε,k ( z ) = ln( | z | + | z − ε | + | z | + | z | k ) , ε ∈ (0 , / . (6.5)Note that every u ε,k is bounded in B and its total MA mass in B (1 /
2) coincides withthose of the function˜ u ε,k := max { u ε,k , s k } where s k = min { ln( | z | + | z | + | z | k ) : z ∈ S (1 / } . Here S (1 /
2) = ∂ B (1 /
2) is the sphere of radius 1 /
2. This fact follows from the Bedford-Taylor definition of the MA mass of a product of bounded psh functions, see [BT]: dd c u ∧ dd c u := dd c ( u dd c u ) and so on by induction. Here the point is, of course, to prove that dd c ( u dd c u ) is again a closed positive current. Now one writes MA B (1 / ( u ε,k ) = Z B (1 / ( dd c u ε,k ) = Z B (1 / dd c u ε,k ∧ ( dd c u ε,k ) = Z ∂ B (1 / d c u ε,k ∧ ( dd c u ε,k ) == Z ∂ B (1 / d c ˜ u ε,k ∧ ( dd c ˜ u ε,k ) = Z B (1 / ( dd c ˜ u ε,k ) = MA B (1 / (˜ u ε,k )because u ε,k = ˜ u ε,k on the sphere S (1 / u ε,k converge uniformly to ˜ u k = max { ln(2 | z | + | z | + | z | k ) , s k } as ε ց MA B (1 / (˜ u k ) = MA B (1 / ( u k ) = 4 k , where ehavior of volumes of graphs under weak and gamma convergence 23 u k = ln(2 | z | + | z | + | z | k ), we shall have that for ε k small enough MA B (1 / ( u ε k ,k ) > k .This finishes the proof. Remark 6.1.
Examples of psh functions with polar singularities and unbounded non-polar MA mass where constructed first by Shiffman and Taylor, see [Si1], and espe-cially simple one by Kiselman, see [Ks]: u ( z , ..., z n ) = (1 − | z n | )( − ln (cid:13)(cid:13) z ′ (cid:13)(cid:13) ) / for z ′ =( z , ..., z n − ). Taking any of these examples and smoothing it by convolutions one gets adecreasing sequences of psh functions converging outside of an analytic set (on any codi-mension) to a psh function, smooth outside of this set with unbounded non-polar MAmass. The remarkable feature of the example of Rashkovskii, just described, is that func-tions in this example have a geometric meaning , their dd c -s are pullbacks of Fubini-Studyform by a meromorphic mappings to the complex projective space, i.e., the sum of theirnon-polar MA masses are the volumes of the corresponding graphs. If { f k } is a Γ-converging sequence of meromor-phic mappings with values in one dimensional complex manifold then it is easy to see thatthe volumes of graphs of f k -s are locally bounded over compacts in the source. Indeed, aone dimensional manifold X either properly imbeds to C n (when X is noncompact) or isprojective and therefore imbeds to P n . In both cases by Theorem 2 we have convergenceof reduced representations to a, may be nonreduced representation of the limit. Inequality(5.5) implies that in an appropriately chosen local coordinates ( z ′ , z n ) one has Vol (Γ f k | ∆ n ) = Z ∆ n ( dd c || z || ) n + Z ∆ n ( dd c || z || ) n − ∧ f ∗ k ω F S Z ∆ n ( dd c || z || ) n ++ Z ∆ n − ( dd c || z || ) n − Z ∂ ∆ z ′ d c ln k F k k const . Next, if the dimension n of the source U is 2 the boundedness of volumes of graphs ofa weakly converging sequence is automatic. This can be seen at least in two ways. First,in projective case this readily follows from the following formula of King, see [Kg]: d (cid:2) d c ln( k f k ) ∧ dd c ln( k f k ) (cid:3) = χ U \ I f h(cid:0) dd c ln( k f k ) (cid:1) i − X j n j [ Z j ] , (6.6)provided I f has pure codimension two. Z j are irreducible components (branches) of theindeterminacy set I f of f . If it has branches of higher codimension then around thesebranches a higher order non-pluripolar masses can be expressed in a similar way. Now if f k weakly converge to f formula (6.6) immediately gives a uniform bound of correspondingMA masses (even together with that concentrated on pluripolar sets I f k ). If n = 2 thenthat’s all we need.Second, using Skoda potentials, or Green functions, as it was done in [Iv2] Theorem2, one can bound non-pluripolar Monge-Amp`ere masses of order two also in the caseof weakly converging sequence with values in disk-convex K¨ahler X . This observationimplies that if X is disk-convex K¨ahler and dim U = 2 then the volumes of graphs ofweakly converging sequences of meromorphic mappings U → X are uniformly boundedover compacts in U .Moreover, it was proved in [Ne] that volumes of weakly converging sequence are boundedalso in the case when X is any compact complex surface. The proof uses Ka¨ahler case separately and then the fact that a non-K¨ahler surface has only finitely many rationalcurves. Remark 6.2.
Let us remark that there is one more important case when the volumes ofgraphs of weakly (even Γ) converging sequence necessarily stay bounded: namely when { f k } is a Γ-converging sequence of meromorphic mappings between projective manifolds X and Y . Indeed the volumes of graphs Γ f k are uniformly bounded as it is straightforwardfrom Besout theorem.
7. Rational connectivity of the exceptional components of the limit
Recall that a rational curve C in a complex manifold X is an image of P in X under a non-constant holomorphic map h : P → X . A chain ofrational curves is a connected union C = S j C j of finitely many rational curves. Definition 7.1.
A closed subset Γ ⊂ X we call rationally connected if for very two points p = q in Γ there exists a chain of rational curves C ⊂ Γ such that p, q ∈ C . One says also that C connects p with q . If Γ is a complex manifold then this propertyis equivalent to the either of the following two ones: • Every two points in X can be connected by a single rational curve. • For any finite set of points F ⊂ X there exists a rational curve C ⊃ F .We refer to [Ar] for these facts. Now let us turn to the proof of Theorem 5 from theIntroduction. It consists from the two following lemmas. Let f k be a weakly or, gamma-converging sequence of meromorphic mappings and f denotes their limit. Let ˆΓ be theHausdorff limit of the graphs, Γ = ˆΓ \ Γ f the corresponding bubble. Set γ := pr (Γ). It isat most a divisor in Γ-case and has codimension > V ∼ = ∆ n − × ∆be a scale adapted to γ in the sense that ( ¯∆ n − × ∆) ∩ γ = ∅ . Lemma 7.1.
Suppose that there exists a dense subset S ⊂ ∆ n − such that the areas ofthe analytic disks Γ f k | ∆ z ′ are uniformly bounded in z ′ ∈ S and k ∈ N then for every point a ∈ γ the fiber Γ a := pr ( pr − ( a )) is rationally connected. Proof.
Here writing f k | ∆ z ′ we mean the restriction of f k to the disk ∆ z ′ := { z ′ } × ∆.Fix a point a ∈ γ and some a , a ∈ Γ a . Suppose a = a , otherwise there is nothingto prove. We need to prove that there exists a chain of rational curves in Γ a connecting a with a . Since Γ f k converge to ˆΓ ⊃ Γ a there exist a k → a and a k → a such that f k ( a k ) → a and f k ( a k ) → a . Perturbing slightly we can take such a ik to be regular ( i.e., not indeterminacy) points of f k for i = 1 ,
2. Take a scale adapted to γ near a in thesense that γ ∩ (∆ n − × ∂ ∆) = ∅ . Denote by ( z ′ , z n ) = ( z , ..., z n − , z n ) the correspondingcoordinates and assume without loss of generality that a = 0.Let b k → ′ and b k → ′ in ∆ n − be such that a k ∈ ∆ b k and a k ∈ ∆ b k . Taking again a ik sufficiently general we can arrange that b ik ∈ S and disks ∆ b k and ∆ b k converge tothe disk ∆ ′ . After taking a subsequence we get that graphs in question converge to thegraph Γ f | ∆0 ′ ∪ C i , where C i ⊂ { a } × X are chains of rational curves. Both these chainscontain the point f | ∆ ′ ( a ). Therefore C := C ∪ C is connected. At the same time byconstruction C i ∋ a i . Lemma is proved. (cid:3) ational connectivity of the exceptional components of the limit 25 Let us first consider the case of Γ-converging sequence ofmeromorphic mappings with values in projective X . Corollary 5.2 gives us the requiredboundedness of ares of analytic disks which makes possible to apply Lemma 7.1 justproved. This gives us the statement of Theorem 5 for Γ-converging sequences of mero-morphic mappings with values in projective manifolds.To treat the case of Gauduchon target manifolds we shall need one more lemma. Lemma 7.2.
Let F be a family of meromorphic mappings from ∆ n to a disk-convexmanifold X, which admits a pluriclosed metric form. Suppose that for some < ǫ < , thefamily F is holomorphic and equicontinuous on the Hartogs figure H nε . Then for every < r < the areas of graphs Γ f z ′ of restrictions f z ′ := f | ∆ z ′ ( r ) of f ∈ F to the disks ∆ z ′ ( r ) := { z ′ } × ∆ r are uniformly bounded in z ′ ∈ ∆ n − r and f ∈ F . Proof.
For f : ∆ n −→ X a meromorphic map, we denote by I f ⊂ ∆ n the set of points ofindeterminacy of f . Since we suppose that all f ∈ F are holomorphic on H nε the sets I f donot intersect ∆ n − × A − ε, . Consider currents T f = f ∗ ω on ∆ n , where ω is a pluriclosedmetric form on X . Write T f = i t α ¯ βf dz α ∧ d ¯ z β , where t α ¯ βf are distributions on ∆ n (in fact measures), smooth on ∆ n \ I f ⊃ H nε . Fix1 − ε < r < r < n − \ π ( I f ) (where π : ∆ n → ∆ n − is the canonicalprojection onto the first factor) the area functions a f given by a f ( z ′ ) = area f z ′ (∆ r ) = Z ∆ z ′ ( r ) T f = i Z ∆ z ′ ( r ) t n ¯ nf dz n ∧ d ¯ z n . (7.1)Functions a f are well-defined and smooth on ∆ n − \ π ( I f ).The proof of Proposition will be done in two steps. Step 1.
Distributions t n ¯ nf are locally integrable in ∆ n . Note that forms T f are smooth on H nε and the family { T f : f ∈ F } is equicontinuous there. The condition that dd c T f = 0implies, in particular, that for all 1 k, l n − ∂ t n ¯ nf ∂z k ∂ ¯ z l + ∂ t k ¯ lf ∂z n ∂ ¯ z n − ∂ t k ¯ nf ∂z n ∂ ¯ z l − ∂ t n ¯ lf ∂z k ∂ ¯ z n = 0 . (7.2)From (7.2) we get that on ∆ n − \ π ( I f ): dd c a f = (cid:18) i (cid:19) n − X k,l =1 Z ∆ z ′ ( r ) ∂ t n ¯ nf ∂z k ∂ ¯ z l dz n ∧ d ¯ z n dz k ∧ d ¯ z l = (7.3)= (cid:18) i (cid:19) n − X k,l =1 Z ∆ z ′ ( r ) ∂ t k ¯ nf ∂z n ∂ ¯ z l + ∂ t n ¯ lf ∂z k ∂ ¯ z n − ∂ t k ¯ lf ∂z n ∂ ¯ z n ! dz n ∧ d ¯ z n · dz k ∧ d ¯ z l == (cid:18) i (cid:19) n − X k,l =1 Z ∂ ∆ z ′ ( r ) ∂t k ¯ nf ∂ ¯ z k d ¯ z n + Z ∂ ∆ z ′ ( r ) ∂t n ¯ lf ∂z k dz n − Z ∂ ∆ z ′ ( r ) ∂t k ¯ lf ∂ ¯ z n d ¯ z n dz k ∧ d ¯ z l =: φ f . Forms φ f are smooth in the whole unit polydisk ∆ n − and equicontinuous there becauseforms T f are smooth in ∆ n − × A − ε, ⊂ H nε and equicontinuous there. Let us find a smoothand equicontinuous family on ∆ nr of solutions ψ f of dd c ψ f = φ f . (7.4)Set h f := a f − ψ f . (7.5)Since a f is positive on ∆ n − \ π ( I f ) and ψ f is smooth on ∆ n − we see that h f is boundedon ∆ n − from below. Also dd c h f = 0 on ∆ n − \ π ( I f ) and therefore h f extends to aplurisuperharmonic function on ∆ n − . This implies that h f ∈ L loc (∆ n − ) see [Ho]. Itfollows that a f and t n ¯ nf are locally integrable. Step 1 is proved. Step 2.
Under the hypotheses of Lemma 7.2 functions a f defined by (7.1) are smooth on ∆ n − and for every fixed r < the family { a f } f ∈F is equicontinuous on ¯∆ n − r . Function h f given by (7.5) is plurisuperharmonic in ∆ n − and pluriharmonic on ∆ n − \ π ( I f ). Thereforeby Siu’s lower semicontinuity of the level sets of Lelong numbers we have dd c h f = − X A irr.comp. of π ( I f ) c A ( f )[ A ] , (7.6)where c A ( f ) > A ] denotes the current of integration over the irreducible component A of π ( I f ) of codimension one. Remark 7.1.
Note that through components of higher codimension a pluriharmonicfunction h f extends (as a pluriharmonic function). Therefore in (7.6) the sum is takenover the components of codimension one only.We need to prove that c A ( f ) = 0. From (7.5) we get dd c a f = dd c ψ f − X A irr.comp. of π ( I f ) c A ( f )[ A ] , (7.7)where dd c from a f is taken in the sense of distributions (as from L loc -function). Let { h A } be equations of A . By Poincar´e formula, see [GK], [ A ] = dd c ln | h A | and therefore (7.7)writes as dd c a f = dd c ψ f − X A irr.comp. of π ( I f ) c A ( f ) dd c ln | h A | (7.8)Take an one dimensional disk ∆ in ∆ n − which intersects π ( I f ) transversely at points { z j } . Then (7.7) gives for restrictions of a f and ϕ f to ∆ (and we shall denote them bythe same letters) the following∆ a f = ∆ ψ f − X z j ∈ π ( A f ) c j ( f ) δ z j ( f ) . (7.9)Fix δ > δ, z j ) are pairwise disjoint. Let ϕ be a test function on ∆ withsupport in ∆( δ, z j ) for some fixed j . The coordinate on ∆ denote as z .Set a ǫf ( z ) = i Z ∆ z ( r ) t n ¯ nf,ǫ dz n ∧ d ¯ z n , atou components 27 where t n ¯ nf,ǫ is the smoothing of t n ¯ nf by convolution. Since t n ¯ nf,ǫ → t n ¯ nf in L loc we get by FubiniTheorem that a ǫf → a f in L loc . Therefore using (7.3) for dimension two we obtain < ∆ a ǫf , ϕ > = i Z ∆( δ,z j ) ϕ ( z ) Z ∆ z ( r ) ∂ t n ¯ nf,ǫ ∂z ∂ ¯ z dz n ∧ d ¯ z n dz ∧ d ¯ z == i Z ∆( δ,z j ) ϕ ( z ) i Z ∂ ∆ z ( r ) ∂t f,ǫ ∂ ¯ z d ¯ z + i Z ∂ ∆ z ( r ) ∂t f,ǫ ∂z dz dz ∧ d ¯ z −− i Z ∆( δ,z j ) ϕ ( z ) Z ∂ ∆ z ( r ) ∂t f,ǫ ∂ ¯ z d ¯ z dz ∧ d ¯ z −→ < φ f , ϕ > as ǫ −→
0. Therefore, ∆ a f = φ f in ∆ in the sense of distributions. By regularity of theLaplacian a f ∈ C ∞ on ∆ and therefore c A ( f ) = 0 for all A and all f . Therefore a f aresmooth on ∆ n − and a f = ψ f + h f there. ψ f -s are equicontinuous and h f are pluriharmoniceverywhere and uniformly bounded from below. Moreover a f are equicontinuous on ∆ n − ε by assumption. Therefore h f are equicontinuous on ∆ n − ε . This implies equicontinuity of h f on compacts of ∆ n − , and therefore the equicontinuity of a f . Step 2 and therefore ourLemma are proved. (cid:3) Lemmas 7.1 and 7.2 obviously imply the Theorem 5 from the Introduction for thecase of weakly converging sequences of meromorphic mappings with values in disk-convexGauduchon manifolds.
8. Fatou components
First let us prove two lemmas.
Lemma 8.1.
Suppose that a weakly converging sequence { f k } of meromorphic mappingsfrom a two-dimensional domain U to a compact complex surface X doesn’t convergestrongly. Then X is bimeromorphic to P . Proof.
Indeed, in that case there exists a point a ∈ U and a neighborhood V ∋ a suchthat f k converge uniformly on compacts of V \ { a } but Γ f k do not converge to Γ f , where f : U → X is the limit map. Vol (Γ f k ) are uniformly bounded. Indeed, for K¨ahler X it wasproved in [Iv2] using Skoda’s potentials. In [Ne] its was proved for non-K¨ahler X usingthat fact that such X can contain only finitely many rational curves as well as existenceof certain dd c -exact (2 , lim Γ f k contains Γ f plus { a } × X (with some multi-plicity). But this is a bubble and therefore X is rationally connected by Theorem 5. Fromthe classification of surfaces, see [BPV], we know that such X must be bimeromorphic to P . (cid:3) For a meromorphic map f : U → X denote by D f := pr (cid:0) { ( z, x ) ∈ U × X : dim ( z,x ) pr − ( x ) > } (cid:1) the set of degeneration of f . f : U → X is degenerate if D f = U . Lemma 8.2.
Let f : X → X be a non-degenerate meromorphic self-map of a compactcomplex surface X and let z ∈ Φ s (resp. Φ w ). Then for every l > one has f l [ z ] \ f l | D fl ( D f l ) ⊂ Φ s (resp. Φ w ) . (8.1) Proof.
Take some a ∈ f l [ z ] \ f l | D fl ( D f l ). Since f l doesn’t contract any curve to a thereexist neighborhoods V ∋ z and U ∋ a such that pr : ( V × U ) ∩ Γ f l → U is proper.That means that ( f l ) − : U → V is well defined as a multivalued holomorphic map.Now let { f n k } ⊂ { f n } be a subsequence. By assumption from the sequence { f n k + l } wecan subtract a strongly/weakly converging on V subsequence { f n kj + l } . That means that { f n kj = f n kj + l ◦ f − l } will converge in an appropriate sense on U . (cid:3) Let us turn to the proof of Corollary 2 from the Introduction. Since every compactcomplex surface admits a dd c -closed metric form Theorem 3 applies in our case and giveslocal pseudoconvexity of the weak Fatou set Φ w . Suppose now that Φ s = Φ w . a ) By Lemma 8.1 X ⋍ P . b ) There exists a point p ∈ X , a ball B centered at p , a subsequence of iterates { f n k } ,which uniformly converges on compacts of ¯ B \ { p } to a meromorphic map f ∞ : ¯ B → X ,holomorphic on B \{ p } , but not converges strongly on any neighborhood of p . In particularthis means that p ∈ I ( f ∞ ) by Rouch´e Principle of [Iv2] and, moreover, C = f ∞ [ p ] is a chainof rational curves S Ni =1 C i . As it was said Vol (Γ f nk ) are uniformly bounded on ¯ B . So Γ f nk converge (after going to a subsequence) in cycle topology to Γ f ∞ ∪ d ( { p } × X ) for someinteger d >
1. In particular f cannot be degenerated in this case. Take a point q ∈ X \ C .Then for k ≫ q ∈ f n k ( B \ { p } ). If moreover q f n k ( D ( f n k )) then q ∈ Φ w .But S k f n k ( D ( f n k )) is at most countable set of points and Φ w is Levi-pseudoconvex. SoΦ w ⊃ X \ C . Again from pseudoconvexity of Φ w it follows that if Φ w intersects someirreducible component of C then it contains this component minus the rest of C . I.e.,Φ w = P \ { some components of C } . c ) Take a point ( p, x ) ∈ { p } × X such that x ∈ C . Suppose that Γ f ∞ ∩ ( X × { x } ) has( p, x ) as isolated point. Then we can find neighborhoods W ∋ p and V ∋ x such that( ∂W × ¯ V ) ∩ Γ f ∞ = ∅ . Therefore ( ∂W × ¯ V ) ∩ Γ f nk = ∅ for k big enough. This meansthat Φ w ⊃ V as before and, moreover, Φ w contains the component of C passing through x minus the rest of C .To finish the proof let us distinguish two cases. Case 1.
Every component of C contains such a point. In this case our sequence { f n k } strongly converges on X \ { finite set } . Furthermore, Vol (Γ f nk ) are uniformly bounded.Since they can’t be less than ( deg f ) n k · Vol ( X ), we see that f has degree one, and f ∞ isa degenerate map to C because Γ f ∞ has zero volume in this case. Moreover f ∞ cannotbe holomorphic near p , otherwise f n k would converge strongly in a neighborhood of p . C in this case should consist only from one component as a meromorphic image of anirreducible variety. Case 2.
There exists a component C of C such that for all points x ∈ C \ ∪ i =1 C i dim ( p,x ) Γ f ∞ ∩ ( X × { x } ) >
0. Then f ∞ is a degenerate mapping of X onto this C andtherefore again C is a single component of C . Indeed, any other component C of C should contain a point x as above, because the image of f ∞ should be irreducible. I.e., inboth cases C consists from one rational curve only. atou components 29 (cid:3) The following simple example shows that the situation described in part (b) of thisCorollary can really happen. Let X = P and f : [ z : z : z ] → [ z : 2 z : 2 z ]. Then forthis f we have the phenomena described above with p = [1 : 0 : 0] and C = { z = 0 } . Let us give one more example relevant to the Fatou sets.
Example 8.1.
Consider the following rational self-map of P : f : [ z : z : z ] → [ z z : z : z z ] . (8.2)By induction one easily checks that f k : [ z : z : z ] → [ z k z k − : z k +1 − : z k +1 − z ] . (8.3) pq r ** l ll Ω Φ ∆ * Figure 1.
Mapping f contracts the line l := { z = 0 } to the first of itspoints of indeterminacy q = [0 : 0 : 1], line at infinity l := { z = 0 } to theregular point r = [0 : 1 : 0] and do not contracts anything to its second pointof indeterminacy p = [1 : 0 : 0]. Levi flat cone = Julia set for f is markedby two punctured lines.Cover P by three standard affine charts U i = { z i = 0 } with coordinates u = z z , u = z z , v = z /z , v = z /z and w = z /z , w = z /z respectively. Mapping f : U → U writes as f : ( u , u ) → ( u , u /u ) . (8.4)We see from here that f has degree 2. Furthermore f k writes as f k : ( u , u ) → ( u k , u /u k − ) . (8.5)In the charts f : U → U our iterate writes as f k : ( v , v ) → ( v k , v k +1 − v ) → (0 ,
0) = r on {| v | < } . (8.6)I.e., we see that Φ = {| v | < } = {| u | > } is a component of the Fatou set of f , inall senses, because all f k are holomorphic there and converge uniformly on compacts to aconstant map to r = [0 : 1 : 0].Levi flat cone L := {| z | = | z |} is a Julia set of f . It contains one of two indeterminacypoints of f , namely q = [0 : 0 : 1]. A connected component Ω of P \ L different from Φ carries a more interesting information about f k . First of all remark that as a mappingfrom U to U our iterate writes as f k : ( w , w ) → w k − w k − , w k +1 − w k +1 − ! → (0 ,
0) = q on {| w | < | w |} = {| u | < , u = 0 } . (8.7)Therefore the second component Φ s of the strong Fatou set contains the domain Ω \{ u =0 } . Since it is easy from (8.5) to see that f k on compacts in Ω \ { u = 0 } convergeto q , and on the puncture disk ∆ ∗ := { u = 0 , < | u | < } to p , we conclude thatΦ s = Φ w = Ω \ { u = 0 } . Remark that the second component Φ of f in the sense of[FS] is smaller, namely it is equal to Ω \ ( { u = 0 } ∪ { u = 0 } ), because the projective line l := { z = 0 } is the preimage of I f (and of all I f k ) under f . Now let us turn to the secondcomponent Φ of the Γ-Fatou set of f . Lemma 8.3.
For a fixed < ε < the volumes of graphs of f k over the bidisk ∆ ε ⊂ U centered at p are uniformly bounded. In particular Φ = Ω . Proof.
To estimate the volume of Γ f k over a neighborhood of p we use coordinates u , u and representation (8.5). In these coordinates ∆ ε = { u : k u k < ε } . Since f k preserves the vertical lines { u = const } we can simplify our computations assumingthat f takes values in ∆ × P , the last being equipped with the Hermitian metric form ω = ω + ω = i dz ∧ d ¯ z + i dz ∧ d ¯ z (1+ | z | ) . Now we get( f k ) ∗ ω = i " k | u | k +1 − + (1 − k ) | u | k +1 − ( | u | k +1 − + | u | ) du ∧ d ¯ u + i | u | k +1 − du ∧ d ¯ u ( | u | k +1 − + | u | ) + i − k )¯ u − k | u | k +2 − du ∧ d ¯ u ( | u | k +1 − + | u | ) + i − k ) u − k | u | k +2 − du ∧ d ¯ u ( | u | k +1 − + | u | ) . Therefore Z ∆ ε ( f k ) ∗ ω ∧ dd c k u k == 4 π ε Z ε Z " k r k +1 − + (1 − k ) r k +1 − ( r k +1 − + r ) + r k +1 − ( r k +1 − + r ) dr r dr π ε k +1 k +1 − k + 2 π k ε Z ε Z r k +1 − t ( r k +1 − + t ) dtdr + 2 π k ε Z ε Z r k +1 − ( r k +1 − + t ) dtdr π k ε Z ε Z r k +1 − r k +1 − + t dtdr + 2 π ε Z − r k +1 − + t (cid:12)(cid:12)(cid:12) ε r k +1 − ! dr π k +1 ε Z r k +1 − ln 1 r dr + 2 π ε Z r dr . atou components 31 The second term is bounded and doesn’t tend to zero as k → + ∞ . The first for 0 < ε < π k +1 ε Z r k +1 − dr = π k +1 k +1 − ε k +1 − → k → + ∞ . And finally Z ∆ ε ( f k ) ∗ ω = Z ∆ ε k | u | k +2 − ( | u | k +1 − + | u | ) + (2 k − | u | k +2 − | u | ( | u | k +1 − + | u | ) ++ 2(2 k − | u | k +3 − | u | | u | k +1 ( | u | k +1 − + | u | ) d m ( u ) Z ∆ ε k +2 | u | k +2 − ( | u | k +1 − + | u | ) d m ( u ) ≈≈ k +2 ε Z ε Z r k +2 − ( r k +1 − + t ) dtdr k +2 ε Z r k +1 − dr = 2 k +2 k +1 − ε k +1 − → k → + ∞ . Therefore the lemma is proved. (cid:3) Remark that the integral of ( f k ) ∗ ω degenerates, as it should be, because f k Γ-convergeon Ω to a constant map. And to the contrary the integral of ( f k ) ∗ ω doesn’t degenerate,moreover has order ε . That means that bubbling takes place over all points of the disk∆ ∗ = { u = 0 , < | u | < } . We see that for our map one hasΦ ⊂ Φ w = Φ s ⊂ Φ Γ , and inclusions are strict.Finally let us see what is going on over the indeterminacy point p = (0 ,
0) in coordinates( u , u ). Blowing up 2 k − f k stays degenerate on all exceptionalcurves except the last one, which it send onto l = { u = 0 } . In appropriate coordinates v , v on the last blow up f k writes as ( u = v k u = v . Therefore the dynamical picture over p can be described as follows. Let ˆΩ be an infiniteblow up of Ω over p and let C = S ∞ i =1 C i be the Nori string of rational curves on ˆΩ over p .Then every f k lifts to a holomorphic map ˆ f k : ˆΩ → P which is constantly equal to q onevery C i except C k − . The last curve it sends bijectively onto the line l . At the sametime in the sense of divisors (currents) f k ( C k − ) = 2 k l . Remark 8.1. a)
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