Dynamics of Meromorphic Maps with Small Topological Degree I: From Cohomology to Currents
We consider the dynamics of a meromorphic map on a compact Kähler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which iterates of the map expand the cohomology class of a Kähler form. Our goal in this article and its sequels is to carry out a program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, using the linear action of the map on cohomology to construct and analyze invariant currents with special geometric structure. We also give some examples and consider in more detail the special cases where the surface is irrational or the self-intersections of the invariant currents vanish.
Key words and phrases:complex dynamics, meromorphic maps, complex surfaces, positive closed currents
1991 Mathematics Subject Classification:37F10, 32H50, 14E07
Throughout this paper we consider the dynamics of a meromorphic map on a compact connected Kähler surface . Various categories of such maps have been studied from a dynamical point of view for more than twenty years now, beginning in particular with holomorphic self-maps [FoSi] of the projective plane and polynomial automorphisms of [bs1, fs-henon, BLS]. Gradually, there has emerged a clear conjectural picture concerning the ergodic behavior of generic [G5]. The reader is referred to the surveys [Sib, G6] for a more comprehensive discussion.
Though it might not be continuously defined at all points, the meromorphic map induces natural pullback and pushforward actions on the cohomology groups of . A well-known idea of Gromov [gromov] shows that the topological entropy of is bounded above by . Conjecturally, equality holds. The action on cohomology can be seen as a way of keeping track of how fast the volumes of compact subvarieties are expanded by iterates of . In particular, meromorphic maps on surfaces fall into two classes: those with ‘large topological degree’ that expand points faster, i.e. for which is the dominant action; and those with ‘small topological degree’ that expand curves more quickly, i.e. for which predominates.
To state the distinction more precisely, we let denote the topological degree of , that is, the number of preimages of a generic point; and we let denote the (first) dynamical degree. Then we say that has small topological degree if . A delicate point which must be underlined here is that on , the equality is not true in general [FoSi, Sib]. This is due to the fact that our mappings have indeterminacy points. We say that is 1-stable if equality holds for all .
The reverse of Gromov’s inequality for entropy has been completely justified for maps with large topological degree [briend-duval, DS, G1]. The idea is that equidistributing Dirac masses over the iterated preimages of a generic point gives rise to a convergent sequence of measures, whose limit has maximal entroy (among other good properties). For maps with small topological degree, one hopes to arrive at an interesting invariant measure by choosing two generic curves and considering something like the sequence of measures
The wedge product can be understood here as a sum of Dirac masses at intersection points. Of course, the analysis and geometry of such measures is much more involved than those obtained by pulling back points. The present work and its sequels [part2, part3] are largely devoted to overcoming this extra difficulty.
Our approach follows one used in the invertible (i.e. bimeromorphic) case. A bimeromorphic map has small topological degree as soon as . A broad class of such maps has been successfully analyzed (see [BLS, Ca, DF, BD, Du4]) in the following fashion.
Step 1: find a birational model of where (the conjugate of) becomes 1-stable.
Step 2: analyze the action on cohomology and construct a (resp ) invariant and ‘attracting’ current (resp. ) with special geometric properties.
Step 3: give a reasonable meaning to the wedge product , both from the analytic and the geometric points of view. This results in a positive measure .
Step 4: study the dynamical properties of .
The only step which remains incomplete in the bimeromorphic setting is Step 3.
In this paper and its sequels, we will completely carry out Steps 2 (this paper) and 4 [part3] for arbitrary mappings of small topological degree, and achieve Step 3 [part2] for a class of meromorphic maps that goes beyond what has previously been considered even in the bimeromorphic case. In each step, going from to arbitrary brings up serious difficulties.
We stress that we will not address Step 1, which remains open in general. Rather, we take -stability as a standing hypothesis on our maps. However, Favre and Jonsson [FJ2] have recently shown that on passing to an iterate, any polynomial map of with small topological degree becomes -stable on some compactification of . Moreover, our results in [part2] suffice completely for Step 3 in the polynomial case. Hence for polynomial maps of , our results and those in [FJ2] can be viewed as a maximum possible generalization of the work completed in [BLS] for polynomial automorphisms.
Let us review the results of this paper in more detail. The reader may consult [part2, part3] for more about Steps 3 and 4.
As already noted, the main purpose of this paper is to construct invariant currents and prove convergence theorems. To appreciate the level of generality in our results, one should note that even if we were to begin with a map of , the need for -stability might lead us to a new rational surface with much more complicated geometry. In section 1 we consider in detail the spectral behavior of the action on . It is known that when is -stable and of small topological degree, there is a unique (up to scale) nef class such that and that all other eigenvalues of are dominated by .
We break our first new ground by looking for a good positive current to represent . If belongs to the interior of the nef cone, it is represented by a Kähler form and therefore much easier to deal with. Finding a suitable representative for a class on the boundary of the nef cone is an important problem in complex geometry and can be quite difficult. Demailly et al (see e.g. [Dem2, Section 2.5] and references therein) have paid much attention to this issue. We resolve the problem for in a fashion that is, to our knowledge, new.
Let be a 1-stable meromorphic map of small topological degree . Then the invariant class is represented by a positive closed current with bounded potentials. The same is true for the analogous class invariant under .
Positive representatives for and with bounded potentials will serve as a starting point for the sequel [part2] to this paper. Here they give us a convenient way to construct the invariant currents and referred to in Step 2 from the outline above. Actually, we prove a somewhat more general version (Theorem 1.6) of Theorem 1 in which -stability is unnecessary, and the hypothesis is needed only to deal with .
Section 2 is devoted to constructing and analyzing the current . Over the course of the section, we prove
Let be a -stable meromorphic map with small topological degree . There is a positive closed current representing such that for any Kähler form on ,
for some . In particular, , and has minimal singularities among all such invariant currents.
If moreover is projective, then is a laminar current.
Versions of Theorem 2 have been previously obtained (e.g. [Sib, DF, G4, DG, Ca]) under restrictions on the surface , the map , or the class . The main innovation here is that even when is not a Kähler class, we recover the current as a limit of pullbacks of a Kähler form. Our proof of laminarity in for projective depends on this. To get the desired convergence, we work in two stages. We first prove it when the Kähler form is replaced by the positive representative with bounded potentials from Theorem 1. Then we employ some delicate volume estimates from [G2] and a precise understanding of the singularities of to get convergence for arbitrary Kähler forms.
Concerning the notion of laminarity, we refer readers to §2.4 for background. We point out that Theorem 4 below shows that the projectivity assumption is an issue only when has Kodaira dimension zero.
In Section LABEL:sec:T-, we consider the pushforward operator . Pushforward of currents is harder to control, but by taking advantage of the fact that is dual via intersection to , we reduce some of the more difficult questions about to corresponding features of . The end result is a nearly exact analogue of Theorem 2.
Let be a meromorphic map with small topological degree . There is a positive closed current representing such that for any Kähler form on ,
for some . In particular, , and has minimal singularities among all such invariant currents.
If is projective, is a woven current.
Currents of this sort for non-invertible maps have been considered in e.g. [DTh1, fs98, G4]. The fact that is woven is essentially due to Dinh [Dinh]. Wovenness is weaker than laminarity in that the curves that one averages to approximate are allowed to intersect each other. This allowance is necessary for maps which are not invertible. Another point to stress is that, while the current exists even for maps with large topological degree, small topological degree is essential for the construction of . If , one does not generally expect that there is a single current playing the role of .
In section LABEL:sec:examples, we present several interesting examples of meromorphic maps with small topological degree. A central theme of the section is that examples are plentiful on rational surfaces but much rarer on others. In particular, by adapting arguments from [Ca, DF] we classify those surfaces which admit maps with small topological degree.
Let be a compact Kähler surface, supporting a meromorphic self map of small topological degree . Then either is rational or has Kodaira dimension zero. In the latter case, by passing to a minimal model and a finite cover, one may assume that is a torus or a K3 surface and that the map is -stable.
We also show that invariant currents, etc. associated to maps on irrational surfaces must behave somewhat better than they do in the rational setting.
Finally, in section LABEL:sec:self, we consider maps for which the self-intersection of either or vanishes. The general idea here is that such a map must be quite close to holomorphic.
Let be a -stable meromorphic map with small topological degree . If vanishes then so does . And if the latter vanishes, then there is modification , where is a (possibly singular) surface under which descends to a holomorphic map .
This generalizes a result of [DF] and, as we explain before Proposition LABEL:zero, has an interesting natural interpretation in terms of the Riemann-Zariski formalism developed in [BFJ].
1. Meromorphic maps, cohomology, and positive currents
1.1. Meromorphic maps
Let be a compact Kähler surface with distinguished Kähler form . Our goal is to analyze the dynamics of a meromorphic map . The term ‘map’ is applied rather loosely here, since is technically only a correspondence. That is, there is an irreducible subvariety with projections , and . The projection is required to be a modification of : there is a (possibly empty) exceptional curve such that maps onto a finite set of points and biholomorphically onto the complement of this set. We will require, among other things, that our map be dominating, i.e. that the projection onto the range is surjective. It is often advantageous, and for our purposes never a problem (see, however, the proof Theorem 2.12) to replace the graph of with its minimal desingularization. Hence we do this implicitly, assuming throughout that is smooth.
We let denote the indeterminacy locus of . The set theoretic image of each is a connected curve. If is a curve, then we adopt the convention that is the (reduced) ‘strict transform’ of under . In particular belongs to the exceptional locus if is zero dimensional. The exceptional locus is included in turn in the critical locus , which also contains curves where is ramified. Finally, for convenience, we name the sets and . These are, morally speaking, the indeterminacy and exceptional loci for .
The above terminology extends trivially to the more general case of a meromorphic surface map with inequivalent domain and range. To the extent that iteration is not required, the discussion in the next two subsections will also apply to meromorphic maps generally. However, for simplicity, we continue to discuss only the given map . In keeping with our abuse of the term ‘map’ we will usually forego the more correct symbol ‘’ in favor of ‘’ when introducing meromorphic maps.
1.2. Action on cohomology and currents
Suppose is a smooth form on . We define the pullback and pushforward of by to be
where the action , is understood in the sense of currents. Both currents and are actually forms with coefficients. Indeed is smooth away from , whereas is continuous away from and smooth away from .
The topological degree of
is the number of preimages of a generic point.
The first dynamical degree of is
We say that has small topological degree if .
Most often we use , as a shorthand for . Dynamical degrees are discussed at greater length in [RS, DF, G1]. As the terminology suggests, always exists and is independent of the choice of . Furthermore, it is invariant under bimeromorphic conjugacy and satisfies the inequality . Another observation is that the spectral radius of the action on or is dominated by [Dinh, Proposition 5.8], so we could replace with in the definition of .
Both and classically induce operators . These really only interest us in two bidegrees. When , is just multiplication by the topological degree . When , the operators and can be quite subtle. Both preserve the real subspace , and we will generally only use their restrictions to this subspace. We denote by the intersection (cup) product on , and by the self intersection of a class. The operators are adjoint with respect to intersection.
There is also a ‘push-pull’ formula for [DF, Theorem 3.3]. The precise statement of the latter is a bit cumbersome, so we’ll only state those consequences of the push-pull formula that are important to us (Propositions LABEL:pushpull1 and 1.3).
An important point is that pullback and pushforward might not behave well under composition.
Definition 1.2 ([FoSi, Sib]).
We say that is 1-stable if for all .
This property is equivalent (see [FoSi] or [DF, Theorem 1.14]) to the condition that for all . If is 1-stable, then .
It is known [DF, Theorem 0.1] that when , then one can always find a bimeromorphic map that lifts to a map that is 1-stable. Much more recently, similar results (see below) have been obtained by Favre and Jonsson [FJ2, Theorem A] for a meromorphic maps obtained by compactifying polynomial maps of . It remains an open problem to determine whether such results hold for arbitrary meromorphic surface maps of small topological degree. Notice that we do not use the 1-stability assumption until Section 2.
Much of the geometry of can be described in terms of positive closed currents. Recall that the pseudoeffective cone is the set of cohomology classes of positive closed currents. It is ‘strict’ in the sense that it contains no non-trivial subspaces. The cone dual to via intersection is , whose interior is precisely equal to the set of Kähler classes. Clearly .
Any effective divisor on is naturally a positive closed current that acts by integration on smooth test forms.111By ‘divisor’, in this paper, we will always mean -divisor, i.e. we allow coefficients to be real numbers rather then just integers. Most often we use the same letter for a curve and the associated reduced effective divisor, for a divisor and the associated current of integration, and for a current and its cohomology class. The context should make the point of view clear in each instance.
Given any positive closed current on , we may write where is a smooth closed form cohomologous to and is a -plurisubharmonic function determined up to an additive constant by and . We call a potential for relative to . The definitions of pushforward and pullback given in (1) may be applied to , once we declare that for either projection . Thus defined, and are positive closed currents that vary continuously with in the weak topology on currents. In particular, they do not depend on the choice of and .
It is immediate from adjointness that and preserve and .
The following consequence of the pushpull formula from [DF] will be important to us. Notice that there is a similar statement for cohomology classes rather than currents.
For any positive closed current , we have
where is an effective divisor supported on .
Furthermore, from the precise expression of the , given , if all intersections with curves are non-negative, then is effective. If, additionally, one such intersection is positive, then charges all of .
It is useful to know how , act on curves.
Suppose that is an irreducible curve. Then , where , is the local degree of near a generic point of , and is an effective divisor with support exactly equal to those curves in that map to . On the other hand , where is an effective divisor with .
1.3. Spectral analysis of and
Let be any norm on , and let
be the spectral radius of . In general with equality if is 1-stable.
Theorem 1.5 ([Df]).
Suppose . Then is a simple root of the characteristic polynomial of (resp ), and the corresponding eigenspace is generated by a nef class (resp ), and . The subspace is the unique invariant subspace complementary to , and there is a constant such that for every we have
The corresponding result holds for .
For convenience, we normalize the invariant classes and the distinguished Kähler form so that . This is essentially [DF, Theorem 5.1], where it is shown that is a simple root of the characteristic polynomial and that all other roots have magnitude no greater than . It suffices for establishing Theorem 1.5 to show further that eigenspaces associated to a root with magnitude equal to is generated by eigenvectors. The arguments from [DF] are easily modified to do this. An alternative approach to the second assertion in the theorem may be found in the more recent paper [BFJ], where it is shown that we can bypass 1-stability to obtain interesting information about the cohomological behaviour of meromorphic maps.
1.4. Positive currents with bounded potentials
We now prove Theorem 1 in the following slightly more general form:
Let be a meromorphic map such that . Then the invariant class is represented by positive closed currents with bounded potential. If has small topological degree then the same is true of .
The remainder of this subsection is devoted to the proof.
Let be compact complex surfaces and be a proper modification. Let be a smooth closed form such that for every curve . Then potentials for are bounded above.
(see also the proof of [DG, Theorem 2.4]) We write for a smooth closed form and . By hypothesis and Proposition 1.3 applied to , we have
where is an effective divisor. Thus is quasiplurisubharmonic and (in particular) bounded above on . It follows that is bounded above on . ∎
Let be a smooth closed form on such that for every curve . Then any potential for is bounded above. Similarly, if for every curve , then any potential for is bounded above.
Consider first . For each irreducible , we have that is either trivial or an irreducible curve in . Hence . The assertion thus follows from Lemma 1.7 applied to and .
Now consider . We recall (see e.g. the paragraph before Lemma 2.4 in [BFJ]) that there exists a modification that lifts to a meromorphic map with and that . We claim that , where is a bounded function and is a smooth form satisfying for all . Given the claim, we can apply Lemma 1.7 with , obtaining
Hence has potentials that are bounded above.
It remains to prove the claim. Let be the minimal desingularization of the graph of , and , be projections onto domain and range. Since collapses no curves, we have . In particular, for each connected component , the image is a point. We write on a neighborhood and obtain that is -exact on a neighborhood of . Therefore, if is any point—even a point in the image of , there is a neighborhood such that is -exact on each connected component of . This gives us that
has bounded potentials near . Since is arbitrary, the claim is established. ∎
Now let , and fix smooth closed forms whose cohomology classes form a basis for . Then for each positive closed current on , we have a unique decomposition
where and . Using the weak topology on the set of positive closed currents, we have that both and depend continuously on . As the dependence is also linear, the decomposition extends naturally to any difference of positive closed currents. In particular, it extends to all smooth closed forms on and to their images under pushforward and pullback by meromorphic maps.
We give the norm . The following is essentially a restatement of [BD, Lemma 2.2].
There is a constant such that for every representing a nef class.
The difficult point here is that the form is not itself positive. So despite the positivity of the class and the normalization of potentials, we cannot directly apply compactness theorems for positive closed currents.
We work only with pullbacks, the proof being identical for pushforwards. Let and . Then is a compact convex subset of that avoids . Since any representing a nef class may be rescaled to give an element in , it suffices to find satisfying for all .
Let . Then is defined by finitely many linear inequalities and contains . Hence we can find finitely many elements whose (compact) convex hull contains . By Lemma 1.8, we have such that for . Since the function is convex on , we have for every in the convex hull of . ∎
For any class , we set
and we let
be the convex cone of classes represented by positive closed currents with bounded potentials. While depends on our choice of , the cone does not. To our knowledge, this cone has not been previously considered.
For any Kähler surface , we have . There exist for which both inclusions are strict. Hence is neither open nor closed in general.
Kähler forms have smooth local potentials, so Kähler classes belong to by definition. On the other hand, if is bounded for a given , then it is well-known [BT] that is a well-defined positive measure for any other positive closed current on . In particular , which implies that represents a nef class.
Finally, [DPS, Example 1.7] exhibits a bundle over an elliptic curve for which . Moreover, the pullback to of any Kähler form on is smooth and positive and represents a class with zero self-intersection. This shows that is larger than the interior of . ∎
There is a constant such that
Thus and preserve .
We deal only with . The only difference in the pushforward case comes from the fact that for functions bounded above on , one has . Let be a positive closed current representing such that . Then
Note that is smooth off .
Let be open neighborhoods of small enough that each form can be expressed as for some smooth . Writing , we let . Then is a potential for on . So for large enough, the function
is well-defined and bounded. Indeed, paying more careful attention, one finds that
suffices here. The current represents and agrees with outside . Since is a potential for on , we see that on all of . Hence , with .
In the other direction, our choice of gives
The final term is estimated as above. Writing , we control the middle term by Thus we arrive at
The proof is ended by taking the infimum of the right side over all representing . ∎
2. The canonical -invariant current
We now construct and analyze the invariant current . There are of course many precedents (see e.g. [Sib, Fav2, DG]) for this. The novelty here concerns the level of generality in which we are working.
2.1. Construction of
Suppose that is -stable and that . Then there is a positive closed current representing such that and for any smooth form representing , we have weak convergence
The latter holds more generally for (non-smooth) representatives with bounded local potentials.
This theorem is proven with a different argument in [DG]. Here we give only those details of the proof that are different and/or important for the sequel. An advantage to the present approach is that it works equally well for pushforwards (see Theorem LABEL:thm:cv-).
By the -lemma, , where is uniquely determined by the normalization . We pull this equation back by and get
We claim that the sequence converges. The main point is that is bounded above, so that the sequence is essentially decreasing. Given the claim, convergence follows from a (by now standard) argument of Sibony [Sib], whose details we omit. On the level of currents, we obtain , where is a priori a difference of positive closed current, and represents .
To prove the claim, we apply Theorem 1.6 to get a positive representative for with potential . Thus
Since is positive, it follows that is bounded above. Since is bounded, we conclude that itself is bounded above.
Furthermore, we see that
from which we infer that is positive. From continuity of on positive closed currents, we finally conclude that . ∎
It easily follows from the second part of the proof that has minimal singularities among invariant currents: that is, let be a positive closed current satisfying , rescaled so that is cohomologous to . Hence for . From invariance and our construction of it follows that . As Fornæss and Sibony [FoSi] have observed, this implies that is extremal among -invariant currents, which is a form of ergodicity.
2.2. Lelong numbers of
It is important for us have a good control on singularities, i.e. Lelong numbers, of . The first proposition gives some information about how Lelong numbers of a positive closed current transform under pullback.
Proposition 2.3 (Theorem 2 and Proposition 5 in [Fav]).
Let be a positive closed current on . Then there is a constant such that implies that
If also , then may be taken to be the local topological degree of at .
The argument for the following result is due to Favre [Fav2]. We include it for convenience.
Assume that is -stable and has small topological degree. Suppose is such that for every . Then the Lelong number of vanishes at . In particular does not charge curves.
Suppose additionally that for any . Then Proposition 2.3 gives
The Lelong numbers of are moreover uniformly bounded above by a constant depending only the cohomology class . Since , we conclude that . Indeed the weaker upper bound in (3) implies the same even if for finitely many .
On the other hand implies that lies in the finite set . So if for infinitely many , it follows that is preperiodic. Since is finite away from , it follows that is finite at . So . ∎
The pullback of a positive closed current tends to have non-zero Lelong numbers at points in , even if itself is smooth. In order to strengthen the convergence in Theorem 2.1, we need a precise version of this assertion.
There is a constant such that for any positive closed current that does not charge and any ,
Throughout the proof we will use to denote equality up to a positive multiple that depends only on .
Fixing , we factor the projection from the graph of onto its domain as where is an ordinary point blowup with exceptional curve . Since is the minimal desingularization of the graph of , it follows that . Otherwise we could replace with , with and with and obtain a ‘smaller’ desingularization of the graph. Hence does not charge .
Applying Proposition 1.3 to and tel