Decay of volumes under iteration of meromorphic mappings

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Here bl ( f ~ ) ~ À3 means that is bounded from above.
The study of volume estimates has a long history in complex dynamics. It has been used to construct and characterize invariant currents (see [G 03a], [FJ 03], [CG 04] for references). A partial result was obtained in [G 03a] in this direction. The present one has several advantages: it concerns a more general frame ([G 03a] dealt with case X = in which we do not assume either algebraic stability, or integrability of the logarithm of the jacobian. We also avoid the use of the Green function made in [G 03a]. In fact, the existence of the Green current is not known in this quite general context and one may hope that volume estimates will help to construct it. Indeed after establishing our main result in section l, we give such a construction in section 2 under a mild cohomological assumption (theorem 2.2). In the last section 3 we discuss our hypotheses and related open questions.
Acknowledgement. -Part of this work was done while the author was visiting Kyoto University. I would like to express my warm thanks to the department of Mathematics and especially to professor M.Shishikura for his friendly welcome and support. I am also grateful to the referee whose suggestions helped to clarify the exposition.
In the whole paper we fix a Kahler form cv on X normalized by = 1, where k denotes the (complex) dimension of X. All volumes are computed with respect to the probability volume form wk.
Given a smooth real form 9 of bidegree (l, l), the pull-back of 9 by f is defined in the following way: let r f C X x X denote the graph of f and consider a desingularization r f of r f . We  Proof. Let Jac( f ) denote the complex jacobian of f with respect to wk , defined by It satisfies the chain rule A straightforward computation shows that log IJac(f)1 I can be written as a difference of quasiplurisubharmonic (qpsh) functions. Let us recall that a qpsh function is an upper semi-continuous (u.s.c.) function cp E Ll (X, R U {2013oo}) which is locally given as the sum of a psh and a smooth function. Thus is a well defined real current of bidegree (1,1) on X which is bounded from below by a smooth form, in particular for some large A &#x3E; 0. Here d = 8 + 8 and dC [8 -8]. Write log I Jac (f ) I = uv, where u, v both are qpsh. Since u, v are u.s.c. on X which is compact, we can assume without loss of generality that u, v 0. By the change of variables formula, we get where the last inequality follows from the concavity of the logarithm. Using the chain rule and log u, we obtain Our lower bound is thus a consequence of the next proposition which is of independent interest. 0 PROPOSITION 1.3. -There exists C &#x3E; 0 such that for all qpsh function cp, -cv, and for all n E N, Proof. Let cp be a qpsh function such that dd~cp &#x3E; -cv. We can assume without loss of generality 0.
Fix 6 a smooth probability measure whose support is concentrated near a point a which is neither critical, nor a point of indeterminacy. Thus f is a local biholomorphism from a neighborhood of a (containing the support of O) onto a neighborhood of f (a) (containing the support of f * O) .
Therefore is yet another smooth probability measure on X. Since X is Kahler, we can find a smooth form R of bidegree ( l~ -1, k -1) on X such that f*8 = 6 + ddcr. Adding a large multiple of we can further assume Iterating the previous functional equation yields is an increasing sequence of positive currents of bidimension ( 1,1 ) on X whose is controlled by It follows from Stokes theorem and (1) that since wand Rn &#x3E; 0. Now 6 and wkare both smooth probability measures, so we can fix S, a smooth form of bidegree (k -1, k -1) on X, such that = e + and 0 S'

We infer
The first term is controlled by (2) and (3). The second can be estimated by using Stokes theorem again, Using (2), (3), (4), (5) we obtain for some large constant C3 &#x3E; 0. This concludes the proof. 0 We now specialize the previous estimates when the behaviour of sequence (~i(/~)) is under control. Our assumptions will be discussed in section 3.
Our volume estimates (theorem 1.4) are sharp in the sense that (VolO)ÀJ for many mappings. Here is an elementary example: .À ~ 2. Then f gives rise to an holomorphic endomorphism of X = p2 with À = ~1 ( f ) . Simple computations show that where 02 (r) denotes the bidisk of radius (r, r) centered at the origin in ~C2 . Our goal is now to show how volume estimates can be used to construct the Green current of an algebraically stable mapping f. Hl ~ l (X, R) .
Let us recall that one can define the pull-back of any positive closed current ~S' of bidegree (1, 1) on X. The mapping Sf *,S' is continuous for the weak topology of (positive) currents (see [S 99]). The induced property on cohomology classes reads where R) denotes the cone generated by pseudoeffective cohomology classes, i.e. classes that can be represented by a positive closed current. This cone is closed and strict (i.e. = ~0~) . It follows from Perron-Frobenius theory that the spectral radius ~1 ( f ) of f* : H1,1 (~~ ~) ~ H1,1 (X, JR) is an eigenvalue of f * which dominates all the other eigenvalues, and that there exists an eigenvector a E R) with f *a = .À1 (f)a.
Our aim is now to construct a canonical positive closed current Ta such that a and Fix 6 a smooth closed real (1, l)-form with 101a and consider is u.s.c. and This is an extremal function with respect to the family of 0-psh functions normalized by sup x p x 0 (see [GZ 04]). The function v is not necessarily upper semi-continuous (u.s.c.), so we consider its upper semi-continuous regularization, The current Omin := 0 + CLv 0min is a positive closed current cohomologous to 0 with "minimal singularities" (see theorem 1.5 in [DPS 01]): if ,S' is any positive closed current cohomologous to 0, then S' writes ,S' = 0 + dd'w, where w -vo is bounded from above, so that w is more singular than min v0 min Similarly if T a positive closed current cohomologous to 0 which is invariant, f *T = we say that T has minimal singularities among such invariant currents if whenever ,S' is another invariant positive closed current cohomologous to 0, the potential of S is dominated from above by that of T (up to an additive constant).
We make the following assumption on the cohomology class a: Vt &#x3E; 0, This is clearly an assumption on a rather than on 0: if 8' is another smooth closed real (1, 1)-form such that = a, then 0' -0 + dd'u with u smooth hence bounded, so that vminis bounded on X (see [GZ 04]). The assumption (H~) will be discussed in section 3. Observe that the integrability condition is satisfied if h grows fast enough to infinity, as t ~ +00, e.g. if h(t) ~--[log(l + t)~l+~, ~ &#x3E; 0. The current T, has minimal singularities among invariant closed currents whose cohomology class is a. It is extremal within the cone of positive closed invariant (1,1)-currents whose cohomology class belongs to the ray I~a.
Proof -Set for simplicity A = Al(f). Let 9 be a smooth closed real (1, I)-form such that a. Let 9min = 9 + be the corresponding positive current with minimal singularities. Since X is Kahler, the invariance relation /*c~ = ~a reads where y E L1 (X ) is locally given as the sum of a psh function and We normalize y by requiring Ix = 0. Observe that y -vmin + C for some constant C E R. Since f is algebraically stable, we can pull-back (6) by f n and obtain this way We want to show that (qn) converges in L~(~C). Observe that and vn = are both decreasing sequences of L1-functions.

We show in lemma 2.3 below that (vn ) is a convergent sequence in
It is therefore sufficient to get a lower bound on f X 'rnwk. We establish it in the same vein as what was done in the proof of proposition 1.3. Observe first that it is sufficient to evaluate qn against a properly chosen smooth probability measure 6. Indeed we can write w k = 8 + ddc S with S &#x3E; 0 smooth, so that We choose 6 as in the proof of proposition 1.3 and use (1). Recall that Rj = increasing sequence of positive currents of bidimension ( 1,1 ) on X such that (fi),,6 -6 + ddrj. Therefore Adding these inequalities and observing that 0min A Rn &#x3E; 0, we infer It follows that in L1 (X ), hence The current T~ is positive (as a limit of positive currents) and invariant since = ÀBn+1. It is obviously closed and cohomologous to 0min hence = a. Observe also that A-~~)~ -~ Ta since o f n -~ ~ (by lemma 2.3).
We now claim that Ta is the invariant current with minimal singularities within the compact set of positive invariant closed ( 1,1 )-currents ,S' with {S} = a. Indeed let ,S' = 0min + 0 be such a current. Then w x ~1 by definition of 2Jmin . Shifting w, we can assume C1 = 0. The assumption &#x3E; 1 is therefore dynamically significant.
We moreover assumed 61(fi) )B1 (f)j. In general there may be a further polynomial growth, 61(f3) as the following example shows: Example 3.1. Let X = l~l x pI and f be the compactification of the following polynomial endomorphism of ~2, The linear action induced by f * on H1,1(X, 1R) ~ R2 is given by the 2-by-2 matrix and we obtain in this case Observe that f has topological degree dt ( f ) _ .À2 &#x3E; ~ = Al(f).
However in all known examples such that s &#x3E; 1, the mapping f preserves a fibration. This motivates the following QUESTION 3.2. -Assume f does not preserve any fibration. Is it true then The question makes sense even when = 1. When dimc X -2, J.Diller and C.Favre [DF 01] gave a positive answer to this question when f is bimeromorphic. They also observed that a bimeromorphic mapping can not preserve a fibration when al ( f ) &#x3E; 1. More generally one can ask the following QUESTION 3.3. -Assume ~1 ( f ), the first dynamical degree, strictly dominates all the other dynamical degrees. Is it true then that ~i(/~) ã l(f)3 ? When f preserves a fibration, one can show that the first dynamical degree is not the largest dynamical degree. The construction of the invariant current Ta is of crucial importance precisely when the first dynamical degree is the largest.

Properties of invariant cohomology classes.
We have considered in section 2 a psef class a such that f**a = Al ( f )a. There may be several (linearly independent) eigenvectors ai , a2,... : Exam pl e 3.4. -Let be the direct product of two rational mappings gi : Pl --4 P1 of the same degree A &#x3E; 2.
Then ~1 ( f ) = a and there are two eigenvectors ozi, a2 associated to ~1 ( f ) which are given by the fibers of the two natural fibrations 7ri : I~1 x Pl (projection onto the ith factor). The corresponding invariant current Ta2 is then the pull-back of the Lyubich measure of gi : I~1 -~ I~1 under the projection Observe that here again the topological degree dt(f) _ .À2 strictly dominates the first dynamical degree = A.
However when dimC X = 2 and the first dynamical degree is the largest dynamical degree (i . e. ~ 1 ( f ) &#x3E; dt ) , J.Diller and C.Favre have proved that the eigenspace associated to Ai(/) is one-dimensional (see remark 5.2 in [DF 01]). One expects similar results to hold true in higher dimension.
We now discuss further positivity properties of the invariant class a.
Let H,,,,f(X, R) and H big (X, R) denote the cones generated respectively by nef and big cohomology classes. Recall that a class a is numerically eventually free (nef for short) if a + Efwl is a Kdhler class for all E &#x3E; 0. The class a is big if it contains a Kahler current, i.e. if there exists a positive closed (1, I)-current T on X such that {T} = a and T &#x3E; EOW for some Eo &#x3E; 0. These notions coincide with the corresponding classical notions in algebraic geometry when X is projective and a C H2 (X, ~) . We refer the reader to [D 90] for more information on these positivity conditions. PROPOSITION 3.5. -The cone is preserved by f * .
If dime X = 2, then is also preserved by f * .
The proof is an easy application of proposition 4.12 in [B 02] and proposition 1.11 in [DF 01].
Since the cone R) is closed and strict, it follows from the Perron-Frobenius theory that the invariant class a is nef when dimr X = 2.
The same argument does not apply to (X, R) because the latter is not closed. It seems however reasonable to expect a being big when X is e.g. rational. For 2-dimensional bimeromorphic mappings, a is not big precisely when f is conjugate to an automorphism (see theorem 0.4  where h : JR+ -7 R+ is such that h(t) = +oo. Observe that this is always satisfied when p is a potential with minimal singularities of a big and nef cohomology class. LEMMA 3.6. -be a smooth closed real (1,1)-form such that a = f 01 E Then vmin has zero Lelong number at every point. This is well-known to complex geometers, at least when a = cl (L) is the first Chern class of a big and nef holomorphic line bundle L on X (see proposition 1.6 in [DPS 01]). We nevertheless include a proof for the reader's convenience.
Proof. -Since cx is big, we can fix a positive closed current ,S' of bidegree ( 1, 1 ) on X such that f Sl = a and S &#x3E; coW for some Eo &#x3E; 0. Thus S -coW is still a positive current. Fix Q E an u.s.c. function such that S -coW = 0 -coW + 0; we normalize 0 by 0.
Since a is nef, Na + Eofw I is Kahler for all N e N. Fix 1/J N smooth functions such that &#x3E; 0 and normalized by sUPx 1/JN = 0.
We set These are u.s.c L1-functions such that whence vmin' is smooth so that We infer 0 for all x E X.

D
Our hypothesis (H,) asks for more precise information than lim h(t) _ +oo. It would be satisfied if e.g. h(t) ~-(log[1 + t~)1+a, S &#x3E; 0. This is trivially true in all cases considered so far: When a is semi-positive (i.e. when it can be represented by a smooth closed non-negative form), then (Ha) is trivially satisfied: indeed x E 2013~} is empty for t &#x3E; 0 large enough, hence h(t) =-for t » 1. This is the case considered in [S 99], [FG 01], [G 02]: when X is a complex homogeneous manifold (i.e. when the group of biholomorphisms Aut(X) acts transitively on X), every psef class is actually semi-positive (X = I~~ in [S 99] and X = pk, x ~ ~ ~ x pks in [FG 01]). There are psef classes that are not semi-positive on a Hirzebruch surface (the situation considered in [G 02]), however every nef class is semi-positive on these minimal rational surfaces. When f is holomorphic (i.e. when If = 0), a admits a positive closed representative with continuous potential, so (Ha ) is trivially satified again (this is the situation considered in [Ca 01]). Indeed one can write in this case where 0 is smooth. Therefore