A NULLSTELLENSATZ FOR AMOEBAS

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536 BOUCKSOM, FAVRE, and JONSSON where h 1 ∈ H. Inductively, (3.3)gives F n∗ θ ∗ = ( λ2 λ 1 ) nθ∗ + λ n 1 ( 1 − ( λ2 λ 2 1 ) n ) (θ 2 ∗ )θ ∗ + h n , (3.4) where h n+1 = F ∗ h n + (λ 2 /λ 1 ) n h 1 ∈ H. Using the fact that ‖F ∗ h‖ 2 = λ 2 ‖h‖ 2 on H, weget‖h n+1 ‖≤λ 1/2 2 ‖h n ‖+(λ 2 /λ 1 ) n ‖h 1 ‖, which is easily seen to imply that ‖h n ‖=O(λ n/2 2 ) since ∑ k (λ1/2 2 /λ 1 ) k < +∞. This concludes the proof of (i). Let us now turn to the pushforward operator. The first two equations are clear. As θ ∗ may not be an eigenvector for F ∗ , H need not be invariant by F ∗ , but since F ∗ h is orthogonal to θ ∗ for any h ∈ H, we can write F∗ nh = a nθ ∗ +g n with a n = (F n∗ θ ∗ ·h) and g n ∈ H. We have seen that F n∗ θ ∗ = h n modulo θ ∗ ,θ ∗ with ‖h n ‖=O(λ n/2 2 ); thus |a n |=|(h n · h)| ≤Cλ n/2 2 ‖h‖. On the other hand, we have (gn 2) = (F n∗ g n · h); thus ‖g n ‖ 2 ≤ λ n/2 2 ‖g n ‖‖h‖, and this shows that ‖F∗ nh‖≤Cλn/2 2 ‖h‖. Remark 3.7 It follows from the proof of the main theorem that there exist nef classes α ∗ ,α ∗ ∈ H 1,1 R (X) such that for any Kähler classes ω, ω′ on X, wehave ( deg ω (F n ) deg ω ′(F n ) = (α∗ · ω) X (α ∗ · ω) (λ2 ) ) n/2 X + O . (α ∗ · ω ′ ) X (α ∗ · ω ′ ) X Indeed, we can take α ∗ and α ∗ as the incarnations in X of θ ∗ and θ ∗ , respectively. Remark 3.8 When F is bimeromorphic, we have θ ∗ (F ) = θ ∗ (F −1 ); hence (θ∗ 2 ) = 0. However,in general, we may have (θ∗ 2 ) > 0. For example, let F be any polynomial map of C2 whose extension to P 2 is not holomorphic but does not contract any curve. If ω is the class of a line on P 2 ,thendeg ω (F ) > √ λ 2 > 1. On the other hand, F ∗ ω = deg ω (F )ω by Corollary 2.6,soλ 1 = deg ω (F ), θ ∗ = ω and (θ∗ 2) = 1. Remark 3.9 The case when θ ∗ (or θ ∗ ) is Cartier is very special. For example, when F is bimeromorphic, it follows from [DF, Theorem 0.4] that θ ∗ (or, equivalently, θ ∗ ) is Cartier if and only if F is biholomorphic in some birational model. In the general noninvertible case, similar rigidity results are expected (see [C1] for work in this direction). Note also that F being algebraically stable in some birational model does not imply that the eigenclasses are Cartier. We do not know whether having a Cartier eigenclass implies algebraic stability in some model, but having a Cartier eigenclass has many of the same consequences as stability: λ 1 is an algebraic integer, and the sequence of degrees (deg ω F n ) ∞ 1 satisfies a linear recurrence relation. λ 2 1
DEGREE GROWTH OF MEROMORPHIC SURFACE MAPS 537 Acknowledgments. We thank Serge Cantat and Jeff Diller for many useful remarks and the referees for careful readings of the article. References [BHPV] W. P. BARTH, K. HULEK, C. A. M. PETERS,andA. VAN DE VEN, Compact Complex Surfaces, 2nd ed., Ergeb. Math. Grenzgeb. (3) 4, Springer, Berlin, 2004. MR 2030225 521 [BK1] E. BEDFORD and K. H. KIM, On the degree growth of birational mappings in higher dimension, J. Geom. Anal. 14 (2004), 567 – 596. MR 2111418 520 [BK2] ———, Periodicities in linear fractional recurrences: Degree growth of birational surface maps, Michigan Math. J. 54 (2006), 647 – 670. MR 2280499 520 [BF] A. M. BONIFANT and J. E. FORNAESS, Growth of degree for iterates of rational maps in several variables, Indiana Univ. Math. J. 49 (2000), 751 – 778. MR 1793690 520 [B] S. BOUCKSOM, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), 45 – 76. MR 2050205 528 [BFJ] S. BOUCKSOM, C. FAVRE,andM. JONSSON, Differentiability of volumes of divisors and a problem of Teissier, to appear in J. Algebraic Geom., preprint, arXiv:math/0608260v2 [math.AG] 521 [C1] S. CANTAT, Caractérisation des exemples de Lattès et de Kummer, preprint, 2006. 521, 536 [C2] ———, Sur les groupes de transformations birationnelles des surfaces, preprint, 2007. 520 [DF] J. DILLER and C. FAVRE, Dynamics of bimeromorphic maps of surfaces,Amer.J. Math. 123 (2001), 1135 – 1169. MR 1867314 520, 521, 531, 532, 536 [DS] T.-C. DINH and N. SIBONY, Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math. (2) 161 (2005), 1637 – 1644. MR 2180409 520 [F] C. FAVRE, Les applications monomiales en deux dimensions, Michigan Math. J. 51 (2003), 467 – 475. MR 2021001 520 [FJ1] C. FAVRE and M. JONSSON, The Valuative Tree, Lecture Notes in Math. 1853, Springer, Berlin, 2004. MR 2097722 521, 524 [FJ2] ———, Valuations and multiplier ideals,J.Amer.Math.Soc.18 (2005), 655 – 684. MR 2138140 521 [FJ3] ———, Valuative analysis of planar plurisubharmonic functions, Invent. Math. 162 (2005), 271 – 311. MR 2199007 521 [FJ4] ———, Eigenvaluations, Ann. Sci. École Norm. Sup. (4) 40 (2007), 309 – 349. MR 2339287 521 [Fo] G. B. FOLLAND, Real Analysis: Modern Techniques and Their Applications, 2nd ed., Pure Appl. Math. (N.Y.), Wiley, New York, 1999. MR 1681462 522 [FS] J.-E. FORNAESS and N. SIBONY, “Complex dynamics in higher dimension, II” in Modern Methods in Complex Analysis (Princeton, 1992), Ann. of Math. Stud. 137, Princeton Univ. Press, Princeton, 1995, 135 – 182. MR 1369137 520 [Fr] S. FRIEDLAND, Entropy of polynomial and rational maps, Ann. of Math. (2) 133 (1991), 359 – 368. MR 1097242 520
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448 DAVID GINZBURG The method we us
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450 DAVID GINZBURG Let ν denote a
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452 DAVID GINZBURG Proof The proof
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454 DAVID GINZBURG correspond to th
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456 DAVID GINZBURG Next, consider t
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458 DAVID GINZBURG To state the fol
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460 DAVID GINZBURG check that up to
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462 DAVID GINZBURG where L τ (¯s)
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464 DAVID GINZBURG To define the li
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466 DAVID GINZBURG cuspidality of t
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468 DAVID GINZBURG immediately foll
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470 DAVID GINZBURG expansion. Here
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472 DAVID GINZBURG Notice that S l
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474 DAVID GINZBURG values of m ′
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476 DAVID GINZBURG For 1 ≤ i ≤
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478 DAVID GINZBURG 2m rows, we have
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480 DAVID GINZBURG left to right. C
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482 DAVID GINZBURG unipotent orbit
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Page 85 and 86: 492 DAVID GINZBURG Proof The proof
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