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Comparison of dynamical degrees for semi-conjugate meromorphic maps. (English) Zbl 1279.32018
Let $$X$$ be a compact Kähler manifold of dimension $$k$$ with a Kähler form $$\omega_X$$, and let $$f: X\rightarrow X$$ be a meromorphic map, which is dominant, i.e., such that the image of $$f$$ contains an open set of $$X$$. Denoting by $$f^n$$ the $$n$$-th iterate of $$f$$, the dynamical degree $$d_p(f)$$ of order $$p$$ is the quantity measuring the growth of the norms of $$(f^n)^*$$ acting on the cohomology groups $$H^{p,p}(X,\mathbb{R})$$ when $$n$$ tends to infinity. Let $$\pi: X\to Y$$ be a dominant meromorphic map from $$X$$ to a compact Kähler manifold $$Y$$ of dimension $$l\leq k$$ with a Kähler form $$\omega_Y$$, and assume that $$f$$ preserves the generic fibers of $$\pi$$. Then $$f$$ induces a dominant meromorphic map $$g: Y\to Y$$ such that $$\pi\circ f = g\circ \pi$$. In this case $$f$$ is said to be semi-conjugate to $$g$$, and it is a natural to study the relations between the dynamical system defined by $$f$$ and the one defined by $$g$$ by comparing the dynamical invariants associated to $$f$$ and $$g$$. In the main result of the paper under review (Theorem 1.1) the authors establish a precise formula relating the dynamical degrees of $$f$$ and $$g$$, proving that, for all $$0\leq p\leq k$$, we have $d_p(f) = \max_{\max\{0, p-k-l\}\leq j\leq \min\{p,l\}} d_j(g)d_{p-j}(f|_\pi),$ where $$d_{p-j}(f|_\pi)$$ measures the growth of $$(f^n)^*$$ acting on the subspace $$H_\pi^{l+p-j,l+p-j}(X,\mathbb{R})$$ of classes in $$H^{l+p-j,l+p-j}(X,\mathbb{R})$$ which can be supported by a generic fibre of $$\pi$$.
If the map $$\pi$$ is generically finite, that is $$X$$ and $$Y$$ have the same dimension, one obtains that the dynamical degrees of $$f$$ and $$g$$ are equal. This improves an earlier result of [T.-C. Dinh and N. Sibony, Ann. Math. (2) 161, No. 3, 1637–1644 (2005; Zbl 1084.54013)], showing that dynamical degrees are bimeromorphic invariants. The proof of Theorem 1.1 is based on some delicate calculus on positive closed currents.
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The authors then consider the very interesting case where the consecutive dynamical degrees of $$f$$ are distinct, proving that if $$f$$ has this property, then the same holds also for $$g$$ and the consecutive dynamical degrees of $$f$$ with respect to $$\pi$$. Moreover, they prove that if $$X$$ is a projective manifold admitting a dominant meromorphic map with distinct consecutive dynamical degrees, then the Kodaira dimension of $$X$$ can be either 0 or $$-\infty$$.

##### MSC:
 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables 32U40 Currents 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37F05 Dynamical systems involving relations and correspondences in one complex variable
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