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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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