Characterization of the Lattès and Kummer examples.
(Caractérisation des exemples de Lattès et de Kummer.)

*(French)*Zbl 1156.37012Let \(X\) be a compact complex manifold, and let \(f: X \dashrightarrow X\) be a dominant meromorphic transformation of \(X\). We say that \(f\) is a Kummer example if there exists a compact complex torus \(A\), a finite group \(G\) of automorphisms of \(A\) and an affine endomorphism \(F\) inducing an endomorphism \(\overline{F}\) of the quotient \(A/G\) that is birationally equivalent to \(f\), i.e. there exists a birational transformation \(\pi: X \dashrightarrow A/G\) such that \(\pi \circ f= \overline{F} \circ \pi\). We say that it is a Lattès example if moreover the eigenvalues of the linear part of \(F\) are superior or equal to one. The most classical case of such a meromorphic transformation is probably the following: let \(A=\mathbb C^2/\Lambda\) be a compact complex torus of dimension two and let \(G\) be the group generated by the involution \((x,y) \rightarrow (-x,-y)\). The quotient \(A/G\) has exactly sixteen ordinary double points and the blow-up of these points is a smooth surface \(X\), called the Kummer surface of \(A\). A linear endomorphism \(L\) of \(\mathbb C^2\) that preserves the lattice \(\Lambda\) defines an endomorphism \(\overline{F}\) on \(A/G\). The corresponding meromorphic transformation \(f\) of the Kummer surface \(X\) is a Kummer example. It is a Lattès example if the determinant of \(L\) equals \(\rho(L)^2\), where \(\rho(L)\) is the spectral ray of \(L\). The first main theorem of this article characterises Lattès examples among holomorphic endomorphisms satisfying certain dynamical properties.

Theorem: Let \(f: X \rightarrow X\) be a holomorphic endomorphism such that there exist a Kähler class \([\kappa]\) and a real number \(\delta>1\) such that \(f^* [\kappa]= \delta [\kappa]\). If the measure of maximal entropy of \(f\) is absolutely continuous with respect to the Lebesgue measure, then \(f\) is a Lattès example.

The other main results concern meromorphic transformations of surfaces. If \(f: X \dashrightarrow X\) is a dominant meromorphic transformation of a compact complex surface, there are three numbers naturally attached to \(f\):

- the topological degree \(d_t(f)\), that is the number of preimages of a general point;

- the dynamical degree \(\lambda(f)\), that is the superior limit of \(\|(f^k)^*\|^{1/k}\), where \((f^k)^*\) is the linear operator on \(H^{1,1}(X, \mathbb R)\) induced by \(f^k\);

- and the spectral ray \(\rho(f^*)\), that is the superior limit of \(\|(f^*)^k\|^{1/k}\), where \((f^*)^k\) is the \(k\)-th iterate of the linear operator induced by \(f\) on \(H^{1,1}(X, \mathbb R)\). The following statements are proven.

Theorem: Let \(X\) be a compact complex surface and \(f: X \dashrightarrow X\) a dominant meromorphic transformation whose topological degree is strictly larger than one. If the Kodaira dimension of \(X\) is non-negative and \(d_t(f)=\lambda(f)^2\), then \(f\) is a Lattès example. If \(X\) is a Kähler manifold, \(d_t(f)=\rho(f^*)^2\), and the measure of maximal entropy of \(f\) is absolutely continuous with respect to the Lebesgue measure, then \(f\) is a Lattès example.

Theorem: Let \(X\) be a compact Kähler surface and \(f: X \dashrightarrow X\) a dominant meromorphic transformation whose topological degree is strictly larger than one. If the Kodaira dimension of \(X\) is zero and \(f\) commutes with infinitely many automorphisms of \(X\), then \(f\) is a Lattès example.

Moreover it is shown that analogous characterisations do not exist for transformations that are “cohomologically dilatating”, i.e. that satisfy \(d_t(f)>\lambda(f)\): the author constructs a complex projective surface \(X\) of Kodaira dimension zero and a cohomologically dilatating rational transformation \(f\) such that \(\lambda(f)=\rho(f^*)\) and the unique measure of maximal entropy is smooth, but \(f\) is not topologically conjugate to a Kummer example. We refer to the introduction of the article for ample information on the background as well as details on the various techniques used in the proofs of these results.

Theorem: Let \(f: X \rightarrow X\) be a holomorphic endomorphism such that there exist a Kähler class \([\kappa]\) and a real number \(\delta>1\) such that \(f^* [\kappa]= \delta [\kappa]\). If the measure of maximal entropy of \(f\) is absolutely continuous with respect to the Lebesgue measure, then \(f\) is a Lattès example.

The other main results concern meromorphic transformations of surfaces. If \(f: X \dashrightarrow X\) is a dominant meromorphic transformation of a compact complex surface, there are three numbers naturally attached to \(f\):

- the topological degree \(d_t(f)\), that is the number of preimages of a general point;

- the dynamical degree \(\lambda(f)\), that is the superior limit of \(\|(f^k)^*\|^{1/k}\), where \((f^k)^*\) is the linear operator on \(H^{1,1}(X, \mathbb R)\) induced by \(f^k\);

- and the spectral ray \(\rho(f^*)\), that is the superior limit of \(\|(f^*)^k\|^{1/k}\), where \((f^*)^k\) is the \(k\)-th iterate of the linear operator induced by \(f\) on \(H^{1,1}(X, \mathbb R)\). The following statements are proven.

Theorem: Let \(X\) be a compact complex surface and \(f: X \dashrightarrow X\) a dominant meromorphic transformation whose topological degree is strictly larger than one. If the Kodaira dimension of \(X\) is non-negative and \(d_t(f)=\lambda(f)^2\), then \(f\) is a Lattès example. If \(X\) is a Kähler manifold, \(d_t(f)=\rho(f^*)^2\), and the measure of maximal entropy of \(f\) is absolutely continuous with respect to the Lebesgue measure, then \(f\) is a Lattès example.

Theorem: Let \(X\) be a compact Kähler surface and \(f: X \dashrightarrow X\) a dominant meromorphic transformation whose topological degree is strictly larger than one. If the Kodaira dimension of \(X\) is zero and \(f\) commutes with infinitely many automorphisms of \(X\), then \(f\) is a Lattès example.

Moreover it is shown that analogous characterisations do not exist for transformations that are “cohomologically dilatating”, i.e. that satisfy \(d_t(f)>\lambda(f)\): the author constructs a complex projective surface \(X\) of Kodaira dimension zero and a cohomologically dilatating rational transformation \(f\) such that \(\lambda(f)=\rho(f^*)\) and the unique measure of maximal entropy is smooth, but \(f\) is not topologically conjugate to a Kummer example. We refer to the introduction of the article for ample information on the background as well as details on the various techniques used in the proofs of these results.

Reviewer: Andreas Höring (Paris)

##### MSC:

37F99 | Dynamical systems over complex numbers |

37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |

14J28 | \(K3\) surfaces and Enriques surfaces |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14J50 | Automorphisms of surfaces and higher-dimensional varieties |

14E05 | Rational and birational maps |