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Ricci-flat metrics and dynamics on \(K3\) surfaces. (English) Zbl 1464.14042

This paper surveys some recent results on interactions between the geometry of \(K3\) surfaces and the dynamics of \(K3\) automorphisms with positive entropy.
\(K3\) surfaces form a class of compact complex surfaces which has received a tremendous amount of attention in several branches of geometry. For example, \(K3\) surfaces admit Ricci-flat (but non-flat) Kähler-Einstein metrics by the Calabi-Yau theorem, and there is a great interest to understand the degenerations of these metrics. In a seemingly unrelated direction, \(K3\) surfaces have been studied in holomorphic dynamic system. The theory of holomorphic dynamics in one complex variable (on the Riemann sphere) is an enormous research area. In the case of two complex variables, the study of dynamically interesting automorphisms on \(K3\) surfaces was initiated by S. Cantat [Acta Math. 187, No. 1, 1–57 (2001; Zbl 1045.37007)].
This article reviews both aspects of \(K3\) surfaces and explains the recent work by S. Filip and the author [Ann. Inst. Fourier 68, No. 7, 2981–2999 (2018; Zbl 1428.14065); “Kummer rigidity for \(K3\) surface automorphisms via Ricci-flat metrics”, Preprint, arXiv:1808.08673] that exploits the relationship between Ricci-flat Kähler-Einstein metrics and dynamics of \(K3\) automorphisms. Section 2 gives an introduction to \(K3\) surfaces, including examples, and the solutions of the conjectures of Andreotti and Weil. Section 3 discusses the Yau’s Theorem on the existence of Ricci-flat Kähler-Einstein metrics. Section 4 gives an overview of the dynamical study of automorphisms of \(K3\) surfaces, including basic properties and examples. Section 5 reviews the Kummer rigidity theorem of S. Cantat and C. Dupont [J. Eur. Math. Soc. (JEMS) 22, No. 4, 1289–1351 (2020; Zbl 1476.37071)] and S. Filip and the author [“Kummer rigidity for \(K3\) surface automorphisms via Ricci-flat metrics”, Preprint, arXiv:1808.08673], with an emphasis on the application of Ricci-flat Kähler-Einstein metrics. In Section 6 the author shows the applications of dynamics (in particular of Kummer rigidity) to the study of degenerations of Ricci-flat Kähler-Einstein metrics on \(K3\) surfaces [S. Filip and V. Tosatti, Ann. Inst. Fourier 68, No. 7, 2981–2999 (2018; Zbl 1428.14065)]. Finally, Section 7 shows a few open problems.

MSC:

14J28 \(K3\) surfaces and Enriques surfaces
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32J15 Compact complex surfaces
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
32Q25 Calabi-Yau theory (complex-analytic aspects)
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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References:

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