Order 3 symplectic automorphisms on \(K3\) surfaces
Ever since the seminal work of [\textit{V. V. Nikulin}, Tr. Mosk. Mat. O.-va 38, 75--137 (1979; Zbl 0433.14024)], there has been interest in symplectic automorphisms of finite order on \(K3\) surfaces. The situation of such automorphiisms of order 2, the so-called Nikulin involutions, is understood quite well, see, e.g., [\textit{B. van Geemen} and \textit{A. Sarti}, Math. Z. 255, No. 4, 731--753 (2007; Zbl 1141.14022)]. In a thorough manner, the article under review progresses to the study of symplectic automorphisms of order 3. A \(K3\) surface together with a symplectic automorphism \(\sigma\) of order \(3\) on \(S\) is constructed in Section 2.1. The two subsequent sections are concerned with the action of \(\sigma\) on the Néron-Severi group \(\mathrm{NS}(S)\) and the lattice \(H^2(S;\mathbb Z)\). This builds the groundwork for the remaining investivations, for, due to Nikulin's results, the action of the automorphism on the \(K3\) lattice -- which is derived in Section 3.1 -- depends only on the order and is otherwise independent from the automorphism (and the surface acted upon). Given a symplectic automorphism \(\sigma\) on a \(K3\) surface \(X\), the minimal resolution \(Y\) of the quotient \(X/\sigma\) is a \(K3\) surface as well and the quotient map lifts to a rational map \(\pi\colon X\dashrightarrow Y\). As the action of the automorphism on the \(K3\) lattice is mostly independent of the automorphism, so is the induced action of this rational map on the \(K3\) lattice, \(\pi_*\), which is provided and intensely studied in Section 3. The families \(\mathcal S\), parametrising \(K3\) surfaces admitting a symplectic automorphism of order \(3\), and \(\mathcal T\), parametrising (the minmal resolutions of) their quotients by said involutions, are described in Section 4, cf. Corollary 4.5 and Corollary 4.10, respectively. The relationship between those families, by means of \(\pi_*\) and its dual \(\pi^*\), is clarified in Section 5. The remaining sections are concerned with applications of the above results, including the proof that there exists an infinite tower of isogenies of \(K3\) surfaces as a sequence of quotients of quotients by symplectic automorphisms of order 3 (Section 7.1) and a suitable generalisation of the notion of the so-called Shioda-Inose structures to order 3, cf. Section 7.2. Section 6 concretely describes certain components of the above-mentioned families of \(K3\) surfaces and their relationships.
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