SYMPLECTIC ACTION OF GROUPS OF ORDER FOUR ON K3 SURFACES AND K3^[2]-TYPE MANIFOLDS

An irreducible holomorphic symplectic (IHS) manifold is a compact, simply connected complex manifold X such that H^{2,0}(X) = CCω_X, where ω_X is a symplectic (nowhere degenerate) form. The rich geometry of IHS manifolds is contrasted by the scarcity of examples: only K3 surfaces in dimension 2, then in every even dimension two deformation classes, K3^[n] and Km_n, already described in the same paper by Beauville in which the concept was introduced; the only other known examples are O’Grady’s manifolds in dimension 6 and 10. In recent years therefore the theory of symplectic varieties, that have been introduced as generalizations of IHS manifolds admitting some mild singularities, has become its own research subject. One way to obtain symplectic varieties is to take terminalizations of quotients by groups of symplectic automorphisms: an automorphism α of X is symplectic if α^∗(ω_X) = ω_X. Moreover, if H ⊂ G is a normal subgroup, the symplectic variety obtained as terminalization of X/H admits a symplectic action of G/H. We study the action of groups G of order 4 on K3^[2]-type manifolds, providing the first examples of non simple symplectic actions on IHS manifolds. We start with G acting on a K3 surface S; we give a lattice-theoretic characterization of Z˜, the K3 surface that arises as terminalization of S/i, with i ∈ G of order 2, and compare its moduli space to that of S and the K3 terminalization Y˜ of S/G. In the projective case, the many different irreducible components of the moduli spaces of S and Y˜ are in bijection, but this does not extend to Z˜. If G acts on a K3^[2]-type manifold X, the role of Z˜ is taken by the Nikulin orbifold Y, the terminalization of X/i: we establish the correspondence between the moduli spaces of X and Y in the projective case, and we describe the two induced involutions on Y. Moreover, we give lattice theoretic conditions under which each of these involutions persist under deformations Y˜ of Y. We also explore the relation between manifolds with the same transcendental lattice. By Lefschetz’s theorem, algebraic classes are identified with H^{1,1}(X) ∩ H^2(X, ZZ): therefore the Hodge structure on H^2(X, ZZ), which encodes most of the geometric information, is always trivial on the algebraic sublattice, but not on the transcendental T(X):=(H{1,1}(X) ∩ H2(X, ZZ))^⊥. In chapter 4 we study generalizations of Shioda-Inose structures to the order 4. Given any abelian surface A, if X is a K3 surface such that T(X) and T(A) are Hodge isometric, then X has a symplectic involution ι such that the resolution of singularities of X/ι is isomorphic to Kum(A) (and the converse also holds). We prove that, if A has a symplectic automorphism of order 4, the same condition on T(X) is not enough to get a symplectic action of a group of order 4 on X, and we propose some partial generalizations of the standard construction. The last chapter presents a joint work with A´ngel David R´ıos Ortiz, about IHS manifolds whose transcendental lattice is Hodge isometric to that of a K3 or an abelian surface, and whether they are birationally equivalent to moduli spaces over said surfaces or not. We find that this holds if X is an IHS manifold of K3^[n]-type or Km_n-type, but not always if X is of OG6-type or OG10-type: in these cases, we give some partial results using lattice theory.