CLASSIFICATION OF POLARIZED SYMPLECTIC AUTOMORPHISMS OF FANO VARIETIES OF CUBIC FOURFOLDS

Lie Fu (1, 2)

1 Algèbre et Géométrie (DMA)
2 DMA - Département de Mathématiques et Applications - ENS Paris

We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci. 0. Introduction The purpose of this paper is to classify the polarized symplectic automorphisms of the irreducible holomorphic symplectic projective varieties constructed by Beauville and Donagi [4], namely, the Fano varieties of (smooth) cubic fourfolds. Finite order symplectic automorphisms of K3 surfaces have been studied in detail by Nikulin in [14]. A natural generalization of K3 surfaces to higher dimensions is the notion of irreducible holomorphic symplectic manifolds or hyper-Kähler manifolds (cf. [2]), which by definition is a simply connected compact Kähler manifold with H 2,0 generated by a symplectic form (i.e. nowhere degenerate holomorphic 2-form). Initiated by Beauville [1], some results have been obtained in the study of automorphisms of such manifolds. Let us mention [3], [6], [5], [7]. In [4], Beauville and Donagi show that the Fano varieties of lines of smooth cubic fourfolds provide an example of a 20-dimensional family of irreducible holomorphic symplectic projective fourfolds. We propose to classify the polarized symplectic automorphisms of this family. Our result of classification is shown in the table below 1. We firstly make several remarks concerning this table: • As is remarked in §1, such an automorphism comes from a (finite order) automorphism of the cubic fourfold itself. Hence we express the automorphism in the fourth column as an element f in PGL 6. • In the third column, n is the order of f , which is primary (i.e. a power of a prime number). The reason why we only listed the automorphisms with primary order is that every finite order automorphism is a product of commuting automorphisms with primary orders, by the structure of cyclic groups. See Remark 3.3. • We give an explicit basis of the family in the fifth column. • In the last column, we work out the fixed loci for a generic member. For geometric descriptions of the fixed loci, see §4. • The Family I in our classification has been discovered in [13]. • The Family V-(1) in our classification has been studied in [7], where the fixed locus and the number of moduli are calculated. • The classification of prime order automorphisms of cubic fourfolds has been done in [11]. I am also informed by G. Mongardi that he classifies the prime order symplectic automorphisms of hyper-Kähler varieties which are of K3 [n]-deformation type in his upcoming thesis.

Domaines: Géométrie algébrique [math.AG]

Lie Fu. CLASSIFICATION OF POLARIZED SYMPLECTIC AUTOMORPHISMS OF FANO VARIETIES OF CUBIC FOURFOLDS. Glasgow Mathematical Journal, 2016, 58 (1), pp.17-37. ⟨10.1017/S001708951500004X⟩. ⟨hal-01465215⟩