ON THE ACTION OF SYMPLECTIC AUTOMORPHISMS ON THE CH 0 -GROUPS OF SOME HYPER-KAHLER FOURFOLDS

Lie Fu (1)

1 Algèbre et Géométrie (DMA)

We prove that for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch-Beilinson conjecture. 0. Introduction In this paper we are interested in an analogue of Bloch's conjecture for the action on 0-cycles of a symplectic automorphism of a irreducible holomorphic symplectic variety. First of all, let us recall Bloch conjecture and the general philosophy of Bloch-Beilinson conjecture which motivate our result. The Bloch conjecture for 0-cycles on algebraic surfaces states the following (cf. [6, Page 17]): Conjecture 0.1 (Bloch). Let Y be a smooth projective variety, X be a smooth projective surface and Γ ∈ CH 2 (Y × X) be a correspondence between them. If the cohomological correspondence [Γ] * : H 2,0 (X) → H 2,0 (Y) vanishes, then the Chow-theoretic correspondence Γ * : CH 0 (Y) alb → CH 0 (X) alb vanishes as well, where CH 0 (•) alb := Ker (alb : CH 0 (•) hom → Alb(•)) denotes the group of the 0-cycles of degree 0 whose albanese classes are trivial. The special case in Bloch's conjecture where X = Y is a surface S and Γ = ∆ S ∈ CH 2 (S × S) states: if a smooth projective surface S admits no non-zero holomorphic 2-forms, i.e. H 2,0 (S) = 0, then CH 0 (S) alb = 0. This has been proved for surfaces which are not of general type [7], for surfaces rationally dominated by a product of curves (by Kimura's work [18] on the nilpotence conjecture, cf. [23, Theorem 2.2.11]), and for Catanese surfaces and Barlow surfaces [20], etc.. What is more related to the present paper is another particular case of Bloch's conjecture: let S be a smooth projective surface with irregularity q = 0. If an automorphism f of S acts on H 2,0 (S) as identity, i.e. it preserves any holomorphic 2-forms, then f also acts as identity on CH 0 (S). This version is obtained from Conjecture 0.1 by taking X = Y = S a surface and Γ = ∆ S − Γ f ∈ CH 2 (S × S), where Γ f is the graph of f. Recently Voisin [25] and Huybrechts [15] proved this conjecture for any symplectic automorphism of finite order of a K3 surface (see also [16]): Theorem 0.2 (Voisin, Huybrechts). Let f be an automorphism of finite order of a projective K3 surface S. If f is symplectic, i.e. f * (ω) = ω, where ω is a generator of H 2,0 (S), then f acts as identity on CH 0 (S). The purpose of the paper is to investigate the analogous results in higher dimensional situation. The natural generalizations of K3 surfaces in higher dimensions are the so-called irreducible holomorphic sym-plectic varieties or hyperkähler manifolds (cf. [3]), 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). We can conjecture the following vast generalization of Theorem 0.2: 1

Domaines: Géométrie algébrique [math.AG]

Lie Fu. ON THE ACTION OF SYMPLECTIC AUTOMORPHISMS ON THE CH 0 -GROUPS OF SOME HYPER-KAHLER FOURFOLDS. Mathematische Zeitschrift, 2015. ⟨hal-01465214⟩