Finiteness of polarized K3 surfaces and hyperkähler manifolds

Huybrechts, Daniel

Annales Henri Lebesgue, Tome 1 (2018), pp. 227-248.

Dans l’espace de modules des variétés polarisées $(X,L)$ la variété (non-polarisée) $X$ peut apparaître plus d’une fois. Néanmoins, pour les surfaces K3, les variétés hyperkähleriennes compactes et les variétés abéliennes il est connu que l’orbite de la variété $X$, i.e. l’ensemble $\{(X_i,L_i)\mid X_i\cong X\}$, est fini, ce qui peut être vu comme une conséquence de la conjecture du cône de Kawamata–Morrison. Nous donnons ici une démonstration de la finitude qui ne repose pas sur la conjecture du cône et qui n’utilise même pas le théorème de Torelli global. La finitude de l’orbite se déduit plutôt de la géométrie de l’espace de modules des variétés polarisées et d’arguments à la Baily–Borel. Le problème de la finitude pour la famille de twisteurs associée à une surface K3 à multiplication complexe est également traité.

In the moduli space of polarized varieties $(X,L)$ the same unpolarized variety $X$ can occur more than once. However, for K3 surfaces, compact hyperkähler manifolds, and abelian varieties the ‘orbit’ of $X$, i.e. the subset $\{(X_i,L_i)\mid X_i\cong X\}$, is known to be finite, which may be viewed as a consequence of the Kawamata–Morrison cone conjecture. In this note we provide a proof of this finiteness not relying on the cone conjecture and, in fact, not even on the global Torelli theorem. Instead, it uses the geometry of the moduli space of polarized varieties to conclude the finiteness by means of Baily–Borel type arguments. We also address related questions concerning finiteness in twistor families associated with polarized K3 surfaces of CM type.

Classification : 14J28, 14J32, 14J15
Mots clés : K3 surfaces, cone conjecture, moduli spaces

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