Huybrechts, Daniel
Finiteness of polarized K3 surfaces and hyperkähler manifolds
Annales Henri Lebesgue, Volume 1 (2018), p. 227-246

KeywordsK3 surfaces, cone conjecture, moduli spaces

### Abstract

In the moduli space of polarized varieties $\left(X,L\right)$ 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 $\left\{\left({X}_{i},{L}_{i}\right)\mid {X}_{i}\cong X\right\}$, 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.

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