Finiteness of polarized K3 surfaces and hyperkähler manifolds

Abstract

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 {(Xi, Li) | Xi ≃ 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.The paper studies the connection between moduli spaces of polarized varieties, on the one hand, and the shape of the ample cone on a fixed variety, on the other hand. To illustrate our point of departure, let us revue a few well-known results. 0.1. The classical Torelli theorem shows that two complex smooth projective curves C and C ′ are isomorphic if and only if their polarized Hodge structures are isomorphic, i.e. there exists a Hodge isometry H 1 (C, Z) ≃ H 1 (C, Z), or, equivalently, if their principally polarized Jacobians J(C) ≃ J(C ′ ) are isomorphic. Dropping the compatibility with the polarizations, so only requiring isomorphisms of unpolarized Hodge structures or unpolarized abelian varieties, the geometric relation between C and C ′ becomes less clear. In moduli theoretic terms, one may wonder about the geometric nature of the quotient map M g / / M g / ∼ . Here, M g denotes the moduli space of genus g curves and C ∼ C ′ if and only if J(C) ≃ J(C ′ ) unpolarized.Similarly, two polarized K3 surfaces (S, L) and (S ′ , L ′ ) are isomorphic if and only if there exists a Hodge isometry H 2 (S, Z) ≃ H 2 (S ′ , Z) that maps L to L ′ . Dropping the latter condition has a clear geometric meaning and corresponds to considering isomorphisms between unpolarized surfaces S and S ′ . Therefore, dividing out by the resulting equivalence relation yields a map M d / / M d / ∼ from the moduli space of polarized K3 surfaces (S, L) of degree d to the space of isomorphism classes of K3 surfaces that merely admit a polarization of this degree. Considering only isomorphisms of Hodge structures without any further compatibilities leads to the analogue of the aforementioned question for curves. At this time, there is no clear picture of what the existence of an unpolarized isomorphism of Hodge structures could mean for the geometry of the two K3 surfaces, but finiteness has recently been established in [Ef17].The author is supported by the SFB/TR 45 'Periods, Moduli Spaces and Arithmetic of Algebraic Varieties' of the DFG (German Research Foundation) and the Hausdorff Center for Mathematics.1 2 D. HUYBRECHTS Other types of varieties, like abelian varieties, Calabi-Yau or hyperkähler varieties, can be discussed from the same perspective. 0.2. Let us move to the cone side. For a K3 surface S, t...

The global moduli theory of symplectic varieties

“…Remark 8.11. Some ideas similar to those appearing in the proof of Theorem 8.2 have also been used recently by Huybrechts [Huy18] to prove some finiteness results for hyperkähler manifolds, and these arguments can likely be adapted to the singular setting.…”