6. Higher dimensional case [04YU]

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6. Higher dimensional case

We expect that our results for K3 surfaces naturally extend to higher dimensional compact hyperKähler manifolds. Let us focus on algebraic case in this notes. We set up as follows. Fix any connected moduli $M$ of polarized $2n$ -dimensional irreducible holomorphic symplectic manifolds $(X,L)$ whose second cohomology $H^{2}(X,\mathbb{Z})$ is isomorphic (as a lattice) to $\Lambda$ . By [Ver13, Mark11] ([GHS13, Theorem 3.7]), it is a Zariski open subset of a Hermitian locally symmetric space of orthogonal type $\Gamma\backslash\mathcal{D}_{M}$ .

Then (a rough version of) our conjecture for algebraic case (in [OO]) is as follows:

Conjecture 6.1.

There is a continuous map $\Psi$ (call “geometric realization map”) from the Satake compactification $(M\subset)\overline{\Gamma\backslash\mathcal{D}_{M}}^{\rm Sat,\tau_{ad}}$ with respect to the adjoint representation to the Gromov-Hausdorff compactification of $M$ , extending the identity map on $M$ . The $(b_{2}(X)-4)$ -dimensional boundary strata of $\overline{\Gamma\backslash\mathcal{D}_{M}}^{\rm Sat,\tau_{ad}}$ parametrize via $\Psi$ the projective space $\mathbb{P}^{n}$ with special Kähler metrics in the sense of [Freed99] and the metric space parametrized by $0$ -dimensional cusps are all homeomorphic to the closed ball of dimension $n$ .

At the moment of writing this notes, the authors have only succeeded in proving that $(M\subset)\Gamma\backslash\mathcal{D}_{M}$ is the moduli of polarized symplectic varieties with continuous (non-collapsing) weak Ricci-flat Kähler metrics, and making some progress on the necessary algebro-geometric preparations in particular for the case of K3^[n]-type.

Remark 6.2 (Calabi-Yau case).

In [OO], we also propose an extension of Conjecture 4.3 for general Calabi-Yau varieties under some technical conditions, although there are much fewer evidences in that case.

Acknowledgement We appreciate for giving us the chances to talk on [OO] in various countries and cities. The first was at a talk by the first author at a Clay conference held at Oxford in September 2016, when Theorems 4.4 and 5.1 were only partially proved and claimed, whose confirmation in the form of this notes has taken long time. In particular, we appreciate Kenji Hashimoto, Shouhei Honda, Radu Laza, Daisuke Matsushita, Shigeru Mukai, Yoshinori Namikawa, Bernd Siebert, Cristiano Spotti, Song Sun, Yuichi Nohara, Kazushi Ueda, Ken-ichi Yoshikawa for helpful discussions. There are plans of some lecture series by the first author on this topic during the next fall semester in Nagoya, Tokyo. The first author is partially supported by JSPS Grant-in-Aid (S), No. 16H06335, Grand-in-Aid for Early-Career Scientists No. 18K13389. The second author is partially supported by JSPS KAKENHI Grant No. 16K17562.