Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen type

Abstract

We study certain Morgan-Shalen type compactifications for locally Hermitian symmetric spaces and identify them with Satake compactifications for the adjoint representation.Then, via such compactification theory, we provide a moduli-theoretic framework for the collapsing of Ricci-flat-Kähler metrics. In other words, we give geometric meaning to the Satake compactifications for the adjoint representations or the Morgan-Shalen type compactifications, for certain cases.More precisely, we apply the compactification to moduli spaces of compact hy-perKähler manifolds, and state our main conjectures that they parametrize the Gromov-Hausdorff limits of the rescaled hyperKähler metrics with fixed diameters. Before partially proving the conjectures mainly for K3 surfaces, we first establish abelian varieties version, refining the previous work of the first author. A benefit of our conjecture is that, once it is confirmed, then for quite general sequences which are not even necessarily "maximally degenerating", we can determine the Gromov-Hausdorff limits.Our partial confirmation of the conjectures provides, for example, a proof of [KS06, Conjecture 1] and [GroWil00, Conjecture 6.2] at least for K3 surfaces, when applied to one parameter maximally degenerating family. In this case, the obtained Gromov-Hausdorff limits are metrized spheres S 2 , studied in [GroWil00, KS06], which have natural integral affine structures with singularities and are often regarded as tropical analogue of K3 surfaces. We also identify such limits along one parameter holomorphic families from the monodromy information.Trying to prove our conjectures for higher dimensional hyperKähler manifolds, in this paper, we solve the non-collapsing part and make certain progress on algebrogeometric preparations for the collapsing part. More precise form of the former part states that, for a fixed deformation class of polarized irreducible symplectic manifolds, the set of all Q-Gorenstein degenerations as symplectic varieties with ample Q-line bundles are bounded, and further the corresponding partial compactification of the moduli space becomes an orthogonal locally symmetric variety as in K3 surfaces case. Finally, we discuss possible extension of our collapsing picture for general Ricci-flat Kähler metrics of K-trivial varieties.Since the first author presented the main conjecture 4.3 at Oxford in the Clay conference "Algebraic geometry -new and old-" in September 2016 (resp., Conjecture 6.2 in the Mirror symmetry international conference at Kyoto university in December 2016), we have talked and discussed on the topic at Singapore, Levico Terme, Ann Arbor, Shanghai, Xiamen and various cities in Japan. We thank the organizers and the hosts for making such enjoyable chances which encouraged us.We also would like to thank our colleagues and friends, who have given helpful comments during our project;

Stable pair compactification of moduli of K3 surfaces of degree 2

“…These monodromy formulas verify the prediction of topological SYZ mirror symmetry that the Picard-Lefschetz transformation is cup product with a section of the SYZ fibration. See also [OO18,Cor. 4.24].…”

“…Gross-Wilson [GW00] proved that the Gromov-Hausdorff collapse of a K3 degeneration is a metric sphere with 24 singular points. See also a more recent work of Odaka-Oshima [OO18].…”

Moduli of elliptic $K3$ surfaces: monodromy and Shimada root lattice strata

“…As we shall see, both points of view are relevant for our purposes. An interesting result of Odaka and Oshima [OO,Theorem 7.9] even says that the GIT compactification and the Baily-Borel compactification of this space coincide, i.e. F GIT ∼ = F BB .…”

Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

“…The partial second-order estimate is motivated by the study of collapsing problems in Kähler geometry (see, e.g., [1,6,14,22,23,28,29,32,33,34,35,41,42,47,59,53,54,55,56,58,63,64,66,67,68,69,71,74,80]), as well as the study of canonical metrics in Kähler geometry and the behavior of the Kähler-Ricci flow. More specifically, the estimate in Theorem A plays a crucial role in establishing the following conjectural picture for collapsed Gromov-Hausdorff limits of Ricci-flat Kähler metrics which were originally proposed in [66,67], inspired by [24,38,39] and has been intensively studied since:…”