Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces

Abstract

We exhibit families of Ricci-flat Kähler metrics on K3 surfaces which collapse to an interval, with Tian-Yau and Taub-NUT metrics occurring as bubbles. There is a corresponding continuous surjective map from the K3 surface to the interval, with regular fibers diffeomorphic to either 3-tori or Heisenberg nilmanifolds. Contents 1. Introduction 1 2. The Gibbons-Hawking ansatz and the model space 10 3. The asymptotic geometry of Tian-Yau spaces 17 4. Liouville theorem for harmonic functions 22 5. Liouville theorem for half-harmonic 1-forms 41 6. Construction of the approximate hyperkähler triple 45 7. Geometry and regularity of the approximate metric 52 8. Weighted Schauder estimate 68 9. Perturbation to genuine hyperkähler metrics 74 References 93

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

“…In fact, we believe that even in K3 case we lack enough conceptual understanding. See [HSVZ18] for the recent related development in K3 case, and we strongly wish to discuss and develop our understanding further on this problem in near future. Also, it has been expected by experts that the dual intersection complex of maximally degenerating irreducible symplectic manifolds is homeomorphic to the complex projective space of half real dimension.…”

“…In a similar vein, as L. Foscolo, S. Sun and J. Viaclovsky kindly pointed out to us in June of 2018 after our [OO18], the continuity of Φ around the boundary component M K3 (b 2 ) (resp., M K3 (c 2 )) should fit with collapsing of the K3 surfaces constructed in Chen-Chen [CC16] by gluing along cylindrical metrics, to the segment (resp., the collapsing of glued K3 surfaces of very recent Hein-Sun-Viaclovsky-Zhang [HSVZ18] to the segment.) We thank them for the discussions, which seem to provide more evidences to above Conjecture 6.2.…”

Global higher order estimates for collapsing Calabi-Yau metrics on elliptic K3 surfaces

“…Our method also works in other situations. For example, in [2] Hein-Sun-Viaclovsky-Zhang studied other types of degenerations of Calabi-Yau metrics on K3 surfaces. From their construction of approximation metrics, one can see that our proof also works in their situation, therefore also gives similar global higher order estimates.…”

“…As an application of Theorem 1.1, and also motivated by the work [2], we study the blow-up limit of ωǫ at singular fibers. We have the following result: (The precise definition of the coordinates u, y 1 , y 2 is given in section 4.…”

Small fiberwise oscillation of the eigenfunctions of collapsing Einstein manifolds

“…However, with many natural examples of Einstein manifolds collapsing to lower dimensional metric spaces without a priori curvature bounds -for instance, the collapsing of Ricci flat K3 surfaces constructed by Gross and Wilson [20] and recently by Hein, Sun, Viaclovsky and Zhang [21] -one wonders if there should be any sort of extension of Fukaya's Key Lemma to Einstein manifolds that are pointed Gromov-Hausdorff close to lower dimensional metric spaces at a given scale, without assuming (1.1). To explain the basic setup in this situation, let us focus on a small piece of a very collapsed Riemannian manifold with almost non-negative Ricci curvature, and recall the following fundamental theorem due to Cheeger and Colding (see [7,Theorem 1.2] and [9, Lemma 1.21]): Theorem 1.2 (Cheeger-Colding's Almost Splitting Theorem).…”