Large Complex Structure Limits of K3 Surfaces

2000/08/02 by Mark Gross, Gross, Mark, P. M. H. Wilson +2 · 10 citations
Mathematics · #14J32 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Base (topology) #Conjecture #Differential Geometry (math.DG) #FOS: Mathematics #Fibration #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Homotopy #Mathematical analysis #Mathematics #Mirror symmetry #Pure mathematics #Torus #math.AG #math.DG #msc:14J32

paper · pdf · doi:10.48550/arxiv.math/0008018

published in arXiv (Cornell University) (Cornell University) · 65 pages; Final version, to appear in Journal of Differential Geometry

openalex publication_date 2000/08/02 · arxiv created 2001/04/03 · arxiv updated 2016/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08

Abstract

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsevich, Soibelman and Todorov. Roughly stated, the conjecture says that, if the metrics are normalized to have constant diameter, then this limit is the base of the conjectural special lagrangian torus fibration associated with the large complex structure limit, namely an n-sphere, and that the metric on this Sⁿ is induced from a standard (singular) Riemannian metric on the base, the singularities of the metric corresponding to the discriminant locus of the fibration. This conjecture is trivially true for elliptic curves; in this paper we prove it in the case of K3 surfaces. Using the standard description of mirror symmetry for K3 surfaces and the hyperkähler rotation trick, we reduce the problem to that of studying Kähler degenerations of elliptic K3 surfaces, with the Kähler class approaching the wall of the Kähler cone corresponding to the fibration and the volume normalized to be one. Here we are able to write down a remarkably accurate approximation to the Ricci-flat metric -- if the elliptic fibres are of area ε>0, then the error is O(e^-C/ε) for some constant C>0. This metric is obtained by gluing together a semi-flat metric on the smooth part of the fibration with suitable Ooguri-Vafa metrics near the singular fibres. For small ε, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

Cited by

• Compactification of metric moduli space of K3 surfaces
• Complex structures of the Gibbons-Hawking ansatz with infinite topological type
• Non-archimedean cylinder counts are logarithmic Gromov-Witten invariants
• On multiscale aspects of Kähler-Einstein metrics and algebraic geometry
• Physics and Geometry of Complex Structure Limits in Type IIB Calabi-Yau Compactifications
• Constructive Proof of the Hodge Conjecture for K3 Surfaces via Nodal Degenerations
• Gromov-Hausdorff limits of collapsing Calabi-Yau fibrations
• Degeneration of Calabi-Yau metrics and canonical basis
• Gromov–Hausdorff collapsing of Calabi–Yau manifolds
• Compact Manifolds with Unbounded Nilpotent Fundamental Groups and Positive Ricci Curvature

Related

• The Kummer Construction of Calabi-Yau and Hyper-Kähler Metrics on the K3 Surface, and Large Families of Volume Non-collapsed Limiting Compact Hyper-Kähler Orbifolds
• Collapsing K3 Surfaces and Moduli Compactification
• Asymptotic curvature of moduli spaces for Calabi-Yau threefolds
• Collapsing Ricci-flat metrics on elliptic K3 surfaces
• On K3 Surfaces with Large Complex Structure