Large Complex Structure Limits of K3 Surfaces

By Mark Gross, Pelham M. H. Wilson. 2000

319 citations recorded by the source corpus.

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 K\"ahler 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 S^n 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 hyperk\"ahler rotation trick, we reduce the problem to that of studying K\"ahler degenerations of elliptic K3 surfaces, with the K\"ahler class approaching the wall of the K\"ahler 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 $\epsilon >0$, then the error is $O(e^{-C/\epsilon})$ 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 $\epsilon$, this is a sufficiently good approximation that the above conjecture is then an easy consequence.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dbbc899c8a6b148e5b9d5e31f7296ab42c5eb84e

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Collapsing of abelian fibred Calabi-Yau manifolds (2011, methodology): “The collapsing of Riemannian manifolds in the Gromov-Hausdorff sense has been extensively studied from different viewpoints, see for example [2, 5, 6, 7, 8, 12, 18, 27, 28, 33] and the reference therein.”
Gromov-Hausdorff collapsing of Calabi-Yau manifolds (2013, unknown): “These last two facts are intimately related to the Strominger-Yau-Zaslow picture of mirror symmetry [32], and to a conjecture of Gross-Wilson [13, Conjecture 6.2], [1, Conjecture 7.32], Kontsevich-Soibelman [18, Conjecture 1] and Todorov [22, p. 66] which predicts that collapsed Gromov-Hausdorff lim”
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014, background): “Theorem 1.3 was first proved by Gross-Wilson [20] in the case when X is a K3 surface, B is CP1 and the only singular fibers of π are of type I1.”
Triples, Fluxes, and Strings (2001, methodology): “a metric on B — A whose asymptotics near A are described by the Greene-Shapere-Vafa-Yau ansatz [63] (which can be seen as a limit of metrics on Y as the area of the elliptic fiber approaches zero [64]).”
Torsional Heterotic Geometries (2009, background): “Indeed the most general heterotic dual is not a classical geometry but will be a background that requires quantum patching conditions involving T-duality.”
Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface (2018, methodology): “s ALE and ALF spaces of Ak−1 type. In particular, see [Min11] for a complete theory of ALF-Ak−1 spaces. Example 2.2. Let U= S1 ×R2 and let V be a Green’s function with exactly one pole on S1 ×{0}. In [GW00] V is constructed by passing to the universal cover Ue = R3, where the lifted function Ve is a”
Survey on Affine Spheres (2008, background): “The structure of the singular locus S = M\M reg is more complicated, but the picture is largely understood in dimension two due to the work of Gross-Wilson [33].”
Tropical and non-Archimedean limits of degenerating families of volume forms (2016, background): “, GTZ13a, GTZ13b, HT14, TWY14]. By performing a \hyper-Kahler rotation", this implies a version of the Kontsevich{Soibelman conjecture for special cases of Type III degenerations of K3 surfaces [GW00]. Theorems A and C indicate a possible approach to the Kontsevich{Soibelman conjecture. Indeed,”
Tropical Geometry and its applications (2006, unknown): “Already the tropical K3-surfaces form a very sophisticated geometric object, see [14] and [9].”
Affine manifolds, log structures, and mirror symmetry (2002, background): “Finally, let us underline the context of the error estimate in terms of the ideas of [10].”
Almost toric symplectic four-manifolds (2003, unknown): “As a byproduct we provide a geometric explanation for why a generic Lagrangian fibration over the two-sphere must have 24 singular fibers.”
Emergent strings from infinite distance limits (2019, unknown): “Evidently a natural interpretation would be in terms of a tower of D3-branes wrapping some special Lagrangian 3-torus fiber on X [85–87].”