Abstract
Motivated by the study of collapsing Calabi–Yau 3-folds with a Lefschetz K3 fibration, we construct a complete Calabi–Yau metric on {mathbb {C}}^3 with maximal volume growth, which in the appropriate scale is expected to model the collapsing metric near the nodal point. This new Calabi–Yau metric has singular tangent cone at infinity {mathbb {C}}^2/{mathbb {Z}}_2 times {mathbb {C}}, and its Riemannian geometry has certain non-standard features near the singularity of the tangent cone, which are more typical of adiabatic limit problems. The proof uses an existence result in H-J. Hein’s Ph.D. thesis to perturb an asymptotic approximate solution into an actual solution, and the main difficulty lies in correcting the slowly decaying error terms.
Highlights
This work grows out of the attempt to model the collapsing behaviour of Calabi–Yau metrics on a K3 fibred compact Calabi–Yau manifold (i.e. Kähler Ricci-flat with parallel nonvanishing holomorphic volume form) over a Riemann surface, where the Kähler class has very small volume on the K3 fibres
It is natural to write down the so called semi-Ricci flat metric as a first approximation, which restricts to the Stenzel metrics on the quadric fibres of f = z12 + z22 + z32 : C3 → C
We carefully examine the error terms involved in the approximate metric, and apply standard techniques for the Monge–Ampère equation on noncompact manifolds to perturb this into a genuine Calabi–Yau metric
Summary
This work grows out of the attempt to model the collapsing behaviour of Calabi–Yau metrics on a K3 fibred compact Calabi–Yau manifold (i.e. Kähler Ricci-flat with parallel nonvanishing holomorphic volume form) over a Riemann surface, where the Kähler class has very small volume on the K3 fibres. It is natural to write down the so called semi-Ricci flat metric as a first approximation, which restricts to the Stenzel metrics on the quadric fibres of f = z12 + z22 + z32 : C3 → C This is a singular metric admitting a large symmetry group, suggesting that the construction of Calabi–Yau metric is a cohomogeneity two problem. The behaviour of the Laplacian is likewise non-standard in this region; the cohomogeneity two problem of inverting the Laplacian effectively reduces to solving ODEs, in contrast to the Euclidean intuition, which suggests a genuine dependence of the Laplacian on both variables Another context in which our metric is expected to arise is related to Joyce’s construction of G2 manifolds [7]. Some informal discussions on the moduli question of classifying complete Calabi–Yau metrics on Cn with maximal volume growth will be given at the end
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