Communications in Analysis and Geometry

Volume 28 (2020)

Number 8

The Second of Two Special Issues in Honor of Karen Uhlenbeck’s 75th Birthday

Special-Issue Editors: Georgios Daskalopoulos (Brown University), Kefeng Liu, Chuu-Lian Terng (U. of Cal. Irvine), and Shing-Tung Yau

Collapsing Ricci-flat metrics on elliptic K3 surfaces

Pages: 2019 – 2133

DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n8.a9

Authors

Gao Chen (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Jeff Viaclovsky (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)

Ruobing Zhang (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)

Abstract

For any elliptic K3 surface $\mathfrak{F} : \mathcal{K} \to \mathbb{P}^1$, we construct a family of collapsing Ricci-flat Kähler metrics such that curvatures are uniformly bounded away from singular fibers, and which Gromov–Hausdorff limit to $\mathbb{P}^1$ equipped with the McLean metric. There are well-known examples of this type of collapsing, but the key point of our construction is that we can additionally give a precise description of the metric degeneration near each type of singular fiber, without any restriction on the types of singular fibers.

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Received 30 May 2020

Accepted 2 November 2020

Published 8 January 2021