Collapsing Ricci-flat metrics on elliptic K3 surfaces

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.

arXiv comment77 pages

References in corpus (6)

• Gromov-Hausdorff collapsing of Calabi-Yau manifolds
• Structure of fundamental groups of manifolds with Ricci curvature bounded below
• Gravitational instantons with faster than quadratic curvature decay (II)
• Nilpotent structures and collapsing Ricci-flat metrics on K3 surfaces
• Complex structure degenerations and collapsing of Calabi-Yau metrics
• The space of hyperkähler metrics on a 4-manifold with boundary

Cited by in corpus (4)

• Polystable log Calabi-Yau varieties and Gravitational instantons
• The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
• Twisted Kähler-Einstein Metrics and Collapsing
• Spectral convergence in geometric quantization on $K3$ surfaces