Weak Fan Structures for Two-Parameter Degenerations of K3 Surfaces

TL;DR

This paper explicitly computes the weak fan structure for a two-parameter degeneration of K3 surfaces, illustrating a framework for understanding boundary components of period domains in complex geometry.

Contribution

It provides a concrete example of weak fan construction in a non-Hermitian setting, expanding the understanding of degenerations of K3 surfaces and their period maps.

Findings

Explicit weak fan computation for K3 degenerations

Clarification of nilpotent cone compatibility conditions

Insight into boundary components of period domains

Abstract

In this note, we provide an explicit computation of the weak fan associated with a two-parameter degeneration of K3 surfaces. This example serves as a concrete illustration of the general framework developed by Robles and Deng (2023) for the compactification of period maps via nilpotent orbits in the non-Hermitian case. We describe the associated nilpotent cones, examine their compatibility conditions, and construct the weak fan governing the degeneration behavior. This computation contributes to the broader understanding of boundary components of period domains and their relation to limiting mixed Hodge structures.