Extend the generalized Kummer height computation to deformation types

Determine the quasi-$F$-split heights of generalized Kummer varieties throughout their deformation types, extending the computation for generalized Kummer varieties themselves beyond the cases directly treated by the existing methods.

Background

The paper discusses a computation of quasi-$F$-split heights for generalized Kummer varieties carried out using methods from Yobuko's work. It explicitly states that this computation was not extended to the deformation types of generalized Kummer varieties, and that the computation is not included in the paper.

This is an unresolved extension problem concerning whether the height dichotomy established for the standard generalized Kummer varieties can be determined uniformly across their deformation types.