On type II degenerations of hyperk\"ahler manifolds

TL;DR

This paper proves Nagai's conjecture for type II degenerations of hyperk"ahler manifolds and extends the proof to arbitrary degree under additional assumptions, completing the conjecture's verification for all degeneration types.

Contribution

It provides a simple proof for Nagai's conjecture in type II degenerations and extends the result to arbitrary degree with extra assumptions, unifying the understanding across all degeneration types.

Findings

Nagai's conjecture is proven for type II degenerations.

Under extra assumptions, the conjecture is proved in all degrees.

The approach relates to recent work on good degenerations.

Abstract

We give a simple argument to prove Nagai's conjecture for type II degenerations of compact hyperk\"ahler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai's conjecture in general, as it was proved already for type I degenerations by Koll\'ar, Laza, Sacc\`a, and Voisin and independently by Soldatenkov, while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder proving similar results for the restrictive class of good degenerations.