Avoiding variational Hodge theory in the non-compact locally symmetric case

Determine whether, in the non-compact case of locally symmetric varieties, the nefness of the tautological line bundle on the projectivized logarithmic differential forms and the Chern–Weil-theoretic step in the proof of the equality between the augmented base locus and the higher-degree characteristic subvariety can be established without using variational Hodge theory.

Background

The paper studies positivity properties of logarithmic differential-form bundles on smooth projective compactifications of locally symmetric varieties. In the non-compact setting, the proof that the augmented base locus equals the higher-degree characteristic subvariety uses variational Hodge theory in two specific ways: to prove nefness of the relevant tautological line bundle and to justify a Chern–Weil identity for singular metrics in the proof of Theorem \ref{thm:lsav}. The authors explicitly identify avoiding variational Hodge theory at both points as unresolved.