  We generalize a compactification technique due to C. Simpson in the context
of $\mathbb{G}_m$-actions over the ground field of complex numbers, to the case
of a universally Japanese base ring. We complement this generalized
compactification technique so that it can sometimes yield projectivity results
for these compactifications. We apply these projectivity results to the Hodge,
de Rham, and Dolbeault moduli spaces for curves, with special regards to ground
fields of positive characteristic.
