Projective completion of moduli of \(t\)-connections on curves in positive and mixed characteristic (Q2131774)

Let \(\pi: C\to B\) be a projective and smooth family of geometrically connected curves, where \(B\) is a scheme of finite type over a Nagata ring \(J\). Then there exists a quasi-projective coarse moduli space \(M_{\mathrm{Hod}} (C/B)/B\) of slope semistable \(t\)-connections of fixed rank \(r\) and degree \(d\) on \(C/B\). This moduli space comes with a morphism \(M_{\mathrm{Hod}} (C/B)\to {\mathbb A} ^1_B\) sending a \(t\)-connection to \(t\). One of the main aims of the paper is to construct a natural \({\mathbb G} _m\)-equivariant projective completion \(\overline{M_{\mathrm{Hod}} (C/B)}\to {\mathbb A} ^1_B\) of this morphism. If \(J\) is a field of positive characteristic \(p\) they also extend the so called Hodge-Hitchin morphism \(M_{\mathrm{Hod}} (C/B)\to A(C/B, \omega_{C/B}^p)\times {\mathbb A} ^1_B\), which is roughly speaking a deformation of the Hitchin morphism in characteristic \(p\). This implies the properness of the Hodge-Hitchin morphism that generalizes an earlier result of [\textit{Y. Laszlo} and \textit{C. Pauly}, Int. Math. Res. Not. 2001, No. 3, 129--143 (2001; Zbl 0983.14004)], who dealt with the nilpotent \(t\)-connections. Another approach to properness of the Hodge-Hitchin morphism can be found in a recent paper of the reviewer (see [\textit{A. Langer}, ``Moduli spaces of semistable modules over Lie algebroids'', Preprint, \url{arXiv:2107.03128}]), where one can find a more general result working also in higher dimensions over any noetherian base \(B\). Tha authors approach to the above theorems is through a generalization of compactification techniques for quotients by a \({\mathbb G} _m\)-action. Roughly speaking, the authors consider a scheme \(X\) with a \({\mathbb G} _m\)-action and an invariant projective morphism \(X\to S\). Then they try to find a \({\mathbb G} _m\)-invariant open subset \(U\subset X\) with a proper (or projective) quotient \(U/{\mathbb G} _m \to S\). Similar problems where first considered (also in an arbitrary characteristic) by \textit{A. Bialynicki-Birula} and \textit{J. Swiecicka} [Lect. Notes Math. 956, 10--22 (1982; Zbl 0493.14027)] but only when \(S\) is a point. Later \textit{C. Simpson} [Proc. Symp. Pure Math. 62, 217--281 (1997; Zbl 0914.14003)] considered the relative case, but only over complex numbers. Reviewer's remark: The authors work with noetherian schemes of finite type over a universally Japanese ring \(J\). However, the notion of ``universally Japanese'' that they use seems to be the same as in \textit{C. S. Seshadri}'s paper [Adv. Math. 26, 225--274 (1977; Zbl 0371.14009)], where the author also assumes that \(J\) is noetherian (this is certainly the case when they quote the reviewer's results). Nowadays, such rings are known as a ``Nagata ring'' (see [Stacks project, Tag 032R]) and a scheme of finite type over such a ring is automatically noetherian.