We present a paradifferential calculus adapted to the study of nonlinear partial differential equations in Gevrey classes. We give an application concerning Gevrey microregularity of the solutions of fully nonlinear equations at elliptic points.

We calculate the Morava $K$-theory of the spaces in the Omega spectra for $BP\langle q\rangle$. They fit into an exotic array of short and long exact sequences of Hopf algebras. We apply this to calculate the $p$-adically completed Brown-Peterson cohomology, as well as all of the intermediary cohomology theories, $E$, of these spaces. We give two descriptions of the answer, both of which turn out to be surprisingly nice. One part of our first description is just the image in the $E$ cohomology of the corresponding space in the Omega spectrum for $BP$, which is as big as it could possibly be and which we show how to calculate. The other part is just the $E$ cohomology of several copies of Eilenberg-MacLane spaces, something which is already known. Our second description is inductive and gives us a new way of looking at the Brown-Peterson cohomology of Eilenberg-Mac Lane spaces. The Brown-Comenetz dual of $BP\langle q\rangle$ shows up in our calculations and so we take up the study of this spectrum as well. It was already known that the Morava $K$-theory of the spaces in the Omega spectrum for the Brown-Comenetz dual of $BP\langle q\rangle$ made it look like a product of Eilenberg-Mac Lane spaces and we find, somewhat to our surprise, that the same is true for the BP cohomology. In order to state our answers we set up the foundations for the category of completed Hopf algebras.

In this paper, we will consider a generalization of Bogomolov’s inequality and Cornalba-Harris-Bost’s inequality to the case of semistable families of arithmetic varieties under the idea that geometric semistability implies a certain kind of arithmetic positivity. The first one is an arithmetic analogue of the relative Bogomolov’s inequality in [22]. We also establish the arithmetic Riemann-Roch formulae for stable curves over regular arithmetic varieties and generically finite morphisms of arithmetic varieties.