Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userPerfectoid spaces arising from arithmetic jet spaces
102
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $delta$-morphisms appearing in the theory such as the $delta$-characters of elliptic curves and the $delta$-period map on modular curves.
rate research
Read More
We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications we give a new interpretation of arithmetic Laplacians and we discuss the de Rham cohomology of some specific arithmetic jet spaces.
We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topologically tame, i.e. homeomorphic to the interior of a compact manifold with boundary. We also construct domains of discontinuity (not necessarily with a compact quotient) in a much more general setting.
We link small modifications of projective varieties with a ${mathbb C}^*$-action to their GIT quotients. Namely, using flips with centers in closures of Bia{l}ynicki-Birula cells, we produce a system of birational equivariant modifications of the original variety, which includes those on which a quotient map extends from a set of semistable points to a regular morphism. The structure of the modifications is completely described for the blowup along the sink and the source of smooth varieties with Picard number one with a ${mathbb C}^*$-action which has no finite isotropy for any point. Examples can be constructed upon homogeneous varieties with a ${mathbb C}^*$-action associated to short grading of their Lie algebras.
We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a well-pointed topological monoid of the homotopy type of a CW complex and $G$ a well-pointed grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G$. This fact has several applications. As the first application, we show that the connecting map $Grightarrowmathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $mathrm{Map}_0(B_nG,BG)rightarrowmathrm{Map}(B_nG,BG)rightarrow BG$ is delooped. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups, $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon and homotopy pullback of $A_n$-maps. In particular, we show that the $T_k^f$-space and the $C_k^f$-space are exactly the same concept and give some new examples of $T_k^f$-spaces.
The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
