In this paper, we study the <i></i>-ordinary locus of a Shimura variety with parahoric level structure. Under the axioms in [12], we show that <i></i>-ordinary locus is a union of certain maximal EkedahlKottwitzOortRapoport strata introduced in [12] and we give criteria on the density of the <i></i>-ordinary locus.

No abstract uploaded!

Abstract Kontsevich and Soibelman defined DonaldsonThomas invariants of a 3d CalabiYau category equipped with a stability condition [41]. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Given a stability condition, the DT-transformation allows to recover the DT-invariants. Let S be an oriented surface with punctures and a finite number of special points on the boundary, considered modulo isotopy. It gives rise to a moduli space X P G L m, S, closely related to the moduli space of P G L m-local systems on S, which carries a canonical cluster Poisson variety structure [13]. For each puncture of S, there is a birational Weyl group action on the space X P G L m, S. We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution of X P G L m, S

In the present paper we extend the Riemann–Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and push-forwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann–Roch theorem for moment graphs. As an application, we obtain the Riemann–Roch type theorem for the equivariant K‑theory of some Kac–Moody flag varieties.

We study rationality properties of quadric surface bundles over the projective plane. We exhibit families of smooth projective complex fourfolds of this type over connected bases, containing both rational and non-rational fibers.