We prove the existence of integral canonical models of unitary Shimura varieties in arbitrary unramified mixed characteristic. Errata to [Va1] are also included.

Denote by ed_K(G) the essential dimension of G over K. If K is an algebraically closed field with char K = 0, Buhler and Reichstein determine explicitly all finite groups G with ed_K(G) = 1. We will prove a generalization of this theorem when K is an arbitrary field.

In this article, we prove a theorem comparing the dihedral angles of simplexes in the hyperbolic, spherical and Euclidean geometries.

The Gieseker-Uhlenbeck morphism from the moduli space of Gieseker semistable rank-2 sheaves over an algebraic surface to the Uhlenbeck compactification was constructed by Jun Li. We prove that if the anti-canonical divisor of the surface is effective and the first Chern class of the semistable sheaves is odd, then the Gieseker-Uhlenbeck morphism is crepant

This article provides a complete description of the differential Gerstenhaber algebras of all nilpotent complex structures on any real six-dimensional nilpotent algebra. As an application, we classify all pseudo-Kählerian complex structures on six-dimensional nilpotent algebras whose differential Gerstenhaber algebra is quasi-isomorphic to that of the symplectic structure. In a weak sense of mirror symmetry, this gives a classification of pseudo-Kähler structures on six-dimensional nilpotent algebras whose mirror images are themselves.

In this paper we prove an equivariant version of the McKay correspondence for the elliptic genus on open varieties with a torus action. As a consequence, we will prove the equivariant DMVV formula for the Hilbert scheme of points on C².