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Denote by edK(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 edK(G) = 1. We will prove a generalization of this theorem when K is an arbitrary ﬁeld.
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.