Showing papers in "Duke Mathematical Journal in 1981"

Strong rigidity of compact quotients of exceptional bounded symmetric domains

Abstract: In this paper we complete the proof of the Main Theorem by doing the cases of the two exceptional bounded symmetric domains. The proof for the cases of the four types of classical bounded symmetric domains consists of using harmonic maps, deriving a Bochner type formula, and a tedious part involving complicated linear algebra manipulations to verify that the curvature conditions from the Bochner type formula are satisfied by the classical bounded symmetric domains. In this paper we reformulate the curvature conditions so that they can be verified directly from the root system without using any explicit expression of the curvature tensor in terms of the coordinates of a realization of the bounded symmetric domain (see 2 and 3). We prove the strong rigidity for the two exceptional cases by using this reformulation (see 4 and 5). This reformulation of the curvature condition, when applied to the cases of the first three classical types, can yield a proof of the strong rigidity for these cases which is simpler than that given in [5] (see the Remark in 4). (The strong rigidity proof given in [5] for the fourth classical type is already very simple.) We now know an alternative method of deriving the Bochner type formula for harmonic maps between compact Khler manifolds. This alternative method does not use the special trick of considering the wedge product instead of the square norm of the differential of a harmonic map. It is more natural and we give this alternative derivation in this paper (see 1). At the end of this paper ([}6) we discuss a conjecture concerning the analyticity of harmonic maps with sufficiently high rank into compact quotients of bounded symmetric domains. We would like to thank the referee for suggesting the present simple proof of Proposition (4.1) which replaces our original lengthy case-by-case verification and for suggesting the alternative proof of Proposition (5.1). In the meantime,

On equations defining arithmetically Cohen-Macaulay schemes, II

Abstract: In this paper we continue the study of the simplest arithmetically Cohen-Macauly (CM for short) schemes (cf [ 111 and [ 12, I]) We describe the Hilbert function of an arbitrary CM scheme in P’ of given degree e lying on a hypersurface of given degree k and such that its general curvilinear section has minimal genus (Proposition 13) We present a geometric construction of all reduced irreducible codimension two CM varieties whose general curvilinear sections have minimal genus with respect to degree (Theorem 17) We also prove the following assertion essentially due to Severi: there exist smooth nonspecial CM curves of degree n and genus g in p’ iff r < y1= g + r d T(Y + 1)/2 (Proposition 16) Any such curve has minimal genus with respect to degree In the last section we consider d-dimensional CM schemes of degree e mvd = (“-j,Jed) in P” whose general curvilinear sections have minimal genus These are the most accessible CM schemes (cf [ 12, I]) We prove that the corresponding part of the Hilbert scheme is a connected smooth variety naturally birationally equivalent to nd+ ’ SymemVd( p’), provided v = d+ 2 (Theorem 26) This was known for (m, v, d) = (2,3, 1) (cf [9]) Our proof relies on an explicit description of the infinitesimal deformations of a thick subspace of arbitrary codimension (see 24) We work over an algebraically closed field of characteristic 0 By the genus of a CM curve we mean its arithmetic genus