Showing papers in "Mathematische Annalen in 1991"

The Moment Problem on Compact Semi-Algebraic Sets

Abstract: In this chapter we begin the study of the multidimensional moment problem. The passage to dimensions d ≥ 2 brings new difficulties and unexpected phenomena. In Sect. 3.2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions. It seems to be natural to look for similar characterizations in higher dimensions as well. This leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi-algebraic sets. In this chapter we treat this approach for basic closed compact semi-algebraic subsets of \(\mathbb{R}^{d}\). It turns out that for such sets there is a close interaction between the moment problem and Positivstellensatze for strictly positive polynomials.

Hyperkähler and quaternionic Kähler geometry

Abstract: A quaternion-Hermitian manifold, of dimension at least 12, with closed fundamental 4-form is shown to be quaternionic Kahler A similar result is proved for 8-manifolds HyperKahler metrics are constructed on the fundamental quaternionic line bundle (with the zero-section removed) of a quaternionic Kahler manifold (indefinite if the scalar curvature is negative) This construction is compatible with the quaternionic Kahler and hyperKahier quotient constructions and allows quaternionic Kahler geometry to be subsumed into the theory of hyperKahler manifolds It is shown that the hyperKahler metrics that arise admit a certain type of SU (2)- action, possess functions which are Kahler potentials for each of the complex structures simultaneously and determine quaternionic Kahler structures via a variant of the moment map construction Quaternionic Kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperKahler metrics with circle actions on complex line bundles over Kahler-Einstein (complex) contact manifolds Nilpotent orbits in a complex semi-simple Lie algebra, with the hyperKahler metrics defined by Kronheimer, are shown to give rise to quaternionic Kahler metrics and various examples of these metrics are identified It is shown that any quaternionic Kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds The twistor space structure of the projectivised nilpotent orbits is studied

The Capelli identity, the double commutant theorem, and multiplicity-free actions

Abstract: 0. The Capelli identity [Cal-3; W, p. 39] is one of the most celebrated and useful formulas of classical invariant theory [W; D; CL; Z]. The double commutant theorem [W, p. 91] is likewise a basic result in the general theory of associative algebras. Both play key roles in Weyl's book: The classical groups. The main purpose of this paper is to demonstrate a close connection between the two, in the context of multiplicity-free actions EKJ of groups on vector spaces. The focus of our discussion will be the structure of the differential operators which commute with a multiplicity-free action. Applications include a new derivation of some formulas of Shimura [Shl ] and Rubenthaler and Schiffmann [RS] for b-functions associated to Hermitian symmetric spaces, and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin. We also give a detailed discussion of certain aspects of multiplicity-free representations. Here is an overview of the contents of the paper. In Sect. 1 we review the classical Capelli identities. We observe how their existence is predicted by a double commutant result (1.6) and (1.8), and we show how they can be used to compute b-functions. We remark that Capelli understood that his operators generate the center of the algebra generated by the polarization operators (see [B1, p. 77]). Also Capelli's motivation in introducing his operators was to "explain" a formula of Cayley essentially the computation of a b-function. Capelli's point of view was remarkably modern and structuralist, in certain ways more modern even than that of Weyl. Sections 2 through 9 provide a conceptual context for understanding the Capelli identities as a feature of multiplicity-free actions. They contain a general discussion of the structure of ~ G , the algebra of polynomial coefficient differential operators which commute with a given group G of linear transformations. The results of the discussion are summarized in Theorem 9.1, which says in particular that the polynomial coefficient differential operators commuting with a

The inverse Galois problem and rational points on moduli spaces

Abstract: We reduce the regular version of the Inverse Galois Problem for any finite group G to finding one rational point on an infinite sequence of algebraic varieties. As a consequence, any finite group G is the Galois group of an extension L/P (x) with L regular over any PAC field P of characteristic zero. A special case of this implies that G is a Galois group over Fp(x) for almost all primes p.

Projective manifolds whose tangent bundles are numerically effective.

Abstract: In 1979, Mori [Mo] proved the so-called Hartshorne-Frankel conjecture: Every projective n-dimensional manifold with ample tangent bundle is isomorphic to the complex projective space P,. A differential-geometric analogon assuming the existence of a K/ihler metric on X with positive holomorphic bisectional curvature is independently due to Siu-Yau [SY]. Thus it seems natural to classify projective manifolds X whose tangent bundle Tx satisfy a degenerate condition of ampleness: numerical effectivity (abbreviated by "nef'). This means that the tautological quotient line bundle d~(1) on F(Tx) is numerically effective, i.e. C_>_0

Uniformly distributed orbits of certain flows on homogeneous spaces

Abstract: is non-compact. For a co-compact lattice, this result was strengthened by F/irstenberg [1 lJ proving that every orbit is uniformly distributed with respect to a G-invariant measure. For non-uniform lattices in SL(2, R), using a classification of invariant measures obtained by Dani in [21, Dani and Smillie [31 proved that every non-periodic orbit is uniformly distributed. There are also various results obtained on orbit closures and invariant measures etc. of larger subgroups consisting of unipotent elements, especially the horospherical subgroups. Recently, there was a spurt in the area initiated by Margulis' proof (cf. [151, see also [7]) of Oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows. The reader is referred to the survey articles by Dani [41 and Margulis [14J for various related developments. We now note some conjectures expected to hold for orbits of a unipotent flow, namely the U-action on