Showing papers in "Annals of Mathematics in 1972"

On the first variation of a varifold

Abstract: Suppose M is a smooth m dimensional Riemannian manifold and k is a positive integer not exceeding m. Our purpose is to study the first variation of the k dimensional area integrand in M. Our main result is a regularity theorem for weakly defined k dimensional surfaces in M whose first variation of area is summable to a power greater than k. A natural domain for any k dimensional parametric integral in M, among which the simplest is the k dimensional area integral, is the space of k dimensional varifolds in M introduced by Almgren in [AF 1]. Such a varifold is defined to be a Radon measure on the bundle over M whose fiber at each point p of M is the Grassmann manifold of k dimensional linear subspaces of the tangent space to M at p. If V is a varifold in M, let I I V I I be the Radon measure on M obtained from V by ignoring the fiber variables. Naturally injected in the space of k dimensional varifolds in M is the set of k dimensional rectifiable subsets of M, which includes the set of k dimensional submanifolds of M as well as more general k dimensional surfaces in M. A k dimensional varifold in M is said to be rectifiable (integral) if it can be strongly approximated by a positive real (integral) linear combination of varifolds corresponding to continuously differentiable k dimensional submanifolds of M. To any classical k dimensional geometric object in M there corresponds a k dimensional integral varifold in M. If N is a smooth Riemannian manifold and F: M-e N is smooth, then F induces in a natural way a strongly continuous mapping F# of the k dimen-

The intermediate Jacobian of the cubic threefold

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Quotient Spaces Modulo Reductive Algebraic Groups

Abstract: This conjecture is known to be true when the base field is of characteristic zero. In fact we can then find a linear F and this is an immediate consequence of the complete reducibility of finite dimensional rational G-modules (theorem of H. Weyl). In this paper we present an attempt at the solution of the above conjecture. We have partial success. To state our main result, one first observes (cf. Remark 1.1) that (A) is equivalent to the following stronger assertion.

A Defect Relation for Equidimensional Holomorphic Mappings Between Algebraic Varieties

Abstract: 0. Introduction 1. Notations, terminology, and sign conventions (a) Line bundles and Chern classes (b) Currents and forms in C0 2. Construction of a volume form 3. A second main theorem for non-degenerate maps 4. The defect relation (preliminary form) 5. The first main theorem and defect relation 6. Variants and applications (a) Schottky-Landau theorems (b) Remarks on the case ci(L) + ci(Kv) ? 0 (c) Holomorphic mappings with growth conditions Bibliography

Orbital Integrals in Reductive Groups

Abstract: Let G be a connected reductive linear algebraic group defined over a (nondiscrete) locally compact field k of characteristic zero, and G be its group of k-rational points. If O(x) = {yxy': y e G} is the orbit of x, then it is known2 that O(x) is locally compact (in the Hausdorff topology of G) and homeomorphic to G/G,. The isotropy subgroup Gx is known to be unimodular ([2], p. 235) and so the space G/GX carries a G-invariant Radon measure dy*(y* =yGx). A question that seems to be of some importance in harmonic analysis is, whether this measure, transplanted to the orbit O(x), is a Radon measure in G, i.e., is finite for compact subsets of G. If this is the case then the integral

On algebraically closed groups

Abstract: In this paper we use model-theoretic techniques to analyze the structure of algebraically closed groups. The notion of algebraically closed group first appeared in W. R. Scott's paper [24] in 1951. The intention must surely have been to provide for grouptheory an analogue of that central notion of field theory, the notion of algebraically closed field. A group G is said to be algebraically closed if every consistent finite system of equations, with parameters in G, is solvable in G. A system of equations is said to be consistent over G if it has a solution in a group extending G. A group G is said to be existentially closed if every consistent finite system of equations and inequations, with parameters in G, is solvable in G.

Implicit Function Theorems and Isometric Embeddings

Abstract: This is a somewhat rewritten and expanded version of a Ph. D. dissertation at the Courant Institute. The author wishes to express his gratitude to his thesis advisor, Professor Jurgen Moser, for his assistance and encouragement. This paper was prepared for publication while the author was supported by a grant from the National Science Foundation (GP-7952X2) made to the Institute for Advanced Study.

The Adams-Novikov Spectral Sequence for the Spheres

Abstract: The Adams spectral sequence for the stable homotopy of the spheres has been extensively studied since its introduction in 1958 [1]. Variants of this sequence using extraordinary cohomology theories have exciting possibilities, but are not well understood. In this paper we consider the MU-cobordism version first studied by Novikov [13]. We find that for odd primes this sequence is drastically simpler than the classical Adams spectral sequence; while it does not collapse, as Novikov first thought it would, we have been able to find only one family of nonzero differentials. For the prime 2 the sequence is more complicated and harder to compute in low dimensions, but still displays several interesting new patterns (see Tables 2 and 3). Our results also comprise vanishing lines for the E2-term of the MUspectral sequence and periodicity phenomena near this edge. We have not yet determined the full extent of this periodicity, but in a range of dimensions the diagonal "towers" parallel to the line t = 2s eventually stabilize for p = 2; for odd primes these leaning towers form jagged lines at smaller angles. We also find recurring families of elements in E,' *. For p = 2 one of these families corresponds to the classical Arf-invariant elements h2 [8]. Our basic method is simply construction of economical resolutions over the algebra of operations BP*(BP), which is the analogue of the classical Steenrod algebra in the cohomology theory given by the p-primary BrownPeterson spectrum pBP. Using Quillen's results [14] about the structure of BP*(BP), we calculate E2 in the range t - s ? 17 for p = 2 and t - s < 45 for p = 3, as well as the multiplicative structure in a somewhat smaller range. We can then independently determine all differentials in this range, with the exception of the 3-primary differential needed to kill the ephemeral element a/. In this way we can recover the corresponding 2-primary and 3-primary stable stems up to group extension. In the course of the work we

A Theorem on Semi-Simple P-adic Groups

Abstract: Let k be a non-archimedean local field and G a connected, semi-simple algebraic group defined over k. If G = G(k) denotes the group of k-rational points of G, then G, with its natural topology, is locally compact. Let G' and V denote respectively the sets of regular and unipotent elements of G, and let Cc(G) denote the space of locally constant, complex valued functions on G having compact support. For x E G' U V, let G(x) denote the conjugacy class of G containing x. G(x) carries an essentially unique G-invariant measure Pa (? 1.2). Fix a normalization of pa. For f E Cc(G) and x as above, let

On the Values of Zeta and L-Functions: I

Abstract: In this paper we begin a study of the relationship between values of zeta and L-functions on the one hand, and Euler characteristics of sheaves for the etale topology, on the other hand. We start with the following conjecture: Let F be a totally real number field, p a prime, and n a negative odd integer. Let A be the ring of integers of F, S the set of primes of F over p, X = SpecA, and X, = X S. Let j be the natural inclusion of SpecF in X, Let F be the algebraic closure of F, and GF the Galois group of F over F. Let W be the GF-module consisting of all p-power roots of unity in F, let T(W) be the Tate-module of W, and let W(m' be the GF-module W ? T(W)xm('". Then W(m' may be identified with a sheaf for the etale topology on SpecF, and we may take the direct image j* W(m' on X, If M is a finite group, let #(M) denote the number of elements of M.

On K 2 and Some Classical Conjectures in Algebraic Number Theory

Abstract: Let k be any field, and let kx be the multiplicative group of k. One of the several equivalent definitions of K2k is that K2k = (kx ?z kx)/J, where J is the subgroup of the tensor product generated by all elements a 0 b with a + b = 1. Assume now that k is a finite algebraic extension of the rational field Q. An important theorem of Garland [2] implies that, in this case, K2k is a torsion group. Garland's proof, however, does not give precise information about the structure of K2k, nor does it show how this group is related to more classical invariants of k. The aim of the present paper has been to establish more precise results. Motivated by recent work of Tate [13] on the function field analogue, we show that, for each prime number 1, the i-primary subgroup of K2k is isomorphic to a certain group that arises naturally in Iwasawa's [6] theory of Z,-extensions of number fields. The precise statement of our result is given later in the paper (see Theorems 4 and 7). We then discuss various consequences of this description of K2k. In particular, we show that a classical conjecture in the theory of cyclotomic fields is equivalent to a conjecture about K2k (see Theorem 7), which, unfortunately, also seems very difficult to settle. Further, it will be apparent that our results lead, in a natural way, to a module which may possibly be an analogue for k of the Jacobian variety of a curve of genus > 1 defined over a finite field. The arguments we use rely heavily on Iwasawa's [6] theory of Z,-extensions of number fields and on cohomological methods due to Tate [13], which, in turn, were inspired by conjectures of Lichtenbaum [7]. In addition, Tate's work, and therefore indirectly ours also, depends essentially on the work of Bass [1], Garland [2], Matsumoto [8], Moore [9], and others. Finally, I wish to heartily thank J. Tate for many stimulating suggestions on the questions discussed in this paper.