Showing papers in "Annals of Mathematics in 1980"

The torsion of the group of $0$-cycles modulo rational equivalence

Abstract: In this work we continue the study of rational equivalence of O-cycles on nonsingular projective varieties over an algebraically closed field k, which we began in [13], [14]. The main result of this work is the calculation of the group of finite order O-cycles modulo rational equivalence. Except possibly for p-torsion if char k = p > 0, this group is equal to the group of points of finite order on the Albanese variety. We shall always use the following notation (we do not suppose here that X is nonsingular): Z0(X)-the group of O-cycles on the projective variety X; ZO(X) c Z0(X) the subgroup of cycles of degree 0; Z+(X) c Z0(X) the semigroup of effective cycles; S"X the n-fold symmetric product of X, SzX= Xn/=n; S"(X/S)-the n-fold relative symmetric product of X ->

Discrete series for semisimple symmetric spaces

Abstract: We give a sufficient condition for the existence of minimal closed G-invariant subspaces of L2(G/H) for a semisimple symmetric space G/H. As a semisimple Lie group with finite center may always be considered as a symmetric space, this provides, in particular, a new and elementary proof of Harish-Chandra's result that G has a discrete series if rank (G) = rank (K), where K is a maximal compact subgroup. Let G be a connected noncompact semisimple Lie group, let z be an involution on G, and let H be the connected component of the fixed-point group Gr containing the identity. Then G/H is a semisimple symmetric space, and the group G acts by left translation on C*(G/H) and L2(G/H). In the introduction we will, for simplicity, assume that G has a finite center. By the discrete series for G/H we shall mean the set of equivalence classes of the representations of G on minimal closed invariant subspaces of L2(G/H). Let a be a Cartan involution commuting with z. The fixed-point group K for a is a maximal compact subgroup. Our main result is THEOREM 1.1. The discrete series for G/H is nonempty and infinite if

Automorphisms of trivalent graphs

Abstract: In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a vertex is transitive on adjacent vertices, then its order divides 3 . 24. As observed by Sims [5], the hypothesis is equivalent to the following group-theoretic conditions: a) G is a finite group generated by a pair of subgroups {P1, Pd, b) i Pi: Pi n P2 1= 3 for i = 1, 2, c) no non-trivial normal subgroup of G is contained in P1 A, d) P1 and P2 are G-conjugate. What happens if we drop condition d) or, what is essentially the same thing, replace "vertex transitive" by "edge transitive"? This question is primarily motivated by the examples afforded by the rank 2 BN pairs over GF(2). In this case, the trivalent graph mentioned above is the so-called "building" associated to the BN pair [6]. In this paper, we classify all pairs of subgroups (P1, P2) for which hypotheses a), b) and c) are satisfied. There are precisely fifteen such pairs, and in particular, we find that P1 n P2 has order dividing 27. In order to describe the results more completely, let us define an amalgam to be a pair of group monomorphisms (5, 02) with the same domain: P, 1 B P2. We will say that (01, 02) is finite if both co-domains P1, P2 are finite. In this case, we define the index of the amalgam to be the pair of indices (I Pl: im s1 I, P2: im 02 1). By a completion of the amalgam we mean a pair of homomorphisms (*1, '2) to some group G making the obvious diagram commute, i.e., such that 01* = 102'2 (in right-hand notation). By abuse of notation, we may say that G is a completion of the amalgam. Of course, we always have the trivial completion g1 = g2 = 1We also always have the universal completion, usually known as the amalgamated product, from

On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface

Abstract: kbstract In this paper, we look for periodic solutions, with prescribed energy h C R, of Hamilton's equations: (H) a H (x, p), p aH (x, p). ap Ax It is assumed that the Hamiltonian H is convex on R" x R", and that the origin (0, 0) is an isolated equilibrium. It is also assumed that some ball B around the origin can be found such that the energy surface H'(h) lies outside B but inside v'2 B. Under these assumptions, we prove that there are at least n distinct periodic orbits of the Hamiltonian flow (H) with energy level h.

Strong rigidity for ergodic actions of semisimple Lie groups

Abstract: The aim of this paper is to prove an analogue of the strong rigidity theorems for lattices in semisimple Lie groups of Mostow and Margulis in the context of ergodic actions of semisimple Lie groups and ergodic foliations by symmetric spaces. If G is a locally compact group and S is an ergodic G-space with finite invariant measure, one can attempt to study the action by means of the equivalence relation on S which it defines. Two actions (of possibly different groups) are called orbit equivalent if they define isomorphic equivalence relations. Of course for two actions of the same group, conjugacy of the actions (or more generally, automorphic conjugacy, i.e., conjugacy modulo an automorphism of the group) implies orbit equivalence. However in general, orbit equivalence is, to a surprising extent, a far weaker notion. This

Moduli of abelian varieties

Abstract: On extra components in the functorial compactification of Ag.- On Mumford's uniformization and Neron models of Jacobians of semistable curves over complete rings.- Torelli theorem via Fourier-Mukai transform.- On the Andre-Oort conjecture for Hilbert modular surfaces.- Toroidal resolutions for some matrix singularities.- Formal Brauer groups and moduli of abelian surfaces.- Isogeny classes of abelian varieties with no principal polarizations.- Igusa's modular form and the classification of Siegel modular threefolds.- Mirror symmetry and quantization of abelian varieties.- Group schemes with additional structures and Weyl group cosets.- Moduli space of elliptic curves with Heisenberg level structure.- Singularities of the height strata in the moduli of K3 surfaces.- A stratification of a moduli space of abelian varieties.- Newton polygon strata in the moduli space of abelian varieties.- The dimension of Oort strata of Shimura varieties of PEL-type.- Hyperelliptic Jacobians and modular representations.- Windows for displays of p-divisible groups.

A complete minimal surface in R3 between two parallel planes

Abstract: E. Calabi has asked if it is possible to have a complete minimal surface in RX entirely contained in a half-space. We answer his question in the affirmative. In fact, we prove considerably more, namely, we show how to produce complete minimal surfaces contained in slabs of R3. The proof was motivated by Remmert's ingenious idea of using Runge's theorem to exhibit proper analytic embeddings of the unit disc D c( C in C3, as explained in [11,

A compact Kähler surface of negative curvature not covered by the ball

Abstract: The uniformization theorem for Riemann surfaces says that a simply connected Riemann surface must be the Riemann sphere, the whole complex plane, or the open unit disc. In the higher dimensional case there is no such simple trichotomy, because generic slight perturbations of the ball give rise to complex manifolds no two of which are biholomorphic [1]. However, it is conjectured that with reasonable curvature assumptions a similar trichotomy exists in the higher dimensional case. Corresponding to the case of the Riemann sphere, one has the Frankel conjecture that a compact Kihler manifold of positive sectional curvature must be biholomorphic to the complex projective space. This conjecture was proved in the dimension 2 case by Andreotti-Frankel [2] and in the dimension 3 case by Mabuchi [5]. Very recently the general case was proved independently by Mori [6] using algebraic geometry of positive characteristic and by Siu-Yau [11] using the complex-analyticity of harmonic maps. (Mori's result is stronger than the result of Siu-Yau. Mori's result assumes only that the manifold has ample tangent bundle, whereas the result of Siu-Yau assumes that the manifold has positive holomorphic bisectional curvature.) Corresponding to the case of the complex plane, one has the conjecture that a noncompact complete Kahler manifold of positive sectional curvature must be biholomorphic to some C". Or, more generally, a noncompact simply connected complete Kahler manifold with sectional curvature K > -A/r'+ or even with the weaker assumption K ? -lk(r) with k(r) > 0 and

Twisted homological stability for general linear groups

Abstract: Suppose that R is a principal ideal domain and that, for each n, p" is a module over the general linear group GL, (R). The major purpose of this paper is to show that for very general choices of {p,} the homology groups Hk(GL, (R), pa) stabilize with respect to n, i.e., for each fixed k assume a constant value as n becomes large. A secondary purpose is to show that this stability phenomenon has interesting consequences for Waldhausen's "algebraic K-theory of topological spaces" and ultimately for geometric topology. The main stability theorem (Theorem 2.2) is explained in detail at the beginning of Section 2; it has an inductive statement too elaborate to summarize here. A sample application of 2.2 is the following. Let Ab be the category of abelian groups, and let Ad, (R) be the abelian group of n x n matrices over R, considered as a GL,, (R)-module by conjugation. Note that if T: Ab -> Ab is a functor, there is a natural action of GLUM (R) on T(AdJ(R)). 1.1. PROPOSITION. If T: Ab -* Ab is a functor of finite degree (see ? 3 and [31) then the homology groups Hk(GL, (R), T(Adn(R))) stabilize with respect to n.

Some general algorithms. I: Arithmetic groups

Abstract: An algebraic matrix group of degree n, defined over Q, is a subgroup of GLn(C) which is the set of common zeros in GLn(C) of finitely many polynomials, with rational coefficients, in the n2 matrix entries. We shall also call such a group a Q-group, of degree n. We say that the Q-group is given explicitly if these polynomials are explicitly given. If G is such a group and R is a subring of C, put

Le theoreme fondamental de la K-theorie hermitienne

Abstract: Cet article est la suite naturelle de [9] dont nous reprenons les notations et la terminologie. Rappelons en d'abord les definitions essentielles. Soit A un anneau "hermitien", c'est-a-dire un anneau unitaire muni d'une antiinvolution notee z v-z et soit s un element du centre de A tel que se = 1. Pour tout entier r, on definit le "groupe orthogonal" O7,,(A) comme le groupe forme des matrices 2r x 2r dont l'ecriture par blocs de matrices r x r est

On the fundamental group of the complement of a node curve

Abstract: where dl, * dt are the degrees of the irreducible components of C, would follow if one knew that the intersection of all subgroups of finite index in r(P 2C) were trivial. Zariski's original goal of determining all finite coverings of P2 C follows from the theorem stated here: the profinite completions of the displayed groups are the same, and similarly for the tame fundamental group and the prime-to-p completion in characteristic p (cf. [11, [31). The theorem also implies the same conclusion for a hypersurface in PI whose generic plane sections are node curves. Zariski's proof relied on the assertion of Enriques and Severi that any node curve can be degenerated to lines in general position. This assertion remains unproved, however. Abhyankar has proved many special cases of this result. A recent strengthening of Bertini's connectedness theorem, proved jointly with J. Hansen [21, yields a lemma which completes Abhyankar's beautiful argument. Let f: X -+ Y be a finite morphism of projective surfaces over an algebraically closed field, with Y non-singular and X normal. Assume f is a

The arithmetic of certain zeta functions and automorphic forms on orthogonal groups

Abstract: Q(z) = ,Q?(&2 of an integral weight > 0; k is a positive integer; * is an embedding of K into C; r is an element of K0 such that Id2 is its only positive conjugate; 4D-b(* + Ap) + ,cpgp, where b and c, are non-negative integers, p is the complex conjugation, and {qp} is the set of all embeddings of K into C other than * and Ap. It will be shown that the series is convergent for sufficiently large Re (s) and can be continued to a meromorphic function on the whole plane. Now one of our main results will assert that the values ?D(4a) for certain integers , are algebraic numbers times wkPK(*i *)2k I, P (Tv, ,)-2c. where PK(q', A) is a complex number depending only on q and A. We shall actually prove such an algebraicity for the series defined in a similar way with a Hilbert modular form in place of Q (Theorem 9.2). If [K: QI = 2, 9) is essentially of the type

Propagation of singularities for semilinear hyperbolic equations in one space variable

Abstract: In this paper we investigate the propagation of singularities for general semi-linear hyperbolic equations in two variables. In order to make clear the motivation for this work, we give a little history. In [91, Reed showed that in one space dimension solutions ju = f (u, Du) are C~o except at points Kx, t> where the backward characteristics through the point intersect singular points of the initial data at t = 0; thus, as in the linear case, the singularities lie on rays issuing from singularities at t = 0. The counterexample of Lascar [61 and the theorems of Rauch [71 show that when the number of space dimensions is greater than one, the solutions u of Elu = f (u) may have other singularities too. Roughly what is true is the following. Suppose that the solution u is in Hk and that the backward characteristics through Kx, t> do not intersect the singular support of the initial data. Then u will be in HP at Kx, t> where p > k depends on k and the number of space dimensions. Such results are valid for higher order equations and more strongly nonlinear equations. In this paper we reexamine the result of Reed to understand what is special about one space dimension. First, we prove that the C~o result holds in one space dimension for all semilinear hyperbolic equations of second order. Second, we construct a third order semi-linear counterexample in which the propagation of singularities is not that predicted by the highest order linear part. A new singularity is created when two singularities cross and the new singularity propagates in the direction of the third characteristic. Our example makes it clear that the same phenomenon will also occur for higher order equations. Thus, we have the following general picture: the only semi-linear hyperbolic equations of order >2 for which the propagation of singularities is like the linear case are second order