Showing papers in "Annals of Mathematics in 1985"

The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin's entropy formula

Abstract: Soit M une variete de Riemann compacte, soit f:M→M un diffeomorphisme, et soit m une mesure de probabilite de Borel f-invariante sur M. On identifie les mesures pour lesquelles l'inegalite de Margulis et Ruelle qui relie l'entropie aux exposants de Lyapunov atteint l'egalite. On demontre une formule valable pour toutes les mesures invariantes

Stability of the homology of the mapping class groups of orientable surfaces

Abstract: The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the identity on dF and fix the s punctures. When r = 0 and 2g + s > 3, an important feature of F is that it acts properly discontinuously on the Teichmiller space US; the quotient is the moduli space .#f of isometry classes of complete hyperbolic metrics of finite area on F. Since US is homeomorphic to Euclidean space and F is virtually torsion-free, we have

Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains

Abstract: On demontre un resultat dynamique sur l'iteration d'une application analytique complexe sur la sphere de Riemann

Unique continuation and absence of positive eigenvalues for Schrodinger operators

Abstract: On considere l'operateur de Laplace Δ sur R n et une fonction V(x) sur un sous-ensemble connexe, ouvert Ω de R n . On montre que si n≥3, une propriete de prolongement unique est vraie pour V∈L loc n/2 (R n ) dans l'espace de Sobolev H loc 2,q (R n ) avec q=2n/(n+2)

The metric entropy of diffeomorphisms Part II: Relations between entropy, exponents and dimension

Abstract: On considere f: (M,m)→(M,m) un C 2 -diffeomorphisme f d'une variete de Riemann compacte M preservant une mesure de probabilite de Borel m. Soit hm(f) l'entropie metrique de f et λ 1 (x)>...>λ r(x) (x) les exposants de Lyapunov distincts en x. On etablit une formule qui relie h m (f) et les λ r(x) (x)

Frobenius splitting and cohomology vanishing for Schubert varieties

Abstract: Let k be an algebraically closed field of characteristic p > 0 and X be a projective variety over k. We then have the absolute Frobenius F: X -> X and an injection Ox -* F *6x given by f fP, f e O9x. If this makes Ox a direct summand in F* Ox (as an (x-module) we call X a Frobenius split variety. For such a variety the vanishing theorem for ample line bundles follows trivially from Serre's vanishing theorem. For, tensoring Ox -F*Ox by an ample line bundle L and noting that L ? F * (x = F * F *L = F * LP (projection formula) we get that the map in cohomology H'(X, L) -+ H'(X, F*LP) = H'(X, LP) is an injection. Iterating this we see that H'(X, L) injects into H'(X, LP) for every P. But, for large v the latter is zero! Thus Frobenius split varieties have quite pleasant properties. It also turns out that using duality for the Frobenius morphism, or equivalently, the Cartier operator, one can give a manageable criterion for Frobenius splitting. The point here is the local nature of duality. The relevance of the compatibility of local and global duality was suggested to us by Grothendieck's proof of H'(X, Y) = 0 for a noncomplete variety X of dimension n ([4], Theorem 6.9) and by Kempf's paper [10]. By this criterion and the Bott-Samelson-Demazure desingularisation of Schubert varieties, it follows very easily that Schubert varieties in characteristic p are Frobenius split. The following vanishing theorem is then an immediate consequence. Let G be a reductive group over the field k (of arbitrary characteristic, zero or positive). Let Q be a parabolic subgroup and X c G/Q a Schubert variety. Let L be an ample line bundle on G/Q. Then HI(X, L) = 0 for i > 0 and the restriction map H0(G/Q, L) -+ H0(X, L) is surjective. If char k > 0 this is a consequence of the compatible Frobenius splitting of X in G/Q and the char k = 0 case is handled by semicontinuity. The above result for special Schubert varieties has been proved by SeshadriMusili-Lakshmibai [12], [13] and by Kempf [9] by using characteristic free methods. For X = G/Q, Andersen [1] and Haboush [5] have given simple proofs using characteristic p methods.

On Iterations of 1 - ax 2 on (- 1,1)

Abstract: On etudie des iterations de F(x';a)=1-ax u 2, -1

Positive harmonic functions on complete manifolds of negative curvature

Abstract: This paper studies the interaction between the geometry of complete Riemannian manifolds of negative curvature and some aspects of function theory on these spaces. The study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction; we recall some aspects of this below. There is a well-known representation of positive harmonic functions on the unit disc U. due to Herglotz [13], in terms of positive Borel measures tt on the circle S1:

Partial desingularisations of quotients of nonsingular varieties and their Betti numbers

Abstract: When a reductive group G acts linearly on a nonsingular complex projective variety X one can define a projective "quotient" variety X//G using Mumford's geometric invariant theory. If the condition that every semistable point be (properly) stable is satisfied, this quotient is the ordinary topological quotient of an open subset Xss of the variety by the group. In [K] a formula is obtained for the rational cohomology of X//G under this condition. The formula involves the rational cohomology of X and various linear sections of X, together with the rational cohomology of the classifying spaces of G and certain reductive subgroups of G. In many interesting examples the condition required in [K] is not satisfied. Thus the question arises as to what information we can obtain in general. The quotient X//G can now have serious singularities in contrast to the good case considered in [K] where the only singularities are those caused by finite isotropy groups. It will be shown in this paper that there is a systematic way of blowing up X along a sequence of nonsingular subvarieties to obtain a variety X with a linear action of G such that every semistable point of X is stable. The only assumption that we have to make is that there exists at least one stable point of X. Then X//G is almost a resolution of singularities of X1/G, in the sense that the most serious singularities have been resolved. Moreover there is a formula for the rational cohomology of X//G again involving the rational cohomology of X and certain linear sections of X, together with the rational cohomology of the classifying spaces of G and some reductive subgroups of G (see Theorem 8.14). For convenience we shall assume throughout that G is connected. However the construction of X works in general, and it is straightforward to modify the cohomological formulas to apply to the general case. The construction of X//G can also be modified to apply in some cases when X has no stable points. The layout of the paper is as follows. Section 1 is a review of the basic facts of geometric invariant theory which will be needed and Section 2 describes the relationship of geometric invariant theory with symplectic geometry and the moment map. In Section 3 semistability and stability in a blow-up of X along a nonsingular C-invariant subvariety are related to semistability and stability in X,

Unipotent representations of complex semisimple groups

Abstract: algebraic group over R or C. In this paper, we restrict attention to C. We generalize Arthur's definition slightly (or perhaps simply make it more precise). All of the resulting representations, except for a finite set, are then unitarily induced from representations of the same kind on proper parabolic subgroups. We call the finite set remaining special unipotent representations; a precise definition will be given later (Definition 1.17). Our main result (Theorem III of this introduction) is a character formula for any special unipotent representation. Of course such a formula can be deduced from the Kazhdan-Lusztig conjecture (cf. [V3]). The advantages of Theorem III are that it is in closed form, and that it lends itself to verification of some conjectures of Arthur in [A]. So let G be a connected complex semisimple Lie group, and q its Lie algebra. Choose

Structure of manifolds of nonpositive curvature

Abstract: Clearly 1 < rank(M) < dim(M). We may regard rank(M) as a measurement of the flatness of M since a parallel field X along Yv and perpendicular to Yv is a Jacobi field if and only if the sectional curvatures of the planes i'(t) A X(t) are zero for all t E R. In particular, M is flat if rank(M) = dim(M). Hence a compact surface of nonpositive curvature has rank one unless it is a torus or a Klein bottle. It also follows that rank(M1 x M2) = rank(Ml) + rank(M2). Finally, it is important that the definition of rank(M) above coincides with the usual one if M is locally symmetric.

Torsion points on curves and $p$-adic Abelian integrals

Abstract: THEOREM A. Let f: C --* J be an Albanese morphism defined over a number field K, of a s-mooth curve of genus g into its Jacobian. Suppose J has potential complex multiplication. Let S denote the set of primes p of K satisfying i) p does not divide 2 or 3. ii) K/Q is unramified at P. iii) C has good ordinary reduction over K. Then the set T of torsion points of A defined over an algebraic closure of K which lie on the image of C is defined over an extension of K unramified above S. Moreover #To pg

Normal forms for smooth Poisson structures

Abstract: On classe, localement, les tenseurs de Poisson C ∞ de rang variable, soumis a une condition de non-degenerescence

Nest Algebras and Similarity Transformations

Abstract: In recent years the theory of algebras of operators on Hilbert space has been stimulated by developments in the theory of quasitriangularity. Andersen [1] has shown that up to unitary equivalence there is only one "continuous" quasitrianguilar algebra. We use this to provide the following answer to a question posed by J. R. Ringrose approximately 20 years ago: Similar continuous nests on separable Hilbert space can fail to be unitarily equivalent (Theorem 2.2). A consequence is the existence of a nonhyperintransitive compact operator (Corollary 2.3), which answers a question of Kadison and Singer [12] and of Gohberg and Krein [11]. We extend our initial theorem to show that arbitrary continuous nests are similar (Theorem 2.10), and that every maximal nest is similar to a multiplicity-one nest (Theorem 2.11). A consequence is that every compact operator is similar to a hyperintransitive compact operator (Corollary 2.12). The similarity transformation can be induced by an arbitrarily small compact perturbation of a unitary operator. The methods of Section 2 apply only to the continuous parts of nests. For general results the atomic core part must be dealt with. In Section 3 different methods are developed for this purpose, again utilizing Andersen's results. These are used in Section 4 to prove that a complete nest X admits an Arveson factorization for every positive invertible operator if and only if X is countable as a family of subspaces (Theorem 4.7). A consequence is that if X is an uncountable nest with atomic core then some similarity transformation of X has a continuous part. This could not be deduced from Section 2. These methods also yield a weak factorization result which concludes the paper. It should be noted that a negative resolution to the Ringrose question was conjectured in recent years by several mathematicians, including W. Arveson and J. Ringrose. Also, many of the results presented in this paper were announced in an A.M.S. Bulletin article [16]. Finally, we wish to thank the referee for suggesting that the original manuscript could be condensed and improved.

Tate's conjecture for K3 surfaces of finite height

Abstract: This paper extends the proof [16] of the Tate conjecture for ordinary K3 surfaces over a finite field to the more general case of all K3's of finite height. As in [16], our method is to find a lifting of the K3 to characteristic zero with sufficiently many Hodge cycles. In the ordinary case, the so-called "canonical lifting" of Deligne and Illusie [7] did the job, and a study of the Galois action on p-adic etale cohomology revealed the Hodge cycles. Here we use more general " quasi-canonical liftings," and the action of the crystalline Weil group on

Varietes stablement rationnelles non rationnelles

Abstract: La question qui nous interesse ici est souvent appelee probleme de Zariski. Son expression algebrique est tres simple. Une extension F d'un corps k est dite pure si elle est k-isomorphe 'a un corps de fractions rationnelles k(t1, ... , td), et stablement pure si elle devient pure par addition d'un nombre fini de variables: F(u1,..., Urn) k(v1,... , vn). En ces termes, la question est la suivante: une extension stablement pure est-elle pure? De fait cette question est de nature geometrique et il vaut mieux l'exprimer comme suit. Soit X une k-variete algebrique geometriquement integre, de dimension d: on dit qu'elle est k-rationnelle si elle est k-birationnelle 'a l'espace affine Adk, et stablement k-rationnelle si, pour m convenable, le produit X x kAk est une variete k-rationnelle. En ces termes, la question devient: une k-variete stablement k-rationnelle est-elle une variet k-rationnelle? On retrouve l'expression initiale en considerant le corps des fonctions rationnelles F = k(X) de la variete X, auquel cas n = m + d. Cette question a ete posee par Zariski pour k = C (voir [22]). Elle lui est encore attribuee, pour k quelconque, par plusieurs auteurs, dont Voskresenskii qui s'est particulierement interesse au cas oui X est un tore algebrique [26], [27]. Ce probleme est evidemment tres naturel du point de vue de la geometrie birationnelle: c'est le probleme de la simplification par l'espace affine. Mais il intervient aussi sous d'autres formes et en d'autres occasions. Comme l'a montre Demazure [10], il est etroitement lie 'a l'etude des tores maximaux du groupe de Cremona Crn k en n variables sur le corps k: une reponse negative signifie l'existence de sous-tores deployes maximaux de dimension < n. II se pose aussi tout naturellement dans le cas de varietes algebriques dont on parvient a etablir qu'elles sont stablement k-rationnelles sans savoir si elles sont k-rationnelles. C'est le cas sur C des varietes de modules A#g des courbes de genre g pour 3 < g < 6 (voir [17]). C'est aussi le cas, si C est une courbe algebrique complexe projective et lisse de genre ? 2, des varietes de modules Yo/c(r, d) des fibres vectoriels

Interaction of nonlinear progressing waves for semilinear wave equations

Abstract: On etudie l'interaction transversale de 3 ondes planes progressives pour une equation d'onde semi-lineaire a 3 dimensions

A characterization of seven compact Kaehler submanifolds by holomorphic pinching

Abstract: From the results of Simons [9] and Chern, Do Carmo and Kobayashi [2], we know that, in the class of compact minimal submanifolds of a sphere, the totally geodesic submanifolds are isolated and that some simple minimal submanifolds can be characterized by suitable pinching on their curvatures. These ideas are extended naturally to compact Kaehler submanifolds of a complex projective space. The problem of the best pinching for the above submanifolds was studied later by several authors, e.g. Yau [11] and Ogiue [4]. For surfaces and hypersurfaces, the problem is completely resolved. However in the general case we have only partial results. Let M' be a compact Kaehler submanifold, of complex dimension n, immersed in the complex projective space CPtm(1) endowed with the Fubini-Study metric of constant holomorphic sectional curvature 1. Let H and K be the holomorphic sectional curvature and the sectional curvature of Mn respectively. Ogiue conjectured the following: (1) If H > , or (2) If n ? 2 and K> , or (3) If m-n n(n + ) and K > 0, 2 then Mn is a linear subvariety of CPm(1). Recently, using natural arguments at the minimum of the function H defined on the unit tangent bundle of MW, the author [7] and Verstraelen and the author [8] resolved the conjectures (1) and (2) respectively. In this paper we obtain the following complete solution of the pinching problem in the Kaehlerian case.

On the general linear group and Hochschild homology

Abstract: Our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring. Before stating the result we establish some notation and conventions. Rings are associative and with unit. If A is a ring then GL(A)= Uk?0GLk(A) is its infinite general linear group. An A-bimodule is an abelian group B which is both a left A-module and a right A-module and satisfies (alb)a2 = al(ba2) for ai eA, be B (for example B = A). If B is an A-bimodule, then M(B) = Uk?oMk(B) is the infinite additive group of matrices with entries in B. Conjugation defines an action (the adjoint action) of GL(A) on M(B). Note that an A ? Q-bimodule is just an A-bimodule which is also a rational vector space. If B is an A ? Q-bimodule, then HJ(A C) Q; B) denotes the Hochschild homology of A ? Q with coefficients in B. Our main result (it appears in slightly more detailed form as Theorem V.3) is: