Showing papers in "Annals of Mathematics in 1994"

Demography of Cirsium vulgare in a grazing experiment.

Abstract: The complete demographies of 16 populations of Cirsium vulgare were followed in a replicated experiment. The experiment was a factorial combination of two intensities of sheep grazing in each of three seasons - winter (grazed or ungrazed), spring (grazed or ungrazed), and summer (light or heavy grazing) - giving eight treatments in two blocks. For 6 years from 1987 to 1992 the population sizes of C. vulgare were monitored in each of the 16 paddocks. After 1989 grazing in spring or winter or increased grazing in summer all increased population sizes. Population sizes fluctuated widely between years. The effects of the grazing treatments and plant sizes on the transitions between nine life-history stages were determined (...)

The Elementary Theory of Restricted Analytic Fields with Exponentiation

Abstract: numbers with exponentiation is model complete. When we combine this with Hovanskii's finiteness theorem [9], it follows that the real exponential field is o-minimal. In o-minimal expansions of the real field the definable subsets of R' share many of the nice structural properties of semialgebraic sets. For example, definable subsets have only finitely many connected components, definable sets can be stratified and triangulated, and continuous definable maps are piecewise trivial (see [5]). In this paper we will prove a quantifier elimination result for the real field augmented by exponentiation and all restricted analytic functions, and use this result to obtain o-minimality. We were led to this while studying work of Ressayre [13] and several of his ideas emerge here in simplified form. However, our treatment is formally independent of the results of [16], [17], [9], and [13].

A positive answer to the Busemann-Petty problem in three dimensions

Abstract: We prove that in E3 the Busemann-Petty problem, concerning cen- tral sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.

Self-dual instantons and holomorphic curves

Abstract: A gradient flow of a Morse function on a compact Riemannian manifold is said to be of Morse-Smale type if the stable and unstable manifolds of any two critical points intersect transversally. For such a Morse-Smale gradient flow there is a chain complex generated by the critical points and graded by the Morse index. The boundary operator has as its (x, y)-entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the Morse indices is 1. The homology of this chain complex agrees with the homology of the underlying manifold M and this can be used to prove the Morse inequalities (cf. [33], [26]). Around 1986, Floer generalized this idea to infinite-dimensional variational problems in which every critical point has infinite Morse index but the moduli spaces of connecting orbits form finite-dimensional manifolds for every pair of critical points. The dimensions of these spaces give rise to a relative Morse index and the boundary operator is defined by counting connecting

The pinwheel tilings of the plane

Abstract: "prototiles") in Euclidean n-space, En, for n > 2. The prototiles are usually required to be rather nice topologically, at least homeomorphs of the closed unit ball. One then makes arbitrarily many congruent copies, called "tiles," of these prototiles, and considers all ways (called "tilings") that such tiles may provide a simultaneous covering and packing of En; a tiling is thus an unordered collection of tiles for which the union is all of En, but such that the interiors of each pair of tiles do not intersect. Wang's original problem was to determine if it was possible to have a finite set of prototiles, with associated tilings of space, for which all the tilings were nonperiodic. (A tiling is "nonperiodic" if it is not invariant under any single simultaneous translation of its tiles.) This was settled in the affirmative in the thesis of Wang's student Robert Berger [1].

Homotopy fixed-point methods for Lie groups and finite loop spaces

Abstract: A loop space X is by definition a triple (X, BX, e) in which X is a space, BX is a connected pointed space, and e: X -QBX is a homotopy equivalence from X to the space QBX of based loops in BX. We will say that a loop space X is finite if the integral homology H*(X, Z) is finitely generated as a graded abelian group, i.e., if X appears at least homologically to be a finite complex. In this paper we prove the following theorem.

Metrics of negative Ricci curvature

Abstract: One of the most natural and important topics in Riemannian geometry is the relation between curvature and global structure of the underlying manifold. For instance, complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic. Furthermore, it seemed to be a natural principle that a (closed) manifold cannot carry two metrics of different signed curvatures, as it is a basic fact that this is true for sectional curvature. But it turned out to be wrong (much later and from a strongly analytic argument) for the scalar curvature S, since each manifold M', n > 3, admits a complete metric with S _-1 (cf. Aubin [A] and Bland, Kalka [BIK]). Hence the situation for Ricci curvature Ric, lying between sectional and scalar curvature, seemed to be quite delicate. Up to now, the most general results concerning Ric < 0 were proved by Gao, Yau [GY] and Brooks [Br] using Thurston's theory of hyperbolic threemanifolds, viz.: Each closed three-manifold admits a metric with Ric < 0. This is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities; Gao and Yau (resp. Brooks) smoothed these singularities to get a regular metric with Ric < 0. These methods extend to three-manifolds of finite type and certain hyperbolic orbifolds. In any case, the arguments rely on exploiting some extraordinary metric structures, whose existence is neither obvious nor conceptually related to the Ricci curvature problem. Indeed, the existence depends on the assumption that the manifold is three-dimensional and compact. Moreover this approach does not provide insight into the typical behaviour of metrics with Ric < 0 since one is led to very special metrics. In this article we approach negative Ricci curvature using a completely different and new concept (which will become even more significant in [L2]) as we deliberately produce Ric < 0. Actually we will prove the following results; in these notes Ric(g), resp. r(g), denotes the Ricci tensor, resp. curvature of a smooth metric g:

Stabilizers for ergodic actions of higher rank semisimple groups

Abstract: Let G be a connected semisimple Lie group with finite center and R-rank > 2. Suppose that each simple factor of G either has R-rank > 2 or is locally isomorphic to Sp(l, n) or F4(-20). We prove that any faithful, irreducible, properly ergodic, finite measure-preserving action of G is essentially free. We extend the result to reducible actions and actions of lattices.

A Harish-Chandra homomorphism for reductive group actions

Abstract: Consider a semisimple complex Lie algebra g and its universal enveloping algebra U(g). In order to study unitary representations of semisimple Lie groups, Harish-Chandra ([HC1] Part III) established an isomorphism between the center Z(g) of U(g) and the algebra of invariant polynomials C[t] . Here, t ⊆ g is a Cartan subspace and W is the Weyl group of g. This is one of the most basic results in representation theory. Later on ([HC2] Thm. 1), he found a similar isomorphism for a symmetric space X = G/K. Here, instead of Z(g), he considered the algebra D(X) of invariant differential operators on X and showed, that it is isomorphic to the ring of invariants of the little Weyl group WX attached to X. This isomorphism is very important for analyzing the action of G on various function spaces on X. Actually, it is a generalization of the former result if one considers the natural G×G-action on X = G. The proofs of these theorems relied very much on the very special structure theory of symmetric spaces. Therefore, it may be surprising that such an isomorphism can be constructed for every algebraic variety X carrying an action of a connected reductive group G. Because invariants and centers behave well under field extensions, we assume from now on, that the base field k is algebraically closed of characteristic zero. Let me first explain the case, where X is a smooth, affine G-variety. Here, I obtained the most complete results. This covers all linear actions on a vector space as well as all homogeneous spaces G/H where H is reductive, in particular symmetric varieties. Consider the algebra of (algebraic) linear differential operators D(X). We are interested in the subalgebra D(X) of invariant operators and in particular, in its center Z(X).

Models of degenerate Fourier integral operators and Radon transforms

Abstract: Fourier integral operators whose Lagrangians project with singularities arise frequently in many areas of analysis and geometry [3], [5], [9], [10], [18], [20]. However, there are as yet few analytic tools available for their study, and even their simplest regularity properties with respect to the singularities of the Lagrangian are still obscure [4], [5], [13], [16]. In this paper we study a class of oscillatory integrals TA and Fourier integral operators R which can be expected to model the higher order singularities of the Lagrangian. They have homogeneous polynomial phases in two variables of order n, and indeed the case n = 3 is the model for Lagrangians which project with Whitney folds [3], [10], [11]. The main difficulty which sets the higher order degeneracy cases n > 4 apart from the the lower ones n = 2, 3 is that the critical varieties which arise there are usually not smooth manifolds. A systematic study of these oscillatory integrals was begun in [14]. The main idea in that work was to treat the critical varieties as smooth manifolds away from a lower-dimensional subvariety, and to keep track of the distance to this lower-dimensional subvariety. To achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points. The method gave sharp bounds on the size of the kernel K(x, y) of TAT*, but no information on its phase and the resulting cancellations. It does not seem possible to refine this approach to the phase level, and this suggests looking instead for decompositions of the operators TA which can incorporate indirectly the required cancellations. The main goal of this paper is to introduce such decompositions. These decompositions, which we will try to describe momentarily, reflect the singular nature of the critical points, and are powerful enough to yield sharp bounds for the above oscillatory integrals and Radon transforms. Although the models under consideration are very

Combinatorics, geometry and attractors of quasi-quadratic maps

Abstract: The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $\omega(c)$ of a "non-renormalizable" map. It is proven that the scaling factors characterizing the geometry of this set go down to 0 at least exponentially. This resolves the problem of the non-linearity control in small scales. The proofs strongly involve ideas from renormalization theory and holomorphic dynamics.

Zero cycles and projective modules

Abstract: Let A be a reduced affine k-algebra of dimension n over an algebraically closed field k. Let FnKo(A) denote the sub-group of Ko(A) generated by the images of all the residue fields of all smooth maximal ideals of height n. For a module of finite projective dimension M, let (M) denote the image of M in Ko(A). For a projective A-module P of rank n, we define the nth Chern class of P to be: Cn(P) = E(-l)i(AiP*), where P* is the dual of P. Suppose FnKo(A) has no (n 1)! torsion; then our main result (Th. 3.8) is that P has a free direct summand of rank one if and only if Cn(P) = 0. When the characteristic of k is zero or A is normal and n > 3, it is known that FnKo(A) is torsion-free ([Le], [Sr]). Hence our theorem is applicable in these cases. Also when A is regular, FnKo(A) coincides with CHn(X) the Chow group of zero cycles of Spec A and Cn(P) coincides with the usual nth Chern class as defined by Grothendieck (see [Fu]). When n < 3 and A is regular, this result was proved in [MKM]. When n = 3 and the characteristic of k is not equal to two, this result in [MKM] was extended to the singular case by M. Levine. In this paper, we extend the results of [MKM] to all dimensions. With the assumption stated above on torsion in FnKo(A), we first show that (Cor. 3.4) if I C A is a local complete intersection of height n, then (A/I) is zero in Ko(A) if and only if I is a complete intersection. Corollary 3.4 together with a result of Mohan Kumar ([MK], Cor. 1.9 here) at once gives Theorem 3.8. In Section 1, we give some preliminaries and generalize results in [MK2]. These results are crucially used in the rest of the paper. The basic ideas in Section 1 are all taken from [MK2]. In Section 2, we prove Theorem 2.2, which strengthens a result of Boratynski [Bo] for local complete intersections with trivial co-normal bundle. Theorem 2.2 is one of the crucial ingredients in the proofs of the results in Section 3. As an amusing application of Theorem 2.2, we give a new proof of a theorem of Srinivas ([Sr]) about torsion in zero cycles

Cohomological and cycle-theoretic connectivity

Abstract: One of the themes in algebraic geometry is the study of the relation between the ``topology'' of a smooth projective variety and a (``general'') hyperplane section. Recent results of Nori produce cohomological evidence for a conjecture that a general hypersurface of sufficently large degree should have no ``interesting'' cycles. We compute precise bounds for these results and show by example that there are indeed interesting cycles for degrees that are not high enough. In a different direction Esnault, Nori and Srinivas have shown connectivity for intersections of small multidegree. We show analogous cycle-theoretic connectivity results.

Homotopy groups of the complements to singular hypersurfaces, II

Abstract: The homotopy group $\pi_{n-k} ({\bf C}^{n+1}-V)$ where $V$ is a hypersurface with a singular locus of dimension $k$ and good behavior at infinity is described using generic pencils. This is analogous to the van Kampen procedure for finding a fundamental group of a plane curve. In addition we use a certain representation generalizing the Burau representation of the braid group. A divisibility theorem is proven that shows the dependence of this homotopy group on the local type of singularities and behavior at infinity. Examples are given showing that this group depends on certain global data in addition to local data on singularities.

On the rationality homotopy type of configuration spaces

Abstract: In this article we prove that the rational homotopy type of the configuration space of n distinct points on a smooth projective variety X over C is determined by the rational cohomology ring of X together with its canonical orientation class. This strengthens a previous result of Fulton and MacPherson [FM] (see previous issue in this volume). A few words are due on the history of differential graded algebras. By a DGA we shall mean a graded-commutative differential graded algebra. Fulton and MacPherson [FM] gave a model F(n) for the space F(X, n) by resolving the singularities of the inclusion F(X, n) C Xn and using a result of Morgan [M] on the rational homotopy type of smooth quasiprojective varieties over C. Surprisingly the DGA F(n) depended not only on H*X and its orientation class, but also on the Chern classes of X. Fulton suggested that one should be able to find another explicit model of F(X, n) independent of Chern classes. The DGA used here, E(n), was found independently by the author and by Totaro [T]. Totaro's idea was to study the Leray spectral sequence of the inclusion F(X, n) C Xn. By a weight-filtration argument he showed that the only nontrivial differential in this spectral sequence is d2m. The E2 term with the differential d2m is isomorphic to E(n). Thus, by a result of Deligne [D2], H*E(n) ' H*F(X, n) as rings. Finally the relations (1)-(3) of Theorem 1.2 are much older than any of the results mentioned above. They were discovered by F. Cohen [C] (see also [CLMa]) in the study of configuration spaces of IR'. What happens could be described by saying that H*F(X, n) for a smooth projective variety X is related to H*F(lR2m, n) in the "simplest possible way."