Showing papers in "American Journal of Mathematics in 1996"
TL;DR: It is proved that a certain class of semilinear wave equations has global solutions if the initial data is small, and these existence results rely on mixed-norm angular-radial space-time estimates.
Abstract: We prove that a certain class of semilinear wave equations has global solutions if the initial data is small. These existence results rely on mixed-norm angular-radial space-time estimates. More specifically, we consider power nonlinearities, □ u = F p ( u ), where F p ( u ) ~ | u | p and □ = ∂2 t - Δ denotes the D'Alembertian on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. In 1979, John investigated this equation for small initial data in dimension n = 3. He proved that if p 1 + √2 then there is a global solution whereas for p n dimensions, and the critical power p c above which global solutions exist should be the positive root of ( n -1) p 2 c -( n +1) p c - 2 = 0. Sideris proved that for all n the solutions can blow up if p p c . Recently Zhou proved that for n = 4 one has global solutions if p p c = 2. Here we prove that for n ≤ 8 one has global solutions if p p c . Furthermore, we proved that for all n , one has global solutions if p p c and the initial data are spherically symmetric. In the radial case we use L p estimates, derived from the explicit form of the fundamental solution using inequalities from classical analysis. In the nonradial case we write the fundamental solution in polar coordinates and use Fourier integral and special function estimates to handle the angular part.
154 citations
TL;DR: This work shows how the joint distribution of the n-tuple (an>b), and points out a few applications.
Abstract: Let a1, , a n , b 1 , , b n be random variables in a noncommutative probability space, such that a1, , a n is free from b1, , b n . We show how the joint distribution of the n-tuple (a1b1, , a n b n ) can be described in terms of the joint distributions of (a1, , a n )a nd ( b 1 , , b n ), by using the combinatorics of the n-dimensional R-transform. We point out a few applications that can be easily derived from our result, concerning the left-and-right translation with a semicircular element (see Sections 1.6-1.10) and the compression with a projection (see Sections 1.11-1.14) of an n-tuple of noncommutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper. Introduction. The theory of free random variables was developed in a se- quence of papers of D. Voiculescu (see (17), or the recent survey in (16)), as an instrument for approaching free products of operator algebras. Its particular aspect addressed in the present paper is the one concerning the addition and multiplica- tion of free random variables; as shown by Voiculescu in (13), (14), a powerful method in the study of these operations is the use of transforms that convert them (respectively) into addition and multiplication of complex analytic func- tions, or, in an algebraic framework, of formal power series in an indeterminate z. The precise definitions of these transforms (called R-transform for the addition problem and S-transform for the multiplication problem) will be reviewed in the Sections 1.2, 1.3 below. In the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the R- and S-transforms. The first idea is that the connection between the R- and the S-transform is closer than one might suspect at first glance. A way of making this precise was pointed out in our paper (6), in the form of the equation S( )= ( R ( )) for a distribution with nonvan- ishing mean, and where is a combinatorial object with a precise significance ("the Fourier transform for multiplicative functions on noncrossing partitions"). A byproduct of our result in (6) is the remark that the multiplication of free
153 citations
TL;DR: In this paper, the authors study how much regularity of initial data is needed to ensure existence of a local solution to a semi-linear wave equation and give counterexamples to local existence for the typical model equations.
Abstract: In this paper we study how much regularity of initial data is needed to ensure existence of a local solution to a semi-linear wave equation. We give counterexamples to local existence for the typical model equations. The counterexamples we construct are sharp, i.e. one does have a local solution if the data has slightly more regularity.
128 citations
TL;DR: In this paper, the authors established sharp capacitary estimates for Carnot-Caratheodory rings associated to a system of vector fields of Hormander type, which are instrumental to the study of the local behavior of singular solutions of a wide class of nonlinear subelliptic equations.
Abstract: We establish sharp capacitary estimates for Carnot-Caratheodory rings associated to a system of vector fields of Hormander type. Such estimates are instrumental to the study of the local behavior of singular solutions of a wide class of nonlinear subelliptic equations. One of the main results is a generalization of fundamental estimates obtained independently by Sanchez-Calle and Nagel, Stein and Wainger.
126 citations
TL;DR: In this article, the basic Laplacian on Riemannian foliations was studied in terms of the orthogonal projection from square-integrable forms to basic forms.
Abstract: We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square-integrable forms to basic square-integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian. Introduction. Let M be a compact oriented manifold and let be a transver- sally oriented foliation on M. A foliation is a Riemannian foliation if there exists a Riemannian metric on M with the property that the distance from one leaf of to another is locally constant; such a metric is called a bundle-like metric for . Associated to are the space of basic forms: Ω B(M )= Ω B ( M , )= Ω ( M ): i(X) =0 , i ( X ) d = 0 for all X Γ(T ) , where i(X) is the interior product with the vector field X and Γ(T ) denotes the sections of the distribution T associated to . The exterior derivative d maps basic forms to basic forms; let dB denote d restricted to Ω B (M). The basic Laplacian is the operator ΔB = dB B + BdB on basic forms, where B is the adjoint of dB on Ω B (M). The analytic and geometric properties of this operator have been studied by several researchers. In (5), the basic Laplacian was studied as an operator on basic functions (i.e., functions that are constant on leaves of ), and the author proved the existence of the heat kernel in this case. In (13), the existence of the heat kernel on basic forms was proved for the case where the mean curvature form of the foliation is basic. There are also "basic" Hodge theorems, for example (6) and (10). However, the proof of the Hodge theorem in (6) does not yield various estimates that are important in applications, while the theorem proved in (10) has the same restriction as the results in (13), namely that the authors require the mean curvature form to be basic. In this paper, we study the basic Laplacian on forms, without any restriction on the mean curvature. We prove the existence and uniqueness of the heat kernel for ΔB on forms for any Riemannian foliation, and we write down an explicit formula for the heat kernel. We also present a proof of the Hodge theorem for
119 citations
TL;DR: In this paper, the authors studied Schrodinger semigroups in the scale of Sobolev spaces, and showed that eigenfunctions of a semigroup of functions are generically smoother than their potentials.
Abstract: We study Schrodinger semigroups in the scale of Sobolev spaces, and show that, for Kato class potentials, the range of such semigroups in LP has exactly two more derivatives than the potential; this proves a conjecture of B. Simon. We show that eigenfunctions of Schrodinger operators are generically smoother by exactly two derivatives (in given Sobolev spaces) than their potentials. We give applications to the relation between the potential's smoothness and particle kinetic energy in the context of quantum mechanics, and characterize kinetic energies in Coulomb systems. The techniques of proof involve Leibniz and chain rules for fractional derivatives which are of independent interest, as well as a new characterization of the Kato class. 1. Introduction. In this paper we attempt a precise study of the action of Schrodinger semigroups in the scale of Sobolev spaces. Work in this area was begun by B. Simon (Si4) in 1985. He proved a number of positive and negative results on such smoothing, and also presented an open question (in the form of a conjecture) regarding necessary and sufficient conditions on the potential V for smoothing of order s by the semigroup of functions in L2(Rd). Study of Schrodinger semigroups is of interest for a number of reasons (see
98 citations
TL;DR: In this paper, it was shown that a Lie algebra L is 3-gradable if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R.
Abstract: Let R be an irreducible root system. A Lie Algebra L is called graded by R if L is graded with grading group the root lattice of R such that the only nonzero homogeneous subspaces of L have degree 0 or a root in R , the grading is induced by the adjoint action of a split Cartan subalgebra of a finite-dimensional simple Lie subalgebra of L with root system R , and L is generated by the homogeneous subspaces of nonzero degree. This class of Lie algebras was introduced and studied by S. Berman and R. Moody in Invent. Math. 108 (1992), where, in particular, a classification up to central equivalence is given in the simply-laced case. The doubly-laced cases have recently been classified by G. Benkart and E. Zelmanov. Let R be a 3-graded root system, i.e., R is not of type E_8, F_4 or G_2. In this paper, Lie algebras graded by R are described in a unified way, without case-by-case considerations. Namely, it is shown that a Lie algebra L is 3-graded if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R . This result is then used to classify Lie algebras graded by R : we give the classification of Jordan pairs covered by a grid and describe their Tits-Kantor-Koecher algebras. One of the advantages of this approach is that it works over rings containing 1/2 and 1/3, and also for infinite root systems. Another application is the description of Slodowy's intersection matrix algebras arising from multiply-affinized Cartan matrices.
96 citations
TL;DR: For general Sturm-Liouville operators with separated boundary conditions, this article showed that the dimension of the spectral projection P(E,E_2)(H) of H equals the number of zeros of the Wronskian of u_1 and u_2.
Abstract: For general Sturm-Liouville operators with separated boundary conditions, we prove the following: If E_(1,2) ∈ R and if u_(1,2) solve the differential equation Hu_j = E_ju_j, j = 1, 2 and respectively satisfy the boundary condition on the left/right, then the dimension of the spectral projection P(E_1,E_2)(H) of H equals the number of zeros of the Wronskian of u_1 and u_2.
95 citations
TL;DR: The generalized Busemann-petty problem as discussed by the authors has been shown to be NP-hard in the case of projection bodies, polar projection bodies and their central sections, which is related to the maximal slice problem.
Abstract: The generalized Busemann-Petty problem asks: If the volume of i -dimensional central section of a centrally symmetric convex body in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is smaller than that of another such body, is the volume of the body also smaller? It is proved that the answer is negative if 2 i n . The case of a 2-dimensional section remains open. The proof uses techniques in functional analysis and Radon transforms on Grassmannians. It also requires the notion of an i -intersection body which generalizes the notion of an intersection body. Inequalities among the volumes of projection bodies, polar projection bodies and their central sections are proved. They are related to the maximal slice problem.
88 citations
TL;DR: Ohshika as mentioned in this paper showed that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the linear algebraic limit.
Abstract: Troels Jorgensen conjectured that the algebraic and geometric limits of an algebraically convergent sequence of isomorphic Kleinian groups agree if there are no new parabolics in the algebraic limit. We prove that this conjecture holds in ‘most’ cases. In particular, we show that it holds when the domain of discontinuity of the algebraic limit of such a sequence is non-empty. We further show, with the same assumptions, that the limit sets of the groups in the sequence converge to the limit set of the algebraic limit. As a corollary, we verify the conjecture for finitely generated Kleinian groups which are not (non-trivial) free products of surface groups and infinite cyclic groups. These results are extensions of similar results for purely loxodromic groups. Thurston previously established these results in the case when the Kleinian groups are freely indecomposable. Using different techniques from ours, Ohshika has proven versions of these results for purely loxodromic function groups.
70 citations
TL;DR: In this article, it was shown that the Seshadri constant of an ample line bundle is at least one if and only if the polarized abelian variety splits as a product of a principally polarized elliptic curve and a polarized subvariety of codimension one.
Abstract: We show that on a complex abelian variety of dimension two or greater the Seshadri constant of an ample line bundle is at least one. Moreover, the Seshadri constant is equal to one if and only if the polarized abelian variety splits as a product of a principally polarized elliptic curve and a polarized abelian subvariety of codimension one. We also examine the case when the Seshadri constant is not one and obtain lower bounds when the dimension of the abelian variety is small.
TL;DR: In this article, the Weyl integration formula, local constancy of the spherical character on the θ regular set, Howe's conjecture for θ-groups, the germ expansion for spherical characters at the origin, and a spherical version of Howe's Kirillov theory for compact p-adic groups are presented.
Abstract: Symmetric spaces over local fields, and the harmonic analysis of their class one representations, arise as the local calculation of Jacquet's theory of the relative trace formula. There is an extensive literature at the real place, but few general results for p-adic fields are known. The objective here is to carry over to symmetric spaces as much as possible of Queens and the prerequisite results of Howe, and to provide counterexamples for those things which do not generalize. We derive the Weyl integration formula, local constancy of the spherical character on the θ-regular set, Howe's conjecture for θ-groups, the germ expansion for spherical characters at the origin, and a spherical version of Howe's Kirillov theory for compact p-adic groups. We find that the density property of regular orbital integrals fails. Some of the basic ideas are nascent in Hakim's thesis (written under Herve Jacquet).
TL;DR: In this article, it was shown that the Manin obstruction to the Hasse principle and weak approximation is the only one provided there are only few "degenerate" fibres (usually two, three in some cases), and that "sufficiently many" smooth fc-fibres satisfy weak approximation.
Abstract: We formulate a general set-up for the descent method of J.-L. Colliot-Thelene and J. J. Sansuc applied to varieties fibred over the projective line over a number field. This makes it possible to prove that the Manin obstruction to the Hasse principle and weak approximation is the only one provided there are only few "degenerate" fibres (usually two, three in some cases), and that "sufficiently many" smooth fc-fibres satisfy the Hasse principle and weak approximation. We introduce a new concept of what should be called a "degenerate" fibre, the so called "split" fibres, the property which depends only on the generic fibre and not on the choice of a particular model. This yields a simpler and more general approach to the previous results of that kind. 0. Introduction. Motivated by Hasse's theorem on quadratic forms one says that the Hasse principle holds for a family of (geometrically integral, smooth and proper) varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a ^-rational point. The fact that the Hasse principle holds for some family signif icantly simplifies the study of its arithmetic, because the problem of determining whether or not there is a ^-rational point is then reduced to a finite number of
TL;DR: In this article, the authors studied the upper bound of the reduction number of an ideal in a Noetherian local ring with infinite residue field k, where k is the Krull dimension of the fiber ring.
Abstract: 1. Introduction. Let R be a Noetherian local ring with infinite residue field k, and let / be an R-ideal. When studying algebraic properties of the blow-up of Spec(/?) along V(l), it is often important to have good upper bounds for the "reduction number" of the ideal /. Recall that an ideal / c / is a reduction of / if the extension of Rees algebras R[Jt] C R[It] is module finite, or equivalently, if ir+l = J\r for some r > 0 ([32]). The least such r is denoted by rj(I). A reduction is minimal if it is minimal with respect to inclusion, and the reduction number r(l) of / is defined as min{o(/) | J a minimal reduction of/}. Finally, the analytic spread ?(I) of / is the Krull dimension of the fiber ring R[lt] &)R k, or equivalently, the minimal number of generators jbi(J) of any minimal reduction J of 1 ([32]). Philosophically speaking, J is a "simplification" of /, with the reduction number r(l) measuring how closely the two ideals are related. Of the numerous recent results about Rees algebras and associated graded rings of ideals having "small" reduction number ([18], [19], [38], [35], [12], [13], [14], [4], [3],
TL;DR: In this paper, the homology spectral sequence of a cosimplicial space was studied and conditions for strong convergence were given for the Eilenberg-Moore spectral sequence and for mapping spaces.
Abstract: We produce new convergence conditions for the homology spectral sequence of a cosimplicial space by requiring that each codegree of the cosimplicial space has finite type mod p homology. Specifically, we find conditions which ensure strong convergence if and only if the total space has p -good components. We also find exotic convergence conditions for cosimplicial spaces not covered by the strong convergence conditions. These results give new convergence conditions, for example, for the Eilenberg-Moore spectral sequence and for mapping spaces.
TL;DR: In this article, the definition algebrique du front d'onde d'une representation lisse d'un groupe p -adique, donnee par Kawanaka, was rappelle.
Abstract: Dans cet article, on rappelle la definition algebrique du front d'onde d'une representation lisse d'un groupe p -adique, donnee par Kawanaka. On en donne quelques proprietes; en particulier pour les groupes classiques p -adiques, on montre que ce front d'onde ne contient que des orbites unipotentes speciales. On termine en montrant que si ce front d'onde est induit, dans certains cas, cela entraine que la representation est elle aussi induite.
TL;DR: In this article, it was shown that the subset of the maximal ideal space of H ∞ formed by the trivial Gleason parts is totally disconnected, and that the set U 2 (H ∞ ) of corona pairs is dense.
Abstract: We show that the subset of the maximal ideal space of H ∞ formed by the trivial Gleason parts is totally disconnected. A pair ( ƒ, g ) ∈ H ∞ × H ∞ is called a corona pair if [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /][inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] 0, where [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is the open unit disc. We prove that the set U 2 ( H ∞ ) of corona pairs is dense in H ∞ × H ∞ .
TL;DR: In this article, it was shown that the higher secant ideals of rational normal scrolls are determinantal ideals and that there is always a rational normal scroll X such that ai(X) = A^ for any sequence A^ satisfying the two conditions above.
Abstract: Let X C P be an irreducible closed subvariety of projective space. The kth higher secant variety of X, denoted X , is the closure of the union of all linear spaces spanned by k points of X. Let a^iX) = dim(X*) ? dim(X*_l). It is known that the sequence -ty(X) is nonincreasing, and that there may be at most one 1 in such a sequence. We prove that these two conditions are also sufficient conditions for an arbitrary sequence A^ of nonnegative integers to be the fl^(X)'s for an irreducible variety X. To this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals. This allows us to compute their dimensions, showing that there is in fact always a rational normal scroll X such that ak(X) = A^ for any sequence A^ satisfying the two conditions above. Adlandsvik (Al), (A2), and Flenner and Vogel (F-V). Much of this work has centered on the behavior of the differences ak(X) = dimX* ? dimX*-1 of the dimensions of consecutive higher secant varieties. It is well known that this sequence is nonincreasing, and that there is at most one 1 in the sequence. Palatini claimed, moreover, that once ak(X) 1 were counterexamples (A2). We answer here the question of what sequences actually appear as the ai(X)'s of some variety X. Some of the guiding examples of varieties with degenerate higher secant varieties come from rational normal scrolls. Explicit equations for the scrolls themselves have been known for some time, (cf. (ACGH)). We give a determi nantal presentation of the ideals of their higher secant varieties generalizing a result for rational normal scrolls in (W). This presentation enables us to compute the dimensions of all the higher secant varieties of a rational normal scroll and gives examples of varieties X with a prescribed nonincreasing ak(X) with at most one 1.
TL;DR: It will be shown in this paper that certain real rank zero C*-algebras which are inductive limits of C-algeses of the form ⊕i M(C([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /])) can be expressed as inductive Limits.
Abstract: It will be shown in this paper that certain real rank zero C*-algebras which are inductive limits of C*-algebras of the form ⊕ i M k i (C([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /])) can be expressed as inductive limits of C*-algebras of the form ⊕ i M k i (C( S 1 )). In particular, if both A and B are of real rank zero and are inductive limits of C*-algebras of the form ⊕ i M k i (C( S 1 )), then also A ⊗ B is an inductive limit of C*-algebras of the form ⊕ i M k i (C( S 1 )). (Hence, A ⊗ B can be classified by its K-theory.) This is a key step in the general classification theory of inductive limit C*-algebras.
TL;DR: In this paper, the Ricci curvature is replaced by a lower bound on the sectional curvature, which is a much weaker assumption than the one made in this paper.
Abstract: The concept of best constants for Sobolev embeddings appeared to be crucial for solving limiting cases of some partial differential equations. A striking example where it has played a major role is given by the very famous Yamabe problem. While the situation is well understood for compact manifolds, things are less clear when dealing with complete manifolds. Aubin proved in '76 that optimal Sobolev inequalities are valid for complete manifolds with bounded sectional curvature and positive injectivity radius. We prove here that the result still holds if the bound on the sectional curvature is replaced by a lower bound on the Ricci curvature (a much weaker assumption). We also get estimates for the remaining constants.
TL;DR: In this article, the authors investigated the zeta function associated to a nilpotent group Γ and gave an explicit finite form for it in terms of the combinatorial data of the root system of G and information about the weights of various representations of G.
Abstract: We investigate in this paper the zeta function [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] associated to a nilpotent group Γ introduced in [GSS]. This zeta function counts the subgroups H ≤ Γ whose profinite completion Ĥ is isomorphic to the profinite completion [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i"/]. By representing [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i"/] as an integral with respect to the Haar measure on the algebraic automorphism group G of the Lie algebra associated to Γ and by generalizing some recent work of Igusa [I], we give, under some assumptions on Γ, an explicit finite form for [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i"/] in terms of the combinatorial data of the root system of G and information about the weights of various representations of G . As a corollary of this finite form we are able to prove (1) a certain uniformity in p confirming a question raised in [GSS]; and (2) a functional equation that the local factors satisfy [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="05i"/]. This functional equation is perhaps the most important result of the paper as it is a new feature of the theory of zeta functions of groups.
TL;DR: In this article, the authors characterize the unique K3 surface of discriminant 3 or 4, called the most algebraic K3 surfaces by Vinberg, in terms of the fixed locus of an automorphism on it.
Abstract: We shall characterize the unique K3 surface of discriminant 3 or 4, called the most algebraic K3 surfaces by Vinberg, in terms of the fixed locus of an automorphism on it. Based on this result, we show that there is, up to isomorphisms, only one rational log Eriques surface of Type D 19 and one of Type A 19 .
TL;DR: In this paper, a general theory of mixed twisted Segre classes was proposed, and a general mixed-multiplicity formula was given, including a positivity theorem, which is closely related to the additivity formula, and showed that it's closely related, surprisingly, to another fundamental formula.
Abstract: We prove the results about mixed Buchsbaum-Rim multiplicities announced in (6, (9.10)(ii), p. 224), including a general mixed-multiplicity formula. In addition, we identify these multiplicities as the coefficients of the "leading form" of the appropriate Buchsbaum-Rim polyno mial in three variables, and we prove a positivity theorem. In fact, we define the multiplicities as the degrees of certain zero-dimensional "mixed twisted" Segre classes, and we develop an encom passing general theory of these new rational equivalence classes in all dimensions. In parallel, we develop a theory of pure "twisted" Segre classes, and we recover the main results in (6) about the pure Buchsbaum-Rim multiplicities, the polar multiplicities, and so forth. Moreover, we identify the additivity theorem (6, (6.7b)(i), p. 205) as giving a sort of residual-intersection formula, and we show its (somewhat unexpected) connection to the mixed-multiplicity formula. Also, we work in a more general setup than in (6), and we develop a new approach, based on the completed normal cone. 1. Introduction. The theory of mixed multiplicities of primary ideals was introduced by Teissier in his study of complex hypersurface germs with isolated singularities. A decade later, Gaffney began extending Teissier's work to com plete intersections, and was led to conjecture a theory of generalized multiplicities of submodules of finite colength in a free module, including an important mixed multiplicity formula for the product of an ideal and a submodule. It turned out that these generalized multiplicities are nothing but the multiplicities introduced a decade before Teissier's work by Buchsbaum and Rim, who established many of their fundamental properties, but no mixed-multiplicity formula. Recently, the authors gave a general treatment of the Buchsbaum-Rim multiplicity, based on blowups and intersection numbers, in (6) (that paper also contains a more exten sive history of the subject). On p. 225, the authors announced a mixed-multiplicity formula for an arbitrary pair of submodules. Here we prove an even more general formula, and show that it's closely related, surprisingly, to another fundamental formula, the additivity formula. We also simplify, generalize, and advance the previous treatment via a new approach. Sections 2 and 3 study two preliminary notions, module transforms and dis tinguished subsets. Section 4 studies a "twisted" version ofthe usual Segre classes of a subscheme. The degrees of these classes yield the Buchsbaum-Rim multi plicities. Moreover, the usual theory of Segre classes leads to simpler and more
TL;DR: In this paper, the universal enveloping algebra of a reductive Lie algebra g is shown to be free over the center of the cone of the nilpotent cone, which is a generalization of Kostant's theorem.
Abstract: In this paper we present a simple proof of the fundamental result by B. Kostant which claims that the universal enveloping algebra of a reductive Lie algebra g is free over its center. We also indicate how this result allows to simplify the proof of another important result of B. Kostant—the description of the algebra of functions on the nilpotent cone. We use this technique to prove some generalizations of Kostant's theorem. We also deduce from it a way to check which subalgebras of g are "centrally free."
TL;DR: In this article, the authors give a condition on the images of two morphisms to a Grassmannian that ensures that their fibered product is connected, thereby extending connectedness results of Fulton and Hansen.
Abstract: We give a numerical condition on the images of two morphisms to a Grassmannian (or a product of projective spaces) that ensures that their fibered product is connected, thereby extending connectedness results of Fulton and Hansen. This result is valid over any algebraically closed field; it yields a condition on the class of an irreducible subvariety of a Grassmannian that implies that it is simply connected. This applies in particular to Fano varieties of certain hypersurfaces in a projective space.
TL;DR: In this paper, it was shown that a complex analytic isolated hypersurface singularity admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite dimensional nonnegatively graded algebra.
Abstract: A moduli algebra A ( V ) of hypersurface singularity ( V , 0) is a finite dimensional C -algebra. In 1982, Mather and Yau proved that two germs of complex analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebra are isomorphic. It is a natural question to ask for a necessary and sufficient condition for a complex analytic isolated hypersurface singularity to be quasi-homogeneous in terms of its moduli algebra. In this paper we prove that ( V , 0) admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite dimensional nonnegatively graded algebra. In 1983, Yau introduced a finite dimensional Lie algebra L ( V ) to an isolated hypersurface singularity ( V , 0). L ( V ) is defined to be the algebra of derivations of the moduli algebra A ( V ) and is finite dimensional. We prove that ( V , 0) is quasi-homogeneous singularity if (1) L ( V ) is isomorphic to a nonnegatively graded Lie algebra without center, (2) There exists E in L ( V ) of degree zero such that [ E, D i ] = i D i for any D i in L ( V ) of degree i , and (3) For any element a ∈ m - m 2 where m is the maximal ideal of A ( V ), aE is not in degree zero part of L ( V ).
TL;DR: In this article, the authors give a sufficient condition for G( I ) to be a Cohen-Macaulay local ring, which is described in terms of the depths of A/I n for finitely many n, the reduction numbers of I Q for certain prime ideals Q and the Artin-Nagata property of I in the sense of Ulrich ( Contemp. Math. vol. 159, pp. 373-400).
Abstract: Let G( I ) be the associated graded ring of an ideal I in a Cohen-Macaulay local ring A . We give a sufficient condition for G( I ) to be a Cohen-Macaulay ring. It is described in terms of the depths of A/I n for finitely many n , the reduction numbers of I Q for certain prime ideals Q and the Artin-Nagata property of I in the sense of Ulrich ( Contemp. Math. vol. 159, pp. 373-400). The main theorem unifies several results which are already known in this aspect of the theory. Although the statement of the main theorem is rather complicated, we restate the conditions under special situations, so that they are practical and simple.
TL;DR: In this paper, the authors studied the dynamics of an oriented hypersurface Γ(t )i n N that evolves in time t 0 continuously, and established a stability criterion about a stationary hypersuranface in the form of an eigenvalue problem.
Abstract: The dynamics of a moving hypersurface in a domain D N is studied. It is assumed that the hypersurface moves depending on its curvature, normal vector and position with the boundary that intersects D with a constant contact angle. A stability criterion about a stationary hypersurface is established in the form of an eigenvalue problem, which includes geometrical information of D and the stationary hypersurface. 1. Introduction. In this paper, we study the dynamics of an oriented hy- persurface Γ(t )i n N that evolves in time t 0 continuously. Here, we call such an oriented hypersurface an interface. We assume that the motion of Γ(t) depends on its curvature, normal vector and position x Γ(t). Then we are led to the equation V(x )= F ( ( x), (x), x), x Γ(t), t 0, (1.1) where (x )a nd ( x ) are the mean curvature and the outward unit normal vector of Γ(t )a t x Γ ( t ), respectively, and V(x) is the normal velocity in the out- ward direction at x Γ(t). We assume that the function F depends smoothly on ( , , x) 1 N 1 N and satisfies
TL;DR: In this paper, the authors studied projective homogeneous spaces viewed as G/H, where H is any subgroup scheme containing a Borel subgroup, and showed that G can be identified with the orbit of the highest weight line in the projective space over a simple G -representation L (λ) of a certain highest weight λ.
Abstract: Let G be a reductive linear algebraic group. The simplest example of a projective homogeneous G -variety in characteristic p , not isomorphic to a flag variety, is the divisor x 0 y p 0 + x 1 y p 1 + x 2 y p 2 = 0 in P 2 × P 2 , which is SL 3 modulo a nonreduced stabilizer containing the upper triangular matrices. In this paper embeddings of projective homogeneous spaces viewed as G/H , where H is any subgroup scheme containing a Borel subgroup, are studied. We prove that G/H can be identified with the orbit of the highest weight line in the projective space over the simple G -representation L (λ) of a certain highest weight λ. This leads to some strange embeddings especially in characteristic 2, where we give an example in the C 4 -case lying on the boundary of Hartshorne's conjecture on complete intersections. Finally we prove that ample line bundles on G/H are very ample. This gives a counterexample to Kodaira type vanishing with a very ample line bundle, answering an old question of Raynaud.
TL;DR: In this article, an operateur d'entrelacement explicite for π1 and π2 is presented, which evite cette difficulte, but not the convergence of certain integrale.
Abstract: Soit G un groupe de Lie exponentiel, d'algebre de Lie g. Soit f un point du dual g* de g. Si h1 et h2 sont des polarisations en f qui satisfont la condition de Pukanszky, alors les representations π i = ind G H i χ f ( i = 1, 2, H i = exp h i , χ f ( exp X ) = e if ( X ) pour X dans g) sont irreductibles et equivalentes. Le principal obstacle a la construction d'un operateur d'entrelacement pour ces representations est de prouver la convergence d'une certaine integrale. Dans cet article, nous presentons un operateur d'entrelacement explicite pour π1 et π2 qui evite cette difficulte.