Showing papers in "Tohoku Mathematical Journal in 1995"

The functor of a smooth toric variety

Abstract: (1) Y 7→ {line bundle quotients of O Y } . This is easy to prove since a surjection O Y → L gives n + 1 sections of L which don’t vanish simultaneously and hence determine a map Y → Pnk . The goal of this paper is to generalize this representation to the case of an arbitrary smooth toric variety. We will work with schemes over an algebraically closed field k of characteristic zero, and we will fix a smooth n-dimensional toric variety X determined by the fan ∆ in NR = R. As usual, M denotes the dual lattice of N and ∆(1) denotes the 1-dimensional cones of ∆. We will use ∑

Système générateur minimal, diviseurs essentiels et $G$-désingularisations de variétés toriques

Abstract: We give a geometric interpretation of the minimal generating system of the semi-group defined by a rational polyhedral cone in any dimension, via a natural bijection with the set of essential divisors of equivariant desingularizations of the toric variety associated to the cone. We prove, for varieties of dimension three, the existence of a desingularization associated to a regular fan whose edges contain the elements of the minimal generating system, its uniqueness for canonical toric varieties of index at least two, and the uniqueness in general up to flops. We give an example of non-existence of such desingularizations in dimension four. Introduction. L'etude des varietes algebriques munies de Γaction d'un tore, entreprise par Demazure [5] en introduisant les eventails et developpee ensuite par beaucoup d'autres pour les varietes singulieres, repose sur un dictionnaire entre des objets geometriques et combinatoires. Nous poursuivons cette etude avec une interpretation geometrique du systeme generateur minimal G du semi-groupe defϊni par un cone polyedral rationnel σ: il existe une bijection naturelle entre G et les diviseurs essentiels des desingularisations equivariantes de la variete torique Vσ associee a σ sur un corps k, en dimension quelconque (theoreme 1.10). En dimension trois on demontre, en utilisant la theorie de Mori, que les diviseurs essentiels equivariants correspondant aux elements de G sont aussi essentiels pour toute desingularisation de Vσ, non necessairement equivariante (theoreme 2.5). On appelle G-desingularisation de Vσ une desingularisation equivariante definie par une subdivision de σ en un eventail dont les aretes portent les elements de G (G-subdivision reguliere), i.e. telle que tout diviseur exceptionnel soit essentiel. On demontre, en dimension trois, Γexistence des G-desingularisations par une methode constructive a partir d'un modele terminal minimal (theoreme 2.9), une caracterisation des G-subdivisions regulieres par la minimalite de leur volume (proposition 2.12), et le fait que toute G-desingularisation domine un modele terminal minimal (theoreme 2.22). Ce dernier resultat et Γunicite de la G-desingularisation d'une variete torique canonique d'indice > 1 (theoreme 2.23) impliquent Γunicite en general Mots-clέs: Varietes toriques, diviseurs essentiels, desingularisations, theorie de Mori, systemes generateurs de semi-groupes. 1991 Mathematics Subject Classification. Primary 14E15; Secondary 14L30, 14J30. 126 c. BOUVIER ET G. GONZALEZ-SPRINBERG a flops pres (corollaire 2.24). En dimension supέrieure ou egale a quatre, il n'existe pas necessairement des G-desingularisations de varietes toriques, comme le montre le contre-exemple 3.1. Une partie de ces resultats a ete annoncee dans [1]. Dans le premier paragraphe on fait des rappels, on donne les definitions sur les eventails et les varietes toriques utilises dans la suite et on demontre Γinterpretation geometrique du systeme generateur minimal. La deuxieme partie porte sur les resultats pour les varietes toriques en dimension trois: les diviseurs essentiels, Γexistence et la construction des G-desingularisations, ainsi que les proprietes d'unicite. Le dernier paragraphe est un exemple d'une variete torique terminale d'indice un de dimension quatre qui n'admet pas de G-desingularisation. Nous remercions Monique Lejeune-Jalabert et Michel Brion pour des discussions fructueuses, Arlette Guttin-Lombard pour la grande qualite de frappe dont elle est coutumiere. Le deuxieme auteur remercie Γinvitation de JAMS a Ohnuma et du Tokyo Institute of Technology pour un sejour en Aoύt 1993 pendant lequel il a pu faire avancer la redaction de cet article. Les editeurs nous signalent Γexistence de recoupements entre certains resultats de cet article et de Γarticle \"An algorithmic desingularization of 3-dimensional toric varieties\" de Stefano Aguzzoli et Daniele Mundici a paraϊtre dans cette revue. (The editor's note: It appeared in Tόhoku Math. J. 46, No. 4 (1994), 557-572.) 1. Rappels et definitions. Interpretation geometrique du systeme generateur minimal. Soit d un en tier positif, N un Z-module libre de rang d et M=Homz(JV, Z) son dual. Les β-espaces vectoriels N(x)z Q et M(χ)z Q sont notes respectivement NQ et MQ. Un cone dans NQ est toujours suppose convexe et polyedral: pour un tel cone σ, il existe un en tier s et des points vh \\«1=0 oun2 = 0} des elements irreductibles du semi-groupe <τn7V\\{0}. PROPOSITION 1.2. Lensemble Gσ engendre le semi-groupe σn7V\\{0} et il est contenu dans tout systeme generateur de σn7V\\{0}. En particulier, Gσ estfini. G-DESINGULARISATIONS DE VARIETES TORIQUES 127 DEMONSTRATION. Le cone σ etant fortement convexe, il existe un element m de M tel que σ soit contenu dans le demi-espace m~{Q) de NQ et tel que Γintersection de σ avec Γhyperplan m\" 1 ^) soit la face {0} de σ. Un element non nul x de σnN est alors somme d'au plus m(x) elements de σnN\\{0}. Soit * = Σf= 1 n{ une ecriture de x comme somme d'elements de σnΛf\\{0} avec p maximal. Alors les nh l<ί

Characterizing a class of totally real submanifolds of $S^6$ by their sectional curvatures

Abstract: The first author introduced in a previous paper an important Riemannian invariant for a Riemannian manifold, namely take the scalar curvature function and subtract at each point the smallest sectional curvature at that point. He also proved a sharp inequality for this invariant for submanifolds of real space forms. In this paper we study totally real submanifolds in the nearly Kahler six-sphere that realize the equality in that inequality. In this way we characterize a class of totally real warped product immersions by one equality involving their sectional curvatures.

FLAT TOTALLY REAL SUBMANIFOLDS OF CPn AND THE SYMMETRIC GENERALIZED WAVE EQUATION

Abstract: In this paper we show that the projection of the Hopf fibration establishes a one-to-one correspondence between the set of symmetric flat submanifolds in Euclidean sphere and the set of totally real flat submanifolds in complex projective space with the same codimension. We also show that any complete totally real submanifold in complex projective space with mean curvature of constant length and equal dimension and codimension is a flat torus.

Root strings with two consecutive real roots

Abstract: For a Kac-Moody Lie algebra we study pairs of real roots the sum of which is a real root. More precisely, we study in which way the existence of such pair of roots determines the existence of certain subroot system within the root system. 0. Introduction. The study of pairs of real roots {yuγ2} of a Kac-Moody Lie algebra g whose sum is a real root was initiated by Morita in [3] and [4] (though [4] contains a mistake as pointed out in [5]). Morita put this information to good use to derive information about K2 in the case of Kac-Moody groups. Morita looks at the case when <γ1? y2 > = — 1 and = — a where a= 1, 2, 3. (There are also some results if a>3 but only under some strong assumptions on the Cartan matrix.) Morita assumes that yί9 γ2 are positive and that y^— y2 is not a root (a Morita pair in our terminology). His key observation is that a determines the existence of certain entries in the corresponding Cartan matrix A of g (and hence that A somehow sheds information about the existence of such pairs of roots). Our own interest in this problem came out from trying to understand the nilpotency degree of certain subalgebras of g (Conjecture 1 below). We will deal with a above arbitrary and show how a determines a sequence of entries in A with certain properties. 1. Notation and some basic facts about root systems of Kac-Moody Lie algebra. We begin by recalling some well-known objects related to Kac-Moody Lie algebras. Our running reference for this will be [9, Ch. 4, 5]. Most of this material is also covered in [1]. A=(Aij)ijeI will throughout denote a generalized Cartan matrix. (The index set / is allowed to be infinite.) Let (I), 77, Π v ) be a realization of A. Thus

Stability Properties and Integrability of the Resolvent of Linear Volterra Equations

Abstract: Integrability of the resolvent and the stability properties of the zero solution of linear Volterra integrodifferential systems are studied. In particular, it is shown that, the zero solution is uniformly stable if and only if the resolvent is integrable in some sense. It is also shown that, the zero solution is uniformly asymptotically stable if and only if the resolvent is integrable and an additional condition in terms of the resolvent and the kernel is satisfied. Finally, the integrability of the resolvent is obtained under an explicit condition.

Logarithmic divergence of heat kernels on some quantum spaces

Abstract: Asymptotic behaviour of the heat kernels on some explicitly known quantum spaces are studied. Then the heat kernels are shown to be logarithmically divergent. These results suggest to us that the \"dimensions\" of these quantum spaces would not be zero but less than one so that these quantum spaces look almost like \"discrete spaces\". Introduction. In spectral geometry, there is a famous asymptotic formula of McKean and Singer [4] which relates the spectrum of the Laplacian and differential geometrical data (volume, the integration of the scalar curvature etc.) of a compact closed manifold. Namely, let {λί9 λ2,...} be the spectrum of the Laplacian Δ (including the multiplicity) of a given compact closed manifold M of dimension n. Then the asymptotic behaviour of the heat kernel H(ί): = Traced\") = £ e~* for 110 is given by 1 Vr 4πt) where ao = Volume(M) is the volume of the manifold M and aγ is the amount given in terms of the scalar curvature κ(x) of the manifold M by

Complex structures on partial compactifications of arithmetic quotients of classifying spaces of hodge structures

Abstract: Usui, Sampei. Complex structures on partial compactifications of arithmetic quotients of classifying spaces of Hodge structures. Tohoku Math. J. (2) 47 (1995), no. 3, 405--429.

A differentiable sphere theorem by curvature pinching, II

Abstract: Introduction. Let (Mn, g) be a complete, simply connected and ƒÂ-pinched Riemannian n-manifold. In this paper we prove that if ƒÂ=0.654, then M is diffeomorphic to the standard sphere Sn. For a ƒÂ(>1/4)-pinched Riemannian n-manifold, an orientation preserving diffeomorphism f of Sn-1 is naturally defined, and is used in the proof of the differentiable sphere theorem [3, 4]. In fact, if there exists a diffeotopy from f to an isometry f1 of Sn-1, then M is diffeomorphic to the standard sphere. In order to find the minimum of such ƒÂ's it is important to construct a diffeotopy in as many different ways as possible. In this paper, we propose a new construction of a diffeotopy. The statement of our diffeotopy theorem and the construction of diffeotopy in it are fairly simple in comparison with these in [4]. Furthermore, by giving new estimates concerning f and its differential df we prove the differentiable sphere theorem above. In this paper we use the same notation as in [4, • ̃2-• ̃6]. The author would like to thank the referees for careful reading of the previous versions of this paper and for valuable suggestions for improvements.

The mordell property of hyperbolic fiber spaces with noncompact fibers

Abstract: We show that under some boundary conditions a hyperbolic fiber space with noncompact complete hyperbolic fibers has at most finitely many essentially nontrivial sections. Introduction. Let % and & be irreducible, reduced complex spaces and p : 3C -* M a surjective holomorphic mapping with connected fibers. We call (% /?, &) a complex fiber space. We shall say that (#*, /?, J*) is Mordellίc or has the Mordell property if it has at most finitely many meromorphic sections (essentially nontrivial sections) except for the ones which come from trivial sections of meromorphically trivial fiber subspaces of (&,p, &) modulo base change. The term \"meromorphically trivial\" means that it is bimeromorphic to a trivial fiber subspace as fiber spaces. In 1974, Lang [11, p. 781] conjectured the following analog of MordelPs conjecture (cf. Manin [17] and Grauert [5]); an algebraic family of compact hyperbolic complex spaces is Mordellic. Noguchi [24, Theorem B] proved that if a hyperbolic fiber space ($£, p, $) with compact fibers is hyperbolically imbedded into its compactification (% p, $) along d$, then it is Mordellic. In this paper, we study a noncompact version of Noguchi's result. Let 3C and $ be irreducible, reduced compact complex spaces and p\\ &'^>M a surjective holomorphic mapping. Let 9£ and & be Zariski open subsets of 3C and ^ , respectively and p=p\\βί the restriction over ΘC which is a surjection from ^ t o &. We consider the fiber space (β, p, ^) with hyperbolic fibers, which we call a hyperbolic fiber space. Let be the union of all irreducible components of d% = 9C\\9£ which are not contained where d& = MAIN THEOREM. Let {% p, J*) and (St, p, # ) be as above. Assume that (#\", /?, &) is hyperbolically imbedded in {% /?, $) along the fibers and that 3£\\dh3£ is locally complete hyperbolic in 9C. Then (β£, p, $) is Mordellic. See Definition 1.1 in §1 for the assumptions. This extend the result in the case of trivial fiber spaces obtained in [28]. Some higher dimensional cases were considered by Riebesehl [27], Noguchi [20], 1991 Mathematics Subject Classification. Primary 32H20; Secondary 32L05, 14H15, 11G99.

The local theta correspondence of irreducible type $2$ dual reductive pairs

Abstract: The explicit Howe duality correspondence is partially solved in the case of irreducible type 2 dual reductive pairs defined over a non-Archimedean local field. Introduction. Let (GLn, GLm) be an irreducible type 2 dual reductive pair defined over a non-Archimedean local field F. The Weil representation ωnm of GLn(F) x GLm(F) on the Schwartz-Bruhat space ϊf{Mnm(F)) is given by ωΛim(Λ, g)f(x) = \\dQth\\-^ \\ det g \\f(h \" 'xg) (h e GLn(F), g e GLm(F)). Then a problem on the (explicit) Howe correspondence for (GLn, GLm) is stated as follows. For a given irreducible admissible representation σ of GLn{F), determine an irreducible admissible representation σ' oΐGLm(F) such that Hom G L n ( F ) x GLm(io( n,m' σ ® σ)φ0. The purpose of this paper is to study this problem in the case where m = n+l and σ is generic. Our starting point is a global theta series lifting of a cusp form on the adele group GLn(A). For a cusp form φ on GLn(A) and a Schwartz-Bruhat function fe<9 ?(Mnm(A)), we define a theta series lifting φ}, where s is a complex parameter with Re(V)«0. This φsf is an automorphic form on GLm(A). In Section 1, we calculate a Whittaker function WφS of φ s f and prove that WφS is identically zero if mφn, n+ 1. In the case where m = « or m = n+ 1, the function WφS is represented by a convolution of the Whittaker function Wφ of φ and a certain function Φm(f) related to /. More precisely, we have a formula of the form Wψ,f{g) = Wφ(h) I det h } s AΦm(ωn,m(g)f)(h)dh , (m = n,n+\\). J Un(Λ) \\GLn(Λ) On the basis of this formula, we can define a local theta series lifting of a local Whittaker function. This is the reason why we study the Howe correspondence in the case where m — n+\\ and σ is generic. The case m = n will be investigated in another paper [17]. We state the results of this paper. Let σ be an irreducible generic representation of GLn(F). By using a local analogue of the formula mentioned above, one can construct Partly supported by the Grants-in-Aid for Encouragement of Young Scientists, The Ministry of Education, Science and Culture, Japan. 1991 Mathematics Subject Classification. Primary 11F70; Secondary 11F27, 22E50.

On q-structures of quasisymmetric domains

Abstract: We will give a complete classification of (^-structures of quasisymmetric domains. In the standard case, it will be shown that there are only very natural g-structures coming from semisimple g-algebras with positive involutions. As is shown in the Appendix, when the domain is symmetric, any g-structure of it as a quasisymmetric domain can uniquely be extended to one as a symmetric domain. The purpose of this note is to determine the (^-structures of quasisymmetric domains. The notion of a quasisymmetric domain was introduced in [S3] (cf. also [S6, Ch. V]). It was shown that, among Siegel domains (of the second kind), the symmetric domains were characterized by three conditions (i), (ii), (iii). A Siegel domain is called quasisymmetric if it satisfies the conditions (i), (ii). It is known that any symmetric domain Q) with a fixed boundary component J* has a natural structure of a fiber space (a Siegel domain of the third kind) over ^ , in which the fiber over each point of 3F is a quasisymmetric domain. All quasisymmetric domains of \"standard\" type are obtained in this form (see §4), while there are quasisymmetric domains of non-standard (quadratic) type that are not obtained in this manner. A quasisymmetric domain Sfj is defined by a data (U, V, A, <&, /), where U, (V, I) are real and complex vector spaces of finite dimension, / denoting a complex structure on V. %> is a self-dual homogeneous cone in U (condition (i)) and A is an alternating bilinear map Vx V-^U such that A(v, Iv) (υ, υ'eV) is \"^-positive\" (see 1.1). In §§1, 2 we summarize basic definitions and properties concerning quasisymmetric domains. Here we give the condition (ii) in the form independent of the complex structure /, viewing / as a point in the parameter space © = ®(K, A, €). To give a g-structure of &Ί is, roughly speaking, equivalent to giving a β-structure of (U, V) such that the affine automorphism group Aff Sfι is defined over Q. By virtue of the complete reducibility of quasisymmetric domains (see 2.5), our problem of determining (J-structures of 9>ι is reduced to the g-irreducible case. A general method of determining β-irreducible (J-structures of 9?ι with K/0 is given in §3. In particular, it will be shown that a g-structure of £f ι is essentially determined by that of the enveloping algebra of the representation of LieAut^ on V, which is a (β-simple) β-algebra with positive involution. Applying this method to the standard and non-standard cases, in §§4, 5, 1991 Mathematics Subject Classification. Primary 32M15; Secondary 11E39, 11F55, 20G20.