Showing papers in "Inventiones Mathematicae in 1998"

Automorphic forms with singularities on Grassmannians

Abstract: We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some of the applications are as follows. We construct families of holomorphic automorphic forms which can be written as infinite products, which give many new examples of generalized Kac-Moody superalgebras. We extend the Shimura and Maass-Gritsenko correspondences to modular forms with singularities. We prove some congruences satisfied by the theta functions of positive definite lattices, and find a sufficient condition for a Lorentzian lattice to have a reflection group with a finite volume fundamental domain. We give some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.

On a reverse form of the brascamp-lieb inequality

Abstract: We prove a reverse form of the multidimensional Brascamp-Lieb inequality. Our method also gives a new way to derive the Brascamp-Lieb inequality and is rather convenient for the study of equality cases.

Knots, links, and 4-manifolds

Abstract: In this paper we investigate the relationship between isotopy classes of knots and links in S and the di€eomorphism types of homeomorphic smooth 4-manifolds. As a corollary of this initial investigation, we begin to uncover the surprisingly rich structure of di€eomorphism types of manifolds homeomorphic to the K3 surface. In order to state our theorems we need to view the Seiberg-Witten invariant of a smooth 4-manifold as a multivariable (Laurent) polynomial. To do this, recall that the Seiberg-Witten invariant of a smooth closed oriented 4-manifold X with b2 …X † > 1 is an integer valued function which is de®ned on the set of spinc structures over X , (cf. [W], [KM], [Ko1], [T1]). In case H1…X ;Z† has no 2-torsion (which will be the situation in this paper) there is a natural identi®cation of the spinc structures of X with the characteristic elements of H2…X ;Z†. In this case we view the Seiberg-Witten invariant as

Invariance of plurigenera

Abstract: In this paper we give a proof of the following long conjectured result on the invariance of the plurigenera. Main Theorem. Let p : X ! D be a smooth projective family of compact complex manifolds parametrized by the open unit 1-disk D. Assume that the ®bers Xt ˆ py1…t†, t 2 D, are of general type. Then for every positive integer m the plurigenus dimC C…Xt;mKXt† is independent of t 2 D, where KXt is the canonical line bundle of Xt.

Quantum groups and quantum shuffles

Abstract: Let U q + be the “upper triangular part” of the quantized enveloping algebra associated with a symetrizable Cartan matrix We show that U q + is isomorphic (as a Hopf algebra) to the subalgebra generated by elements of degree 0 and 1 of the cotensor Hopf algebra associated with a suitable Hopf bimodule on the group algebra of Z n This method gives supersymetric as well as multiparametric versions of U q + in a uniform way (for a suitable choice of the Hopf bimodule) We give a classification result about the Hopf algebras which can be obtained in this way, under a reasonable growth condition We also show how the general formalism allows to reconstruct higher rank quantized enveloping algebras from U q sl(2) and a suitable irreducible finite dimensional representation

Operator Algebras and Conformal Field Theory III. Fusion of positive energy representations of LSU(N) using bounded operators

Abstract: I Positive energy representations of LSU(N) 477 II Local loop groups and their von Neumann algebras 491 III The basic ordinary di€erential equation 505 IV Vector and dual vector primary ®elds 513 V Connes fusion of positive energy representations 525 References 536

Counting plane curves of any genus

Abstract: We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is proved by studying suitable degenerations of plane curves.

Birational automorphisms of Fano hypersurfaces

Abstract: v : V y! V 0 between smooth three-dimensional quartics V ˆ V4, V 0 ˆ V 0 4 in P is a projective isomorphism. This is the original wording of their theorem; in fact, the arguments of [IM] give a much stronger result, namely, they actually prove that smooth quartics are birationally superrigid. The concept of birational rigidity is not visual. Being rather technical, it makes little impression. It is immediate implications of the property of being birationally (super)rigid that are really impressive. For a projective variety X, smooth in codimension 1, and a linear system jDj, free from ®xed components, we de®ne the threshold of canonical adjunction (or, brie y, just threshold) of the pair …X;D†, setting

Simple geodesics and a series constant over Teichmuller space

Abstract: We investigate the Birman Series set in a neighborhood of a cusp on a punctured surface, showing that it is homeomorphic to a Cantor set union countably many isolated points cross a line. The local topology of the Cantor set is shown to be related in a simple way to the global behavior of simple geodesics. From this we deduce that a certain series is constant across the Teichmuller space.

Discrete decomposability of the restriction of Aq(λ) with respect to reductive subgroups III. Restriction of Harish-Chandra modules and associated varieties

Abstract: Let H⊂G be real reductive Lie groups and π an irreducible unitary representation of G. We introduce an algebraic formulation (discretely decomposable restriction) to single out the nice class of the branching problem (breaking symmetry in physics) in the sense that there is no continuous spectrum in the irreducible decomposition of the restriction π| H . This paper offers basic algebraic properties of discretely decomposable restrictions, especially for a reductive symmetric pair (G,H) and for the Zuckerman-Vogan derived functor module , and proves that the sufficient condition [Invent. Math. '94] is in fact necessary. A finite multiplicity theorem is established for discretely decomposable modules which is in sharp contrast to known examples of the continuous spectrum. An application to the restriction π| H of discrete series π for a symmetric space G/H is also given.

An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg

Abstract: We present a direct analytic approach to the Guillemin-Sternberg conjecture [GS] that `geometric quantization commutes with symplectic reduction', which was proved recently by Meinrenken [M1], [M2] and Vergne [V1], [V2] et al. Besides providing a new proof of this conjecture, our methods also lead immediately to further extensions in various contexts.

Representations p-adiques surconvergentes

Abstract: Ce travail s'inscrit dans le cadre de la classification des representations p-adiques du groupe de galois absolu d'un corps local par des modules (verifiant certaines proprietes, notamment l'action d'un operateur de frobenius), realise par j. -m. Fontaine dans le grothendieck festschrift. Une representation est dite surconvergente lorsque son module associe provient d'un module sur un anneau de series bornees et convergentes en dehors d'un voisinage de 0. Dans le cas de caracteristique mixte, on montre que les representations p-adiques surconvergentes forment une sous-categorie tannakienne, stable par extension et changement de base, contenant les representations abeliennes. On montre ensuite que l'on peut retrouver le module des periodes d'une representation p-adique v surconvergente a partir de son module. Dans le cas de caracteristique p, on etablit, en exhibant un contre-exemple, que les conditions de ramification ne suffisent pas a assurer la surconvergence d'une representation. Dans une derniere partie, on retrouve et precise des resultats de bloch, kato et perrin-riou sur l'exponentielle de bloch et kato relatif a la representation v = q#p(r)

Non-uniform hyperbolicity and universal bounds for s-unimodal maps

Abstract: An S-unimodal map f is said to satisfy the Collet-Eckmann condition if the lower Lyapunov exponent at the critical value is positive. If the infimum of the Lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure. We prove that an S-unimodal map satisfies the Collet-Eckmann condition if and only if it has a uniform hyperbolic structure. The equivalence of several non-uniform hyperbolicity conditions follows. One consequence is that some renormalization of an S-unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the Collet-Eckmann condition is satisfied. The proof uses new universal bounds that hold for any S-unimodal map without periodic attractors.

Non-vanishing of quadratic twists of modular L-functions

Abstract: ;kƒhave been the subject of much study, both because of their intrinsicinterest and because of the prominent role they have played inKolyvagin’s work on the Birch and Swinnerton-Dyer Conjecture (see[B-F-H], [I], [J], [Ko], [Ma-M], [M-M1], [O-S], [P-P]).Waldspurger proved a fundamental theorem [The´ore`me 1, W1]relating these central critical values to the Fourier coe†cients of half-integral weight cusp forms. For notational convenience, if D is afundamental discriminant of a quadratic number field, then define D

Operator Algebras and Conformal Field Theory

Abstract: We report on a programme to understand unitary conformal field theory (CFT) from the point of view of operator algebras. The earlier stages of this research were carried out with Jones, following his suggestion that there might be a deeper “subfactor” explanation of the coincidence between certain braid group representations that had turned up in subfactors, statistical mechanics, and conformal field theory. (Most of our joint work appears in Section 10.) The classical additive theory of operator algebras, due to Murray and von Neumann, provides a framework for studying unitary Lie group representations, although in specific examples almost all the hard work involves a quite separate analysis of intertwining operators and differential equations. Analogously, the more recent multiplicative theory provides a powerful tool for studying the unitary representations of certain infinite–dimensional groups, such as loop groups or Diff S1. It must again be complemented by a detailed analysis of certain intertwining operators, the primary fields, and their associated differential equations.

Borel-weil-bott theory on the moduli stack of g-bundles over a curve

Abstract: Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in [Gro2]. A self-contained account of the “uniformization theorem” of [LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial theory of stacks is outlined in Appendix B.

Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients

Abstract: The problem of the uniqueness in the Cauchy problem for linear di€erential operators has been widely investigated during the last years (see [Z] for references). It is now well understood in the analytic framework, with Holmgren's theorem, where uniqueness always holds (at least for non characteristic surfaces) and in the C1 case, with HoE rmander's theorem ([H1], IV, chap. 28) where the uniqueness is governed by principal normality and pseudo-convexity. The purpose of this work is to ®ll the gap between these two theorems by considering operators with C1 and partly analytic coecients. In particular one of our results will contain both the theorems mentioned above. Let us be more precise. Let na, nb be two non negative integers with n ˆ na ‡ nb 1. We shall set Rn ˆ Rna Rnb and, for x or n in Rn, x ˆ …xa; xb†, n ˆ …na; nb†. Let P ˆ P …x;D† ˆ P …xa; xb;Dxa ;Dxb† be a linear di€erential operator of arbitrary order m, with principal symbol pm. We shall assume that

On the subgroup structure of classical groups

Abstract: In this paper we establish a result on the subgroup structure of classical groups over algebraically closed ®elds, and use this to give a new proof of a fundamental theorem of M. Aschbacher on subgroups of ®nite classical groups. Let V be a ®nite-dimensional vector space over an algebraically closed ®eld K, and let G be one of the classical algebraic groups SL…V †; Sp…V † or SO…V †. Our result is a reduction theorem concerning the subgroups of G: we de®ne a certain collection C of natural proper subgroups of G, and prove that any closed (®nite or in®nite) subgroup of G either lies in a member of C, or is, roughly speaking, a simple group acting irreducibly on V . Aschbacher's result is an analogous reduction theorem for subgroups of ®nite classical groups. We obtain this as a relatively easy consequence of our main result by taking ®xed points under the action of a Frobenius morphism, using a standard process involving Lang's theorem. The proof of the main result uses elementary linear algebra, together with a few basic facts from the theory of algebraic groups. Various complications which arise in the ®nite group setting in [As] become much more straightforward in the algebraic group setting; in particular, questions involving extension ®elds do not occur, and issues of conjugacy are easily settled. When we descend to ®nite Invent. math. 134, 427 ± 453 (1998)

Plane curves of minimal degree with prescribed singularities

Abstract: We prove that there exists a positive α such that for any integer d≥3 and any topological types S1,…,Sn of plane curve singularities, satisfying $$$$

Compact 3-Sasakian 7-manifolds with arbitrary second Betti number

Abstract: Amongst all Riemannian geometries the class of Einstein metrics stands out as perhaps the most natural and interesting [Bes]. Even so there are still many open questions about the relation between the topology of a compact manifold and the possible existence of Einstein metrics. In dimensions bigger than four almost nothing seems to be known in general. Yet, Einstein metrics on compact manifolds are relatively rare and they usually appear as part of additional geometric structure which makes their study tractable. In recent years the ®rst three authors [BGM1, BGM2] have studied a class of Riemannian manifolds known as 3-Sasakian manifolds which have proven to be a remarkable source of compact Einstein manifolds of positive scalar curvature. In view of this work several seemingly unrelated questions regarding the possible breakdown of ®niteness and Betti number bounds come to mind: