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The paper deals with relations between the Hard Lefschetz property, (non)vanishing of Massey products and the evenness of odd-degree Betti numbers of closed symplectic manifolds. It is known that closed symplectic manifolds can violate all these properties (in contrast with the case of Kaehler manifolds). However, the relations between such homotopy properties seem to be not analyzed. This analysis may shed a new light on topology of symplectic manifolds. In the paper, we summarize our knowledge in tables (different in the simply-connected and in symplectically aspherical cases). Also, we discuss the variation of symplectically harmonic Betti numbers on some 6-dimensional manifolds.
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arxiv:math/0002071
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Finding examples of tangentially degenerate submanifolds (submanifolds with degenerate Gauss mappings) in an Euclidean space $R^4$ that are noncylindrical and without singularities is an important problem of differential geometry. The first example of such a hypersurface was constructed by Sacksteder in 1960. In 1995 Wu published an example of a noncylindrical tangentially degenerate algebraic hypersurface in $R^4$ whose Gauss mapping is of rank 2 and which is also without singularities. This example was constructed (but not published) by Bourgain. In this paper, the authors analyze Bourgain's example, prove that, as was the case for the Sacksteder hypersurface, singular points of the Bourgain hypersurface are located in the hyperplane at infinity of the space $R^4$, and these two hypersurfaces are locally equivalent.
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arxiv:math/0002092
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In this paper, we prove that the dimension of the space spanned by the characters of the symmetric powers of the standard n-dimensional representation of the symmetric group S_n is asymptotic to n^2/2. This is proved by using generating functions to obtain formulas for upper and lower bounds, both asymptotic to n^2/2, for this dimension. In particular, for n>6, these characters do not span the full space of class functions on S_n.
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arxiv:math/0002106
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We prove a Riemann-Roch formula for deformation quantization of complex manifolds and its corollary, an index theorem for elliptic pairs conjectured by Schapira and Schneiders.
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arxiv:math/0002115
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Given a Radon measure $\mu$ on $R^d$, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties that hold when $\mu$ is doubling remain valid for the space BMO introduced in this paper, without assuming $\mu$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from $H^1$ into $L^1$ are also bounded on $L^p$, $1<p<\infty$. This result gives a new proof of the T(1) theorem for the Cauchy transform with non doubling measures. Finally, a result about commutators is obtained.
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arxiv:math/0002152
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Using the classical S.Lie method we obtain a complete description of infinitesimal symmetries of a holomorphic PDE system defining the Segre family of a real analytic hypersurface. This gives a new proof of some well known results of CR geometry.
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arxiv:math/0002197
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Let Y be a projective non-singular curve of genus g, X a projective manifold, both defined over the field of complex numbers, and let f:X ---> Y be a surjective morphism with general fibre F. If the Kodaira dimension of X is non-negative, and if Y is the projective line we show that f has at least 3 singular fibres. In general, for non-isotrivial morphisms f, one expects that the number of singular fibres is at least 3, if g=0, or at least 1, if g=1. Using the strong additivity of the Kodaira dimension, this is verified, if either F is of general type, or if F has a minimal model with a semi-ample canonical divisor. The corresponding result has been obtained by Migliorini and Kovacs, for families of surfaces of general type and for families of canonically polarized manifolds, and by Oguiso-Viehweg for families of elliptic surfaces. As a byproduct we obtain explicit bounds for the degree of the direct image of powers of the dualizing sheaf, generalizing those obtained by Bedulev-Viehweg for families of surfaces of general type.
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arxiv:math/0002203
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Dijkgraaf, Pasquier and Roche introduced twisted quantum doubles of a finite group in the context of conformal field theory. We study equivalences that arise among the braided monoidal categories associated to these quantum doubles, especially in the commutative case. This involves a close study of the cohomology of various complexes introduced by Eilenberg-MacLane. We show among other things that an equivalence of braided monoidal categories is the same as gauge equivalence of the corresponding quantum doubles, and that an invariant for an equivalence class is given by a metabolic quadratic space.
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arxiv:math/0002246
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An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.
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arxiv:math/0003030
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In this paper we give the projective generation of congruences of order 1 of r-dimensional projective spaces in P^N from their focal loci. In a natural way, this construction shows that the corresponding surfaces in the grassmannian are the Veronese surface, and rational ruled surfaces eventually with singularities. We characterize when these surfaces are smooth, recovering and generalizing a Ziv Ran's result.
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arxiv:math/0003048
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Is a Verma module transformed into another Verma module by a selfequivalence? The answer is affirmative and the proof suggests a notion of standard object in the category of Harish-Chandra modules that coincides often, but not always, with the usual one.
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arxiv:math/0003069
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Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove this theorem, we'll prove the Gherardelli linkage theorem, which asserts that a partial intersection of two hypersurfaces is subcanonical if and only if its residual intersection is, scheme-theoretically, the intersection of the two hypersurfaces with a third.
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arxiv:math/0003075
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We prove relations among the classes of certain divisors on the moduli spaces of curves with marked points, generalizing the Brill-Noether Ray Theorem of Eisenbud and Harris.
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arxiv:math/0003104
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We report in this survey some new results concerning noncommutative Chern characters: construction and the cases when they are exactly computed. The major result indicates some clear relation of these noncommutative objects and their commutative counterparts. This survey can be considered as the second part of the previous survey (J. of Lie Theory vol. 3 (1993), 149-176.
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arxiv:math/0003108
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It is well-known that the automorphism towers of infinite centreless groups of cardinality kappa terminate in less than (2^{kappa})^+ steps. But an easy counting argument shows that (2^{kappa})^+ is not the best possible bound. However, in this paper, we will show that it is impossible to find an explicit better bound using ZFC.
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arxiv:math/0003120
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Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which was left unsolved in both papers was the number m of times one must `cycle' (resp. `decycle') in order to increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the given conjugacy class. Our main result is to prove that m is bounded above by n-2 in the situation of the second algorithm and by ((n^2-n)/2)-1 in the situation of the first. As a corollary, we show that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, as defined by Charney, and so we also obtain a polynomial-time algorithm for computing this geodesic length.
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arxiv:math/0003125
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An L-embedded Banach spaace is a Banach space which is complemented in its bidual such that the norm is additive between the two complementary parts. On such spaces we define a topology, called an abstract measure topology, which by known results coincides with the usual measure topology on preduals of finite von Neumann algebras (like $L_1([0,1])$). Though not numerous, the known properties of this topology suffice to generalize several results on subspaces of $L_1([0,1])$ to subspaces of arbitrary L-embedded spaces.
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arxiv:math/0003154
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For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group $G$ of the canonical discrete Lagrange 2-form $\omega_\mathbb{L}$ on $G \times G$. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example.
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arxiv:math/0004018
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A very short and direct proof along the lines of the Kamae-Katznelson-Weiss approach.
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arxiv:math/0004070
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Let X be a complex Fano-manifolds with second Betti-number 1 which carries a contact structure. It follows from previous work that such a manifold can always be covered by lines. Thus, it seems natural to consider the geometry of lines in greater detail. In this brief note we show that if x in X is a general point, then all lines through x are smooth. If X is not the projective space, then the tangent spaces to lines generate the contact distribution at x. As a consequence we obtain that the contact structure on X is unique, a result previously obtained by C. LeBrun in the case that X is a twistor space.
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arxiv:math/0004103
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A general approach to the well-behaved unbounded *-representations of a *-algebra X is proposed. Let B be a normed *-algebra equipped with a left action |> of X on B such that (x |> a)^+ b=a^+(x^+ |> b) for a,b\in B and x\in X. Then the pair (X,B) is called a compatible pair. For any continuous non-degenerate *-representation \rho of B there exists a closed *-representation \rho' of X such that \rho'(x)\rho(b)=\rho(x |> b), where x\in X and b\in B. The *-representations \rho' are called the well-behaved *-representations associated with the compatible pair (X,B). A number of examples are developed in detail.
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arxiv:math/0004163
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We prove that, with probability one, eventually there are no more than three favourite (i.e. most visited) sites of simple random walk. This partially answers a relatively long standing question of Pal Erdos and Pal Revesz.
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arxiv:math/0004164
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In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang-Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) which may not be ambient isotopic. We illustrate our results with explicit computations using solutions of the trigonometric Yang-Baxter equation associated with the one-parameter family of minimal typical representations of the quantum superalgebra U_q[gl(2|1)]. We have implemented Mathematica code to evaluate the invariants for all prime knots up to 10 crossings.
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arxiv:math/0004169
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We show that counting functions of covers of $\mathbb{C}^\times$ are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves by Dijkgraaf.
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arxiv:math/0004178
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Let C be a connected noetherian hereditary abelian Ext-finite category with Serre functor over an algebraically closed field k, with finite dimensional homomorphism and extension spaces. Using the classification of such categories from math.RT/9911242, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.
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arxiv:math/0005100
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A refinement of the q-trinomial coefficients is introduced, which has a very powerful iterative property. This ``T-invariance'' is applied to derive new Virasoro character identities related to the exceptional simply-laced Lie algebras E_6,E_7 and E_8.
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arxiv:math/0005123
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We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone cohomology, such that the cohomology group in dimension one is the group of equivalence classes of extensions.
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arxiv:math/0005134
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There has been much recent interest in the satisfiability of random Boolean formulas. A random k-SAT formula is the conjunction of m random clauses, each of which is the disjunction of k literals (a variable or its negation). It is known that when the number of variables n is large, there is a sharp transition from satisfiability to unsatisfiability; in the case of 2-SAT this happens when m/n --> 1, for 3-SAT the critical ratio is thought to be m/n ~ 4.2. The sharpness of this transition is characterized by a critical exponent, sometimes called \nu=\nu_k (the smaller the value of \nu the sharper the transition). Experiments have suggested that \nu_3 = 1.5+-0.1, \nu_4 = 1.25+-0.05, \nu_5=1.1+-0.05, \nu_6 = 1.05+-0.05, and heuristics have suggested that \nu_k --> 1 as k --> infinity. We give here a simple proof that each of these exponents is at least 2 (provided the exponent is well-defined). This result holds for each of the three standard ensembles of random k-SAT formulas: m clauses selected uniformly at random without replacement, m clauses selected uniformly at random with replacement, and each clause selected with probability p independent of the other clauses. We also obtain similar results for q-colorability and the appearance of a q-core in a random graph.
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arxiv:math/0005136
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Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration defines a perfect pairing between de Rham cohomology with values in the connection and homology with values in the dual connection.
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arxiv:math/0005137
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The paper consists of two sections. The first section provides a new definition of mirror symmetry of abelian varieties making sense also over $p$-adic fields. The second section introduces and studies quantized theta-functions with two-sided multipliers, which are functions on non-commutative tori. This is an extension of an earlier work by the author. In the Introduction and in the Appendix the constructions of this paper are put into a wider context.
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arxiv:math/0005143
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We study 3-valent maps $M_n(p,q)$ consisting of a ring of $n$ $q$-gons whose the inner and outer domains are filled by $p$-gons, for $p,q \ge 3$. We describe a domain in the space of parameters $p$, $q$, and $n$, for which such a map may exist. With four infinite sequences of maps - prisms $M_p(p \ge 3,4)$, $M_4(4,q \ge 4)$, $M_4(5,5t+2 \ge 7)$, $M_4(5,5t+3 \ge 8)$, we give 20 sporadic ones. The maps whose $p$-gons form two paths are first two infinite sequences and 5 maps: $M_{28}(7,5)$, $M_{12}(6,5)$, $M_{10}(5,6)$, $M_{20}(5,7)$, $M_{2}(3,6)$
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arxiv:math/0005269
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In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics.
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arxiv:math/0005306
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The "quantum-event / prime ideal in a category/ noncommutative-point" alternative to "classical-event / commutative prime ideal/ point" is suggested. Ideals in additive categories, prime spectra and representation of quivers are considered as mathematical tools appropriate to model quantum mechanics. The space-time framework is to be reconstructed from the spectrum of the path category of a quiver. The interference experiment is considered as an example.
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arxiv:math/0006024
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Let $X$ be a smooth projective curve over the complex numbers. To every representation $\rho\colon \GL(r)\lra \GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that there be an integer $\alpha$ with $\rho(z \id_{\C^r})=z^\alpha \id_V$ for all $z\in\C^*$ we associate the problem of classifying triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line bundle on $X$, and $\phi\colon E_\rho\lra L$ is a non trivial homomorphism. Here, $E_\rho$ is the vector bundle of rank $\dim V$ associated to $E$ via $\rho$. If we take, for example, the standard representation of $\GL(r)$ on $\C^r$ we have to classify triples $(E,L,\phi)$ consisting of $E$ as before and a non-zero homomorphism $\phi\colon E\lra L$ which includes the so-called Bradlow pairs. For the representation of $\GL(r)$ on $S^2\C^3$ we find the conic bundles of Gomez and Sols. In the present paper, we will formulate a general semistability concept for the above triples which depends on a rational parameter $\delta$ and establish the existence of moduli spaces of $\delta$-(semi)stable triples of fixed topological type. The notion of semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the main reason that such a general approach becomes feasible. In the known examples (the above, Higgs bundles, extension pairs, oriented framed bundles) we show how to recover the "usual" semistability concept. This process of simplification can also be formalized. Altogether, our results provide a unifying construction for the moduli spaces of most decorated vector bundle problems together with an automatism for finding the right notion of semistability and should therefore be of some interest.
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arxiv:math/0006029
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This is the manuscript for Proceedings of International Conference and Workshop on Valuation Theory held at University of Saskachewan, Canada in 1999. I have succeeded in showing that any two-dimensional hypersurface singularities of germs of varieties in any characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties. The new proof for the two-dimensional case depends on new ideas. Ideas are essentially different from Abhyankar's one in 1956 and Lipman's one in 1978. It seems to be possible to generalize the new proof into higher dimensional cases, if we add several ideas further. In this article I try to explain my new ideas rather than partial result I explained at the conference.
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arxiv:math/0006071
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We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincare' decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a discrete group.
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arxiv:math/0006082
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We construct and prove a diagrammatic version of the Duflo isomorphism between the invariant subalgebra of the symmetric algebra of a Lie algebra and the center of the universal enveloping algebra. This version implies the original for metrized Lie algebras (Lie algebras with an invariant non-degenerate bilinear form). As an application of this isomorphism, we will compute the Kontsevich integral of the unknot and the Hopf link to all orders. At the core of the proof, we use an elementary property of the Hopf link which can be summarized by the equation ``1+1=2'' in abacus arithmetic: doubling one component of the Hopf link is equivalent to taking the connected sum of two Hopf links. This property of the Hopf link turns out, when suitably interpreted, to be exactly the property required for the Duflo map to be multiplicative. To compute the Kontsevich integral of the unknot, we use a property of the unknot that can be summarized by ``n * 0 = 0'': the n-fold connected cabling of the unknot is again an unknot.
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arxiv:math/0006083
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Toral automorphisms, represented by unimodular integer matrices, are investigated with respect to their symmetries and reversing symmetries. We characterize the symmetry groups of GL(n,Z) matrices with simple spectrum through their connection with unit groups in orders of algebraic number fields. For the question of reversibility, we derive necessary conditions in terms of the characteristic polynomial and the polynomial invariants. We also briefly discuss extensions to (reversing) symmetries within affine transformations, to PGL(n,Z) matrices, and to the more general setting of integer matrices beyond the unimodular ones.
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arxiv:math/0006092
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Let g_0(N) be the genus of the modular curve X_0(N). We record several properties of the sequence {g_0(N)}. Even though the average size of g_0(N) is 1.25N/pi^2, a random positive integer has probability zero of being a value of g_0(N). Also, if N is a random positive integer then g_0(N) is odd with probability one. The smallest non-negative integer not occuring as g_0(N) for any N is 150; the smallest such odd integer is 49267.
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arxiv:math/0006096
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We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.
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arxiv:math/0006144
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Let X be a smooth projective variety. Using modified psi classes on the stack of genus zero stable maps to X, a new associative quantum product is constructed on the cohomology space of X. When X is a homogeneous variety, this structure encodes the characteristic numbers of rational curves in X, and specialises to the usual quantum product upon resetting the parameters corresponding to the modified psi classes. For X = P^2, the product is equivalent to that of the contact cohomology of Ernstrom-Kennedy.
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arxiv:math/0006148
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Let $M$ be a simply-connected closed manifold of dimension $\geq 5$ which does not admit a metric with positive scalar curvature. We give necessary conditions for $M$ to admit a scalar-flat metric. These conditions involve the first Pontrjagin class and the cohomology ring of $M$. As a consequence any simply-connected scalar-flat manifold of dimension $\geq 5$ with vanishing first Pontrjagin class admits a metric with positive scalar curvature. We also describe some relations between scalar-flat metrics, almost complex structures and the free loop space.
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arxiv:math/0006149
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For transcendental values of q the quantum tangent spaces of all left-covariant first order differential calculi of dimension less than four on the quantum group $\SLq 2$ are given. All such differential calculi $\Gamma $ are determined and investigated for which the left-invariant differential one-forms $\omega (u^1_2)$, $\omega (u^2_1)$ and $\omega (u^1_1-u^2_2)$ generate $\Gamma $ as a bimodule and the universal higher order differential calculus has the same dimension as in the classical case. Important properties (cohomology spaces, *-structures, braidings, generalized Lie brackets) of these calculi are examined as well. Keywords: quantum groups, noncommutative differential calculus, quantum tangent space
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arxiv:math/0006211
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How much of the combinatorial structure of a pointed polyhedron is contained in its vertex-facet incidences? Not too much, in general, as we demonstrate by examples. However, one can tell from the incidence data whether the polyhedron is bounded. In the case of a polyhedron that is simple and "simplicial," i.e., a d-dimensional polyhedron that has d facets through each vertex and d vertices on each facet, we derive from the structure of the vertex-facet incidence matrix that the polyhedron is necessarily bounded. In particular, this yields a characterization of those polyhedra that have circulants as vertex-facet incidence matrices.
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arxiv:math/0006225
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We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist.
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arxiv:math/0007040
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New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold $X$ are exhibited. When the cohomology of $X$ is generated by divisor classes and classes ``with vanishing one-point invariants,'' the relations determine many-point invariants in terms of one-point invariants.
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arxiv:math/0007082
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Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. Orbifold elliptic genus is given in terms of the fixed point sets of the action. We show that the generating function for this orbifold elliptic genus $\sum Ell_{orb}(X^n,\Sigma_n)p^n$ for symmetric groups $\Sigma_n$ acting on $n$-fold products coincides with the one proposed by Dijkgraaf, Moore, Verlinde and Verlinde. Two notions of elliptic genera are conjectured to coincide.
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arxiv:math/0007108
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The nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras. They can form carrier spaces for additional Jordanian twists. The latter can be used to construct sequences of twists for algebras whose root systems contain long series of roots. The corresponding twist for the so(5) algebra is given explicitly.
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arxiv:math/0007182
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Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least two have this property.) Suppose that such a G acts on a compact manifold M by automorphisms of a codimension-one C2 foliation, F. We show that if F has a compact leaf, then some finite-index subgroup of G fixes a compact leaf of F. Furthermore, we give sufficient conditions for some finite-index subgroup of G to fix each leaf of F.
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arxiv:math/0008012
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We show that for some Hopf subalgebras in U_F(so(M)) nontrivially deformed by a twist F it is possible to find the nonlinear primitive copies. This enlarges the possibilities to construct chains of twists. For orthogonal algebra U(so(M)) we present a method to compose the full chains with carrier space as large as the Borel subalgebra B(so(M)). These chains can be used to construct the new deformed Yangians.
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arxiv:math/0008044
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In this paper, we first present a classification theorem of infinite-dimensional simple Novikov algebras over an algebraically closed field with characteristic 0. Then we classify all the irreducible modules of a certain infinite-dimensional simple Novikov algebras with an idempotent element whose left action is locally finite.
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arxiv:math/0008072
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The notion of a virtual knot introduced by L. Kauffman induces the notion of a virtual braid. It is closely related with a welded braid of R. Fenn, R. Rimanyi and C. Rourke. Alexander's and Markov's theorems for virtual knots and braids are proved. Similar results for welded knots and braids are also proved.
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arxiv:math/0008092
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We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations.
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arxiv:math/0008137
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In this note we survey results in recent research papers on the use of Lie groups in the study of partial differential equations. The focus will be on parabolic equations, and we will show how the problems at hand have solutions that seem natural in the context of Lie groups. The research is joint with D.W. Robinson, as well as other researchers who are listed in the references.
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arxiv:math/0008159
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In the 80's M. Cornalba and J. Harris discovered a relation among the Hodge class and the boundary classes in the Picard group with rational coefficients of the moduli space of stable, hyperelliptic curves. They proved the relation by computing degrees of the classes involved for suitable one-parameter families. In the present article we show that their relation can be obtained as the class of an appropriate, geometrically meaningful empty set, thus conforming with C. Faber's general philosophy to finding relations among tautological classes in the Chow ring of the moduli space of curves. The empty set we consider is the closure of the locus of smooth, hyperelliptic curves having a special ramification point.
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arxiv:math/0008176
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The Colombeau algebra of generalized functions allows to unrestrictedly carry out products of distributions. We analyze this operation from a microlocal point of view, deriving a general inclusion relation for wave front sets of products in the algebra. Furthermore, we give explicit examples showing that the given result is optimal, i.e. its assumptions cannot be weakened. Finally, we discuss the interrelation of these results with the concept of pullback under smooth maps.
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arxiv:math/0008206
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In this work, we give a purely analytic introduction to the phenomenon of mirror symmetry for quintic threefolds via classical hypergeometric functions and differential equations for them. Starting with a modular map and recent transcendence results for its values, we regard a mirror map $z(q)$ as a concept generalizing the modular one. We give an alternative approach demonstrating the existence of non-linear differential equations for the mirror map, and exploit both an elegant construction of Klemm-Lian-Roan-Yau and the Ax theorem to prove that the Yukawa coupling $K(q)$ does not satisfy any algebraic differential equation of order less than 7 with coefficients from $\mathbb{C}(q)$.
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arxiv:math/0008237
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The following critical phenomenon was recently discovered. When a memoryless source is compressed using a variable-length fixed-distortion code, the fastest convergence rate of the (pointwise) compression ratio to the optimal $R(D)$ bits/symbol is either $O(\sqrt{n})$ or $O(\log n)$. We show it is always $O(\sqrt{n})$, except for discrete, uniformly distributed sources.
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arxiv:math/0009018
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We give a criterium of holomorphy for some type formal power series. This gives a stronger form of a Rothstein's type extension theorem for a particular ring of holomorphic functions.
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arxiv:math/0009031
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Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin's classical proof of Euler's pentagonal number theorem.
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arxiv:math/0009036
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Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2-dimensional information.
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arxiv:math/0009082
|
We give an explicit way of calculating the set of homotopy classes of morphisms from a Tamsamani n-category A to another one B. This calculation uses a Reedy-cofibrant cosimplicial resolution of A, using a new notion of ``free cofibration'' of n-precats. The free cofibrations of n-precats seem to be the analogue for n-categories of the Bousfield-Kan cofibrations in the theory of diagrams.
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arxiv:math/0009107
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The Blaschke-Lebesgue Theorem states that among all planar convex domains of given constant width B the Reuleaux triangle has minimal area. It is the purpose of the present note to give a direct proof of this theorem by analyzing the underlying variational problem. The advantages of the proof are that it shows uniqueness (modulo rigid deformations such as rotation and translation) and leads analytically to the shape of the area-minimizing domain. Most previous proofs have relied on foreknowledge of the minimizing domain. Key parts of the analysis extend to the higher-dimensional situation, where the convex body of given constant width and minimal volume is unknown.
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arxiv:math/0009137
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It is well-known that the action of a hyperbolic element (``cat map'') of the modular group on the 2-torus has strong chaotic dynamical properties such as mixing and exponential decay of correlations. In this note we study stability of this behaviour with respect to kicks. Our approach is based on geometric group theory, and in particular on a new result on quasimorphisms of the modular group.
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arxiv:math/0009143
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The holonomic rank of the A-hypergeometric system H_A(\beta) is shown to depend on the parameter vector \beta when the underlying toric ideal I_A is a non Cohen Macaulay codimension 2 toric ideal. The set of exceptional parameters is usually infinite.
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arxiv:math/0009148
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Given a del Pezzo surface of degree d between 1 and 6, possibly with rational double points, we construct a "tautological" holomorphic G-bundle over X, where G is a reductive group which is an appropriate conformal form of the simply connected complex linear group whose coroot lattice is isomorphic to the primitive cohomology of the minimal resolution of X. For example, in case d=3 and X is a smooth cubic surface, the rank 27 vector bundle over X associated to the G-bundle constructed above and the standard 27-dimensional representation of E_6 is a direct sum of the line bundles associated to the 27 lines on X. We also discuss the restriction of the G-bundle to smooth hyperplane sections.
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arxiv:math/0009155
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Some aspects of the construction of SW Floer homology for manifolds with non-trivial rational homology are analyzed. In particular, the case of manifolds that are obtained as zero-surgery on a knot in a homology sphere, and for torsion spinc structures. We discuss relative invariants in the case of torsion spinc structures.
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arxiv:math/0009159
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We prove an explicit combinatorial formula for certain structure constants of the T-equivariant cohomology of the flag manifold SLn/B. Our result generalizes the Pieri-type formula in ordinary cohomology proved by Sottile in 1996. Our result also gives a Pieri-type formula for the double Schubert polynomials introduced by Lascoux and Schutzenberger.
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arxiv:math/0009197
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We prove that the value of the quasi-trace on an idempotent element in a AW*-factor of type II_1 is the same as the dimension of its left (or right) support.
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arxiv:math/0009221
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Frenkel-Reshetikhin introduced $q$-characters of finite dimensional representations of quantum affine algebras. We give a combinatorial algorithm to compute them for all simple modules. Our tool is $t$-analogue of the $q$-characters, which is similar to Kazhdan-Lusztig polynomials, and our algorithm has a resemblance with their definition.
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arxiv:math/0009231
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We define an equivariant $K_0$-theory for \textit{Yetter-Drinfeld} algebras over a Hopf algebra with an invertible antipode. We then show that this definition can be generalized to all Hopf-module algebras. We show that there exists a pairing, generalizing Connes' pairing, between this theory and a suitably defined Hopf algebra equivariant cyclic cohomology theory.
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arxiv:math/0009236
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We establish global existence in 3+1 dimensions of small-amplitude solutions of quasilinear Dirichlet-wave equations satisfying the null condition outside of star-shapped obstacles.
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arxiv:math/0009237
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There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these. In this paper we define certain projective varieties V(T,t) whose properties determine whether, if T is in the defining set, the code has minimum distance exceeding t. Thus our attention shifts to the study of these varieties. By investigating them using class field theory and arithmetical geometry, we will prove various new bounds. It is interesting, however, to note that there are cases that existing methods handle, that our methods do not, and vice versa. We end with a number of conjectures.
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arxiv:math/0009257
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For a quantum Lie algebra $\Gamma$, let $\Gamma^\wedge$ be its exterior extension (the algebra $\Gamma^\wedge$ is canonically defined). We introduce a differential on the exterior extension algebra $\Gamma^\wedge$ which provides the structure of a complex on $\Gamma^{\wedge}$. In the situation when $\Gamma$ is a usual Lie algebra this complex coincides with the "standard complex". The differential is realized as a commutator with a (BRST) operator $Q$ in a larger algebra $\Gamma^\wedge[\Omega]$, with extra generators canonically conjugated to the exterior generators of $\Gamma^{\wedge}$. A recurrent relation which defines uniquely the operator $Q$ is given.
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arxiv:math/0010060
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This note briefly reviews the {\it Mirror Principle} as developed in the series of papers \LLYI\LLYII\LLYIII\LLYIV\LCHY. We illustrate this theory with a few new examples. One of them gives an intriguing connection to a problem of counting holomorphic disks and annuli. This note has been submitted for the proceedings of the Workshop on Strings, Duality and Geometry the C.R.M. in Montreal of March 2000.
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arxiv:math/0010064
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We give a construction of a nuclear $C^\ast$-algebra associated with an amalgamated free product of groups, generalizing Spielberg's construction of a certain Cuntz-Krieger algebra associated with a finitely generated free product of cyclic groups. Our nuclear $C^\ast$-algebras can be identified with certain Cuntz-Krieger-Pimsner algebras. We will also show that our algebras can be obtained by the crossed product construction of the canonical actions on the hyperbolic boundaries, which proves a special case of Adams' result about amenability of the boundary action for hyperbolic groups. We will also give an explicit formula of the $K$-groups of our algebras. Finally we will investigate the relationship between the KMS states of the generalized gauge actions on our $C^\ast$ algebras and random walks on the groups.
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arxiv:math/0010097
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This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included.
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arxiv:math/0010114
|
We present a general approach to a modular frame theory in C*-algebras and Hilbert C*-modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and by other bounded modular operators with suitable ranges. We obtain frame representations and decomposition theorems, as well as similarity and equivalence results for frames. Hilbert space frames and quasi-bases for conditional expectations of finite index on C*-algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*-modules over commutative C*-algebras and (F)Hilbert bundles the results find a reintepretation for frames in vector and (F)Hilbert bundles. Fields of applications are investigations on Cuntz-Krieger-Pimsner algebras, on conditional expectations of finite index, on various ranks of C*-algebras, on classical frame theory of Hilbert spaces (wavelet and Gabor frames), and others. 2001: In the introduction we refer to related publications in detail.
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arxiv:math/0010189
|
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds X^3 of rank 2. The authors study focal properties of these submanifolds and prove that they represent examples of different types of tangentially degenerate submanifolds. Namely, the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C, does not have real singular points, the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C^1 P^2, bears two plane singular lines, and finally the submanifold X^3, corresponding in RP^5 to a smooth line \gamma of the projective plane C^0 P^2, bears one singular line.
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arxiv:math/0010192
|
Suppose M is a noncompact connected PL 2-manifold. In this paper we study the topological property of the triple (H(M)_0, H^PL(M)_0, H^PL, c(M)_0), where H(M)_0 is the identity component of the homeomorphism group {\cal H}(M) of M with the compact-open topology, and H^PL(M)_0 and H^PL, c(M)_0 are the identity components of the subgroups consisting of PL-homeomorphisms of M and ones with compact supports. We show that this triple is a (s^infty,sigma^infty,sigma^infty_f)-manifold and determine its topological type. We also study the subgroups of Lipschitz homeomorphisms.
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arxiv:math/0010224
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Bhatwadekar and Raja Sridharan have constructed a homomorphism of abelian groups from an orbit set Um(n,A)/E(n,A) of unimodular rows to an Euler class group. We suggest that this is the last map in a longer exact sequence of abelian groups. The hypothetical group G that precedes Um(n,A)/E(n,A) in the sequence is an orbit set of unimodular two by n matrices over the ring A. If n is at least four we describe a partially defined operation on two by n matrices. We conjecture that this operation describes a group structure on G if A has Krull dimension at most 2n-6. We prove that G is mapped onto a subgroup of Um(n,A)/E(n,A) if A has Krull dimension at most 2n-5.
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arxiv:math/0010226
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S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group of a closed orientable surface of genus 2 is linear.
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arxiv:math/0010267
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In this paper a new connection between the discrete conformal geometry problem of disk pattern construction and the continuous conformal geometry problem of metric uniformization is presented. In a nutshell, we discuss how to construct disk patterns by optimizing an objective function, which turns out to be intimately related to hyperbolic volume. With the use of random Delaunay triangulations we then average this objective function to construct an objective function on the metrics conformal to a fixed one. Finally using this averaged objective function we may reprove the uniformization theorem in two dimensions.
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arxiv:math/0010316
|
It is shown in what way monomial group connects Abelian group Z^{n} and total linear group GL(n). It is shown that any subgroup of Abelian group Z^{n} induces subgroup of monomial group $S_{2}\wr S_n}$, which in its turn induces corresponding subgroup of GL_{n}(R).
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arxiv:math/0010317
|
We study the B-model chiral ring of Calabi-Yau hypersurfaces in Batyrev's mirror construction. The main result is an explicit description of a subring of the chiral ring of semiample regular (transversal to torus orbits) Calabi-Yau hypersurfaces. This subring includes the marginal operators and contains all information about the correlation functions used by physicists. Computation of the chiral ring passes through a description of cohomology of semiample hypersurfaces. Here, we develop the techniques for calculating of the cohomology of resolutions.
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arxiv:math/0010318
|
Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n fixed.
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arxiv:math/0011029
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Bialgebroids, separable bialgebroids, and weak Hopf algebras are compared from a categorical point of view. Then properties of weak Hopf algebras and their applications to finite index and finite depth inclusions of von Neumann algebras are shortly reviewed. A hint is given at a duality between bialgebroid actions and abstract inclusions in 2-categories.
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arxiv:math/0011036
|
This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition, we compute and discuss the differential or variation dV of V, or equivalently the variation of the L^2 norm of the Weyl curvature, on the space of such Einstein metrics.
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arxiv:math/0011051
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We prove a theorem formulated by V. I. Arnold concerning a relation between the asymptotic linking number and the Hopf invariant of divergence-free vector fields. Using a modified definition for the system of short paths, we prove their existence in the general case.
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arxiv:math/0011159
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An infinitary version of the notion of free products has been introduced and investigated by G.Higman. Let G_i (for i in I) be groups and ast_{i in X} G_i the free product of G_i (i in X) for X Subset I and p_{XY}: ast_{i in Y} G_{i}->ast_{i in X} G_{i} the canonical homomorphism for X subseteq Y Subset I. (X Subset I denotes that X is a finite subset of I.) Then, the unrestricted free product is the inverse limit lim (ast_{i in X} G_i, p_{XY}: X subseteq Y Subset I). We remark ast_{i in emptyset} G_i= {e} . We prove: Theorem: Let F be a free group. Then, for each homomorphism h:lim ast G_i-> F there exist countably complete ultrafilters u_0,...,u_m on I such that h = h . p_{U_0 cup ... cup U_m} for every U_0 in u_0, ...,U_m in u_m. If the cardinality of the index set I is less than the least measurable cardinal, then there exists a finite subset X_0 of I and a homomorphism overline {h}: ast_{i in X_0}G_i-> F such that h= overline {h} . p_{X_0}, where p_{X_0}: lim ast G_i->ast_{i in X_0}G_i is the canonical projection.
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arxiv:math/0011231
|
We develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A, which in turn is a faster version of an algorithm due to Lascouz, Leclerc and Thibon (proposed in the setting of Hecke algebras of type A, at roots of unity.)
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arxiv:math/0011245
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We prove that the density of ramified primes in semisimple p-adic representations of Galois groups of number fields is 0. Ravi Ramakrishna has produced examples of such representations that are infinitely ramified.
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arxiv:math/0011272
|
In this paper we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces. Moreover, we point out the relevance in real geometry of the notion of the orbifold fundamental group of a real variety, and we discuss related questions on real varieties $(X, \sigma)$ whose underlying complex manifold $X$ is a $K (\pi, 1)$. Our first result is that if $(S, \sigma)$ is a real hyperelliptic surface, then the differentiable type of the pair $(S, \sigma)$ is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of $(S, \sigma)$, and to prove that they are exactly 78. It follows also as a corollary that there are exactly eleven cases for the topological type of the real part of S. Finally, we show that once we fix the topological type of $(S, \sigma)$ corresponding to a real hyperelliptic surface, the corresponding moduli space is irreducible (and connected). We also give, through a series of tables, explicit analytic representations of the 78 components of the moduli space.
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arxiv:math/0012003
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We describe a family of $q$-series generating the space of weight 1 modular forms coming from indefinite binary quadratic forms and study linear relations between these series.
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arxiv:math/0012005
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Unfortunately, some proofs in the first version of this paper were incorrect. In this revised version, some minor gaps are fixed, one serious mistake found. The main theorem is now claimed only under a restrictive technical assumption. This invalidates the application to quotient singularities by the Weyl group of type $G_2$. Everything else still stands (in particular, the claim that every symplectic resolution is semismall).
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arxiv:math/0012008
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We extend the recent result of T.Tao to wave maps defined from the Minkowski space of dimension >4 to a target Riemannian manifold which possesses a ``bounded parallelizable'' structure. This is the case of Lie groups, homogeneous spaces as well as the hyperbolic spaces. General compact Riemannian manifolds can be imbedded as totally geodesic submanifolds in bounded parallelizable manifolds, and therefore are also covered, in principle, by our result. Compactness of the target manifold, which seemed to play an important role in Tao's result, turns out however to play no role in our discussion. Our proof follows closely that of Tao's recent paper and is based, in particular, on its remarkable microlocal gauge renormalization idea.
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arxiv:math/0012034
|
This work is a survey of relations between Drinfeld modules and higher dimensional fields of positive characteristic. The main new result stated is the expression of vanishing orders of certain modular forms through partial zeta values.
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arxiv:math/0012154
|
We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by VFR Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori-Hecke algebra representations of Artin's braid group of 6 strings, and is defined over integral Laurent polynomials Z[t, t^{-1}]. We substitute the parameter t with -e^{h}, and then expand the powers e^h in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology 3-spheres.
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arxiv:math/0012216
|
The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean curvature tori, respectively energy estimates of harmonic 2-tori, and to Dirac eigenvalue estimates on Riemannian spin bundles in dimension 2 are given.
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arxiv:math/0012238
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For coherent families of crystals of affine Lie algebras of type B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} we describe the combinatorial R matrix using column insertion algorithms for B,C,D Young tableaux.
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arxiv:math/0012247
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