Showing papers in "Asian Journal of Mathematics in 1999"

Anomalies in string theory with D-branes

Abstract: We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.

Mirror Principle II

Abstract: We generalize our theorems in "Mirror Principle I" to a class of balloon manifolds. Many of the results are proved for convex projective manifolds. In a subsequent paper, Mirror Principle III, we will extend the results to projective manifolds without the convexity assumption.

Moment maps and diffeomorphisms

Abstract: Atiyah and Bott pointed out, in [1], that the curvature of a connection on a bundle over a surface can be viewed as the “momentum” corresponding to the action of the gauge group. This observation, together with various extensions, has stimulated a great deal of work and provides a conceptual framework to understand many phenomena in Yang-Mills theory. Our purpose in this paper is to explore some similar ideas in the framework of diffeomorphism groups. We begin by identifying a moment map in a rather general setting, and then see how the ideas work in some more specific situations. We hope to show that the moment map point of view is useful, both in understanding certain established results and also in suggesting new problems in geometry and analysis. While these analytical questions are the main motivation for the work, we will concentrate here on the formal aspects and will not make any serious inroads on the analysis.

Integral canonical models of Shimura varieties of preabelian type

Abstract: We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5. a MSC 2000: Primary 11G10, 11G18, 14B12, 14C30, 14D10, 14D22, 14F20, 14F30, 14G35 and 14K10.

The Moduli space of complex Lagrangian submanifolds

Abstract: Following an earlier paper on the differential-geometric structure of the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold, we follow an analogous approach for compact complex Lagrangian submanifolds of a (K\"ahlerian) complex symplectic manifold. The natural geometric structure on the moduli space is a special K\"ahler metric, but we offer a different point of view on the local differential geometry of these, based on the structure of a submanifold of $V\times V$ (where $V$ is a symplectic vector space) which is Lagrangian with respect to two constant symplectic forms. As an application, we show using this point of view how the hyperk\"ahler metric of Cecotti, Ferrara and Girardello associated to a special K\"ahler structure fits into the Legendre transform construction of Lindstr\"om and Ro\v cek.

Three constructions of Frobenius manifolds: A comparative study

Abstract: The paper studies three classes of Frobenius manifolds: Quantum Cohomology (topological sigma-models), unfolding spaces of singularities (K. Saito's theory, Landau-Ginzburg models), and the recent Barannikov-Kontsevich construction starting with the Dolbeault complex of a Calabi-Yau manifold and conjecturally producing the B--side of the Mirror Conjecture in arbitrary dimension. Each known construction provides the relevant Frobenius manifold with an extra structure which can be thought of as a version of ``non-linear cohomology''. The comparison of thesestructures sheds some light on the general Mirror Problem: establishing isomorphisms between Frobenius manifolds of different classes. Another theme is the study of tensor products of Frobenius manifolds, corresponding respectively to the K\"unneth formula in Quantum Cohomology, direct sum of singularities in Saito's theory, and presumably, the tensor product of the differential Gerstenhaber-Batalin-Vilkovisky algebras. We extend the initial Gepner's construction of mirrors to the context of Frobenius manifolds and formulate the relevant mathematical conjecture.

Calabi-Yau threefolds of quotient type

Abstract: First, we classify Calabi-Yau threefolds with infinite fundamental group by means of their minimal splitting coverings introduced by Beauville, and deduce that the nef cone is a rational simplicial cone and any rational nef divisor is semi-ample if the second Chern class is identically zero. We also derive a sufficient condition for the fundamental group to be finite in terms of the Picard number in an optimal form. Next, we give a concrete structure Theorem concerning $c_{2}$-contractions of Calabi-Yau threefolds as a generalisation and also a correction of our earlier works for simply connected ones. Finally, as an application, we show the finiteness of the isomorphism classes of $c_{2}$-contractions of each Calabi-Yau threefold.

Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

Abstract: 0. Introduction. In this paper, we obtain some new inequalities for integrals of convex functions of the curvature (resp. Wulff curvature) of convex plane curves. We also show that the difference between the two sides of our inequalities are monotone decreasing as the region flows under the unit-speed outward normal (resp. Wulff) flow. Given a bounded plane region K, the unit-speed outward normal flow has been highly studied, and is of interest in many applied problems, e.g. combustion. If instead K grows by varying the outward normal speed to be a function 7(0) of the direction of the unit normal, one has the WulfF flow, which is also of considerable interest, e.g. in studying the growth of crystals [O-M]. When the region K is convex, there is a simple closed-form expression which describes these flows, and the region converges to a disk in the first case and a Wulff shape in the second. The area of the region when the initial region K is convex is a polynomial in t, known respectively as the Steiner polynomial or WulfF-Steiner polynomial. A novel feature of our approach is that we study the roots of the Steiner and WulffSteiner polynomials, which occur at negative values of t. The classical isoperimetric inequality in both cases states that these polynomials have (negative) real roots ti >t2, and that they are distinct if and only if K is not a disc (respectively not a Wulff shape). Bonnesen's inequality states that the inradius and outradius r^ and re lie in the interval [—£1, —£2], and in the open interval if K is not a disk (respectively not a Wulff shape). Our inequalities are most naturally stated and proved in terms of the roots ti and £2We feel that this is a potentially quite fruitful approach to studying convex bodies in higher dimensions. In the context of this new approach, a very natural link between the outward normal and Wulff flows and the curvature integrals of the region appears. In important cases, the quantities that our inequalities state are positive are shown to be monotone decreasing as the region evolves under the flow. Particularly suggestive is the fact that the entropy of the curvature (respectively Wulff curvature) is bounded above in terms of the area and is monotone decreasing with time. The inequalities themselves are quite fascinating. It came as a surprise to us that there are interesting new things to be said about convex plane curves. We state our inequalities here for arbitrary smooth bounded convex plane regions K in the curvature case, and leave the Wulff case to the body of the paper. One of them is due to Gage [G], whose result was a source of inspiration to us. Gage's result is

Equivariant de Rham theory and graphs

Abstract: Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.

On Hessian Riemannian structures

Abstract: In Proposition 4.1 a characterization is given of Hessian Riemannian structures in terms of a natural connection in the general linear group GL(n; R) + , which is viewed as a principal SO(n)-bundle over the space of positive denite symmetric n n-matrices. For n = 2, Proposition 5.3 contains an interpretation of the curvature of a Hessian Riemannian structure at a given point, in terms of an umbilic point of a related surface in R 3 .

Convex curves moving homothetically by negative powers of their curvature

Abstract: We study immersed locally convex curves in R which move homothetically under flow by negative powers of their curvature.

Analytic Torsion and R-Torsion for Manifolds with Boundary

Abstract: We prove a formula relating the analytic torsion and Reidemeister torsion on manifolds with boundary in the general case when the metric is not necessarily a product near the boundary. The product case has been established by W. Lu\"ck and S. M. Vishik. We find that the extra term that comes in here in the nonproduct case is the transgression of the Euler class in the even dimensional case and a slightly more mysterious term involving the second fundamental form of the boundary and the curvature tensor of the manifold in the odd dimensional case.

Two integrable systems related to hyperbolic monopoles

Abstract: Monopoles on hyperbolic 3-space were introduced by Atiyah in 1984. This article describes two integrable systems which are closely related to hyperbolic monopoles: a one-dimensional lattice equation (the Braam-Austin or discrete Nahm equation), and a soliton system in (2+1)-dimensional anti-deSitter space-time.

The symplectic geometry of polygons in hyperbolic 3-space

Abstract: We study the symplectic geometry of the moduli spaces Mr = Mr(H 3 ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B n by the dressing action of SU(2) (here B is the subgroup of the Borel subgroup of SL2(C) defined below). We show that the hyperbolic Gauss map sets up a real analytic isomorphism between the spaces Mr and the weighted quotients of (S 2 ) n by P SL2(C) studied by Deligne and Mostow. We construct an integrable Hamiltonian system on Mr by bending polygons along nonintersecting diagonals. We describe angle variables and the momentum polyhedron for this system. The results of this paper are the analogues for hyperbolic space of the results of (KM2) for Mr(E 3 ), the space of n-gons with fixed side-lengths in E 3 . We prove Mr(H 3 ) and Mr(E 3 ) are symplectomorphic.

Visualizing elements of order three in the Shafarevich–Tate group

Abstract: 1. Introduction. If we wish to write the equations of curves of genus 1 that give elements of the Shafarevich-Tate group of an elliptic curve over a number field K, a choice of ways is open to us. For example, if the element in question is of order 3 the curve of genus 1 corresponding to it occurs as a smooth plane cubic curve over K. In a recent article [C-M] we raised the question of when one can find the curves of genus 1 corresponding to at least some of the elements of Shafarevich-Tate groups as curves in abelian surfaces over K. Adam Logan, using data and programs due to Cremona, studied this question numerically for semistable optimal elliptic curves E over Q of square-free conductor N less than about 3000, and for the odd part of their Shafarevich-Tate group. By an \" optimal \" elliptic curve (or in older terminology, a \" strong Weil \" curve) we mean that there is a modular parametrization φ : J 0 (N) → E where N = the conductor of E, J 0 (N) = the jacobian of the modular curve X 0 (N), and such that the kernel of φ is an abelian variety. Any modular elliptic curve is isogeneous, over Q, to a (unique) optimal elliptic curve, and any optimal elliptic curve of conductor N is isomorphic, over Q, to an elliptic curve in J 0 (N). Logan studied the elements of the Shafarevich-Tate group of such optimal elliptic curves, and sought, in effect, to realize the corresponding curves of genus 1 in question as subcurves defined over Q within abelian surfaces contained in the new part of J 0 (N). If E is an optimal elliptic curve for which we can successfully do this for each of the elements of the Shafarevich-Tate group of E, let us say that we have visualized the Shafarevich-Tate group of E in abelian surfaces in the new part of the modular jacobian.

On the geometry of nilpotent orbits

Abstract: In this paper we obtain various results about the geometry of nilpotent orbits. In particular, we obtain a better understanding of the Kostant-Sekiguchi correspondence and Kronheimer's instanton flow. We utilize the moment map of Ness and the SL(2)-orbit theorem from Hodge theory. The results of this paper are used in the proof of the Barbarsch-Vogan conjecture in math.RT/0005305 (Ann. of Math. (2) 151 (2000), no. 3, 1071-1118).

Haar-Type Multiwavelet Bases and Self-Affine Multi-Tiles

Abstract: Grochenig and Madych showed that a Haar-type wavelet basis of L(R) can be constructed from the characteristic function xn of a compact set O if and only if fl is an integral self-affine tile of Lebesgue measure one. In this paper we generalize their result to the multiwavelet settings. We give a complete characterization of Haar-type scaling function vectors Xn(x) := [XQi ()i • • • J XQr ()]i where ft = (fii, ..., ftr) is an r-tuple of compact sets in R . We call Q a self-affine multi-tile because Q^'s tile R by translation and have the property that each affine image A(Qi) is the union of translates of some fij's. We also construct associated Haar-type multiwavelets , and present examples using various dilation matrices A.

Quantum determinants and quasideterminants

Abstract: The notion of a quasideterminant and a quasiminor of a matrix A=(a_{ij}) with not necessarily commuting entries was introduced recently by I.Gelfand and the second author. The ordinary determinant of a matrix with commuting entries can be written (in many ways) as a product of quasiminors. Furthermore, it was noticed by a number of authors that such well-known noncommutative determinants as the Berezinian, the Capelli determinant, the quantum determinant of the generating matrix of the quantum group U_h(gl_n) and the Yangian Y(gl_n) can be expressed as products of commuting quasiminors. The aim of this paper is to extend these results to a rather general class of Hopf algebras given by the Faddeev-Reshetikhin-Takhtajan type relations -- the twisted quantum groups. Such quantum groups arise when Belavin-Drinfeld classical r-matrices are quantized. Our main result is that the quantum determinant of the generating matrix of a twisted quantum group equals the product of commuting quasiminors of this matrix.

Deforming Calabi-Yau orbifolds

Abstract: A Calabi-Yau 3-fold is a compact complex 3-manifold (X, J) equipped with a Ricci-flat Kähler metric g and a holomorphic volume form Ω which is constant under the Levi-Civita connection of g. Suppose X is a Calabi-Yau 3-fold and G a finite group that acts on X preserving J, g and Ω. Then X/G is a Calabi-Yau orbifold. Often it is possible to find a compact manifold Y that desingularizes X/G, and carries a family of Calabi-Yau structures that converge to the singular Calabi-Yau structure on X/G in a well-defined sense, so that the orbifold metric on X/G may be regarded as the degenerate case in a smooth family of Calabi-Yau metrics on Y . The two main strategies for desingularizing X/G to get Y are called resolution and deformation. A resolution (Y, π) of X/G is called a crepant resolution if KY ∼= π∗(KX/G). To get a Calabi-Yau structure on Y we must choose a crepant resolution. A nonsingular deformation of X/G is called a smoothing. We can also combine the two processes by taking a crepant resolution of a singular deformation of X/G, which we will call a CR-deformation of X/G. Thus, crepant resolutions, smoothings and CR-deformations are three different ways to desingularize a Calabi-Yau orbifold X/G to get a new Calabi-Yau 3-fold Y . As a shorthand we will sometimes use desingularize and desingularization to mean either a crepant resolution, or a smoothing, or a CR-deformation.

The holomorphic kernel of the Rankin–Selberg convolution

Abstract: converges absolutely for complex s with Re(s) > 1, has a meromorphic continuation in s with at most a simple pole at s = 1, and satisfies a functional equation s —> 1 — s. This result was later generalized [L] to more general situations, and in particular, to arbitrary pairs of weights k, £. The proof of the meromorphic continuation and the functional equation of £(5 / ® g) was obtained by expressing L(s, / 0 g) as an inner product of / • g with a nonholomorphic Eisenstein series. We shall give a new proof of this result which does not use Eisenstein series at all, but instead expresses the Rankin-Selberg convolution L-function as an inner product of / with a holomorphic kernel function which depends on g and s. The main result of the paper is the Fourier expansion of the kernel function (when D is squarefree) which is given in Theorem 6.5. In the case where e is a quadratic Dirichlet character (mod .D), a simpler and more explicit version of this result is given in Theorem 9.1. The functional equation of the kernel is stated and proved in various important cases in sections §10, §11. In the special case that g is a theta function attached to the imaginary quadratic extension Q(y/—D), the value of the holomorphic kernel function (or its derivative) at 5 = | coincides with the kernel function computed by Gross and Zagier [G-Z] in their celebrated formula relating the derivative of an L-function of an elliptic curve with the height of a certain Heegner point. Thus, our method simultaneously gives a new simplified proof of the L-value computation in the Gross-Zagier formula together with

Finite type invariants of knots via their Seifert matrices

Abstract: We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial.

Moduli spaces and Fredholm theory for pseudoholomorphic subvarieties associated to self-dual, harmonic 2-forms

Abstract: A compact, oriented Riemannian 4-manifold whose intersection form is not negative definite has a non-trivial closed, self-dual 2-form. Such a form defines a symplectic structure away from its zero set, and the metric can be used to defines a compatible almost complex structure. Although this almost complex structure is singular across the zero set of the given 2-form, one can none-the-less study the associated pseudoholomorphic subvarieties in the compliment of the zero set. In this regard, it is natural to restrict attention to those which have the following three properties: First, they are closed subsets of the compliment of the zero set of the given self-dual 2-form. Second, they are submanifolds except at a set which can be empty, but is at worst countable and non-accumulating. Third, the given self-dual 2-form has finite integral over the sub variety. With regard to the second point, remark that the condition of pseudoholomorphicity is no more nor less than the requirement that the almost complex structure preserve the tangent space at all manifold points. With regard to the third point, remember that a pseudoholomorphic variety is naturally oriented at its manifold points by the restriction of the symplectic form.

Conjectural algebraic formulas for representations of $GL_n$

Abstract: 0.0. Let F be a non-archimedean local field. Due to the recent work of Harris and Taylor [HT], we know that the local Langlands conjecture is true. In other words, for any local field F we know the existence of the one-to-one correspondence <^n: nn —> GLn(F)^ between the set nn of n-dimensional representations of the Galois group 0 = Gal(jF/F) and the set GLn(F)^ of irreducible nondegenerate representations of the group GLn{F). In particular, one can associate an irreducible representation 7rx of the group GLn(F) to a pair (£?,x)> where E is a commutative semisimple algebra over F of degree n and x is a multiplicative character of the group E*. However, we do not know any explicit construction for the representation 7rx. In our paper we propose an explicit \"algebraic\" construction for the representation 7rx at least for n — 4. One can inductively characterize the correspondence (j)n in the following way. Suppose that we know the correspondence n-2Then for any a G nn we can characterize the representation (j>n{a) as the unique representation of GLn(F) such that for any representation p £ Tln-2 we have