Showing papers in "Mathematical Research Letters in 1996"

Absolute Continuity of Bernoulli Convolutions, A Simple Proof

Abstract: The distribution νλ">νλνλ of the random series ∑±λn">∑±λn∑±λn has been studied by many authors since the two seminal papers by Erd\H{o}s in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urba\'{n}ski, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author proved a conjecture made by Garsia in 1962, that νλ">νλνλ is absolutely continuous for a.e.\ λ∈(1/2,1)">λ∈(1/2,1)λ∈(1/2,1). Here we give a considerably simplified proof of this theorem, using differentiation of measures instead of Fourier transform methods. This technique is better suited to analyze more general random power series.

Four-Manifolds without Einstein Metrics

Abstract: It is shown that there are infinitely many compact simply con- nected smooth 4-manifolds which do not admit Einstein metrics, but nev- ertheless satisfy the strict Hitchin-Thorpe inequality 2 !> 3|" |. The exam- ples in question arise as non-minimal complex algebraic surfaces of general type, and the method of proof stems from Seiberg-Witten theory.

Instantons and affine algebras I: The Hilbert scheme and vertex operators

Abstract: This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.

The Seiberg-Witten equations and four-manifolds with boundary

Abstract: An early result by Donaldson says that if Z is closed and JZ is negative definite then JZ is isomorphic to some diagonal form 〈−1〉 ⊕ · · · ⊕ 〈−1〉. More generally one may ask which negative definite forms can occur if Z is allowed to have some fixed oriented rational homology sphere Y as boundary. The main purpose of the present paper is to apply the equations recently introduced by Seiberg and Witten [W] to prove a finiteness result about the definite forms associated to an arbitrary Y . It is useful to consider the more general situation where the boundary of Z is a disjoint union of rational homology spheres: ∂Z = Y1 ∪ · · · ∪ Yl. (Of course, ∪jYj and #jYj bound the same intersection forms, since the standard cobordism connecting them has no rational homology in dimension 2.) Let JZ = m〈−1〉 ⊕ JZ , where JZ has no elements of square −1. Note that

Some New applications of General Wall Crossing Formula, Gompf’s Conjecture and its Applications

Abstract: As early as the birth of Seiberg Witten invariants [W1], the positive scalar curvature metrics on four dimensional manifolds have played a very important role. It was Witten [W] who first noticed that assuming b2 > 1 then one could easily derive the vanishing result of Seiberg Witten Invariants for those manifolds which carried the positive scalar curvature (psc) metrics. Combined with Taubes’ nonvanishing result [T2] for symplectic four manifolds, one could easily conclude that if a symplectic four manifold carries some psc metric, then its b2 must be equal to one. In addition the same vanishing result was used by P. Kronheimer and T. Mrowka [K.M.] in proving the Thom conjecture, and by C.H. Taubes in his “more constraints of symplectic forms on CP ” [T3]. Furthermore R. Friedman and J. Morgan [FM] systematically discussed which kind of Kahler manifold can carry the psc metrics. Based on surface classification theory, Friedman and Morgan could argue that the Kahler surfaces carrying psc metrics are either rational, rational ruled or irrational ruled. Recently, the same vanishing result has been used by Taubes [T4] to show that CP 2 has a unique symplectic structure. At this point it is interesting to ask about the possibilities of classifying symplectic four manifolds carrying psc metrics. One of the purposes of this paper is to address this possibility. Another goal of this paper is to address the question raised by Gompf [Gom] which states that:

The classification of ruled symplectic $4$-manifolds

Abstract: Let M be an oriented S2-bundle over a compact Riemann surface Σ. We show that up to diffeomorphism there is at most one symplectic form on M in each cohomology class. Since the possible cohomology classes of symplectic forms on M are known, this completes the classification of symplectic forms on these manifolds. Our proof relies on a simplification of our previous arguments and on the equivalence between Gromov and Seiberg-Witten invariants that we apply twice.

Almost complex structures and geometric quantization

Abstract: We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin-c quantization. We prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semiclassically. We also introduce a new quantization scheme, based on a rescaled Laplacian, for which we are able to prove strong semiclassical properties. The two quantizations are shown to be close semiclassically.

Logarithmic Harnack Inequalities

Abstract: Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for random walks on graphs [5, 8]. The problem of bounding log-Sobolev constants tends to be harder than estimating eigenvalues. Logarithmic Harnack inequalities provide a direct approach for estimating the log-Sobolev constant. We will derive lower bounds for log-Sobolev constants for Riemannian manifolds and for large classes of graphs.

Hitchin systems, higher Gaudin operators and $r$-matrices

Abstract: We adapt Hitchin's integrable systems to the case of a punctured curve. In the case of $\CC P^{1}$ and $SL_{n}$-bundles, they are equivalent to systems studied by Garnier. The corresponding quantum systems were identified by B. Feigin, E. Frenkel and N. Reshetikhin with Gaudin systems. We give a formula for the higher Gaudin operators, using results of R. Goodman and N. Wallach on the center of the enveloping algebras of affine algebras at the critical level. Finally we construct a dynamical $r$-matrix for Hitchin systems for a punctured elliptic curve, and $GL_{n}$-bundles, and (for $n=2$) the corresponding quantum system.

Cocycles' stability for partially hyperbolic systems

Abstract: In this paper we establish Livshitz-type theorems for partially hyperbolic systems. To be more precise, we prove that for a large class of partially hyperbolic transformations and flows the subspace of Hoelder coboundaries is closed and can be described by some natural geometric conditions. This class includes an open, in C2 topology, neighborhood of the time-one maps of contact Anosov flows (for example, the geodesic flows on manifolds of negative curvature). Along the way we prove several results on the transitivity of the pair of stable and unstable foliations for partially hyperbolic systems. In particular, we establish the transitivity property for the time-one maps of contact Anosov flows and their small perturbations, which has important applications to the stable ergodicity of the time-one maps of geodesic flows on the manifolds of negative curvature. 1. Stability of cocycle spaces Let f be a transformation of a space X. The cohomological equation corresponding to a fixed function φ (also called a cocycle) is the following equation, with an unknown function h, φ(x) − h(x) + h(f(x)) = 0, x ∈ X. If this equation has solutions then φ is called a coboundary, and h is called a transfer function. Two functions are called cohomologous if their difference is a coboundary. Depending on the classes of regularity of cocycles and transfer functions allowed one associates with the transformation f various cohomology spaces ([10], [13]). A number of well known results describes the spaces of coboundaries for different types of maps f. The most celebrated results of this kind are the Livshitz-type theorems for hyperbolic diffeomorphisms and flows ([4], [7], [8], [10] (Section 19.2), [13], [14], [15], [16]). However some non-hyperbolic systems also display certain regularity for cocycles. These include Diophantine translations of a torus, affine maps ([12], Section 10.5), partially Received December 27, 1995. The first author is partially supported by NSF Grant DMS 9404061.

Heat kernel and moduli spaces II

Abstract: In this paper we describe a proof of the formulas of Witten [W1], [W2] about the symplectic volumes and the intersection numbers of the moduli spaces of principal bundles on a compact Riemann surface. It is known that these formulas give all the information needed for the Verlinde formula. The main idea of the proof is to use the heat kernel on compact Lie groups, in a way very similar to the heat kernel proof of the Atiyah-Singer index formula and the Atiyah-Bott fixed point formula. The Reidemeister torsion comes into the picture, through a beautiful observation of Witten, as the symplectic volume of the moduli space. It plays the role similar to that played by the Ray-Singer torsion in the path-integral computations on the space of connections. The basic idea is as follows. Consider a smooth map between two compact smooth manifolds f : M → N . Let H(t, x, x0) be the heat kernel of the Laplace-Beltrami operator on N with x0 a fixed regular value of f . Because of the basic properties of the heat kernel, we know that for any continuous function a(y) on M , when t goes to zero, ∫

The Geometry and Arithmetic of Translation Surfaces with Applications to Polygonal Billiards

Abstract: A translation manifold is a manifold whose transition transformations are translations. There is an important connection between the geometry and arithmetic of translation surfaces and dynamics of polygonal billiards. There are also remarkable relations with automorphic forms. In this note we announce results which further develop these connections.

Geometric Height Inequalities

Abstract: 0. The Results. Let f : X → B be a fibration of a compact smooth algebraic surface over a compact Riemann surface B, denote by g ≥ 2 the genus of a generic fiber of f and by q the genus of B. Let s be the number of singular fibers of f and ωX/B be the relative dualizing sheaf. Let C1, · · · , Cn be n mutually disjoint sections of f , and denote by D the divisor ∑n j=1Cj. Then the main result we are going to prove in this note is the following Theorem 0.1

Lengths of periods and seshadri constants of abelian varieties

Abstract: We start by recalling the definition of Seshadri constants. Let X be a smooth complex projective variety, let L be an ample line bundle on X , and fix a point x ∈ X . Consider the blowing-up f : Y = Blx(X) −→ X of X at x, with exceptional divisor E = f(x) ⊂ Y . Then for 0 < ǫ ≪ 1 the cohomology class fc1(L)− ǫ · [E] will lie in the Kahler cone of Y . As a measure of how positive L is locally near x we ask in effect how large we can take ǫ to be while keeping the class in question positive. More precisely, set

Combinatorial scalar curvature and rigidity of ball packings

Abstract: Let M be a triangulated three-dimensional manifold. In this paper we define a combinatorial analogue of scalar curvature for M, and also a combinatorial analogue of conformal deformation of the metric. We further define a functional S on the combinatorial conformal deformation space, show that S is concave, and show that critical points of S correspond precisely to metrics of constant combinatorial scalar curvature on M. These results are then applied to showing rigidity of ball packings with prescribed combinatorics (the concepts are quite similar to Colin de Verdière’s work on circle packing of surfaces [2]. See also [5] for a related variational argument). The plan of the paper is as follows. In section 1 we define the class of conformal simplices in E, and prove the necessary local versions of our results. In section 3 we extend these techniques to conformal simplices in

A stabilization theorem for Hermitian forms and applications to holomorphic mappings

Abstract: We consider positivity conditions both for real-valued functions of several complex variables and for Hermitian forms. We prove a stabilization theorem relating these two notions, and give some applications to proper mappings between balls in different dimensions. The technique of proof relies on the simple expression for the Bergman kernel function for the unit ball and elementary facts about Hilbert spaces. Our main result generalizes to Hermitian forms a theorem proved by Polya [HLP] for homogeneous real polynomials, which was obtained in conjunction with Hilbert's seventeenth problem. See [H] and [R] for generalizations of Polya's theorem of a completely different kind. The flavor of our applications is also completely different.

Weak Hironaka theorem

Abstract: The purpose of this note is to give a simple proof of the following theorem: Let $X$ be a normal projective variety over an algebraically closed field $k$, $\op{char} k = 0$ and let $D \subset X$ be a proper closed subvariety of $X$. Then there exist a smooth projective variety $M$, a strict normal crossings divisor $R \subset M$ and a birational morphism $f : M \to X$ with $f^{-1} D = R$. The method of proof is inspired by A.J. de Jong alteration ideas. We also use a multidimensional version of G.Belyi argument which allows us to simplify the shape of a ramification divisor. By induction on the dimension of $X$ the problem is reduced to resolving toroidal singularities. This process however is too crude and does not permit any control over the structure of the birational map $f$. A different proof of the same theorem was found independently by D. Abramovich and A.J. de Jong. The approach is similar in both proofs but they seem to be rather different in details.

K3 surfaces, Lorentzian Kac–Moody algebras and Mirror Symmetry

Abstract: We consider the variant of Mirror Symmetry Conjecture for K3 surfaces which relates "geometry" of curves of a general member of a family of K3 with "algebraic functions" on the moduli of the mirror family. Lorentzian Kac--Moody algebras are involved in this construction. We give several examples when this conjecture is valid.

The dimension of the fixed point set on affine flag manifolds

Abstract: Let G be a semisimple simply-connected algebraic group over C, g its Lie algebra. Also, F = C((ε)) is the field of formal Laurent series, A = C[[ε]] is the ring of integers in F . Set ĝ = g ⊗ F , gA = g ⊗ A and Ĝ = G(F ). Let B be the set of all Iwahori subalgebras in ĝ, and X the set of all subalgebras in ĝ which are Ĝ-conjugate to gA. Then B and X have the structure of Ind-algebraic varieties over C (they are unions of increasing system of ordinary projective algebraic varieties over C). They are called the affine flag variety and the affine Grassmanian of G respectively. We have X = Ĝ/G(A) and B = Ĝ/I, where I is an Iwahori subgroup. For any N ∈ ĝ let BN ⊂ B (respectively XN ⊂ X) be the set of all Iwahori subalgebras (respectively, subalgebras conjugate to gA) which contain N . Clearly, BN (XN ) is a closed subvariety of the Ind-variety B (respectively X). The varieties BN , XN were studied by Kazhdan and Lusztig in [KL]. Following their paper let us suppose that N is topologically nilpotent (nilelement in the terminology of [KL]), i.e., ad(N) → 0 in EndF (ĝ) when r → ∞. (The topology on EndF (ĝ) arises from the obvious topology on F .) It was shown in loc. cit. that the Ind-varieties BN , XN are finite dimensional iff the element N is regular semisimple. We will assume from now on that this is the case. Then BN and XN are locally finite unions of finite dimensional projective varieties. Moreover, all components of BN have the same dimension, which coincides with the dimension of XN . A precise formula for the dimension of BN was stated in [KL] as a conjecture. The aim of the present note is to give a proof of this conjecture. Let O be the subset of XN defined as follows: if p̂ ∈ XN is a subalgebra conjugate to gA, then p̂ ∈ O iff the image of N in g=̃p̂/εp̂ is a regular nilpotent. Let Z(N) be the centralizer of N in Ĝ; let z(N) be the centralizer of N in ĝ. We also fix a Cartan subalgebra h ⊂ g and denote by W the Weyl group. The result containing the formula for the dimension of BN is the following:

Symplectic biextensions and a generalization of the Fourier-Mukai transform

Abstract: A generalization of the Fourier-Mukai transform is proposed. The construction is based on analogy with the classical picture of representations of the Heisenberg group.

On hele-shaw models with surface tension

Abstract: It is shown that surface tension effects on the free boundary have a regularizing effect for Hele-Shaw models, which implies existence and uniqueness of classicalsolutions for generalinitialdomains. 1. Introduction and main results Recently, N. Alikakos, P. Bates, and X. Chen (1) proved that level sur- faces of solutions to the Cahn-Hilliard equation tend to solutions of the two-phase Hele-Shaw problem with surface tension under the assumption that classical solutions ofthe latter exist. In the present note we are able to guarantee that the above assumption is in fact satisfied. More precisely, our results (11) show the existence ofa unique classical solution to one- and two-phase Hele-Shaw models with surface tension for general initial data. It should be emphasized that even weak solutions to Hele-Shaw models with surface tension were not known to exist in the general setting presented here. In this note we only give the statements ofour results and a brief sketch of their proofs. The full details will appear in (11). We first consider the one-phase problem. Let Ω be a bounded domain in R n and assume that its boundary ∂Ω is ofclass C ∞ . Moreover, assume that ∂Ω consists oftwo disjoint non-empty components J and Γ. Later on, we will model over the exterior componen tΓam oving interface, whereas the interior component J describes a fixed portion ofthe boundary. Let ν denote the outer unit normal field over Γ and fix α ∈ (0, 1). Given a> 0, let

On the Existence of High Multiplicity Interfaces

Abstract: A bstract . In many singularly perturbed Ginzburg–Landau type partial differential equations, such as the Allen–Cahn equation, the nonlocal Allen–Cahn equation, and the Cahn–Hilliard equation, the question arises whether or not the limiting interfaces can have high multiplicity. In other words, do there exist solutions of these PDE’s with many transition layers (where the solution passes rapidly between ±1) which are so close to each other that they collapse to one interface in the limit. In this paper we prove that there exist interfaces with arbitrarily high multiplicity by studying the radially symmetric Allen-Cahn equation. We adapt the energy method of Bronsard-Kohn [BK].

Modules with extremal resolutions

Abstract: Let R be a commutative noetherian local ring R with maximal ideal m and residue field k = R/m. The size of a minimal free resolution of a finite R–module M is given by its Betti numbers β n (M) = rankk Ext ∗ R(M, k). Dually, that of a minimal injective resolution is measured by the Bass numbers μR(M) = rankk Ext ∗ R(k, M). These sizes may be estimated asymptotically on a natural, a polynomial, and an exponential scale. The first scale yields the classical homological dimensions. The second produces known notions of complexity, which distinguish between modules of infinite homological dimensions. The third leads to new concepts of homological curvatures, which discriminate among modules with infinite complexities. It is well known that k has maximal homological dimensions among all finite R–modules. An elementary computation shows that its complexities and curvatures are maximal as well. By analyzing the representations on ExtR(M, k) and Ext ∗ R(k, M) of the homotopy Lie algebra π ∗ of the local ring R, we show that modules with extremal homological invariants occur with unexpected frequency.

The symplectic Floer homology of a Dehn twist

Abstract: We compute examples of symplectic Floer homology in the lowest dimension i.e. for surfaces.

Dispersive smoothing for Schrödinger equations

Abstract: The phenomenon of the global (in time) dispersive smoothing for the “free” Schrödinger evolution can be described as follows: For any distribution f of compact support, the solution ψ(t, x) of the Cauchy problem ( 1i ∂ ∂t − ∆) ψ(t, x) = 0, t > 0, ψ(0, x) = f(x), x ∈ R, is infinitely differentiable with respect to t and x, when t > 0 and x ∈ R. This is equivalent to saying that the corresponding fundamental solution (= the solution S0(t, x, y) of the initial value problem with f(x) = δ(x − y)) is infinitely differentiable with respect to t, x and y, when t > 0. And we have, indeed, S0(t, x, y) = e−in π 4 (4πt)− n 2 exp{i |x − y|2/4t}, with the only singularity at t = 0. One would expect that dispersive smoothing should survive “small” perturbations of the free Hamiltonian H0 = −∆. The problem is to determine what perturbations are “small”. The case when the perturbed Hamiltonian has the form H = H0 + V with a potential V = V (x), has been examined in [Ze], [OF], [Ki], [CFKS]. The dispersive smoothing takes place, for example, if the potential is infinitely differentiable, and it and all its derivatives are bounded, [Ze], [OF]. On the other hand, if V (x) grows quadratically or faster at infinity, then the singularities may resurrect, as the example of the quantum harmonic oscillator and Mehler’s formula show ([Ze], [We], [CFKS], [MF]). If the perturbation affects the metric of the space, i.e., if H0 = −∆ is replaced by H = − n ∑