Showing papers in "Duke Mathematical Journal in 2006"

Toric degenerations of toric varieties and tropical curves

Abstract: We show that the counting of rational curves on a complete toric variety which are in general position relative to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory

The rational points of a definable set

Abstract: Let $X\R^n$ be a set that is definable in an o-minimal structure over $R$. This article shows that in a suitable sense, there are very few rational points of $X$ which do not lie on some connected semialgebraic subset of $X$ of positive dimension

Birkhoff normal form for partial differential equations with tame modulus

Abstract: We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrodinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the d-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions

Global wellposedness of KdV in H−1(T,R)

Abstract: By the inverse method we show that the Korteweg–de Vries equation (KdV) ∂tv(x,t)=-∂x3v(x,t)+6v(x,t)∂xv(x,t) (x∈T,t∈R) is globally (in time) wellposed in the Sobolev space of distributions Hβ(T,R) for any β≥−1

A link invariant from the symplectic geometry of nilpotent slices

Abstract: We define an invariant of oriented links in S 3 using the symplectic geometry of certain spaces which arise naturally in Lie theory. More specifically, we present a knot as the closure of a braid, which in turn we view as a loop in configuration space. Fix an affine subspaceSm of the Lie algebra sl2m(C) which is a transverse slice to the adjoint action at a nilpotent matrix with two equal Jordan blocks. The adjoint quotient map restricted to Sm gives rise to a symplectic fibre bundle over configuration space. An inductive argument constructs a distinguished Lagrangian submani

Planar dimers and Harnack curves

Abstract: In this article we study the connection between dimers and Harnack curves discovered in [15]. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the space of Harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates. We characterize Harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their Ronkin function with given boundary conditions

Hom-stacks and restriction of scalars

Abstract: Fix an algebraic space S, and let X and Y be separated Artin stacks of finite presentation over S with finite diagonals (over S). We define a stack HomS(X,Y) classifying morphisms between X and Y. Assume that X is proper and flat over S, and assume fppf locally on S that there exists a finite finitely presented flat cover Z→X with Z an algebraic space. Then we show that HomS(X,Y) is an Artin stack with quasi-compact and separated diagonal

Random symmetric matrices are almost surely nonsingular

Abstract: Let Qn denote a random symmetric (n×n)-matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that Qn is nonsingular with probability 1-O(n-1/8+δ) for any fixed δ>0. The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices

Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

Abstract: A metric space X has Markov type 2, if for any reversible finite-state Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 ) ≤ K 2 t E(D 2) for some K = K(X) 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp.

Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Abstract: Eisenbud Popescu and Walter have constructed certain special sextic hypersurfaces in P 5 as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3 surface (K3) [2] and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is a numerical (K3) [2] , that H is an ample divisor on X of square 2 for Beauville’s quadratic form and that the map X 99K |H| _ is the composition of the quotient X ! Y for an anti-symplectic involution on X followed by an immersion Y ,! |H| _ ; then Y is an EPW-sextic and X ! Y is the natural double cover.

Derived Hall algebras

Abstract: The purpose of this work is to define a derived Hall algebra $\mathcal{DH}(T)$, associated to any dg-category $T$ (under some finiteness conditions). Our main theorem states that $\mathcal{DH}(T)$ is associative and unital. It is shown that $\mathcal{DH}(T)$ contains the usual Hall algebra $\mathcal{H}(T)$ when $T$ is an abelian category. We will also prove an explicit formula for the derived Hall numbers purely in terms of invariants of the triangulated category associated to $T$. As an example, we describe the derived Hall algebra of an hereditary abelian category.

Stability in H1 of the sum of K solitary waves for some nonlinear Schrödinger equations

Abstract: In this article we consider nonlinear Schrodinger (NLS) equations in Rd for d=1, 2, and 3. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let Rk(t,x) be K solitary wave solutions of the equation with different speeds v1,v2,…,vK. Provided that the relative speeds of the solitary waves vk−vk−1 are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the Rk(t) is stable for t≥0 in some suitable sense in H1. To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrodinger equation. This property is similar to the L2 monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of K solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).

Generic Transfer for General Spin Groups

Abstract: We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp4 to GL4 follows from our result as a special case

Prym varieties and Teichmüller curves

Abstract: This article gives a uniform construction of infinitely many primitive Teichmuller curves V⊂Mg for g=2, 3, and 4

Counting Rational Points on Algebraic Varieties

Abstract: For any N≥2, let Z⊂PN be a geometrically integral algebraic variety of degree d. This article is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any e>0, we establish the estimate NZ(B)=Od,e,N(BdimZ+e), provided that d,e. As indicated, the implied constant depends at most on d,e, and N

Painlevé formulas of the limiting distributions for nonnull complex sample covariance matrices

Abstract: In a recent study of large non-null sample covariance matrices, a new sequence of functions gener- alizing the GUE Tracy-Widom distribution of random matrix theory was obtained. This paper derives Painleve formulas of these functions and use them to prove that they are indeed distribution functions. Applications of these new distribution functions to last passage percolation, queues in tandem and to- tally asymmetric simple exclusion process are also discussed. As a part of the proof, a representation of orthogonal polynomials on the unit circle in terms of an operator on a discrete set is presented.

Pullback of the lifting of elliptic cusp forms and Miyawaki's conjecture

Abstract: We construct a lifting from Siegel cusp forms of degree r to Siegel cusp forms of degree r+2n. For r=n=1, our result is a partial solution of a conjecture made by Miyawaki [27, page 307] in 1992. In particular, we can calculate the standard L-function of a cusp form of degree 3 and weight 12, which is in accordance with Miyawaki's conjecture. We give a conjecture on the Petersson inner product of the lifting in terms of certain L-values

The geometry of the Eisenstein-Picard modular group

Abstract: The Eisenstein-Picard modular group ${\rm PU}(2,1;\mathbb {Z}[\omega])$ is defined to be the subgroup of ${\rm PU}(2,1)$ whose entries lie in the ring $\mathbb {Z}[\omega]$, where $\omega$ is a cube root of unity. This group acts isometrically and properly discontinuously on ${\bf H}^2_\mathbb{C}$, that is, on the unit ball in $\mathbb {C}^2$ with the Bergman metric. We construct a fundamental domain for the action of ${\rm PU}(2,1;\mathbb {Z}[\omega])$ on ${\bf H}^2_\mathbb {C}$, which is a 4-simplex with one ideal vertex. As a consequence, we elicit a presentation of the group (see Theorem 5.9). This seems to be the simplest fundamental domain for a finite covolume subgroup of ${\rm PU}(2,1)$

Crystals and coboundary categories

Abstract: Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite-dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similarly to the role of the braid group in braided categories, a group naturally acts on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked, real, genus-zero stable curves

Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends

Abstract: Let M be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, Rn∖B(0,R) for some R>0, each of which carries the standard metric. Our main result is that the Riesz transform on M is bounded from Lp(M)→Lp(M;T*M) for 1 n; the result is new for 2 2 for a more general class of manifolds. Assume that M is an n-dimensional complete manifold satisfying the Nash inequality and with an O(rn) upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on Lp for some p>2 implies a Hodge–de Rham interpretation of the Lp-cohomology in degree 1 and that the map from L2- to Lp-cohomology in this degree is injective

$BC_n$-symmetric abelian functions

Abstract: We construct a family of BC_n-symmetric biorthogonal abelian functions generalizing Koornwinder's orthogonal polynomials, and prove a number of their properties, most notably analogues of Macdonald's conjectures. The construction is based on a direct construction for a special case generalizing Okounkov's interpolation polynomials. We show that these interpolation functions satisfy a collection of generalized hypergeometric identities, including new multivariate elliptic analogues of Jackson's summation and Bailey's transformation.

The minimal lamination closure theorem

Abstract: We prove that the closure of a complete embedded minimal surface M in a Riemannian three-manifold N has the structure of a minimal lamination when M has positive injectivity radius. When N is R3, we prove that such a surface M is properly embedded. Since a complete embedded minimal surface of finite topology in R3 has positive injectivity radius, the previous theorem implies a recent theorem of Colding and Minicozzi in [5, Corollary 0.7]; a complete embedded minimal surface of finite topology in R3 is proper. More generally, we prove that if M is a complete embedded minimal surface of finite topology and N has nonpositive sectional curvature (or is the Riemannian product of a Riemannian surface with R), then the closure of M has the structure of a minimal lamination

Linear and dynamical stability of Ricci-flat metrics

Abstract: We can talk about two kinds of stability of the Ricci flow at Ricci-flat metrics. One of them is a linear stability, defined with respect to Perelman's functional F (see [1, page 5]). The other one is a dynamical stability, and it refers to a convergence of a Ricci flow starting at any metric in a neighborhood of a considered Ricci-flat metric. We show that dynamical stability implies linear stability. We also show that a linear stability together with the integrability assumption implies dynamical stability. As a corollary, we get a stability result for K3-surfaces, part of which has been done in [11, Corollary 4.15, Theorem 4.16]. Our stability result applies to Calabi-Yau manifolds as well

Multibumps analysis in dimension $2$: Quantification of blow-up levels

Abstract: In this article, we describe the asymptotic behavior of sequences of solutions to some semilinear elliptic equations with critical exponential growth in planar domains. We prove, in particular, a result analogous to that of Struwe [12] in higher dimensions and extend the two-dimensional result of Adimurthi and Struwe [3] to arbitrary energies. We thus answer a question explicitly asked in this last article

Agmon-kato-kuroda theorems for a large class of perturbations

Abstract: We prove asymptotic completeness for operators of the form H=-Δ+L on L2(Rd), d≥2, where L is an admissible perturbation. Our class of admissible perturbations contains multiplication operators defined by real-valued potentials V∈Lq(Rd), q∈[d/2,(d+1)/2] (if d=2, then we require q∈(1,3/2]), as well as real-valued potentials V satisfying a global Kato condition. The class of admissible perturbations also contains first-order differential operators of the form a→·∇-∇·a→ for suitable vector potentials a. Our main technical statement is a new limiting absorption principle, which we prove using techniques from harmonic analysis related to the Stein-Tomas restriction theorem

Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink

Abstract: We prove a motivic analogue of Steenbrink's conjecture [25, Conjecture 2.2] on the Hodge spectrum (proved by M. Saito in [21]). To achieve this, we construct and compute motivic iterated vanishing cycles associated with two functions. We are also led to introduce a more general version of the convolution operator appearing in the motivic Thom-Sebastiani formula. Throughout the article we use the framework of relative equivariant Grothendieck rings of varieties endowed with an algebraic torus action

Serre's modularity conjecture: The level one case

Abstract: We prove the level one case of Serre's conjecture. Namely, we prove that any continuous, odd, irreducible representation ρ:GQ→GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in Sk(SL2(Z)) for some integer k≥2. The proof relies on the methods introduced in an earlier joint work with J.-P. Wintenberger [31] together with a new method of weight reduction

A unique extremal metric for the least eigenvalue of the Laplacian on the Klein bottle

Abstract: We prove the following conjecture recently formulated by Jakobson, Nadirashvili and Polterovich \cite{JNP}: on the Klein bottle $\mathbb{K}$, the metric of revolution $$g_0= {9+ (1+8\cos ^2v)^2\over 1+8\cos ^2v} \left(du^2 + {dv^2\over 1+8\cos ^2v}\right),$$ $0\le u <\frac\pi 2$, $0\le v <\pi$, is the \emph{unique} extremal metric of the first eigenvalue of the Laplacian viewed as a functional on the space of all Riemannian metrics of given area. The proof leads us to study a Hamiltonian dynamical system which turns out to be completely integrable by quadratures.

Ricci flow on Kähler-Einstein manifolds

Abstract: This is the continuation of our earlier article [10]. For any Kahler-Einstein surfaces with positive scalar curvature, if the initial metric has positive bisectional curvature, then we have proved (see [10]) that the Kahler-Ricci flow converges exponentially to a unique Kahler-Einstein metric in the end. This partially answers a long-standing problem in Ricci flow: On a compact Kahler-Einstein manifold, does the Kahler-Ricci flow converge to a Kahler-Einstein metric if the initial metric has positive bisectional curvature? In this article we give a complete affirmative answer to this problem

Laminar currents and birational dynamics

Abstract: We study the dynamics of a bimeromorphic map X→X, where X is a compact complex Kahler surface. Under a natural geometric hypothesis, we construct an invariant probability measure, which is mixing, hyperbolic, and of maximal entropy. The proof relies heavily on the theory of laminar currents and is new even in the case of polynomial automorphisms of C2. This extends recent results by E. Bedford and J. Diller