Showing papers in "Duke Mathematical Journal in 2003"

Puzzles and (equivariant) cohomology of Grassmannians

Abstract: The product of two Schubert cohomology classes on a Grassmannian ${\rm Gr}_k (\mathbb{c}^n)$ has long been known to be a positive combination of other Schubert classes, and many manifestly positive formulae are now available for computing such a product (eg, the Littlewood-Richardson rule or the more symmetric puzzle rule from A Knutson, T Tao, and C Woodward [KTW]) Recently, W~Graham showed in [G], nonconstructively, that a similar positivity statement holds for {\em $T$-equivariant} cohomology (where the coefficients are polynomials) We give the first manifestly positive formula for these coefficients in terms of puzzles using an ``equivariant puzzle piece'' The proof of the formula is mostly combinatorial but requires no prior combinatorics and only a modicum of equivariant cohomology (which we include) As a by-product the argument gives a new proof of the puzzle (or Littlewood-Richardson) rule in the ordinary-cohomology case, but this proof requires the equivariant generalization in an essential way, as it inducts backwards from the ``most equivariant'' case This formula is closely related to the one in A Molev and B Sagan [MS] for multiplying factorial Schur functions in three sets of variables, although their rule does not give a positive formula in the sense of [G] We include a cohomological interpretation of their problem and a puzzle formulation for it

Ellipsoids and matrix-valued valuations

Abstract: We obtain a classification of Borel measurable, GL(n) covariant, symmetric-matrix-valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by E. Lutwak, D. Yang, and G. Zhang.

Invariant distributions and time averages for horocycle flows

Abstract: There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.

On the p-adic L-function of a modular form at a supersingular prime

Abstract: In this paper we study the two $p$-adic $L$-functions attached to a modular form $f=\sum a\sb nq\sp n$ at a supersingular prime $p$. When $a\sb p=0$, we are able to decompose both the sum and the difference of the two unbounded distributions attached to $f$ into a bounded measure and a distribution that accounts for all of the growth. Moreover, this distribution depends only upon the weight of $f$ (and the fact that $a\sb p$ vanishes). From this description we explain how the $p$-adic $L$-function is controlled by two Iwasawa functions and by two power series with growth which have a fixed infinite set of zeros (Theorem 5.1). Asymptotic formulas for the $p$-part of the analytic size of the Tate-Shafarevich group of an elliptic curve in the cyclotomic direction are computed using this result. These formulas compare favorably with results established by M. Kurihara in [11] and B. Perrin-Riou in [23] on the algebraic side. Moreover, we interpret Kurihara's conjectures on the Galois structure of the Tate-Shafarevich group in terms of these two Iwasawa functions.

Nonsymmetric Macdonald polynomials and Demazure characters

Abstract: We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of the coefficients of the expansion of the specialized symmetric Macdonald polynomials in the basis formed by the irreducible characters of the associated finite Lie algebra.

Propagation in Hamiltonian dynamics and relative symplectic homology

Abstract: The main result asserts the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

Orbifold cohomology for global quotients

Abstract: Let $X$ be an orbifold that is a global quotient of a manifold $Y$ by a finite group $G$. We construct a noncommutative ring $H\sp \ast(Y, G)$ with a $G$-action such that $H\sp*(Y, G)\sp G$ is the orbifold cohomology ring of $X$ defined by W. Chen and Y. Ruan [CR]. When $Y=S\sp n$, with $S$ a surface with trivial canonical class and $G = \mathfrak {S}\sb n$, we prove that (a small modification of) the orbifold cohomology of $X$ is naturally isomorphic to the cohomology ring of the Hilbert scheme $S\sp {[n]}$, computed by M. Lehn and C. Sorger [LS2].

Cherednik algebras and differential operators on quasi-invariants

Abstract: We develop representation theory of the rational Cherednik algebra ${\rm H}\sb c$ associated to a finite Coxeter group $W$ in a vector space $\mathfrak {h}$, and a parameter "c." We use it to show that, for integral values of "c," the algebra ${\rm H}\sb c$ is simple and Morita equivalent to $\mathscr {D}(\mathfrak {h})\#W$, the cross product of $W$ with the algebra of polynomial differential operators on $\mathfrak {h}$. O. Chalykh, M. Feigin, and A. Veselov [CV1], [FV] introduced an algebra, $Q\sb c$, of quasi-invariant polynomials on $\mathfrak {h}$, such that $\mathbb {C}[\mathfrak {h}]\sp W\subset Q\sb c\subset \mathbb {C}[\mathfrak {h}]$. We prove that the algebra $\mathscr {D}(Q\sb c)$ of differential operators on quasi-invariants is a simple algebra, Morita equivalent to $\mathscr {D}(\mathfrak {h})$. The subalgebra $\mathscr {D}(Q\sb c)\sp W\subset \mathscr {D}(Q\sb c)$ of $W$-invariant operators turns out to be isomorphic to the spherical subalgebra $\mathbf {eH}\sb c\mathbf {e}\subset {\rm H}\sb c$. We show that $\mathscr {D}(Q\sb c)$ is generated, as an algebra, by $Q\sb c$ and its "Fourier dual" $Q\sb c\sp \flat$, and that $\mathscr {D}(Q\sb c)$ is a rank-one projective $(Q\sb c\otimes Q\sb c\sp \flat)$-module (via multiplication-action on $\mathscr {D}(Q\sb c)$ on opposite sides).

Tableau atoms and a new Macdonald positivity conjecture

Abstract: Let $\lambda$ be the space of symmetric functions, and let $V\ sb k$ be the subspace spanned by the modified Schur functions $\{S\sb \lambda[X/(1-t)]\}\sb {\lambda\sb 1\leq k}$. We introduce a new family of symmetric polynomials, $\{A\sp {(k)}\sb \lambda[X;t]\}\sp {\lambda\sb 1\leq k}$, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials $A\sp {(k)}\sb \lambda[X;t]$ form a basis for $V\sb k$ and that the Macdonald polynomials indexed by partitions whose first part is not larger than $k$ expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but also substantially refine it. Our construction of the $A\sp {(k)}\sb \lambda[X;t]$ relies on the use of tableau combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the $A\sp {(k)}\sb \lambda[X;t]$ seem to play the same role for $V\sb k$ as the Schur functions do for $\lambda$. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri-type and Littlewood-Richardson-type coefficients.

Special values of anticyclotomic $L$-functions

Abstract: The purpose of the paper is to extend and refine earlier results of the author on nonvanishing of the $L$-functions associated to modular forms in the anticyclotomic tower of conductor $p\sp \infty$ over an imaginary quadratic field. While the author's previous work proved that such $L$-functions are generically nonzero at the center of the critical strip, provided that the sign in the functional equation is $+1$, the present work includes the case where the sign is $-1$. In that case, it is shown that the derivatives of the $L$-functions are generically nonzero at the center. It is also shown that when the sign is $+1$, the algebraic part of the central critical value is nonzero modulo $\ell$ for certain $\ell$. Applications are given to the mu-invariant of the $p$-adic $L$-functions of M. Bertolini and H. Darmon. The main ingredients in the proof are a theorem of M. Ratner, as in the author's previous work, and a new "Jochnowitz congruence," in the spirit of Bertolini and Darmon.

Relative expanders or weakly relatively Ramanujan graphs

Abstract: Let $G$ be a fixed graph with largest (adjacency matrix) eigenvalue $\lambda\sb 0$ and with its universal cover having spectral radius $rho$. We show that a random cover of large degree over $G$ has its "new" eigenvalues bounded in absolute value by roughly $\sqrt{\lambda\sb 0\rho}$. This gives a positive result about finite quotients of certain trees having "small" eigenvalues, provided we ignore the "old" eigenvalues. This positive result contrasts with the negative result of A. Lubotzky and T. Nagnibeda which showed that there is a tree all of whose finite quotients are not "Ramanujan" in the sense of Lubotzky, R. Phillips, and P. Sarnakand of Y. Greenberg. Our main result is a "relative version" of the Broder-Shamir bound on eigenvalues of random regular graphs. Some of their combinatorial techniques are replaced by spectral techniques on the universal cover of $G$. For the choice of $G$ that specializes our main theorem to the Broder-Shamir setting, our result slightly improves theirs.

The Nash problem on arc families of singularities

Abstract: Nash [21] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution. He asked if the converse also holds: Does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero

Abstract: We determine those $k$-tuples of conjugacy classes of matrices from which it is possible to choose matrices that have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.

The Alexander polynomial of a plane curve singularity via the ring of functions on it

Abstract: We prove two formulae that express the Alexander polynomial $\Delta\sp C$ of several variables of a plane curve singularity $C$ in terms of the ring $\mathscr {O}\sb C$ of germs of analytic functions on the curve. One of them expresses $\Delta\sp C$ in terms of dimensions of some factors corresponding to a (multi-indexed) filtration on the ring $\mathscr {O}\sb C$. The other one gives the coefficients of the Alexander polynomial $\Delta\sp C$ as Euler characteristics of some explicitly described spaces (complements to arrangements of projective hyperplanes).

Elliptic genera of singular varieties

Abstract: The notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced, and the relation between them is studied. The 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. The orbifold elliptic genus is given in terms of the fixed-point sets of the action. We show that the generating function for the orbifold elliptic $\sum {\rm Ell\sp {orb}}(X\sp n,\Sigma\sb n)p\sp n$ for symmetric groups $\Sigma\sb n$ acting on $n$-fold products coincides with the one proposed by R. Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde. The two notions of elliptic genera are conjectured to coincide.

Improved local well-posedness for quasilinear wave equations in dimension three

Abstract: We improve recent results of H. Bahouri and J.-Y. Chemin and of D. Tataru concerning local well-posedness theory for quasilinear wave equations. Our approach is based on the proof of the Strichartz estimates using a combination of geometric methods and harmonic analysis. The geometric component relies on and takes advantage of the nonlinear structure of the equation.

Motivic integration on smooth rigid varieties and invariants of degenerations

Abstract: We develop a theory of motivic integration for smooth rigid varieties. As an application we obtain a motivic analogue for rigid varieties of Serre's invariant for $p$-adic varieties. Our construction provides new geometric birational invariants of degenerations of algebraic varieties. For degenerations of Calabi-Yau varieties, our results take a stronger form.

A polytope calculus for semisimple groups

Abstract: We define a collection of polytopes associated to a semisimple group $\mathsf {G}$. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirkovic and K. Vilonen. These cycles are a canonical basis for the intersection homology of (the closures of the strata of) the loop Grassmannian.

Matroids motives, and a conjecture of Kontsevich

Abstract: We show that a certain class of varieties with origin in physics generates (additively) the Denef-Loeser ring of motives. In particular, this disproves a conjecture of M. Kontsevich on the number of points of these varieties over finite fields.

Billiards in rectangles with barriers

Abstract: In this paper we consider billiards in a square with a barrier. We use Ratner's theorem to compute the asymptotics for the number of closed orbits.

Stationary determinantal processes: Phase multiplicity, Bernoullicity, entropy, and domination

Abstract: We study a class of stationary processes indexed by ℤd that are defined via minors of d-dimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong K-property, a particular strengthening of the usual K (Kolmogorov) property. We show that all of these processes are Bernoulli shifts (isomorphic to independent identically distributed (i.i.d.) processes in the sense of ergodic theory). We obtain estimates of their entropies, and we relate these processes via stochastic domination to product measures.

Low-lying zeros of dihedral L-functions

Abstract: Assuming the grand Riemann hypothesis, we investigate the distribution of the lowlying zeros of the $L$-functions $L(s,\psi)$, where $\psi$ is a character of the ideal class group of the imaginary quadratic field $\mathbb {Q}(\sqrt{-D}) (D\text {squarefree},D>3,D\equiv 3(\mod 4))$. We prove that, in the vicinity of the central point $s = 1/2$, the average distribution of these zeros (for $D\longrightarrow \infty$) is governed by the symplectic distribution. By averaging over $D$, we go beyond the natural bound of the support of the Fourier transform of the test function. This problem is naturally linked with the question of counting primes $p$ of the form $4p = m\sp 2+Dn\sp 2$, and sieve techniques are applied.

Entropy jumps in the presence of a spectral gap

Abstract: It is shown that if X is a random variable whose density satisfies a Poincare inequality, and Y is an independent copy of X, then the entropy of (X + Y )= p 2 is greater than that of X by a fixed fraction of the entropy gap between X and the Gaussian of the same variance. The argument uses a new formula for the Fisher information of a marginal, which can be viewed as a local, reverse form of the Brunn-Minkowski inequality (in its functional form due to Prekopa and Leindler).

Compactifications defined by arrangements, II: Locally symmetric varieties of type IV

Abstract: We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the Proj of an algebra of meromorphic automorphic forms. When that complement has a moduli-space interpretation, then what we get is often a compactification obtained by means of geometric invariant theory. We illustrate this with several examples: moduli spaces of polarized K3 and Enriques surfaces and the semiuniversal deformation of a triangle singularity. We also discuss the question of when a type IV arrangement is definable by an automorphic form.

Squarefree values of multivariable polynomials

Abstract: Given $f\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute the density of $x\in \mathbf {Z}\sp n$ such that $f(x)$ is squarefree, assuming the abc-conjecture. Given $f,g\in \mathbf {Z}[x\sb 1,\ldots x\sb n]$, we compute unconditionally the density of $x\in \mathbf {Z}\sp n$ such that $\gcd(f(x),g(x))=1$. Function field analogues of both results are proved unconditionally. Finally, assuming the abc-conjecture, given $f\in \mathbf {Z}[x]$, we estimate the size of the image of $f(\{1,2,\ldots n\})$ in $(\mathbf {Q}\sp \ast/\mathbf {Q}\sp {\ast 2})\cup \{0\}$.

Discrete gap probabilities and discrete Painlevé equations

Abstract: We prove that Fredholm determinants of the form $\det(1-K\sb s)$, where $K\sb s$ is the restriction of either the discrete Bessel kernel or the discrete $\sb 2F\sb 1$-kernel to $\{s, s + 1,\ldots\}$, can be expressed, respectively, through solutions of discrete Painleve II (dPII) and Painleve V (dPV) equations. These Fredholm determinants can also be viewed as distribution functions of the first part of the random partitions distributed according to a Poissonized Plancherel measure and a $z$-measure, or as normalized Toeplitz determinants with symbols $e\sp {\eta(\zeta+\zeta\sp {-1})}$ and $(1 +\sqrt {\xi}\zeta)\sp z(1 +\sqrt {\xi}/\zeta)\sp {z\sp \prime}$. The proofs are based on a general formalism involving discrete integrable operators and discrete Riemann-Hilbert problems. A continuous version of the formalism has been worked out in [BD].

Compactifications defined by arrangements I: The Ball Quotient case

Abstract: We define a natural compactification of an arrangement complement in a ball quotient. We show that when this complement has a moduli space interpretation, then this compactification is often one that appears naturally by means of geometric invariant theory. We illustrate this with the moduli spaces of smooth quartic curves and rational elliptic surfaces.

Closed orbits for actions of maximal tori on homogeneous spaces

Abstract: Let $G$ be a real algebraic group defined over $\mathbb{Q}$, let $\Gamma$ be an arithmetic subgroup, and let $T$ be any torus containing a maximal $\mathbb{R}$-split torus. We prove that the closed orbits for the action of $T$ on $G/\Gamma$ admit a simple algebraic description. In particular, we show that if $G$ is reductive, an orbit $Tx\Gamma$ is closed if and only if $x^{-1}Tx$ is a product of a compact torus and a torus defined over $\mathbb{Q}$, and it is divergent if and only if the maximal $\mathbb{R}$-split subtorus of $x^{-1}Tx$ is defined over $\mathbb{Q}$ and $\mathbb{Q}$-split. Our analysis also yields the following: · there is a compact $K \subset G/\Gamma$ which intersects every $T$-orbit; · if $\rank_{\mathbb{Q}}G<\rank_{\mathbb{R}} G$, there are no divergent orbits for $T$.

On the Brody hyperbolicity of moduli spaces for canonically polarized manifolds

Abstract: We show that the moduli stack $\mathscr {M}\sb h$ of canonically polarized complex manifolds with Hilbert polynomial $h$ is Brody hyperbolic. Hence if $M\sb h$ denotes the corresponding coarse moduli scheme, and if $U \to M\sb h$ is a quasi-finite morphism, induced by a family, then there are no nonconstant holomorphic maps $\mathbb {C}\to U$.

Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters

Abstract: The Tamagawa number conjecture proposed by S. Bloch and K. Kato describes the "special values" of L-functions in terms of cohomological data. The main conjecture of Iwasawa theory describes a p-adic L-function in terms of the structure of modules for the Iwasawa algebra. We give a complete proof of both conjectures (up to the prime 2) for L-functions attached to Dirichlet characters. We use the insight of Kato and B. Perrin-Riou that these two conjectures can be seen as incarnations of the same mathematical content. In particular, they imply each other. By a bootstrapping process using the theory of Euler systems and explicit reciprocity laws, both conjectures are reduced to the analytic class number formula. Technical problems with primes dividing the order of the character are avoided by using the correct cohomological formulation of the main conjecture.