Showing papers in "Duke Mathematical Journal in 2004"

On two geometric theta lifts

Abstract: The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper we establish for the orthogonal group O(p,2) an adjointness result between Borcherds's singular theta lift and the Kudla-Millson lift. We extend this result to arbitrary signature by introducing a new singular theta lift for O(p,q). On the geometric side, this lift can be interpreted as a differential character, in the sense of Cheeger and Simons, for the cycles under consideration.

Modular invariance of vertex operator algebras satisfying C 2 -cofiniteness

Abstract: We investigate trace functions of modules for vertex operator algebras (VOA) satisfying C2 -cofiniteness. For the modular invariance property, Zhu assumed two conditions in [Z]: (1) A(V) is semisimple and (2) C2 -cofiniteness. We show that C2 -cofiniteness is enough to prove a modular invariance property. For example, if a VOA V= ⊕ m=0 ∞ V m is C2 -cofinite, then the space spanned by generalized characters (pseudotrace functions of the vacuum element) of V -modules is a finite-dimensional $\SL_2(\mathbb{Z})$ SL 2 ( Z) -invariant space and the central charge and conformal weights are all rational numbers. Namely, we show that C2 -cofiniteness implies "rational conformal field theory" in a sense as expected in Gaberdiel and Neitzke [GN]. Viewing a trace map as a symmetric linear map and using a result of symmetric algebras, we introduce "pseudotraces" and pseudotrace functions and then show that the space spanned by such pseudotrace functions has a modular invariance property. We also show that C2 -cofiniteness is equivalent to the condition that every weak module is an N -graded weak module that is a direct sum of generalized eigenspaces of L(0) .

Finite group actions on C*-algebras with the Rohlin property, I

Abstract: Basic properties of finite group actions with the Rohlin property on unital C*-algebras are investigated. A characterization of finite group actions with the Rohlin property on the Cuntz algebra 02 is given in terms of central sequences, which may be considered as an equivariant version of E. Kirchberg and N. C. Phillips's characterization of 02 . A large class of symmetries on 02 are classified in terms of the fixed-point algebras for conjugacy and the crossed products for cocycle conjugacy. Model actions of symmetries of 02 are constructed for given K-theoretical invariants.

Three-dimensional flops and noncommutative rings

Abstract: For $Y,Y^+$ three-dimensional smooth varieties related by a flop, Bondal and Orlov conjectured that the derived categories $D^b({\rm coh}(Y))$ and $D^b({\rm coh}(Y^+))$ are equivalent. This conjecture was recently proved by Bridgeland. Our aim in this paper is to give a partially new proof of Bridgeland's result using noncommutative rings. The new proof also covers some mild singular and higher-dimensional situations (including those occuring in the recent paper by Chen [11]).

Amoebas, Monge-Ampàre measures, and triangulations of the Newton polytope

Abstract: The amoeba of a holomorphic function $f$ is, by definition, the image in $\mathbf{R}^n$ of the zero locus of $f$ under the simple mapping that takes each coordinate to the logarithm of its modulus. The terminology was introduced in the 1990s by the famous (biologist and) mathematician Israel Gelfand and his coauthors Kapranov and Zelevinsky (GKZ). In this paper we study a natural convex potential function $N_f$ with the property that its Monge-Ampere mass is concentrated to the amoeba of $f$ We obtain results of two kinds; by approximating $N_f$ with a piecewise linear function, we get striking combinatorial information regarding the amoeba and the Newton polytope of $f$; by computing the Monge-Ampere measure, we find sharp bounds for the area of amoebas in $\mathbf{R}^n$. We also consider systems of functions $f_{1},\dots,f_{n}$ and prove a local version of the classical Bernstein theorem on the number of roots of systems of algebraic equations.

Integration of twisted Dirac brackets

Abstract: Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form phi on M correspond to maps from the Lie algebroid of G into T* M satisfying an algebraic condition and a differential condition with respect to the phi-twisted Courant bracket. This correspondence describes, as a special case, the global objects associated to phi-twisted Dirac structures. As applications, we relate our results to equivariant cohomology and foliation theory, and we give a new description of quasi-Hamiltonian spaces and group-valued momentum maps.

Stability estimates for the X-ray transform of tensor fields and boundary rigidity

Abstract: We study the boundary rigidity problem for domains in Rn: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform Ig. We prove that the normal operator Ng=I*gIg is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

Hodge cohomology of gravitational instantons

Abstract: We study the space of L2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincare metrics on Q-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kahler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L2 signature formula implied by our result is closely related to the one proved by Dai [25] and more generally by Vaillant [67], and identifies Dai’s -invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [12] and the forthcoming paper of Cheeger and Dai [17]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about L2 harmonic forms in duality theories in string theory.

Crystal bases and two-sided cells of quantum affine algebras

Abstract: Let $\mathfrak{g}$ be an affine Kac-Moody Lie algebra. Let $\mathbf{U}^+$ be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to $\mathfrak{g}$. We construct a basis of $\mathbf{U}^+$ which is related to the Kashiwara-Lusztig global crystal basis (or canonical basis) by an upper-triangular matrix (with respect to an explicitly defined ordering) with 1's on the diagonal and with above-diagonal entries in $q_s^{-1} \mathbf{Z}[q_s^{-1}]$. Using this construction, we study the global crystal basis $\mathscr{B}(\widetilde{\mathbf{U}})$ of the modified quantum enveloping algebra defined by Lusztig. We obtain a Peter-Weyl-like decomposition of the crystal $\mathscr{B}(\widetilde{\mathbf{U}})$ (Th. 4.18), as well as an explicit description of two-sided cells of $\mathscr{B}(\widetilde{\mathbf{U}})$ and the limit algebra of $\widetilde{\mathbf{U}}$ at $q=0$ (Th. 6.44).

The period-index problem for the Brauer group of an algebraic surface

Abstract: In this paper we show that the period equals the index for elements of Brauer groups of (function fields of) surfaces A key idea of the proof is that any Azumaya algebra over a surface can be transformed into an Azumaya algebra that is unobstructed

A characterization of the Anderson metal-insulator transport transition

Abstract: We investigate the Anderson metal-insulator transition for random Schrodinger operators. We define the strong insulator region to be the part of the spectrum where the random operator exhibits strong dynamical localization in the Hilbert-Schmidt norm. We introduce a local transport exponent β(E) and set the weak metallic transport region to be the part of the spectrum with nontrivial transport (i.e., β(E)>0). We prove that these insulator and metallic regions are complementary sets in the spectrum of the random operator and that the local transport exponent β(E) provides a characterization of the metal-insulator transport transition. Moreover, we show that if there is such a transition, then β(E) has to be discontinuous at a transport mobility edge. More precisely, we show that if the transport is nontrivial, then β(E)≥1/(2d), where d is the space dimension. These results follow from a proof that slow transport of quantum waves in random media implies the starting hypothesis for the authors' bootstrap multiscale analysis. We also conclude that the strong insulator region coincides with the part of the spectrum where we can perform a bootstrap multiscale analysis, proving that the multiscale analysis is valid all the way up to a transport mobility edge.

Classification of simple C*-algebras of tracial topological rank zero

Abstract: We give a classification theorem for unital separable simple nuclear C*-algebras with tracial topological rank zero which satisfy the universal coefficient theorem. Let A and B be two such C*-algebras. We prove that A≌B if and only if (K0(A), K0(A)+, [1A], K1(A)) ≌ K0(B), K0(B)+, [1B], K1(B)).

Restriction and Kakeya phenomena for finite fields

Abstract: The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there are some new phenomena in the finite case which deserve closer study.

Compact moduli of plane curves

Abstract: We construct a compactication Md of the moduli space of plane curves of degreed. We regard a plane curveC P 2 as a surface-divisor pair (P 2 ;C) and dene Md as a moduli space of pairs (X;D) where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stackMd is smooth and the degenerate surfaces X can be described explicitly. MSC2000: 14H10, 14J10, 14E30.

Multidimensional boundary layers for a singularly perturbed Neumann problem

Abstract: We continue the study of [34], proving concentration phenomena for the equation − e2 Δu + u = up in a smooth bounded domain Ω ⊆ $\mathbb{R}^n$ and with Neumann boundary conditions. The exponent p is greater than or equal to 1, and the parameter e is converging to zero. For a suitable sequence ej → 0, we prove the existence of positive solutions uj concentrating at the whole boundary of Ω or at some of its components.

Quantum $K$-theory, I: Foundations

Abstract: This work is devoted to the study of the foundations of quantum K -theory, a K -theoretic version of quantum cohomology theory In particular, it gives a deformation of the ordinary K -ring K(X) of a smooth projective variety X , analogous to the relation between quantum cohomology and ordinary cohomology This new quantum product also gives a new class of Frobenius manifolds

Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

Abstract: For a finite subgroup G⊂SL(3,ℂ), Bridgeland, King, and Reid [BKR] proved that the moduli space of G-clusters is a crepant resolution of the quotient ℂ3/G . This paper considers the moduli spaces $\mathcal{M}$θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of geometric invariant theory (GIT) parameter θ. For G Abelian, we prove that every projective crepant resolution of ℂ3/G is isomorphic to $\mathcal{M}$θ for some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier-Mukai transform. We also uncover explicit equivalences between the derived categories of moduli $\mathcal{M}$θ for parameters lying in adjacent GIT chambers.

Holomorphic triangle invariants and the topology of symplectic four-manifolds

Abstract: This article analyzes the interplay between symplectic geometry in dimension $4$ and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in [20]. Specifically, we establish a nonvanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This leads to restrictions on the topology of Stein fillings of such three-manifolds.

Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators

Abstract: In this note we improve the known L p -bounds for Bochner-Riesz operators and their maximal operators.

On the Eisenstein cohomology of arithmetic groups

Abstract: The cohomology $H ^* (\Gamma, E)$ of an arithmetic subgroup $\Gamma$ of a connected reductive algebraic group $G$ defined over some algebraic number field $F$ can be interpreted in terms of the automorphic spectrum of $\Gamma$. With this framework in place, there is a sum decomposition of the cohomology into the cuspidal cohomology (i.e., classes represented by cuspidal automorphic forms for $G$) and the so-called Eisenstein cohomology constructed as the span of appropriate residues or derivatives of Eisenstein series attached to cuspidal automorphic forms on the Levi components of proper parabolic $F$-subgroups of $G$. The main objective of this paper is to isolate a specific structural part in the Eisenstein cohomology. This pertains to regular Eisenstein cohomology classes attached to cuspidal automorphic representations whose archimedean component is tempered. It is shown that the cohomological degree of these classes is bounded from below by the constant $q_0(G(\mathbb{R}))=((1/2) [\dim X_{G(\mathbb{R})}-(\rk (G(\mathbb{R}))-\rk(K_{\mathbb{R}}))]$, where $K_{\mathbb{R}}$ denotes a maximal compact subgroup of the real Lie group $G(\mathbb{R})$, where $X_{G(\mathbb{R})}$ is the associated symmetric space. This investigation has various applications. One of these is a vanishing result for the cohomology in the generic case (i.e., where the representation determining the coefficient system $E$ has regular highest weight) in degrees below $q_0(G(\mathbb{R}))$. This is a sharp bound depending only on the underlying real Lie group $G(\mathbb{R})$ (Corollary 5.6, Proposition 5.8). This result is supplemented by a qualitative structural result in the description of the cohomology in higher degrees by means of regular Eisenstein cohomology classes.

Automorphisms of the pants complex

Abstract: In loving memory of my mother, Batya 1. Introduction In the theory of mapping class groups, \" curve complexes \" assume a role similar to the one that buildings play in the theory of linear groups. Ivanov, Korkmaz, and Luo showed that the automorphism group of the curve complex for a surface is generally isomorphic to the extended mapping class group of the surface. In this paper, we show that the same is true for the pants complex. Throughout, S is an orientable surface whose Euler characteristic χ (S) is negative , while g,b denotes a surface of genus g with b boundary components. Also, Mod(S) means the extended mapping class group of S (the group of homotopy classes of self-homeomorphisms of S). The pants complex of S, denoted C P (S), has vertices representing pants decom-positions of S, edges connecting vertices whose pants decompositions differ by an elementary move, and 2-cells representing certain relations between elementary moves (see Sec. 2). Its 1-skeleton C 1 P (S) is called the pants graph and was introduced by Hatcher and Thurston. We give a detailed definition of the pants complex in Section 2. Brock proved that C 1 P (S) models the Teichmüller space endowed with the Weil-Petersson metric, T W P (S), in that the spaces are quasi-isometric (see [1]). Our results further indicate that C 1 P (S) is the \" right \" combinatorial model for T W P (S), in that Aut C 1 P (S) (the group of simplicial automorphisms of C 1 P (S)) is shown to be Mod(S). This is in consonance with the result of Masur and Wolf that the isometry group of T W P (S) is Mod(S) (see [10]).

Gaps in √n mod 1 and ergodic theory

Abstract: Cut the unit circle $S^1=\mathbb{R}/\mathbb{Z}$ at the points $\{\sqrt{1}\}, \{\sqrt{2}\},\ldots,\{\sqrt{N}\}$, where $\{x\} = x \bmod 1$, and let $J_1, \ldots, J_N$ denote the complementary intervals, or \emph{gaps}, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths $|J_i|/N$ are governed by an explicit piecewise real-analytic distribution $F(t) \,dt$ with phase transitions at $t=1/2$ and $t=2$. The gap distribution is related to the probability $p(t)$ that a random unimodular lattice translate $\Lambda \subset \mathbb{R}^2$ meets a fixed triangle $S_t$ of area $t$; in fact, $p''(t) = -F(t)$. The proof uses ergodic theory on the universal elliptic curve \[ E = \big(\mathrm{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2\big)/ \big(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\big) \] and Ratner's theorem on unipotent invariant measures.

Existence of optimal transport maps for crystalline norms

Abstract: We show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in ℝn, assuming that the initial distribution of mass is absolutely continuous with respect to $\mathcal{L}$n. The proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the Euclidean distance as cost function. Moreover, improving a construction by Larman, we show the existence of a Nikodym set in ℝ3 having full measure in the unit cube, intersecting each element of a family of pairwise disjoint open lines only in one point. This example can be used to show that the regularity of the decomposition in transport rays plays an essential role in Sudakov-type arguments for proving the existence of optimal transport maps.

Jumping coefficients of multiplier ideals

Abstract: We study some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. These jumping coefficients consist of an increasing sequence of positive rational numbers beginning with the log-canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we see that they arise naturally in several different contexts.

K-theory and derived equivalences

Abstract: We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, and for a large class of abelian categories.

Geometric inequalities on locally conformally flat manifolds

Abstract: Through the study of some elliptic and parabolic fully nonlinear partial differential equations, we establish conformal versions of the quermass-integral inequality, the Sobolev inequality, and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on {locally conformally flat} manifolds.

Noncommutative projective curves and quantum loop algebras

Abstract: We show that the Hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding Kac-Moody algebra. In particular, this yields a geometric realization of the quantized enveloping algebra of elliptic (or $2$-toroidal) algebras of types $D_4^{(1,1)}$, $E^{(1,1)}_6$, $E^{(1,1)}_7$, and $E_{8}^{(1,1)}$ in terms of coherent sheaves on weighted projective lines of genus one or, equivalently, in terms of equivariant coherent sheaves on elliptic curves.

Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles

Abstract: The main theme of this paper is a relative version of the almost existence theorem for periodic orbits of autonomous Hamiltonian systems. We show that almost all low levels of a function on a geometrically bounded symplectically aspherical manifold carry contractible periodic orbits of the Hamiltonian flow, provided that the function attains its minimum along a closed symplectic submanifold. As an immediate consequence, we obtain the existence of contractible periodic orbits on almost all low energy levels for twisted geodesic flows with symplectic magnetic field. We give examples of functions with a sequence of regular levels without periodic orbits, converging to an isolated, but very degenerate, minimum. The proof of the relative almost existence theorem hinges on the notion of the relative Hofer-Zehnder capacity and on showing that this capacity of a small neighborhood of a symplectic submanifold is finite. The latter is carried out by proving that the flow of a Hamiltonian with sufficiently large variation has a nontrivial contractible one-periodic orbit when the Hamiltonian is constant and equal to its maximum near a symplectic submanifold and supported in a neighborhood of the submanifold.

Smoothing effect for Schrödinger equations

Abstract: On demontre que l'hypothese de non capture est necessaire pour l'effet regularisant ($H^{1/2}$) pour l'equation de Schrodinger avec conditions aux limites de Dirichlet a l'exterieur d'un domaine de $\mathbb{R}^d$. On donne aussi une classe d'obstacles captifs (l'exemple d'Ikawa) pour lesquels on demontre un effet regularisant affaibli ($H^{1/2 - \varepsilon}$).

Descent Properties of Homotopy K-Theory

Abstract: In this paper, we show that the widely held expectation that Weibel's homotopy K-theory satisfies cdh-descent is indeed fulfilled for schemes over a field of characteristic zero. The main ingredient in the proof is a certain factorization of the resolution of hypersurface singularities. Some consequences are derived. Finally, some evidence for a conjecture of Weibel concerning negative K-theory is given.