Duke Mathematical Journal Articles (Project Euclid)

Crofton measures and Minkowski valuations

Thu, 05 Aug 2010 15:41 EDT

Franz E. Schuster

Source: Duke Math. J., Volume 154, Number 1, 1--30.

Abstract:
A description of continuous rigid motion compatible Minkowski valuations is established. As an application, we present a Brunn-Minkowski type inequality for intrinsic volumes of these valuations

Erratum

Tue, 24 Jul 2012 08:46 EDT

Source: Duke Math. J., Volume 161, Number 11, 2255--2255.

Quadratic tangles in planar algebras

Thu, 06 Sep 2012 08:55 EDT

Vaughan F. R. Jones

Source: Duke Math. J., Volume 161, Number 12, 2257--2295.

Abstract:
In planar algebras, we show how to project certain simple quadratic tangles onto the linear space spanned by linear and constant tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.

Decomposition theorem for perverse sheaves on Artin stacks over finite fields

Thu, 06 Sep 2012 08:55 EDT

Shenghao Sun

Source: Duke Math. J., Volume 161, Number 12, 2297--2310.

Abstract:
We generalize the decomposition theorem for perverse sheaves to Artin stacks with affine stabilizers over finite fields.

Local-global compatibility and the action of monodromy on nearby cycles

Thu, 06 Sep 2012 08:55 EDT

Ana Caraiani

Source: Duke Math. J., Volume 161, Number 12, 2311--2413.

Abstract:
We strengthen the local-global compatibility of Langlands correspondences for $\operatorname{GL}_{n}$ in the case when $n$ is even and $l\neq p$ . Let $L$ be a CM field, and let $\Pi$ be a cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{L})$ which is conjugate self-dual. Assume that $\Pi_{\infty}$ is cohomological and not “slightly regular,” as defined by Shin. In this case, Chenevier and Harris constructed an $l$ -adic Galois representation $R_{l}(\Pi)$ and proved the local-global compatibility up to semisimplification at primes $v$ not dividing $l$ . We extend this compatibility by showing that the Frobenius semisimplification of the restriction of $R_{l}(\Pi)$ to the decomposition group at $v$ corresponds to the image of $\Pi_{v}$ via the local Langlands correspondence. We follow the strategy of Taylor and Yoshida, where it was assumed that $\Pi$ is square-integrable at a finite place. To make the argument work, we study the action of the monodromy operator $N$ on the complex of nearby cycles on a scheme which is locally étale over a product of strictly semistable schemes and we derive a generalization of the weight spectral sequence in this case. We also prove the Ramanujan–Petersson conjecture for $\Pi$ as above.

New outlook on the Minimal Model Program, I

Thu, 06 Sep 2012 08:55 EDT

Paolo Cascini, Vladimir Lazić

Source: Duke Math. J., Volume 161, Number 12, 2415--2467.

Abstract:
We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.

Heisenberg categorification and Hilbert schemes

Thu, 11 Oct 2012 08:59 EDT

Sabin Cautis, Anthony Licata

Source: Duke Math. J., Volume 161, Number 13, 2469--2547.

Abstract:
Given a finite subgroup $\Gamma \subset \mathrm {SL}_{2}(\mathbb{C})$ we define an additive $2$ -category $\mathcal{H}_{\Gamma }$ whose Grothendieck group is isomorphic to an integral form $\mathfrak{h}_{\Gamma }$ of the Heisenberg algebra. We construct an action of $\mathcal{H}_{\Gamma }$ on derived categories of coherent sheaves on Hilbert schemes of points on the minimal resolutions $\widehat{ \mathbb{C}^{2}/\Gamma }$ .

Generalizations of the Kolmogorov–Barzdin embedding estimates

Thu, 11 Oct 2012 08:59 EDT

Misha Gromov, Larry Guth

Source: Duke Math. J., Volume 161, Number 13, 2549--2603.

Abstract:
We consider several ways to measure the “geometric complexity” of an embedding from a simplicial complex into Euclidean space. One of these is a version of “thickness,” based on a paper of Kolmogorov and Barzdin. We prove inequalities relating the thickness and the number of simplices in the simplicial complex, generalizing an estimate that Kolmogorov and Barzdin proved for graphs. We also consider the distortion of knots. We give an alternate proof of a theorem of Pardon that there are isotopy classes of knots requiring arbitrarily large distortion. This proof is based on the expander-like properties of arithmetic hyperbolic manifolds.

A uniform open image theorem for $\ell$ -adic representations, I

Thu, 11 Oct 2012 08:59 EDT

Anna Cadoret, Akio Tamagawa

Source: Duke Math. J., Volume 161, Number 13, 2605--2634.

Abstract:
Let $k$ be a field finitely generated over $\mathbb{Q}$ , and let $X$ be a smooth, separated, and geometrically connected curve over $k$ . Fix a prime $\ell $ . A representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ is said to be geometrically Lie perfect if the Lie algebra of $\rho(\pi_{1}(X_{\overline{k}}))$ is perfect. Typical examples of such representations are those arising from the action of $\pi_{1}(X)$ on the generic $\ell$ -adic Tate module $T_{\ell}(A_{\eta})$ of an abelian scheme $A$ over $X$ or, more generally, from the action of $\pi_{1}(X)$ on the $\ell$ -adic étale cohomology groups $\operatorname{H}^{i}_{\mathrm{\acute{e}t}}(Y_{\overline{\eta}},\mathbb{Q}_{\ell})$ , $i\geq 0$ , of the geometric generic fiber of a smooth proper scheme $Y$ over $X$ . Let $G$ denote the image of $\rho$ . Any $k$ -rational point $x$ on $X$ induces a splitting $x:\Gamma_{k}:=\pi_{1}(\operatorname{Spec}(k))\rightarrow\pi_{1}(X)$ of the canonical restriction epimorphism $\pi_{1}(X)\rightarrow\Gamma _{k}$ so one can define the closed subgroup $G_{x}:=\rho\circ x(\Gamma_{k})\subset G$ . The main result of this paper is the following uniform open image theorem. Under the above assumptions, for every geometrically Lie perfect representation $\rho:\pi_{1}(X)\rightarrow\operatorname{GL}_{m}(\mathbb{Z}_{\ell})$ , the set $X_{\rho}$ of all $x\in X(k)$ such that $G_{x}$ is not open in $G$ is finite and there exists an integer $B_{\rho}\geq1$ such that $[G:G_{x}]\leq B_{\rho}$ for every $x\inX(k)\smallsetminus X_{\rho}$ .

Homogeneity in the free group

Thu, 11 Oct 2012 08:59 EDT

Chloé Perin, Rizos Sklinos

Source: Duke Math. J., Volume 161, Number 13, 2635--2668.

Abstract:
We show that any nonabelian free group $\mathbb {F}$ is strongly $\aleph_{0}$ -homogeneous, that is, that finite tuples of elements which satisfy the same first-order properties are in the same orbit under $\operatorname {Aut}(\mathbb {F})$ . We give a characterization of elements in finitely generated groups which have the same first-order properties as a primitive element of the free group. We deduce as a consequence that most hyperbolic surface groups are not strongly $\aleph_{0}$ -homogeneous.

A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method

Fri, 26 Oct 2012 09:40 EDT

Mattias Dahl, Romain Gicquaud, Emmanuel Humbert

Source: Duke Math. J., Volume 161, Number 14, 2669--2697.

Abstract:
Let $(M,g)$ be a compact Riemannian manifold on which a trace-free and divergence-free $\sigma\in W^{1,p}$ and a positive function $\tau\in W^{1,p}$ , $p\textgreater n$ are fixed. In this paper, we study the vacuum Einstein constraint equations by using the well-known conformal method with data $\sigma$ and $\tau$ . We show that if no solution exists, then there is a nontrivial solution of another nonlinear limit equation on $1$ -forms. This last equation can be shown to be without solutions in many situations. As a corollary, we get the existence of solutions of the vacuum Einstein constraint equation under explicit assumptions which, in particular, hold on a dense set of metrics $g$ for the $C^{0}$ -topology.

$K$ -classes for matroids and equivariant localization

Fri, 26 Oct 2012 09:40 EDT

Alex Fink, David E. Speyer

Source: Duke Math. J., Volume 161, Number 14, 2699--2723.

Abstract:
To every matroid, we associate a class in the $K$ -theory of the Grassmannian. We study this class by using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend the second author’s results concerning the behavior of such classes under direct sum, series and parallel connection, and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult.

Rigidity of min-max minimal spheres in three-manifolds

Fri, 26 Oct 2012 09:40 EDT

Fernando C. Marques, André Neves

Source: Duke Math. J., Volume 161, Number 14, 2725--2752.

Abstract:
In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a three-sphere which has scalar curvature greater than or equal to $6$ and is not round must have an embedded minimal sphere of area strictly smaller than $4\pi$ and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.

Quantization and the Hessian of Mabuchi energy

Fri, 26 Oct 2012 09:40 EDT

Joel Fine

Source: Duke Math. J., Volume 161, Number 14, 2753--2798.

Abstract:
Let $L\to X$ be an ample bundle over a compact complex manifold. Fix a Hermitian metric in $L$ whose curvature defines a Kähler metric on $X$ . The Hessian of Mabuchi energy is a fourth-order elliptic operator $\mathcal{D}^{*}\mathcal{D}$ on functions which arises in the study of scalar curvature. We quantize $\mathcal{D}^{*}\mathcal{D}$ by the Hessian $P^{*}_{k}P_{k}$ of balancing energy, a function appearing in the study of balanced embeddings. $P^{*}_{k}P_{k}$ is defined on the space of Hermitian endomorphisms of $H^{0}(X,L^{k})$ endowed with the $L^{2}$ -inner product. We first prove that the leading order term in the asymptotic expansion of $P^{*}_{k}P_{k}$ is $\mathcal{D}^{*}\mathcal{D}$ . We next show that if $\operatorname {Aut}(X,L)/\mathbb{C}^{*}$ is discrete, then the eigenvalues and eigenspaces of $P^{*}_{k}P_{k}$ converge to those of $\mathcal{D}^{*}\mathcal{D}$ . We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature Kähler metric. As consequences of our results we prove that an estimate of Phong and Sturm is sharp and give a negative answer to a question posed by Donaldson. We also discuss some possible applications to the study of Calabi flow.

Integral points of bounded height on partial equivariant compactifications of vector groups

Thu, 29 Nov 2012 09:10 EST

Antoine Chambert-Loir, Yuri Tschinkel

Source: Duke Math. J., Volume 161, Number 15, 2799--2836.

Abstract:
We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.

Isolatedness of characteristic points at blowup for a 1-dimensional semilinear wave equation

Thu, 29 Nov 2012 09:10 EST

Frank Merle, Hatem Zaag

Source: Duke Math. J., Volume 161, Number 15, 2837--2908.

Abstract:
We consider the 1-dimensional semilinear wave equation with power nonlinearity. We consider an arbitrary blowup solution $u(x,t)$ , the graph $x\mapsto T(x)$ of its blowup points, and $\mathcal {S}\subset \mathbb {R}$ the set of all characteristic points. We show that $\mathcal {S}$ is locally finite.

Embeddability for 3-dimensional Cauchy–Riemann manifolds and CR Yamabe invariants

Thu, 29 Nov 2012 09:10 EST

Sagun Chanillo, Hung-Lin Chiu, Paul Yang

Source: Duke Math. J., Volume 161, Number 15, 2909--2921.

Abstract:
Let $M^{3}$ be a closed Cauchy–Riemann (CR) 3-manifold. In this article, we derive a Bochner formula for the Kohn Laplacian in which the pseudo-Hermitian torsion does not play any role. By means of this formula we show that the nonzero eigenvalues of the Kohn Laplacian have a positive lower bound, provided that the CR Paneitz operator is nonnegative and the Webster curvature is positive. This means that $M^{3}$ is embeddable when the CR Yamabe constant is positive and the CR Paneitz operator is nonnegative. Our lower bound estimate is sharp. In addition, we show that the embedding is stable in the sense of Burns and Epstein.

Residue classes containing an unexpected number of primes

Thu, 29 Nov 2012 09:10 EST

Daniel Fiorilli

Source: Duke Math. J., Volume 161, Number 15, 2923--2943.

Abstract:
We fix a nonzero integer $a$ and consider arithmetic progressions $a\operatorname{mod}q$ , with $q$ varying over a given range. We show that, for certain specific values of $a$ , the arithmetic progressions $a\operatorname{mod}q$ contain, on average, significantly fewer primes than expected. We improve on results of Fouvry, Bombieri, Friedlander, Iwaniec, Granville, Hildebrandt, and Maier.

Minimal length elements of finite Coxeter groups

Thu, 29 Nov 2012 09:10 EST

Xuhua He, Sian Nie

Source: Duke Math. J., Volume 161, Number 15, 2945--2967.

Abstract:
We give a geometric proof that any minimal length element in a (twisted) conjugacy class of a finite Coxeter group $W$ has remarkable properties with respect to conjugation, taking powers in the associated braid monoid and taking the centralizer in $W$ .

On the value distribution of the Epstein zeta function in the critical strip

Mon, 14 Jan 2013 09:01 EST

Anders Södergren

Source: Duke Math. J., Volume 162, Number 1, 1--48.

Abstract:
We study the value distribution of the Epstein zeta function $E_{n}(L,s)$ for $0\textless s\textless \frac{n}{2}$ and a random lattice $L$ of large dimension $n$ . For any fixed $c\in({1}/{4},{1}/{2})$ and $n\to\infty$ , we prove that the random variable $V_{n}^{-2c}E_{n}(\cdot,cn)$ has a limit distribution, which we give explicitly (here $V_{n}$ is the volume of the $n$ -dimensional unit ball). More generally, for any fixed $\varepsilon \textgreater 0$ , we determine the limit distribution of the random function $c\mapstoV_{n}^{-2c}E_{n}(\cdot,cn)$ , $c\in[{1}/{4}+\varepsilon ,{1}/{2}-\varepsilon ]$ . After compensating for the pole at $c=1/2$ , we even obtain a limit result on the whole interval $[1/4+\varepsilon ,1/2]$ , and as a special case we deduce the following strengthening of a result by Sarnak and Strömbergsson concerning the height function $h_{n}(L)$ of the flat torus ${\mathbb {R}}^{n}/L$ : the random variable $n\{h_{n}(L)-(\log(4\pi)-\gamma+1)\}+\log n$ has a limit distribution as $n\to\infty$ , which we give explicitly. Finally, we discuss a question posed by Sarnak and Strömbergsson as to whether there exists a lattice $L\subset {\mathbb {R}}^{n}$ for which $E_{n}(L,s)$ has no zeros in $(0,\infty)$ .

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

Mon, 14 Jan 2013 09:01 EST

Frank Kutzschebauch, Sam Lodin

Source: Duke Math. J., Volume 162, Number 1, 49--94.

Abstract:
We construct holomorphic families of proper holomorphic embeddings of $\mathbb {C}^{k}$ into $\mathbb {C}^{n}$ ( $0\textless k\textless n-1$ ), so that for any two different parameters in the family, no holomorphic automorphism of $\mathbb {C}^{n}$ can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of $\mathbb {C}^{n}$ , we derive the existence of families of holomorphic $\mathbb {C}^{*}$ -actions on $\mathbb {C}^{n}$ ( $n\ge5$ ) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of $\mathbb {C}^{*}$ -actions on $\mathbb {C}^{n}$ (with prescribed linear part at a fixed point).

Nonalgebraic compactifications of quotients of the cylinder

Mon, 14 Jan 2013 09:01 EST

Marco Brunella

Source: Duke Math. J., Volume 162, Number 1, 95--109.

Abstract:
We classify compact complex surfaces which contain a Zariski-open subset whose universal covering is the cylinder ${\mathbb{D}}\times{\mathbb{C}}$ .

Representation zeta functions of compact $p$ -adic analytic groups and arithmetic groups

Mon, 14 Jan 2013 09:01 EST

Nir Avni, Benjamin Klopsch, Uri Onn, Christopher Voll

Source: Duke Math. J., Volume 162, Number 1, 111--197.

Abstract:
We introduce new methods from $\mathfrak{p}$ -adic integration into the study of representation zeta functions associated to compact $p$ -adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of generic members of families of $p$ -adic analytic pro- $p$ groups obtained from a global, “perfect” Lie lattice satisfy functional equations. In the case of “semisimple” compact $p$ -adic analytic groups, we exhibit a link between the relevant $\mathfrak{p}$ -adic integrals and a natural filtration of the locus of irregular elements in the associated semisimple Lie algebra, defined by the centralizer dimension. Based on this algebro-geometric description, we compute explicit formulas for the representation zeta functions of principal congruence subgroups of the groups $\operatorname {SL}_{3}(\mathfrak{o})$ , where $\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$ , and of the groups $\operatorname {SU}_{3}(\mathfrak {O},\mathfrak{o})$ , where $\mathfrak {O}$ is an unramified quadratic extension of $\mathfrak{o}$ . These formulas, combined with approximative Clifford theory, allow us to determine the abscissae of convergence of representation zeta functions associated to arithmetic subgroups of algebraic groups of type $A_{2}$ . Assuming a conjecture of Serre on the congruence subgroup problem, we thereby prove a conjecture of Larsen and Lubotzky on lattices in higher-rank semisimple groups for algebraic groups of type $A_{2}$ defined over number fields.

Errata to “Solutions of super-linear elliptic equations and their Morse indices, I,” Duke Math. J. 94 (1998), 141–157

Mon, 14 Jan 2013 09:01 EST

Salem Rebhi

Source: Duke Math. J., Volume 162, Number 1, 199--200.

Characters of Springer representations on elliptic conjugacy classes

Thu, 24 Jan 2013 09:16 EST

Dan M. Ciubotaru, Peter E. Trapa

Source: Duke Math. J., Volume 162, Number 2, 201--223.

Abstract:
For a Weyl group W, we investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of W in the homology of a Springer fiber. We also give a formula (valid again on elliptic conjugacy classes) of the W-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of W and the Dirac operator for graded affine Hecke algebras play key roles.

Syzygies of Segre embeddings and $\Delta$ -modules

Thu, 24 Jan 2013 09:16 EST

Andrew Snowden

Source: Duke Math. J., Volume 162, Number 2, 225--277.

Abstract:
We study syzygies of the Segre embedding of $\mathbf{P}(V_{1})\times\cdots\times \mathbf{P}(V_{n})$ , and prove two finiteness results. First, for fixed $p$ but varying $n$ and $V_{i}$ , there is a finite list of master $p$ -syzygies from which all other $p$ -syzygies can be derived by simple substitutions. Second, we define a power series $f_{p}$ with coefficients in something like the Schur algebra, which contains essentially all the information of $p$ -syzygies of Segre embeddings (for all $n$ and $V_{i}$ ), and show that it is a rational function. The list of master $p$ -syzygies and the numerator and denominator of $f_{p}$ can be computed algorithmically (in theory). The central observation of this paper is that by considering all Segre embeddings at once (i.e., letting $n$ and the $V_{i}$ vary) certain structure on the space of $p$ -syzygies emerges. We formalize this structure in the concept of a $\Delta$ -module. Many of our results on syzygies are specializations of general results on $\Delta$ -modules that we establish. Our theory also applies to certain other families of varieties, such as tangent and secant varieties of Segre embeddings.

The elliptic Hall algebra and the $K$ -theory of the Hilbert scheme of $\mathbb{A}^{2}$

Thu, 24 Jan 2013 09:16 EST

Olivier Schiffmann, Eric Vasserot

Source: Duke Math. J., Volume 162, Number 2, 279--366.

Abstract:
In this paper we compute the convolution algebra in the equivariant K -theory of the Hilbert scheme of $\mathbb{A}^{2}$ . We show that it is isomorphic to the elliptic Hall algebra and hence to the spherical double affine Hecke algebra of $\mathrm{GL}_{\infty}$ . We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K -theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K -theory of the moduli space of framed torsion free sheaves over $\mathbb{P}^{2}$ , virtual fundamental classes, shuffle algebras, …).

Contracting exceptional divisors by the Kähler–Ricci flow

Thu, 24 Jan 2013 09:16 EST

Jian Song, Ben Weinkove

Source: Duke Math. J., Volume 162, Number 2, 367--415.

Abstract:
We give a criterion under which a solution $g(t)$ of the Kähler–Ricci flow contracts exceptional divisors on a compact manifold and can be uniquely continued on a new manifold. As $t$ tends to the singular time $T$ from each direction, we prove the convergence of $g(t)$ in the sense of Gromov–Hausdorff and smooth convergence away from the exceptional divisors. We call this behavior for the Kähler–Ricci flow a canonical surgical contraction . In particular, our results show that the Kähler–Ricci flow on a projective algebraic surface will perform a sequence of canonical surgical contractions until, in finite time, either the minimal model is obtained, or the volume of the manifold tends to zero.

Sharp inequalities for the Beurling–Ahlfors transform on radial functions

Thu, 24 Jan 2013 09:16 EST

Rodrigo Bañuelos, Adam Osȩkowski

Source: Duke Math. J., Volume 162, Number 2, 417--434.

Abstract:
For $1\leq p\leq2$ , we prove sharp weak-type $(p,p)$ -estimates for the Beurling–Ahlfors operator acting on the radial function subspaces of $L^{p}(\mathbb{C})$ . A similar sharp $L^{p}$ -result is proved for $1\lt p\leq2$ . The results are derived from martingale inequalities which are of independent interest.

A positive density analogue of the Lieb–Thirring inequality

Thu, 14 Feb 2013 15:48 EST

Rupert L. Frank, Mathieu Lewin, Elliott H. Lieb, Robert Seiringer

Source: Duke Math. J., Volume 162, Number 3, 435--495.

Abstract:
The Lieb–Thirring inequalities give a bound on the negative eigenvalues of a Schrödinger operator in terms of an $L^{p}$ -norm of the potential. These are dual to bounds on the $H^{1}$ -norms of a system of orthonormal functions. Here we extend these bounds to analogous inequalities for perturbations of the Fermi sea of noninteracting particles (i.e., for perturbations of the continuous spectrum of the Laplacian by local potentials).

Uniqueness in Calderón’s problem with Lipschitz conductivities

Thu, 14 Feb 2013 15:48 EST

Boaz Haberman, Daniel Tataru

Source: Duke Math. J., Volume 162, Number 3, 497--516.

Abstract:
We use $X^{s,b}$ -inspired spaces to prove a uniqueness result for Calderón’s problem in a Lipschitz domain $\Omega$ under the assumption that the conductivity lies in the space $W^{1,\infty}(\overline{\Omega})$ . For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary $C^{1}$ conductivities.

Collapsing of abelian fibered Calabi–Yau manifolds

Thu, 14 Feb 2013 15:48 EST

Mark Gross, Valentino Tosatti, Yuguang Zhang

Source: Duke Math. J., Volume 162, Number 3, 517--551.

Abstract:
We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi–Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov–Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for $K3$ surfaces, which is motivated by the Strominger–Yau–Zaslow picture of mirror symmetry.

On the smooth locus of aligned Hilbert schemes, the $k$ -secant lemma and the general projection theorem

Thu, 14 Feb 2013 15:48 EST

Laurent Gruson, Christian Peskine

Source: Duke Math. J., Volume 162, Number 3, 553--578.

Abstract:
Let $X$ be a smooth, connected, dimension $n$ , quasi-projective variety embedded in ${\mathbb {P}}^{N}$ . Consider integers $\{k_{1},\ldots,k_{r}\}$ , with $k_{i}\textgreater 0$ , and the Hilbert scheme $H_{\{k_{1},\ldots,k_{r}\}}(X)$ of aligned, finite, degree $\sumk_{i}$ subschemes of $X$ , with multiplicities $k_{i}$ at points $x_{i}$ (possibly coinciding). The expected dimension of $H_{\{k_{1},\ldots,k_{r}\}}(X)$ is $2N-2+r-(\sum k_{i})(N-n)$ . We study the locus of points where $H_{\{k_{1},\ldots,k_{r}\}}(X)$ is not smooth of expected dimension, and we prove that the lines carrying this locus do not fill up ${\mathbb {P}}^{N}$ .

Stability for a GNS inequality and the Log-HLS inequality, with application to the critical mass Keller–Segel equation

Thu, 14 Feb 2013 15:48 EST

Eric A. Carlen, Alessio Figalli

Source: Duke Math. J., Volume 162, Number 3, 579--625.

Abstract:
Starting from the quantitative stability result of Bianchi and Egnell for the $2$ -Sobolev inequality, we deduce several different stability results for a Gagliardo–Nirenberg–Sobolev (GNS) inequality in the plane. Then, exploiting the connection between this inequality and a fast diffusion equation, we get stability for the logarithmic Hardy–Littlewood–Sobolev (Log-HLS) inequality. Finally, using all these estimates, we prove a quantitative convergence result for the critical mass Keller–Segel system.

Convergence of the Abelian sandpile

Fri, 15 Mar 2013 09:55 EDT

Wesley Pegden, Charles K. Smart

Source: Duke Math. J., Volume 162, Number 4, 627--642.

Abstract:
The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^{d}$ , in which sites with at least $2d$ chips topple , distributing one chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of $n$ chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as $n\to\infty$ . However, little has been proved about the appearance of the stable configurations. We use partial differential equation techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as $n\to\infty$ . We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.

Lipschitz regularity for inner-variational equations

Fri, 15 Mar 2013 09:55 EDT

Tadeusz Iwaniec, Leonid V. Kovalev, Jani Onninen

Source: Duke Math. J., Volume 162, Number 4, 643--672.

Abstract:
We obtain Lipschitz regularity results for a fairly general class of nonlinear first-order partial differential equations. These equations arise from the inner variation of certain energy integrals. Even in the simplest model case of the Dirichlet energy the inner-stationary solutions need not be differentiable everywhere; the Lipschitz continuity is the best possible. But the proofs, even in the Dirichlet case, turn out to rely on topological arguments. The appeal to the inner-stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy-minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity, specifically, neo-Hookean-type problems.

Vinogradov’s mean value theorem via efficient congruencing, II

Fri, 15 Mar 2013 09:55 EDT

Trevor D. Wooley

Source: Duke Math. J., Volume 162, Number 4, 673--730.

Abstract:
We apply the efficient congruencing method to estimate Vinogradov’s integral for moments of order $2s$ , with $1\leqslant s\leqslant k^{2}-1$ . Thereby, we show that quasi-diagonal behavior holds when $s=o(k^{2})$ , and we obtain near-optimal estimates for $1\leqslant s\leqslant \frac{1}{4}k^{2}+k$ and optimal estimates for $s\geqslant k^{2}-1$ . In this way we come halfway to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring’s problem is established for sums of $s$ $k$ th powers of natural numbers whenever $s\geqslant 2k^{2}-2k-8$ $(k\geqslant 6)$ .

Definability of restricted theta functions and families of abelian varieties

Fri, 15 Mar 2013 09:55 EDT

Ya’acov Peterzil, Sergei Starchenko

Source: Duke Math. J., Volume 162, Number 4, 731--765.

Abstract:
We consider some classical maps from the theory of abelian varieties and their moduli spaces, and we prove their definability on restricted domains in the o-minimal structure $ \mathbb {R}_{\mathrm {an,\,exp}}$ . In particular, we prove that the projective embedding of the moduli space of the principally polarized abelian variety $\operatorname{Sp}(2g,\mathbb {Z})\backslash \mathbb {H}_{g}$ is definable in $ \mathbb {R}_{\mathrm {an,\,exp}}$ when restricted to Siegel’s fundamental set $\mathfrak {F}_{g}$ . We also prove the definability on appropriate domains of embeddings of families of abelian varieties into projective spaces.

Flexible varieties and automorphism groups

Fri, 15 Mar 2013 09:55 EDT

I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, M. Zaidenberg

Source: Duke Math. J., Volume 162, Number 4, 767--823.

Abstract:
Given an irreducible affine algebraic variety $X$ of dimension $n\ge2$ , we let $\operatorname {SAut}(X)$ denote the special automorphism group of $X$ , that is, the subgroup of the full automorphism group $\operatorname{Aut}(X)$ generated by all one-parameter unipotent subgroups. We show that if $\operatorname {SAut}(X)$ is transitive on the smooth locus $X_{\mathrm{reg}}$ , then it is infinitely transitive on $X_{\mathrm{reg}}$ . In turn, the transitivity is equivalent to the flexibility of $X$ . The latter means that for every smooth point $x\in X_{\mathrm{reg}}$ the tangent space $T_{x}X$ is spanned by the velocity vectors at $x$ of one-parameter unipotent subgroups of $\operatorname{Aut}(X)$ . We also provide various modifications and applications.

Hardy–Petrovitch–Hutchinson’s problem and partial theta function

Fri, 29 Mar 2013 09:16 EDT

Vladimir Petrov Kostov, Boris Shapiro

Source: Duke Math. J., Volume 162, Number 5, 825--861.

Abstract:
In 1907, M. Petrovitch initiated the study of a class of entire functions all whose finite sections (i.e., truncations) are real-rooted polynomials. He was motivated by previous studies of E. Laguerre on uniform limits of sequences of real-rooted polynomials and by an interesting result of G. H. Hardy. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminant inequalities, one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J. I. Hutchinson has shown that an entire function $p(x)=a_{0}+a_{1}x+\cdots+a_{n}x^{n}+\cdots$ with strictly positive coefficients has the property that all of its finite segments $a_{i}x^{i}+a_{i+1}x^{i+1}+\cdots+a_{j}x^{j}$ have only real roots if and only if ${a_{i}^{2}}/{a_{i-1}a_{i+1}}\ge4$ for $i=1,2,\ldots$ . In the present paper, we give sharp lower bounds on the ratios ${a_{i}^{2}}/{a_{i-1}a_{i+1}}$ ( $i=1,2,\ldots$ ) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when $i\to\infty$ equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre–Pólya class $\mathcal {L-P}I$ . We also explain the relation between Newton’s and Hutchinson’s inequalities and the logarithmic image of the set of all real-rooted polynomials with positive coefficients.

On the torsion in the cohomology of arithmetic hyperbolic 3-manifolds

Fri, 29 Mar 2013 09:16 EDT

Simon Marshall, Werner Müller

Source: Duke Math. J., Volume 162, Number 5, 863--888.

Abstract:
In this paper we consider the cohomology of a closed arithmetic hyperbolic $3$ -manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of $\operatorname {SL}(2,\mathbb {C})$ . The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomology grows exponentially as the local system grows. We also consider the twisted Ruelle zeta function of a closed arithmetic hyperbolic $3$ -manifold, and we express the leading coefficient of its Laurent expansion at the origin in terms of the orders of the torsion subgroups of the cohomology.

Subordination by conformal martingales in $L^{p}$ and zeros of Laguerre polynomials

Fri, 29 Mar 2013 09:16 EDT

Alexander Borichev, Prabhu Janakiraman, Alexander Volberg

Source: Duke Math. J., Volume 162, Number 5, 889--924.

Abstract:
Given martingales $W$ and $Z$ such that $W$ is differentially subordinate to $Z$ , Burkholder obtained the sharp inequality $E|W|^{p}\le(p^{*}-1)^{p}E|Z|^{p}$ , where $p^{*}=\max\{p,p/(p-1)\}$ . What happens if one of the martingales is also a conformal martingale? Bañuelos and Janakiraman proved that if $p\geq2$ and $W$ is a conformal martingale differentially subordinate to any martingale $Z$ , then $E|W|^{p}\leq[(p^{2}-p)/2]^{p/2}E|Z|^{p}$ . In this paper, we establish that if $p\geq2$ , $Z$ is conformal, and $W$ is any martingale subordinate to $Z$ , then $\mathbb{E}|W|^{p}\le[\sqrt{2}(1-z_{p})/z_{p}]^{p}\mathbb{E}|Z|^{p}$ , where $z_{p}$ is the smallest positive zero of a certain solution of the Laguerre ordinary differential equation. We also prove the sharpness of this estimate and an analogous one in the dual case for $1\lt p\lt 2$ . Finally, we give an application of our results. Previous estimates on the $L^{p}$ -norm of the Beurling–Ahlfors transform give at best $\|B\|_{p}\lesssim\sqrt{2}p$ as $p\rightarrow\infty$ . We improve this to $\|B\|_{p}\lesssim1.3922p$ as $p\rightarrow\infty$ .

Simple Lie groups without the approximation property

Fri, 29 Mar 2013 09:16 EDT

Uffe Haagerup, Tim de Laat

Source: Duke Math. J., Volume 162, Number 5, 925--964.

Abstract:
For a locally compact group $G$ , let $A(G)$ denote its Fourier algebra, and let $M_{0}A(G)$ denote the space of completely bounded Fourier multipliers on $G$ . The group $G$ is said to have the Approximation Property (AP) if the constant function $1$ can be approximated by a net in $A(G)$ in the weak-∗ topology on the space $M_{0}A(G)$ . Recently, Lafforgue and de la Salle proved that $\operatorname {SL}(3,\mathbb {R})$ does not have the AP, implying the first example of an exact discrete group without it, namely, $\operatorname {SL}(3,\mathbb{Z})$ . In this paper we prove that $\operatorname {Sp}(2,\mathbb {R})$ does not have the AP. It follows that all connected simple Lie groups with finite center and real rank greater than or equal to two do not have the AP. This naturally gives rise to many examples of exact discrete groups without the AP.

Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

Fri, 29 Mar 2013 09:16 EDT

Shigeyuki Morita, Takuya Sakasai, Masaaki Suzuki

Source: Duke Math. J., Volume 162, Number 5, 965--1002.

Abstract:
We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra, and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.

Gabor frames and totally positive functions

Mon, 22 Apr 2013 10:04 EDT

Karlheinz Gröchenig, Joachim Stöckler

Source: Duke Math. J., Volume 162, Number 6, 1003--1031.

Abstract:
Let $g$ be a totally positive function of finite type, that is, $\hat{g}(\xi)=\prod_{\nu=1}^{M}(1+2\pi i\delta_{\nu}\xi)^{-1}$ for $\delta_{\nu}\in\mathbb {R}$ , $\delta_{\nu}\neq0$ , and $M\geq2$ , and let $\alpha,\beta\gt 0$ . Then the set $\{e^{2\pi i\beta lt}g(t-\alpha k):k,l\in\mathbb {Z}\}$ is a frame for $L^{2}(\mathbb {R})$ if and only if $\alpha \beta\lt 1$ . This result is a first positive contribution to a conjecture of Daubechies from 1990. Until now, the complete characterization of lattice parameters $\alpha$ , $\beta$ that generate a frame has been known for only six window functions $g$ . Our main result now yields an uncountable class of window functions. As a by-product of the proof method, we also derive new sampling theorems in shift-invariant spaces and obtain the correct Nyquist rate.

Generalized Heegner cycles and $p$ -adic Rankin $L$ -series

Mon, 22 Apr 2013 10:04 EDT

Massimo Bertolini, Henri Darmon, Kartik Prasanna

Source: Duke Math. J., Volume 162, Number 6, 1033--1148.

Abstract:
This article studies a distinguished collection of so-called generalized Heegner cycles in the product of a Kuga–Sato variety with a power of a CM elliptic curve. Its main result is a $p$ -adic analogue of the Gross–Zagier formula which relates the images of generalized Heegner cycles under the $p$ -adic Abel–Jacobi map to the special values of certain $p$ -adic Rankin $L$ -series at critical points that lie outside their range of classical interpolation.

On Bach-flat gradient shrinking Ricci solitons

Mon, 22 Apr 2013 10:04 EDT

Huai-Dong Cao, Qiang Chen

Source: Duke Math. J., Volume 162, Number 6, 1149--1169.

Abstract:
In this article, we classify $n$ -dimensional ( $n\ge4$ ) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any $4$ -dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat and hence a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^{4}$ or the round cylinder $\mathbb{S}^{3}\times\mathbb{R}$ . More generally, for $n\ge5$ , a Bach-flat gradient shrinking Ricci soliton is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^{n}$ or the product $N^{n-1}\times\mathbb{R}$ , where $N^{n-1}$ is Einstein.

Exceptional bundles associated to degenerations of surfaces

Mon, 22 Apr 2013 10:04 EDT

Paul Hacking

Source: Duke Math. J., Volume 162, Number 6, 1171--1202.

Abstract:
We study degenerations of complex surfaces with no vanishing cycles first described by J. Wahl. Given such a degeneration, we construct an exceptional vector bundle on the general fiber $Y$ in the case $H^{2,0}(Y)=H^{1}(Y)=0$ . For $Y=\mathbb{P}^{2}$ , we show that our construction establishes a bijective correspondence between the possible singular surfaces and the set of exceptional bundles on $Y$ modulo a natural equivalence relation.

Correction to “Discrete fractional Radon transforms and quadratic forms,” Duke Math. J. 161 (2012), 69–106

Mon, 22 Apr 2013 10:04 EDT

Lillian B. Pierce

Source: Duke Math. J., Volume 162, Number 6, 1203--1204.

Area minimizers and boundary rigidity of almost hyperbolic metrics

Fri, 10 May 2013 09:48 EDT

Dmitri Burago, Sergei Ivanov

Source: Duke Math. J., Volume 162, Number 7, 1205--1248.

Abstract:
This paper is a continuation of our paper about boundary rigidity and filling minimality of metrics close to flat ones. We show that compact regions close to a hyperbolic one are boundary distance rigid and strict minimal fillings. We also provide a more invariant view on the approach used in the above-mentioned paper.

Pushing forward matrix factorizations

Fri, 10 May 2013 09:48 EDT

Tobias Dyckerhoff, Daniel Murfet

Source: Duke Math. J., Volume 162, Number 7, 1249--1311.

Abstract:
We describe the pushforward of a matrix factorization along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and we use this construction to study the convolution of kernels defining integral functors between categories of matrix factorizations. We give an elementary proof of a formula for the Chern character of the convolution generalizing the Hirzebruch–Riemann–Roch formula of Polishchuk and Vaintrob.

Donaldson–Thomas theory and cluster algebras

Fri, 10 May 2013 09:48 EDT

Kentaro Nagao

Source: Duke Math. J., Volume 162, Number 7, 1313--1367.

Abstract:
We provide a transformation formula of the (Euler characteristic version of the) non-commutative Donaldson–Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we provide alternative proofs of Fomin–Zelevinsky conjectures on cluster algebras.

Geodesics in the space of Kähler metrics

Fri, 10 May 2013 09:48 EDT

László Lempert, Liz Vivas

Source: Duke Math. J., Volume 162, Number 7, 1369--1381.

Abstract:
Let $(X,\omega)$ be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set $\mathcal{H}_{0}$ of Kähler forms cohomologous to $\omega$ has the natural structure of an infinite-dimensional Riemannian manifold. We address the question of whether any two points in $\mathcal{H}_{0}$ can be connected by a smooth geodesic and show that the answer, in general, is “no.”