Duke Mathematical Journal Articles (Project Euclid)
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The latest articles from Duke Mathematical Journal on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTWed, 01 Jun 2011 09:20 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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Crofton measures and Minkowski valuations
http://projecteuclid.org/euclid.dmj/1279140505
<strong>Franz E. Schuster</strong><p><strong>Source: </strong>Duke Math. J., Volume 154, Number 1, 1--30.</p><p><strong>Abstract:</strong><br/>
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 </p>projecteuclid.org/euclid.dmj/1279140505_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTIntegrality of Hausel–Letellier–Villegas kernelshttps://projecteuclid.org/euclid.dmj/1539137165<strong>Anton Mellit</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 17, 3171--3205.</p><p><strong>Abstract:</strong><br/>
We prove that the coefficients of the generating function of Hausel, Letellier, and Rodriguez-Villegas and its recent generalization by Carlsson and Rodriguez-Villegas, which according to various conjectures should compute mixed Hodge numbers of character varieties and moduli spaces of Higgs bundles of curves of genus $g$ with $n$ punctures, are polynomials in $q$ and $t$ with integer coefficients for any $g,n\geq 0$ .
</p>projecteuclid.org/euclid.dmj/1539137165_20181108220204Thu, 08 Nov 2018 22:02 ESTConserved energies for the cubic nonlinear Schrödinger equation in one dimensionhttps://projecteuclid.org/euclid.dmj/1540540826<strong>Herbert Koch</strong>, <strong>Daniel Tataru</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 17, 3207--3313.</p><p><strong>Abstract:</strong><br/>
We consider the cubic nonlinear Schrödinger (NLS) equation as well as the modified Korteweg–de Vries (mKdV) equation in one space dimension. We prove that for each $s\gt -\frac{1}{2}$ there exists a conserved energy which is equivalent to the $H^{s}$ -norm of the solution. For the Korteweg–de Vries (KdV) equation, there is a similar conserved energy for every $s\ge -1$ .
</p>projecteuclid.org/euclid.dmj/1540540826_20181108220204Thu, 08 Nov 2018 22:02 ESTHigh-frequency backreaction for the Einstein equations under polarized $\mathbb{U}(1)$ -symmetryhttps://projecteuclid.org/euclid.dmj/1542337536<strong>Cécile Huneau</strong>, <strong>Jonathan Luk</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3315--3402.</p><p><strong>Abstract:</strong><br/>
Known examples in plane symmetry or Gowdy symmetry show that, given a $1$ -parameter family of solutions to the vacuum Einstein equations, it may have a weak limit which does not satisfy the vacuum equations, but instead has a nontrivial stress-energy-momentum tensor. We consider this phenomenon under polarized $\mathbb{U}(1)$ -symmetry—a much weaker symmetry than most of the known examples—such that the stress-energy-momentum tensor can be identified with that of multiple families of null dust propagating in distinct directions. We prove that any generic local-in-time small-data polarized $\mathbb{U}(1)$ -symmetric solution to the Einstein–multiple null dust system can be achieved as a weak limit of vacuum solutions. Our construction allows the number of families to be arbitrarily large and appears to be the first construction of such examples with more than two families.
</p>projecteuclid.org/euclid.dmj/1542337536_20181128220332Wed, 28 Nov 2018 22:03 ESTHow large is $A_{g}(\mathbb{F}_{q})$ ?https://projecteuclid.org/euclid.dmj/1542250833<strong>Michael Lipnowski</strong>, <strong>Jacob Tsimerman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3403--3453.</p><p><strong>Abstract:</strong><br/>
Let $B(g,p)$ denote the number of isomorphism classes of $g$ -dimensional Abelian varieties over the finite field of size $p$ . Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ -dimensional Abelian varieties over the finite field of size $p$ . We derive upper bounds for $B(g,p)$ and lower bounds for $A(g,p)$ for $p$ fixed and $g$ increasing. The extremely large gap between the lower bound for $A(g,p)$ and the upper bound for $B(g,p)$ implies some statistically counterintuitive behavior for Abelian varieties of large dimension over a fixed finite field.
</p>projecteuclid.org/euclid.dmj/1542250833_20181128220332Wed, 28 Nov 2018 22:03 ESTEspaces de Banach–Colmez et faisceaux cohérents sur la courbe de Fargues–Fontainehttps://projecteuclid.org/euclid.dmj/1542337535<strong>Arthur-César Le Bras</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3455--3532.</p><p><strong>Abstract:</strong><br/>
We give a new definition, simpler but equivalent, of the abelian category of Banach–Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues–Fontaine curve. One goes from one category to the other by changing the $t$ -structure on the derived category. Along the way we obtain a description of the proétale cohomology of the open disk and the affine space, which is of independent interest.
</p>projecteuclid.org/euclid.dmj/1542337535_20181128220332Wed, 28 Nov 2018 22:03 ESTHypersymplectic 4-manifolds, the $G_{2}$ -Laplacian flow, and extension assuming bounded scalar curvaturehttps://projecteuclid.org/euclid.dmj/1541473293<strong>Joel Fine</strong>, <strong>Chengjian Yao</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 18, 3533--3589.</p><p><strong>Abstract:</strong><br/>
A hypersymplectic structure on a 4-manifold $X$ is a triple $\underline{\omega}$ of symplectic forms which at every point span a maximal positive definite subspace of $\Lambda^{2}$ for the wedge product. This article is motivated by a conjecture by Donaldson: when $X$ is compact, $\underline{\omega}$ can be deformed through cohomologous hypersymplectic structures to a hyper-Kähler triple. We approach this via a link with $G_{2}$ -geometry. A hypersymplectic structure $\underline{\omega}$ on a compact manifold $X$ defines a natural $G_{2}$ -structure $\phi$ on $X\times\mathbb{T}^{3}$ which has vanishing torsion precisely when $\underline{\omega}$ is a hyper-Kähler triple. We study the $G_{2}$ -Laplacian flow starting from $\phi$ , which we interpret as a flow of hypersymplectic structures. Our main result is that the flow extends as long as the scalar curvature of the corresponding $G_{2}$ -structure remains bounded. An application of our result is a lower bound for the maximal existence time of the flow in terms of weak bounds on the initial data (and with no assumption that scalar curvature is bounded along the flow).
</p>projecteuclid.org/euclid.dmj/1541473293_20181128220332Wed, 28 Nov 2018 22:03 ESTMirror symmetry for the Landau–Ginzburg $A$ -model $M=\mathbb{C}^{n}$ , $W=z_{1}\cdots z_{n}$https://projecteuclid.org/euclid.dmj/1545037298<strong>David Nadler</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 1, 1--84.</p><p><strong>Abstract:</strong><br/>
We calculate the category of branes in the Landau–Ginzburg $A$ -model with background $M=\mathbb{C}^{n}$ and superpotential $W=z_{1}\cdots z_{n}$ in the form of microlocal sheaves along a natural Lagrangian skeleton. Our arguments employ the framework of perverse schobers, and our results confirm expectations from mirror symmetry.
</p>projecteuclid.org/euclid.dmj/1545037298_20190103040203Thu, 03 Jan 2019 04:02 ESTCohomologically induced distinguished representations and cohomological test vectorshttps://projecteuclid.org/euclid.dmj/1541646042<strong>Binyong Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 1, 85--126.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a real reductive group, and let $\chi$ be a character of a reductive subgroup $H$ of $G$ . We construct $\chi$ -invariant linear functionals on certain cohomologically induced representations of $G$ , and we show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two Archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain $L$ -functions via modular symbols.
</p>projecteuclid.org/euclid.dmj/1541646042_20190103040203Thu, 03 Jan 2019 04:02 ESTArithmetic theta lifts and the arithmetic Gan–Gross–Prasad conjecture for unitary groupshttps://projecteuclid.org/euclid.dmj/1545037299<strong>Hang Xue</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 1, 127--185.</p><p><strong>Abstract:</strong><br/>
We propose a precise formula relating the height of certain diagonal cycles on the product of unitary Shimura varieties and the central derivative of some tensor product $L$ -functions. This can be viewed as a refinement of the arithmetic Gan–Gross–Prasad conjecture. We use the theory of arithmetic theta lifts to prove some endoscopic cases of it for $\operatorname{U}(2)\times\operatorname{U}(3)$ .
</p>projecteuclid.org/euclid.dmj/1545037299_20190103040203Thu, 03 Jan 2019 04:02 ESTOn the rationality problem for quadric bundleshttps://projecteuclid.org/euclid.dmj/1541646041<strong>Stefan Schreieder</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 187--223.</p><p><strong>Abstract:</strong><br/>
We classify all positive integers $n$ and $r$ such that (stably) nonrational complex $r$ -fold quadric bundles over rational $n$ -folds exist. We show in particular that, for any $n$ and $r$ , a wide class of smooth $r$ -fold quadric bundles over $\mathbb{P}^{n}_{\mathbb{C}}$ are not stably rational if $r\in[2^{n-1}-1,2^{n}-2]$ . In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally $\mathrm{CH}_{0}$ -trivial resolutions of singularities.
</p>projecteuclid.org/euclid.dmj/1541646041_20190122220240Tue, 22 Jan 2019 22:02 ESTLegendrian fronts for affine varietieshttps://projecteuclid.org/euclid.dmj/1547110820<strong>Roger Casals</strong>, <strong>Emmy Murphy</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 225--323.</p><p><strong>Abstract:</strong><br/>
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First, we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several new applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed, exact Lagrangian submanifolds. In particular, we prove that the Koras–Russell cubic is Stein deformation-equivalent to $\mathbb{C}^{3}$ , and we verify the affine parts of the algebraic mirrors of two Weinstein $4$ -folds.
</p>projecteuclid.org/euclid.dmj/1547110820_20190122220240Tue, 22 Jan 2019 22:02 ESTAsymptotics of Chebyshev polynomials, II: DCT subsets of ${\mathbb{R}}$https://projecteuclid.org/euclid.dmj/1547024421<strong>Jacob S. Christiansen</strong>, <strong>Barry Simon</strong>, <strong>Peter Yuditskii</strong>, <strong>Maxim Zinchenko</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 325--349.</p><p><strong>Abstract:</strong><br/>
We prove Szegő–Widom asymptotics for the Chebyshev polynomials of a compact subset of $\mathbb{R}$ which is regular for potential theory and obeys the Parreau–Widom and DCT conditions.
</p>projecteuclid.org/euclid.dmj/1547024421_20190122220240Tue, 22 Jan 2019 22:02 ESTNice triples and the Grothendieck–Serre conjecture concerning principal G-bundles over reductive group schemeshttps://projecteuclid.org/euclid.dmj/1546938026<strong>Ivan Panin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 2, 351--375.</p><p><strong>Abstract:</strong><br/>
The main result of this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved.
</p>projecteuclid.org/euclid.dmj/1546938026_20190122220240Tue, 22 Jan 2019 22:02 ESTNonabelian Cohen–Lenstra momentshttps://projecteuclid.org/euclid.dmj/1548730815<strong>Melanie Matchett Wood</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 3, 377--427.</p><p><strong>Abstract:</strong><br/>
In this article, we give a conjecture for the average number of unramified $G$ -extensions of a quadratic field for any finite group $G$ . The Cohen–Lenstra heuristics are the specialization of our conjecture to the case in which $G$ is abelian of odd order. We prove a theorem toward the function field analogue of our conjecture and give additional motivations for the conjecture, including the construction of a lifting invariant for the unramified $G$ -extensions that takes the same number of values as the predicted average and an argument using the Malle–Bhargava principle. We note that, for even $|G|$ , corrections for the roots of unity in ${\mathbb{Q}}$ are required, which cannot be seen when $G$ is abelian.
</p>projecteuclid.org/euclid.dmj/1548730815_20190207040155Thu, 07 Feb 2019 04:01 ESTThe class of Eisenbud–Khimshiashvili–Levine is the local $\mathbf{A}^{1}$ -Brouwer degreehttps://projecteuclid.org/euclid.dmj/1547607998<strong>Jesse Leo Kass</strong>, <strong>Kirsten Wickelgren</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 3, 429--469.</p><p><strong>Abstract:</strong><br/>
Given a polynomial function with an isolated zero at the origin, we prove that the local $\mathbf{A}^{1}$ -Brouwer degree equals the Eisenbud–Khimshiashvili–Levine class. This answers a question posed by David Eisenbud in 1978. We give an application to counting nodes, together with associated arithmetic information, by enriching Milnor’s equality between the local degree of the gradient and the number of nodes into which a hypersurface singularity bifurcates to an equality in the Grothendieck–Witt group.
</p>projecteuclid.org/euclid.dmj/1547607998_20190207040155Thu, 07 Feb 2019 04:01 ESTOn the $\ell ^{p}$ -norm of the discrete Hilbert transformhttps://projecteuclid.org/euclid.dmj/1548666102<strong>Rodrigo Bañuelos</strong>, <strong>Mateusz Kwaśnicki</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 3, 471--504.</p><p><strong>Abstract:</strong><br/>
Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$ -processes, we prove that its $\ell ^{p}$ -norm, $1\lt p\lt \infty $ , is bounded above by the $L^{p}$ -norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.
</p>projecteuclid.org/euclid.dmj/1548666102_20190207040155Thu, 07 Feb 2019 04:01 ESTThree combinatorial formulas for type $A$ quiver polynomials and $K$ -polynomialshttps://projecteuclid.org/euclid.dmj/1549270814<strong>Ryan Kinser</strong>, <strong>Allen Knutson</strong>, <strong>Jenna Rajchgot</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 505--551.</p><p><strong>Abstract:</strong><br/>
We provide combinatorial formulas for the multidegree and $K$ -polynomial of an arbitrarily oriented type $A$ quiver locus. These formulas are generalizations of three formulas by Knutson, Miller, and Shimozono from the equioriented setting. In particular, we prove the $K$ -theoretic component formula conjectured by Buch and Rimányi.
</p>projecteuclid.org/euclid.dmj/1549270814_20190221220146Thu, 21 Feb 2019 22:01 ESTMetaplectic covers of Kac–Moody groups and Whittaker functionshttps://projecteuclid.org/euclid.dmj/1549594819<strong>Manish M. Patnaik</strong>, <strong>Anna Puskás</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 553--653.</p><p><strong>Abstract:</strong><br/>
Starting from some linear algebraic data (a Weyl group invariant bilinear form) and some arithmetic data (a bilinear Steinberg symbol), we construct a central extension of a Kac–Moody group generalizing the work of Matsumoto. Specializing our construction over non-Archimedean local fields, for each positive integer $n$ we obtain the notion of $n$ -fold metaplectic covers of Kac–Moody groups. In this setting, we prove a Casselman–Shalika-type formula for Whittaker functions.
</p>projecteuclid.org/euclid.dmj/1549594819_20190221220146Thu, 21 Feb 2019 22:01 ESTNon-LERFness of arithmetic hyperbolic manifold groups and mixed $3$ -manifold groupshttps://projecteuclid.org/euclid.dmj/1549270813<strong>Hongbin Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 655--696.</p><p><strong>Abstract:</strong><br/>
We will show that for any noncompact arithmetic hyperbolic $m$ -manifold with ${m\gt 3}$ , and any compact arithmetic hyperbolic $m$ -manifold with $m\gt 4$ that is not a $7$ -dimensional one defined by octonions, its fundamental group is not locally extended residually finite (LERF). The main ingredient in the proof is a study on abelian amalgamations of hyperbolic $3$ -manifold groups. We will also show that a compact orientable irreducible $3$ -manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.
</p>projecteuclid.org/euclid.dmj/1549270813_20190221220146Thu, 21 Feb 2019 22:01 ESTOn topological and measurable dynamics of unipotent frame flows for hyperbolic manifoldshttps://projecteuclid.org/euclid.dmj/1549392546<strong>François Maucourant</strong>, <strong>Barbara Schapira</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 4, 697--747.</p><p><strong>Abstract:</strong><br/>
We study the dynamics of unipotent flows on frame bundles of hyperbolic manifolds of infinite volume. We prove that they are topologically transitive and that the natural invariant measure, the so-called Burger–Roblin measure, is ergodic, as soon as the geodesic flow admits a finite measure of maximal entropy, and this entropy is strictly greater than the codimension of the unipotent flow inside the maximal unipotent flow. The latter result generalizes a theorem of Mohammadi and Oh.
</p>projecteuclid.org/euclid.dmj/1549392546_20190221220146Thu, 21 Feb 2019 22:01 ESTOn the polynomial Szemerédi theorem in finite fieldshttps://projecteuclid.org/euclid.dmj/1548990127<strong>Sarah Peluse</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 749--774.</p><p><strong>Abstract:</strong><br/>
Let $P_{1},\dots,P_{m}\in\mathbb{Z}[y]$ be any linearly independent polynomials with zero constant term. We show that there exists $\gamma\gt 0$ such that any subset of $\mathbb{F}_{q}$ of size at least $q^{1-\gamma}$ contains a nontrivial polynomial progression $x,x+P_{1}(y),\dots,x+P_{m}(y)$ , provided that the characteristic of $\mathbb{F}_{q}$ is large enough.
</p>projecteuclid.org/euclid.dmj/1548990127_20190321040049Thu, 21 Mar 2019 04:00 EDTA gradient estimate for nonlocal minimal graphshttps://projecteuclid.org/euclid.dmj/1551841227<strong>Xavier Cabré</strong>, <strong>Matteo Cozzi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 775--848.</p><p><strong>Abstract:</strong><br/>
We consider the class of measurable functions defined in all of $\mathbb{R}^{n}$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^{n}$ . We establish that the gradient of any such function is bounded in the interior of the ball by a power of its oscillation. This estimate, together with previously known results, leads to the $C^{\infty}$ regularity of the function in the ball. While the smoothness of nonlocal minimal graphs was known for $n=1,2$ —but without a quantitative bound—in higher dimensions only their continuity had been established. To prove the gradient bound, we show that the normal to a nonlocal minimal graph is a supersolution of a truncated fractional Jacobi operator, for which we prove a weak Harnack inequality. To this end, we establish a new universal fractional Sobolev inequality on nonlocal minimal surfaces. Our estimate provides an extension to the fractional setting of the celebrated gradient bounds of Finn and of Bombieri, De Giorgi, and Miranda for solutions of the classical mean curvature equation.
</p>projecteuclid.org/euclid.dmj/1551841227_20190321040049Thu, 21 Mar 2019 04:00 EDTCongruences of $5$ -secant conics and the rationality of some admissible cubic fourfoldshttps://projecteuclid.org/euclid.dmj/1551754840<strong>Francesco Russo</strong>, <strong>Giovanni Staglianò</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 849--865.</p><p><strong>Abstract:</strong><br/>
The works of Hassett and Kuznetsov identify countably many divisors $C_{d}$ in the open subset of ${\mathbb{P}}^{55}={\mathbb{P}}(H^{0}(\mathcal{O}_{{\mathbb{P}}^{5}}(3)))$ parameterizing all cubic fourfolds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family $C_{14}$ . We use congruences of $5$ -secant conics to prove rationality for the first three of the families $C_{d}$ , corresponding to $d=14,26,38$ in Hassett’s notation.
</p>projecteuclid.org/euclid.dmj/1551754840_20190321040049Thu, 21 Mar 2019 04:00 EDTSchwarzian derivatives, projective structures, and the Weil–Petersson gradient flow for renormalized volumehttps://projecteuclid.org/euclid.dmj/1551495708<strong>Martin Bridgeman</strong>, <strong>Jeffrey Brock</strong>, <strong>Kenneth Bromberg</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 867--896.</p><p><strong>Abstract:</strong><br/>
To a complex projective structure $\Sigma $ on a surface, Thurston associates a locally convex pleated surface. We derive bounds on the geometry of both in terms of the norms $\|\phi _{\Sigma }\|_{\infty }$ and $\|\phi _{\Sigma }\|_{2}$ of the quadratic differential $\phi _{\Sigma }$ of $\Sigma $ given by the Schwarzian derivative of the associated locally univalent map. We show that these give a unifying approach that generalizes a number of important, well-known results for convex cocompact hyperbolic structures on $3$ -manifolds, including bounds on the Lipschitz constant for the nearest-point retraction and the length of the bending lamination. We then use these bounds to begin a study of the Weil–Petersson gradient flow of renormalized volume on the space $\operatorname{CC}(N)$ of convex cocompact hyperbolic structures on a compact manifold $N$ with incompressible boundary, leading to a proof of the conjecture that the renormalized volume has infimum given by one half the simplicial volume of $\mathit{DN}$ , the double of $N$ .
</p>projecteuclid.org/euclid.dmj/1551495708_20190321040049Thu, 21 Mar 2019 04:00 EDTApproximation theorems for parabolic equations and movement of local hot spotshttps://projecteuclid.org/euclid.dmj/1551495707<strong>Alberto Enciso</strong>, <strong>MªÁngeles García-Ferrero</strong>, <strong>Daniel Peralta-Salas</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 5, 897--939.</p><p><strong>Abstract:</strong><br/>
We prove a global approximation theorem for a general parabolic operator $L$ , which asserts that if $v$ satisfies the equation $Lv=0$ in a space-time region $\Omega\subset\mathbb{R}^{n+1}$ satisfying a certain necessary topological condition, then it can be approximated in a Hölder norm by a global solution $u$ to the equation. If $\Omega$ is compact and $L$ is the usual heat operator, then one can instead approximate the local solution $v$ by the unique solution that falls off at infinity to the Cauchy problem with a suitably chosen smooth, compactly supported initial datum. We then apply these results to prove the existence of global solutions to the equation $Lu=0$ with a local hot spot that moves along a prescribed curve for all time, up to a uniformly small error. Global solutions that exhibit isothermic hypersurfaces of prescribed topologies for all times and applications to the heat equation on the flat torus are also discussed.
</p>projecteuclid.org/euclid.dmj/1551495707_20190321040049Thu, 21 Mar 2019 04:00 EDTA surface with discrete and nonfinitely generated automorphism grouphttps://projecteuclid.org/euclid.dmj/1552615313<strong>Tien-Cuong Dinh</strong>, <strong>Keiji Oguiso</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 6, 941--966.</p><p><strong>Abstract:</strong><br/>
We show that there is a smooth complex projective variety, of any dimension greater than or equal to $2$ , whose automorphism group is discrete and not finitely generated. Moreover, this variety admits infinitely many real forms which are mutually nonisomorphic over $\mathbb{R}$ . Our result is inspired by the work of Lesieutre and answers questions by Dolgachev, Esnault, and Lesieutre.
</p>projecteuclid.org/euclid.dmj/1552615313_20190328040043Thu, 28 Mar 2019 04:00 EDTIsomonodromy deformations at an irregular singularity with coalescing eigenvalueshttps://projecteuclid.org/euclid.dmj/1552442775<strong>Giordano Cotti</strong>, <strong>Boris Dubrovin</strong>, <strong>Davide Guzzetti</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 6, 967--1108.</p><p><strong>Abstract:</strong><br/>
We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty $ , holomorphically depending on parameter $t$ within a polydisk in $\mathbb{C}^{n}$ centered at $t=0$ , such that the eigenvalues of the leading matrix at $z=\infty $ coalesce along a locus $\Delta $ contained in the polydisk, passing through $t=0$ . Namely, $z=\infty $ is a resonant irregular singularity for $t\in \Delta $ . We analyze the case when the leading matrix remains diagonalizable at $\Delta$ . We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon, and monodromy data as $t$ varies in the polydisk, and their limits for $t$ tending to points of $\Delta$ . When the system also has a Fuchsian singularity at $z=0$ , we show, under minimal vanishing conditions on the residue matrix at $z=0$ , that isomonodromic deformations can be extended to the whole polydisk (including $\Delta $ ) in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at the fixed coalescence point $t=0$ . Conversely, when the system is isomonodromic in a small domain not intersecting $\Delta $ inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that $\Delta $ is not a branching locus for the $t$ -continuation of fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius manifolds is explained. An application to Painlevé equations is discussed.
</p>projecteuclid.org/euclid.dmj/1552442775_20190328040043Thu, 28 Mar 2019 04:00 EDTTwisted moments of $L$ -functions and spectral reciprocityhttps://projecteuclid.org/euclid.dmj/1552615314<strong>Valentin Blomer</strong>, <strong>Rizwanur Khan</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 6, 1109--1177.</p><p><strong>Abstract:</strong><br/>
We establish a reciprocity formula that expresses the fourth moment of automorphic $L$ -functions of level $q$ twisted by the $\ell$ th Hecke eigenvalue as the fourth moment of automorphic $L$ -functions of level $\ell$ twisted by the $q$ th Hecke eigenvalue. Direct corollaries include subconvexity bounds for $L$ -functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic $L$ -functions.
</p>projecteuclid.org/euclid.dmj/1552615314_20190328040043Thu, 28 Mar 2019 04:00 EDTTautological classes on moduli spaces of hyper-Kähler manifoldshttps://projecteuclid.org/euclid.dmj/1556330420<strong>Nicolas Bergeron</strong>, <strong>Zhiyuan Li</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 7, 1179--1230.</p><p><strong>Abstract:</strong><br/>
We study algebraic cycles on moduli spaces $\mathcal{F}_{h}$ of $h$ -polarized hyper-Kähler manifolds. Following previous work of Marian, Oprea, and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces, we first define the tautological ring on $\mathcal{F}_{h}$ . We then study the images of these tautological classes in the cohomology groups of $\mathcal{F}_{h}$ and prove that most of them are linear combinations of Noether–Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3 $^{[n]}$ -type hyper-Kähler manifolds with $n\leq 2$ . Secondly, we prove the cohomological generalized Franchetta conjecture on a universal family of these hyper-Kähler manifolds.
</p>projecteuclid.org/euclid.dmj/1556330420_20190521220352Tue, 21 May 2019 22:03 EDTWeak subconvexity without a Ramanujan hypothesishttps://projecteuclid.org/euclid.dmj/1556848996<strong>Kannan Soundararajan</strong>, <strong>Jesse Thorner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 7, 1231--1268.</p><p><strong>Abstract:</strong><br/>
We describe a new method to obtain weak subconvexity bounds for $L$ -functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic $L$ -functions and (with mild restrictions) the Rankin–Selberg $L$ -functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin–Selberg $L$ -functions.
</p>projecteuclid.org/euclid.dmj/1556848996_20190521220352Tue, 21 May 2019 22:03 EDTNonvanishing for $3$ -folds in characteristic $p\gt 5$https://projecteuclid.org/euclid.dmj/1555574496<strong>Chenyang Xu</strong>, <strong>Lei Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 7, 1269--1301.</p><p><strong>Abstract:</strong><br/>
We prove the nonvanishing theorem for $3$ -folds over an algebraically closed field $k$ of characteristic $p\gt 5$ .
</p>projecteuclid.org/euclid.dmj/1555574496_20190521220352Tue, 21 May 2019 22:03 EDTA local trace formula for the generalized Shalika modelhttps://projecteuclid.org/euclid.dmj/1556330419<strong>Raphaël Beuzart-Plessis</strong>, <strong>Chen Wan</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 7, 1303--1385.</p><p><strong>Abstract:</strong><br/>
We study a local multiplicity problem related to so-called generalized Shalika models . By establishing a local trace formula for these kinds of models, we are able to prove a multiplicity formula for discrete series. As a result, we can show that these multiplicities are, for discrete series, invariant under the local Jacquet–Langlands correspondence and are related to local exterior square $L$ -functions.
</p>projecteuclid.org/euclid.dmj/1556330419_20190521220352Tue, 21 May 2019 22:03 EDTOn the proper moduli spaces of smoothable Kähler–Einstein Fano varietieshttps://projecteuclid.org/euclid.dmj/1556848995<strong>Chi Li</strong>, <strong>Xiaowei Wang</strong>, <strong>Chenyang Xu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 8, 1387--1459.</p><p><strong>Abstract:</strong><br/>
In this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth Kähler–Einstein Fano manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski-open condition, and we establish the uniqueness of the Gromov–Hausdorff limit for a punctured flat family of Kähler–Einstein Fano manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\mathbb{Q}$ -Gorenstein smoothable, K-semistable $\mathbb{Q}$ -Fano varieties, and we verify various necessary properties to guarantee that it is a good moduli space.
</p>projecteuclid.org/euclid.dmj/1556848995_20190527040049Mon, 27 May 2019 04:00 EDTPurity for the Brauer grouphttps://projecteuclid.org/euclid.dmj/1558145274<strong>Kęstutis Česnavičius</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 8, 1461--1486.</p><p><strong>Abstract:</strong><br/>
A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension $\ge2$ . The combination of several works of Gabber settles the conjecture except for some cases that concern $p$ -torsion Brauer classes in mixed characteristic $(0,p)$ . We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover.
</p>projecteuclid.org/euclid.dmj/1558145274_20190527040049Mon, 27 May 2019 04:00 EDTEstimates of solutions of elliptic equations with a source reaction term involving the product of the function and its gradienthttps://projecteuclid.org/euclid.dmj/1558145273<strong>Marie-Françoise Bidaut-Véron</strong>, <strong>Marta García-Huidobro</strong>, <strong>Laurent Véron</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 8, 1487--1537.</p><p><strong>Abstract:</strong><br/>
We study local and global properties of positive solutions of $-\Delta u=u^{p}|{\nabla u}|^{q}$ in a domain $\Omega $ of ${\mathbb{R}}^{N}$ , in the range $p+q\gt 1$ , $p\geq 0$ , $0\leq q\lt 2$ . We first prove a local Harnack inequality and nonexistence of positive solutions in ${\mathbb{R}}^{N}$ when $p(N-2)+q(N-1)\lt N$ . Using a direct Bernstein method, we obtain a first range of values of $p$ and $q$ in which $u(x)\leq c(\operatorname{dist}(x,\partial \Omega ))^{\frac{q-2}{p+q-1}}$ . This holds in particular if $p+q\lt 1+\frac{4}{N-1}$ . Using an integral Bernstein method, we obtain a wider range of values of $p$ and $q$ in which all the global solutions are constants. Our result contains Gidas and Spruck’s nonexistence result as a particular case. We also study solutions under the form $u(x)=r^{\frac{q-2}{p+q-1}}\omega (\sigma )$ . We prove existence, nonexistence, and rigidity of the spherical component $\omega $ in some range of values of $N$ , $p$ , and $q$ .
</p>projecteuclid.org/euclid.dmj/1558145273_20190527040049Mon, 27 May 2019 04:00 EDTThe invariant subspace problem for rank-one perturbationshttps://projecteuclid.org/euclid.dmj/1557907222<strong>Adi Tcaciuc</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 8, 1539--1550.</p><p><strong>Abstract:</strong><br/>
We show that for any bounded operator $T$ acting on an infinite-dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^{*}$ does not consist entirely of eigenvalues, we can find such rank-one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite-rank perturbations of arbitrarily small norm, but not necessarily of rank one.
</p>projecteuclid.org/euclid.dmj/1557907222_20190527040049Mon, 27 May 2019 04:00 EDTTate cycles on some quaternionic Shimura varieties mod $p$https://projecteuclid.org/euclid.dmj/1560326497<strong>Yichao Tian</strong>, <strong>Liang Xiao</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 9, 1551--1639.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a totally real field in which a prime number $p\gt 2$ is inert. We continue the study of the (generalized) Goren–Oort strata on quaternionic Shimura varieties over finite extensions of $\mathbb{F}_{p}$ . We prove that, when the dimension of the quaternionic Shimura variety is even, the Tate conjecture for the special fiber of the quaternionic Shimura variety holds for the cuspidal $\pi$ -isotypical component, as long as the two unramified Satake parameters at $p$ are not differed by a root of unity.
</p>projecteuclid.org/euclid.dmj/1560326497_20190612040156Wed, 12 Jun 2019 04:01 EDTInterior $C^{2}$ regularity of convex solutions to prescribing scalar curvature equationshttps://projecteuclid.org/euclid.dmj/1560326498<strong>Pengfei Guan</strong>, <strong>Guohuan Qiu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 9, 1641--1663.</p><p><strong>Abstract:</strong><br/>
We establish interior $C^{2}$ estimates for convex solutions of the scalar curvature equation and the $\sigma _{2}$ -Hessian equation. We also prove interior curvature estimates for isometrically immersed hypersurfaces $(M^{n},g)\subset \mathbb{R}^{n+1}$ with positive scalar curvature. These estimates are consequences of interior estimates for these equations obtained under a weakened condition.
</p>projecteuclid.org/euclid.dmj/1560326498_20190612040156Wed, 12 Jun 2019 04:01 EDTSubconvex equidistribution of cusp forms: Reduction to Eisenstein observableshttps://projecteuclid.org/euclid.dmj/1560326499<strong>Paul D. Nelson</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 9, 1665--1722.</p><p><strong>Abstract:</strong><br/>
Let $\pi$ traverse a sequence of cuspidal automorphic representations of $\operatorname{GL}_{2}$ with large prime level, unramified central character, and bounded infinity type. For $G\in\{\operatorname{GL}_{1},\operatorname{PGL}_{2}\}$ , let $H(G)$ denote the assertion that subconvexity holds for $G$ -twists of the adjoint $L$ -function of $\pi$ , with polynomial dependence upon the conductor of the twist. We show that $H(\operatorname{GL}_{1})$ implies $H(\operatorname{PGL}_{2})$ .
In geometric terms, $H(\operatorname{PGL}_{2})$ corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, $H(\operatorname{GL}_{1})$ , to the special case in which the relevant sequence of measures is tested against an Eisenstein series.
</p>projecteuclid.org/euclid.dmj/1560326499_20190612040156Wed, 12 Jun 2019 04:01 EDTBirational characterization of Abelian varieties and ordinary Abelian varieties in characteristic $p\gt 0$https://projecteuclid.org/euclid.dmj/1560326500<strong>Christopher D. Hacon</strong>, <strong>Zsolt Patakfalvi</strong>, <strong>Lei Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 9, 1723--1736.</p><p><strong>Abstract:</strong><br/>
Let $k$ be an algebraically closed field of characteristic $p\gt 0$ . We give a birational characterization of ordinary abelian varieties over $k$ : a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if $\kappa _{S}(X)=0$ and $b_{1}(X)=2\dim X$ . We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $\kappa (X)=0$ , and the Albanese morphism $a:X\to A$ is generically finite. Along the way, we also show that if $\kappa _{S}(X)=0$ (or if $\kappa (X)=0$ and $a$ is generically finite), then the Albanese morphism $a:X\to A$ is surjective and in particular $\dim A\leq \dim X$ .
</p>projecteuclid.org/euclid.dmj/1560326500_20190612040156Wed, 12 Jun 2019 04:01 EDTJumps in the Archimedean heighthttps://projecteuclid.org/euclid.dmj/1562572817<strong>Patrick Brosnan</strong>, <strong>Gregory Pearlstein</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 10, 1737--1842.</p><p><strong>Abstract:</strong><br/>
We introduce a pairing on local intersection cohomology groups of variations of pure Hodge structure, which we call the asymptotic height pairing . Our original application of this pairing was to answer a question on the Ceresa cycle posed by R. Hain and D. Reed. (This question has since been answered independently by Hain.) Here we show that a certain analytic line bundle, called the biextension line bundle , and defined in terms of normal functions, always extends to any smooth partial compactification of the base. We then show that the asymptotic height pairing on intersection cohomology governs the extension of the natural metric on this line bundle studied by Hain and Reed (as well as, more recently, by several other authors). We also prove a positivity property of the asymptotic height pairing, which generalizes the results of a recent preprint of J. Burgos Gil, D. Holmes and R. de Jong, along with a continuity property of the pairing in the normal function case. Moreover, we show that the asymptotic height pairing arises in a natural way from certain Mumford–Grothendieck biextensions associated to normal functions.
</p>projecteuclid.org/euclid.dmj/1562572817_20190719040229Fri, 19 Jul 2019 04:02 EDTA tropical motivic Fubini theorem with applications to Donaldson–Thomas theoryhttps://projecteuclid.org/euclid.dmj/1559894420<strong>Johannes Nicaise</strong>, <strong>Sam Payne</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 10, 1843--1886.</p><p><strong>Abstract:</strong><br/>
We present a new tool for the calculation of Denef and Loeser’s motivic nearby fiber and motivic Milnor fiber: a motivic Fubini theorem for the tropicalization map, based on Hrushovski and Kazhdan’s theory of motivic volumes of semialgebraic sets. As applications, we prove a conjecture of Davison and Meinhardt on motivic nearby fibers of weighted homogeneous polynomials, and give a very short and conceptual new proof of the integral identity conjecture of Kontsevich and Soibelman, first proved by Lê Quy Thuong. Both of these conjectures emerged in the context of motivic Donaldson–Thomas theory.
</p>projecteuclid.org/euclid.dmj/1559894420_20190719040229Fri, 19 Jul 2019 04:02 EDTOptimal strong approximation for quadratic formshttps://projecteuclid.org/euclid.dmj/1558339351<strong>Naser T. Sardari</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 10, 1887--1927.</p><p><strong>Abstract:</strong><br/>
For a nondegenerate integral quadratic form $F(x_{1},\dots ,x_{d})$ in $d\geq 5$ variables, we prove an optimal strong approximation theorem. Let $\Omega $ be a fixed compact subset of the affine quadric $F(x_{1},\dots ,x_{d})=1$ over the real numbers. Take a small ball $B$ of radius $0\lt r\lt 1$ inside $\Omega $ , and an integer $m$ . Further assume that $N$ is a given integer which satisfies $N\gg _{\delta ,\Omega }(r^{-1}m)^{4+\delta }$ for any $\delta \gt 0$ . Finally assume that an integral vector $(\lambda _{1},\dots ,\lambda _{d})$ mod $m$ is given. Then we show that there exists an integral solution $\mathbf{x}=(x_{1},\dots ,x_{d})$ of $F(\mathbf{x})=N$ such that $x_{i}\equiv \lambda_{i}\ \mathrm{mod}\ m$ and $\frac{\mathbf{x}}{\sqrt{N}}\in B$ , provided that all the local conditions are satisfied. We also show that $4$ is the best possible exponent. Moreover, for a nondegenerate integral quadratic form in four variables, we prove the same result if $N$ is odd and $N\gg _{\delta ,\Omega }(r^{-1}m)^{6+\epsilon }$ . Based on our numerical experiments on the diameter of LPS Ramanujan graphs and the expected square-root cancellation in a particular sum that appears in Remark 6.8, we conjecture that the theorem holds for any quadratic form in four variables with the optimal exponent $4$ .
</p>projecteuclid.org/euclid.dmj/1558339351_20190719040229Fri, 19 Jul 2019 04:02 EDTMinimal 2-spheres in 3-sphereshttps://projecteuclid.org/euclid.dmj/1562292016<strong>Robert Haslhofer</strong>, <strong>Daniel Ketover</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 10, 1929--1975.</p><p><strong>Abstract:</strong><br/>
We prove that any manifold diffeomorphic to $\mathbb{S}^{3}$ and endowed with a generic metric contains at least two embedded minimal $2$ -spheres. The existence of at least one minimal $2$ -sphere was obtained by Simon and Smith in 1983. Our approach combines ideas from min–max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in $3$ -manifolds. We apply our methods to solve a problem posed by S. T. Yau in 1987 on whether the planar $2$ -spheres are the only minimal spheres in ellipsoids centered about the origin in $\mathbb{R}^{4}$ . Finally, considering the example of degenerating ellipsoids, we show that the assumptions in the multiplicity $1$ conjecture and the equidistribution of widths conjecture are in a certain sense sharp.
</p>projecteuclid.org/euclid.dmj/1562292016_20190719040229Fri, 19 Jul 2019 04:02 EDTThe structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectureshttps://projecteuclid.org/euclid.dmj/1563847673<strong>Terence Tao</strong>, <strong>Joni Teräväinen</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 11, 1977--2027.</p><p><strong>Abstract:</strong><br/>
Let $g_{0},\dots ,g_{k}:\mathbb{N}\to \mathbb{D}$ be $1$ -bounded multiplicative functions, and let $h_{0},\dots ,h_{k}\in \mathbb{Z}$ be shifts. We consider correlation sequences $f:\mathbb{N}\to\mathbb{Z}$ of the form \begin{equation*}f(a):=\mathop{\widetilde{\lim }}_{m\to \infty }\frac{1}{\log \omega_{m}}\sum _{x_{m}/\omega _{m}\leq n\leq x_{m}}\frac{g_{0}(n+ah_{0})\cdots g_{k}(n+ah_{k})}{n},\end{equation*} where $1\leq \omega _{m}\leq x_{m}$ are numbers going to infinity as $m\to \infty $ and $\mathop{\widetilde{\lim }}$ is a generalized limit functional extending the usual limit functional. We show a structural theorem for these sequences, namely, that these sequences $f$ are the uniform limit of periodic sequences $f_{i}$ . Furthermore, if the multiplicative function $g_{0}\cdots g_{k}$ “weakly pretends” to be a Dirichlet character $\chi $ , the periodic functions $f_{i}$ can be chosen to be $\chi $ -isotypic in the sense that $f_{i}(ab)=f_{i}(a)\chi (b)$ whenever $b$ is coprime to the periods of $f_{i}$ and $\chi $ , while if $g_{0}\cdots g_{k}$ does not weakly pretend to be any Dirichlet character, then $f$ must vanish identically. As a consequence, we obtain several new cases of the logarithmically averaged Elliott conjecture, including the logarithmically averaged Chowla conjecture for odd order correlations. We give a number of applications of these special cases, including the conjectured logarithmic density of all sign patterns of the Liouville function of length up to three and of the Möbius function of length up to four.
</p>projecteuclid.org/euclid.dmj/1563847673_20190812040046Mon, 12 Aug 2019 04:00 EDTK-stability of cubic threefoldshttps://projecteuclid.org/euclid.dmj/1562033045<strong>Yuchen Liu</strong>, <strong>Chenyang Xu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 11, 2029--2073.</p><p><strong>Abstract:</strong><br/>
We prove that the K-moduli space of cubic threefolds is identical to their geometric invariant theory (GIT) moduli. More precisely, the K-semistability, K-polystability, and K-stability of cubic threefolds coincide with the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler–Einstein (KE) metrics and provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension $3$ of the volumes of Kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of $3$ -dimensional canonical and terminal singularities, which was established during the study of the explicit $3$ -dimensional minimal model program.
</p>projecteuclid.org/euclid.dmj/1562033045_20190812040046Mon, 12 Aug 2019 04:00 EDTOn the Oberlin affine curvature conditionhttps://projecteuclid.org/euclid.dmj/1562119267<strong>Philip T. Gressman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 11, 2075--2126.</p><p><strong>Abstract:</strong><br/>
We generalize the well-known notions of affine arclength and affine hypersurface measure to submanifolds of any dimension $d$ in $\mathbb{R}^{n}$ , $1\leq d\leq n-1$ . We show that a canonical equiaffine-invariant measure exists and that, modulo sufficient regularity assumptions on the submanifold, the measure satisfies the affine curvature condition of Oberlin with an exponent which is best possible. The proof combines aspects of geometric invariant theory, convex geometry, and frame theory. A significant new element of the proof is a generalization to higher dimensions of an earlier result concerning inequalities of reverse Sobolev type for polynomials on arbitrary measurable subsets of the real line.
</p>projecteuclid.org/euclid.dmj/1562119267_20190812040046Mon, 12 Aug 2019 04:00 EDTHori-mological projective dualityhttps://projecteuclid.org/euclid.dmj/1563328950<strong>Jørgen Vold Rennemo</strong>, <strong>Ed Segal</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 11, 2127--2205.</p><p><strong>Abstract:</strong><br/>
Kuznetsov has conjectured that Pfaffian varieties should admit noncommutative crepant resolutions which satisfy his Homological Projective Duality. We prove half the cases of this conjecture by interpreting and proving a duality of nonabelian gauged linear sigma models proposed by Hori.
</p>projecteuclid.org/euclid.dmj/1563328950_20190812040046Mon, 12 Aug 2019 04:00 EDTStability and invariant random subgroupshttps://projecteuclid.org/euclid.dmj/1565748248<strong>Oren Becker</strong>, <strong>Alexander Lubotzky</strong>, <strong>Andreas Thom</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 12, 2207--2234.</p><p><strong>Abstract:</strong><br/>
Consider $\operatorname{Sym}(n)$ endowed with the normalized Hamming metric $d_{n}$ . A finitely generated group $\Gamma $ is P-stable if every almost homomorphism $\rho _{n_{k}}\colon\Gamma \rightarrow \operatorname{Sym}(n_{k})$ (i.e., for every $g,h\in \Gamma $ , $\lim _{k\rightarrow \infty }d_{n_{k}}(\rho _{n_{k}}(gh),\rho _{n_{k}}(g)\rho _{n_{k}}(h))=0$ ) is close to an actual homomorphism $\varphi _{n_{k}}\colon \Gamma \rightarrow \operatorname{Sym}(n_{k})$ . Glebsky and Rivera observed that finite groups are P-stable, while Arzhantseva and Păunescu showed the same for abelian groups and raised many questions, especially about the P-stability of amenable groups. We develop P-stability in general and, in particular, for amenable groups. Our main tool is the theory of invariant random subgroups, which enables us to give a characterization of P-stability among amenable groups and to deduce the stability and instability of various families of amenable groups.
</p>projecteuclid.org/euclid.dmj/1565748248_20190904040054Wed, 04 Sep 2019 04:00 EDTExceptional isomorphisms between complements of affine plane curveshttps://projecteuclid.org/euclid.dmj/1566612024<strong>Jérémy Blanc</strong>, <strong>Jean-Philippe Furter</strong>, <strong>Mattias Hemmig</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 12, 2235--2297.</p><p><strong>Abstract:</strong><br/>
This article describes the geometry of isomorphisms between complements of geometrically irreducible closed curves in the affine plane $\mathbb{A}^{2}$ , over an arbitrary field, which do not extend to an automorphism of $\mathbb{A}^{2}$ . We show that such isomorphisms are quite exceptional. In particular, they occur only when both curves are isomorphic to open subsets of the affine line $\mathbb{A}^{1}$ , with the same number of complement points, over any field extension of the ground field. Moreover, the isomorphism is uniquely determined by one of the curves, up to left composition with an automorphism of $\mathbb{A}^{2}$ , except in the case where the curve is isomorphic to the affine line $\mathbb{A}^{1}$ or to the punctured line $\mathbb{A}^{1}\setminus\{0\}$ . If one curve is isomorphic to $\mathbb{A}^{1}$ , then both curves are equivalent to lines. In addition, for any positive integer $n$ , we construct a sequence of $n$ pairwise nonequivalent closed embeddings of $\mathbb{A}^{1}\setminus \{0\}$ with isomorphic complements. In characteristic $0$ we even construct infinite sequences with this property.
Finally, we give a geometric construction that produces a large family of examples of nonisomorphic geometrically irreducible closed curves in $\mathbb{A}^{2}$ that have isomorphic complements, answering negatively the complement problem posed by Hanspeter Kraft. This also gives a negative answer to the holomorphic version of this problem in any dimension $n\geq 2$ . The question had been raised by Pierre-Marie Poloni.
</p>projecteuclid.org/euclid.dmj/1566612024_20190904040054Wed, 04 Sep 2019 04:00 EDTSymplectically knotted codimension-zero embeddings of domains in $\mathbb{R}^{4}$https://projecteuclid.org/euclid.dmj/1566612023<strong>Jean Gutt</strong>, <strong>Michael Usher</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 12, 2299--2363.</p><p><strong>Abstract:</strong><br/>
We show that many toric domains $X$ in $\mathbb{R}^{4}$ admit symplectic embeddings $\phi$ into dilates of themselves which are knotted in the strong sense that there is no symplectomorphism of the target that takes $\phi(X)$ to $X$ . For instance $X$ can be taken equal to a polydisk $P(1,1)$ or to any convex toric domain that both is contained in $P(1,1)$ and properly contains a ball $B^{4}(1)$ ; by contrast a result of McDuff shows that $B^{4}(1)$ (or indeed any $4$ -dimensional ellipsoid) cannot have this property. The embeddings are constructed based on recent advances in symplectic embeddings of ellipsoids, though in some cases a more elementary construction is possible. The fact that the embeddings are knotted is proved using filtered positive $S^{1}$ -equivariant symplectic homology.
</p>projecteuclid.org/euclid.dmj/1566612023_20190904040054Wed, 04 Sep 2019 04:00 EDTArithmetic of double torus quotients and the distribution of periodic torus orbitshttps://projecteuclid.org/euclid.dmj/1566612022<strong>Ilya Khayutin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 168, Number 12, 2365--2432.</p><p><strong>Abstract:</strong><br/>
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_{n}$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus.
Using the new invariants we significantly strengthen results toward the equidistribution of packets of periodic torus orbits on higher rank $S$ -arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adèlic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates, and homogeneous dynamics.
</p>projecteuclid.org/euclid.dmj/1566612022_20190904040054Wed, 04 Sep 2019 04:00 EDT