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 EDTThe annihilator of the Lefschetz motivehttp://projecteuclid.org/euclid.dmj/1493344842<strong>Inna Zakharevich</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 1989--2022.</p><p><strong>Abstract:</strong><br/>
In this article, we study a spectrum $K(\mathcal{V}_{k})$ such that $\pi_{0}K(\mathcal{V}_{k})$ is the Grothendieck ring of varieties and such that the higher homotopy groups contain more geometric information about the geometry of varieties. We use the topology of this spectrum to analyze the structure of $K_{0}[\mathcal{V}_{k}]$ and to show that classes in the kernel of multiplication by $[\mathbb{A}^{1}]$ can always be represented as $[X]-[Y]$ , where $[X]\neq[Y]$ , $X\times\mathbb{A}^{1}$ , and $Y\times\mathbb{A}^{1}$ are not piecewise-isomorphic, but $[X\times\mathbb{A}^{1}]=[Y\times\mathbb{A}^{1}]$ in $K_{0}[\mathcal{V}_{k}]$ . Along the way, we present a new proof of the result of Larsen–Lunts on the structure on $K_{0}[\mathcal{V}_{k}]/([\mathbb{A}^{1}])$ .
</p>projecteuclid.org/euclid.dmj/1493344842_20170810040140Thu, 10 Aug 2017 04:01 EDTII $_{1}$ factors with nonisomorphic ultrapowershttp://projecteuclid.org/euclid.dmj/1490666575<strong>Rémi Boutonnet</strong>, <strong>Ionuţ Chifan</strong>, <strong>Adrian Ioana</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2023--2051.</p><p><strong>Abstract:</strong><br/>
We prove that there exist uncountably many separable $\mathrm{II}_{1}$ factors whose ultrapowers (with respect to arbitrary ultrafilters) are nonisomorphic. In fact, we prove that the families of nonisomorphic $\mathrm{II}_{1}$ factors originally introduced by McDuff are such examples. This entails the existence of a continuum of nonelementarily equivalent $\mathrm{II}_{1}$ factors, thus settling a well-known open problem in the continuous model theory of operator algebras.
</p>projecteuclid.org/euclid.dmj/1490666575_20170810040140Thu, 10 Aug 2017 04:01 EDTLocal Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$ -factorshttp://projecteuclid.org/euclid.dmj/1493863448<strong>J. W. Cogdell</strong>, <strong>F. Shahidi</strong>, <strong>T.-L. Tsai</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2053--2132.</p><p><strong>Abstract:</strong><br/>
Let $F$ be a $p$ -adic field, that is, a finite extension of $\mathbb{Q}_{p}$ for some prime $p$ . The local Langlands correspondence (LLC) attaches to each continuous $n$ -dimensional $\Phi$ -semisimple representation $\rho$ of $W'_{F}$ , the Weil–Deligne group for $\overline{F}/F$ , an irreducible admissible representation $\pi(\rho)$ of $\mathrm{GL}_{n}(F)$ such that, among other things, the local $L$ - and $\varepsilon$ -factors of pairs are preserved. This correspondence should be robust and preserve various parallel operations on the arithmetic and analytic sides, such as taking the exterior square or symmetric square. In this article, we show that this is the case for the local arithmetic and analytic symmetric square and exterior square $\varepsilon$ -factors, that is, that $\varepsilon(s,\Lambda^{2}\rho,\psi)=\varepsilon(s,\pi(\rho),\Lambda^{2},\psi)$ and $\varepsilon(s,\operatorname{Sym}^{2}\rho,\psi)=\varepsilon(s,\pi(\rho),\operatorname{Sym}^{2},\psi)$ . The agreement of the $L$ -functions also follows by our methods, but this was already known by Henniart. The proof is a robust deformation argument, combined with local/global techniques, which reduces the problem to the stability of the analytic $\gamma$ -factor $\gamma(s,\pi,\Lambda^{2},\psi)$ under highly ramified twists when $\pi$ is supercuspidal. This last step is achieved by relating the $\gamma$ -factor to a Mellin transform of a partial Bessel function attached to the representation and then analyzing the asymptotics of the partial Bessel function, inspired in part by the theory of Shalika germs for Bessel integrals. The stability for every irreducible admissible representation $\pi$ then follows from those of the corresponding arithmetic $\gamma$ -factors as a corollary.
</p>projecteuclid.org/euclid.dmj/1493863448_20170810040140Thu, 10 Aug 2017 04:01 EDTAlternating links and definite surfaceshttp://projecteuclid.org/euclid.dmj/1493971214<strong>Joshua Evan Greene</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2133--2151.</p><p><strong>Abstract:</strong><br/>
We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait’s conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and of Hirasawa and Sakuma about Seifert surfaces for special alternating links. The appendix, written by Juhász and Lackenby, applies the characterization to derive an exponential time algorithm for alternating knot recognition.
</p>projecteuclid.org/euclid.dmj/1493971214_20170810040140Thu, 10 Aug 2017 04:01 EDTFree Hilbert transformshttp://projecteuclid.org/euclid.dmj/1493344841<strong>Tao Mei</strong>, <strong>Éric Ricard</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 11, 2153--2182.</p><p><strong>Abstract:</strong><br/>
We study Fourier multipliers of Hilbert transform type on free groups. We prove that they are completely bounded on noncommutative $L^{p}$ -spaces associated with the free group von Neumann algebras for all $1\lt p\lt \infty$ . This implies that the decomposition of the free group $\mathbf{F}_{\infty}$ into reduced words starting with distinct free generators is completely unconditional in $L^{p}$ . We study the case of Voiculescu’s amalgamated free products of von Neumann algebras as well. As by-products, we obtain a positive answer to a compactness problem posed by Ozawa, a length-independent estimate for Junge–Parcet–Xu’s free Rosenthal’s inequality, a Littlewood–Paley–Stein-type inequality for geodesic paths of free groups, and a length reduction formula for $L^{p}$ -norms of free group von Neumann algebras.
</p>projecteuclid.org/euclid.dmj/1493344841_20170810040140Thu, 10 Aug 2017 04:01 EDTOn the arithmetic transfer conjecture for exotic smooth formal moduli spaceshttps://projecteuclid.org/euclid.dmj/1496995226<strong>M. Rapoport</strong>, <strong>B. Smithling</strong>, <strong>W. Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 12, 2183--2336.</p><p><strong>Abstract:</strong><br/>
In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of $p$ -divisible groups, associated to a unitary group relative to a ramified quadratic extension of a $p$ -adic field. We prove our conjecture in the case of a unitary group in three variables.
</p>projecteuclid.org/euclid.dmj/1496995226_20170824040044Thu, 24 Aug 2017 04:00 EDTTriangular bases in quantum cluster algebras and monoidal categorification conjectureshttps://projecteuclid.org/euclid.dmj/1495764415<strong>Fan Qin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 12, 2337--2442.</p><p><strong>Abstract:</strong><br/>
We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parameterized by tropical points as expected in the Fock–Goncharov conjecture.
As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorification conjectures of Hernandez and Leclerc and Fomin and Zelevinsky in the corresponding cases: all cluster monomials correspond to simple modules.
</p>projecteuclid.org/euclid.dmj/1495764415_20170824040044Thu, 24 Aug 2017 04:00 EDTCounterexamples to a conjecture of Woodshttps://projecteuclid.org/euclid.dmj/1496995227<strong>Oded Regev</strong>, <strong>Uri Shapira</strong>, <strong>Barak Weiss</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2443--2446.</p><p><strong>Abstract:</strong><br/>
A conjecture of Woods from 1972 is disproved: for $d\geq30$ , there are well-rounded unimodular lattices in ${\mathbb{R}}^{d}$ with covering radius greater than that of ${\mathbb{Z}}^{d}$ .
</p>projecteuclid.org/euclid.dmj/1496995227_20170918220520Mon, 18 Sep 2017 22:05 EDTCM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$https://projecteuclid.org/euclid.dmj/1496995225<strong>Stephan Ehlen</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2447--2519.</p><p><strong>Abstract:</strong><br/>
We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maaß forms of weight $1$ . As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight $1$ studied by Kudla, Rapoport, and Yang.
</p>projecteuclid.org/euclid.dmj/1496995225_20170918220520Mon, 18 Sep 2017 22:05 EDTRepresentation stability and finite linear groupshttps://projecteuclid.org/euclid.dmj/1497924228<strong>Andrew Putman</strong>, <strong>Steven V Sam</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2521--2598.</p><p><strong>Abstract:</strong><br/>
We study analogues of $\operatorname{{\mathtt {FI}}}$ -modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings, and we prove basic structural properties such as local Noetherianity. Applications include a proof of the Lannes–Schwartz Artinian conjecture in the generic representation theory of finite fields, very general homological stability theorems with twisted coefficients for the general linear and symplectic groups over finite rings, and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.
</p>projecteuclid.org/euclid.dmj/1497924228_20170918220520Mon, 18 Sep 2017 22:05 EDTBounded height in pencils of finitely generated subgroupshttps://projecteuclid.org/euclid.dmj/1500364839<strong>F. Amoroso</strong>, <strong>D. Masser</strong>, <strong>U. Zannier</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 13, 2599--2642.</p><p><strong>Abstract:</strong><br/>
In this article we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell–Lang theorem by replacing finiteness with emptiness , for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the result is as follows. Consider the torus scheme ${{\mathbb{G}_{m}^{r}}_{/\mathcal{C}}}$ over a curve $\mathcal{C}$ defined over $\overline{\mathbb{Q}}$ , and let $\Gamma$ be a subgroup scheme generated by finitely many sections (satisfying some necessary conditions). Further, let $V$ be any subscheme. Then there is a bound for the height of the points $P\in\mathcal{C}(\overline{\mathbb{Q}})$ such that, for some $\gamma\in\Gamma$ which does not generically lie in $V$ , $\gamma(P)$ lies in the fiber $V_{P}$ . We further offer some direct Diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.
</p>projecteuclid.org/euclid.dmj/1500364839_20170918220520Mon, 18 Sep 2017 22:05 EDTComplex projective structures: Lyapunov exponent, degree, and harmonic measurehttps://projecteuclid.org/euclid.dmj/1503712833<strong>Bertrand Deroin</strong>, <strong>Romain Dujardin</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2643--2695.</p><p><strong>Abstract:</strong><br/>
We study several new invariants associated to a holomorphic projective structure on a Riemann surface of finite analytic type: the Lyapunov exponent of its holonomy which is of probabilistic/dynamical nature and was introduced in our previous work; the degree which measures the asymptotic covering rate of the developing map; and a family of harmonic measures on the Riemann sphere, previously introduced by Hussenot. We show that the degree and the Lyapunov exponent are related by a simple formula and give estimates for the Hausdorff dimension of the harmonic measures in terms of the Lyapunov exponent. In accordance with the famous Sullivan dictionary, this leads to a description of the space of such projective structures that is reminiscent of that of the space of polynomials in holomorphic dynamics.
</p>projecteuclid.org/euclid.dmj/1503712833_20170929040133Fri, 29 Sep 2017 04:01 EDTSharp phase transitions for the almost Mathieu operatorhttps://projecteuclid.org/euclid.dmj/1495764414<strong>Artur Avila</strong>, <strong>Jiangong You</strong>, <strong>Qi Zhou</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2697--2718.</p><p><strong>Abstract:</strong><br/>
It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya’s conjecture. Together with a previous work by Avila, this gives the sharp description of phase transitions for the AMO for the a.e. phase.
</p>projecteuclid.org/euclid.dmj/1495764414_20170929040133Fri, 29 Sep 2017 04:01 EDTSobolev trace inequalities of order fourhttps://projecteuclid.org/euclid.dmj/1502697757<strong>Antonio G. Ache</strong>, <strong>Sun-Yung Alice Chang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2719--2748.</p><p><strong>Abstract:</strong><br/>
We establish sharp trace Sobolev inequalities of order four on Euclidean $d$ -balls for $d\ge4$ . When $d=4$ , our inequality generalizes the classical second-order Lebedev–Milin inequality on Euclidean $2$ -balls. Our method relies on the use of scattering theory on hyperbolic $d$ -balls. As an application, we characterize the extremal metric of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$ -balls, which surprisingly is not the flat metric on the ball.
</p>projecteuclid.org/euclid.dmj/1502697757_20170929040133Fri, 29 Sep 2017 04:01 EDTA tropical approach to a generalized Hodge conjecture for positive currentshttps://projecteuclid.org/euclid.dmj/1504684816<strong>Farhad Babaee</strong>, <strong>June Huh</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14, 2749--2813.</p><p><strong>Abstract:</strong><br/>
In 1982, Demailly showed that the Hodge conjecture follows from the statement that all positive closed currents with rational cohomology class can be approximated by positive linear combinations of integration currents. Moreover, in 2012, he showed that the Hodge conjecture is equivalent to the statement that any $(p,p)$ -dimensional closed current with rational cohomology class can be approximated by linear combinations of integration currents. In this article, we find a current which does not verify the former statement on a smooth projective variety for which the Hodge conjecture is known to hold. To construct this current, we extend the framework of “tropical currents”—recently introduced by the first author—from tori to toric varieties. We discuss extremality properties of tropical currents and show that the cohomology class of a tropical current is the recession of its underlying tropical variety. The counterexample is obtained from a tropical surface in $\mathbb{R}^{4}$ whose intersection form does not have the right signature in terms of the Hodge index theorem.
</p>projecteuclid.org/euclid.dmj/1504684816_20170929040133Fri, 29 Sep 2017 04:01 EDTErrata for Stephan Ehlen, “CM values of regularized theta lifts and harmonic weak Maaß forms of weight $1$ ,” Duke Math. J., Volume 166, Number 13 (2017), 2447–2519https://projecteuclid.org/euclid.dmj/1506672071<p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 14</p>projecteuclid.org/euclid.dmj/1506672071_20170929040133Fri, 29 Sep 2017 04:01 EDTOn the geometry of thin exceptional sets in Manin’s conjecturehttps://projecteuclid.org/euclid.dmj/1504252913<strong>Brian Lehmann</strong>, <strong>Sho Tanimoto</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2815--2869.</p><p><strong>Abstract:</strong><br/>
Manin’s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set. We study whether the exceptional set in Manin’s conjecture is a thin set using the minimal model program and boundedness of log Fano varieties.
</p>projecteuclid.org/euclid.dmj/1504252913_20171012040515Thu, 12 Oct 2017 04:05 EDTModuli of curves as moduli of $A_{\infty}$ -structureshttps://projecteuclid.org/euclid.dmj/1504836224<strong>Alexander Polishchuk</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2871--2924.</p><p><strong>Abstract:</strong><br/>
We define and study the stack $\mathcal{U}^{ns,a}_{g,g}$ of (possibly singular) projective curves of arithmetic genus $g$ with $g$ smooth marked points forming an ample nonspecial divisor. We define an explicit closed embedding of a natural $\mathbb{G}_{m}^{g}$ -torsor $\widetilde{\mathcal{U}}^{ns,a}_{g,g}$ over $\mathcal{U}^{ns,a}_{g,g}$ into an affine space, and we give explicit equations of the universal curve (away from characteristics $2$ and $3$ ). This construction can be viewed as a generalization of the Weierstrass cubic and the $j$ -invariant of an elliptic curve to the case $g\gt 1$ . Our main result is that in characteristics different from $2$ and $3$ the moduli space $\widetilde{\mathcal{U}}^{ns,a}_{g,g}$ is isomorphic to the moduli space of minimal $A_{\infty}$ -structures on a certain finite-dimensional graded associative algebra $E_{g}$ (introduced by Fisette and Polishchuk).
</p>projecteuclid.org/euclid.dmj/1504836224_20171012040515Thu, 12 Oct 2017 04:05 EDTRank, combinatorial cost, and homology torsion growth in higher rank latticeshttps://projecteuclid.org/euclid.dmj/1504684817<strong>Miklos Abert</strong>, <strong>Tsachik Gelander</strong>, <strong>Nikolay Nikolov</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2925--2964.</p><p><strong>Abstract:</strong><br/>
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of right-angled groups. A group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute.
Most nonuniform lattices in higher rank simple Lie groups are right angled. We provide the first examples of uniform (cocompact) right-angled arithmetic groups in $\mathrm{SL}(n,\mathbb{R})$ , $n\geq3$ , and $\mathrm{SO}(p,q)$ for some values of $p,q$ . This is a class of lattices for which the congruence subgroup property is not known in general. By using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right-angled lattice in a higher rank simple Lie group.
</p>projecteuclid.org/euclid.dmj/1504684817_20171012040515Thu, 12 Oct 2017 04:05 EDTA product for permutation groups and topological groupshttps://projecteuclid.org/euclid.dmj/1502244254<strong>Simon M. Smith</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 15, 2965--2999.</p><p><strong>Abstract:</strong><br/>
We introduce a new product for permutation groups. It takes as input two permutation groups, $M$ and $N$ and produces an infinite group $M\boxtimes N$ which carries many of the permutational properties of $M$ . Under mild conditions on $M$ and $N$ the group $M\boxtimes N$ is simple.
As a permutational product, its most significant property is the following: $M\boxtimes N$ is primitive if and only if $M$ is primitive but not regular, and $N$ is transitive. Despite this remarkable similarity with the wreath product in product action, $M\boxtimes N$ and $M\operatorname{Wr}N$ are thoroughly dissimilar.
The product provides a general way to build exotic examples of nondiscrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups.
We use this to obtain the first construction of uncountably many pairwise nonisomorphic simple topological groups that are totally disconnected, locally compact, compactly generated, and nondiscrete. The groups we construct all contain the same compact open subgroup. The analogous result for discrete groups was proved in 1953 by Ruth Camm.
To build the product, we describe a group $\mathcal{U}(M,N)$ that acts on a biregular tree $T$ . This group has a natural universal property and is a generalization of the iconic universal group construction of Marc Burger and Shahar Mozes for locally finite regular trees.
</p>projecteuclid.org/euclid.dmj/1502244254_20171012040515Thu, 12 Oct 2017 04:05 EDTA metric interpretation of reflexivity for Banach spaceshttps://projecteuclid.org/euclid.dmj/1505959221<strong>P. Motakis</strong>, <strong>T. Schlumprecht</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 16, 3001--3084.</p><p><strong>Abstract:</strong><br/>
We define two metrics $d_{1,\alpha}$ and $d_{\infty,\alpha}$ on each Schreier family $\mathcal{S}_{\alpha}$ , $\alpha\lt \omega_{1}$ , with which we prove the following metric characterization of the reflexivity of a Banach space $X$ : $X$ is reflexive if and only if there is an $\alpha\lt \omega_{1}$ such that there is no mapping $\Phi:\mathcal{S}_{\alpha}\to X$ for which \begin{equation*}cd_{\infty,\alpha}(A,B)\le\Vert \Phi(A)-\Phi(B)\Vert \le Cd_{1,\alpha}(A,B)\quad \text{for all }A,B\in\mathcal{S}_{\alpha}.\end{equation*} Additionally we prove, for separable and reflexive Banach spaces $X$ and certain countable ordinals $\alpha$ , that $\max(\operatorname{Sz}(X),\operatorname{Sz}(X^{*}))\le\alpha$ if and only if $(\mathcal{S}_{\alpha},d_{1,\alpha})$ does not bi-Lipschitzly embed into $X$ . Here $\operatorname{Sz}(Y)$ denotes the Szlenk index of a Banach space $Y$ .
</p>projecteuclid.org/euclid.dmj/1505959221_20171020040059Fri, 20 Oct 2017 04:00 EDTThe Coolidge–Nagata conjecturehttps://projecteuclid.org/euclid.dmj/1499911237<strong>Mariusz Koras</strong>, <strong>Karol Palka</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 16, 3085--3145.</p><p><strong>Abstract:</strong><br/>
Let $E\subseteq\mathbb{P}^{2}$ be a complex rational cuspidal curve contained in the projective plane. The Coolidge–Nagata conjecture asserts that $E$ is Cremona-equivalent to a line, that is, it is mapped onto a line by some birational transformation of $\mathbb{P}^{2}$ . The second author recently analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$ , where $(X,D)\to(\mathbb{P}^{2},E)$ is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in $\mathbb{P}^{2}\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.
</p>projecteuclid.org/euclid.dmj/1499911237_20171020040059Fri, 20 Oct 2017 04:00 EDTK-semistability is equivariant volume minimizationhttps://projecteuclid.org/euclid.dmj/1505527493<strong>Chi Li</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 16, 3147--3218.</p><p><strong>Abstract:</strong><br/>
This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over $\mathbb{Q}$ -Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler–Einstein metrics on Fano varieties. In particular, we prove that for a $\mathbb{Q}$ -Fano variety $V$ , the K-semistability of $(V,-K_{V})$ is equivalent to the condition that the normalized volume is minimized at the canonical valuation $\mathrm{ord}_{V}$ among all $\mathbb{C}^{*}$ -invariant valuations on the cone associated to any positive Cartier multiple of $-K_{V}$ . In this case, we show that $\mathrm{ord}_{V}$ is the unique minimizer among all $\mathbb{C}^{*}$ -invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over $V$ .
</p>projecteuclid.org/euclid.dmj/1505527493_20171020040059Fri, 20 Oct 2017 04:00 EDTAutomatic sequences fulfill the Sarnak conjecturehttps://projecteuclid.org/euclid.dmj/1507169019<strong>Clemens Müllner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 17, 3219--3290.</p><p><strong>Abstract:</strong><br/>
We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Möbius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to $0$ .
</p>projecteuclid.org/euclid.dmj/1507169019_20171116040048Thu, 16 Nov 2017 04:00 ESTEnumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariantshttps://projecteuclid.org/euclid.dmj/1507687598<strong>Penka Georgieva</strong>, <strong>Aleksey Zinger</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 17, 3291--3347.</p><p><strong>Abstract:</strong><br/>
We establish a homology relation for the Deligne–Mumford moduli spaces of real curves which lifts to a Witten–Dijkgraaf–Verlinde–Verlinde (WDVV)-type relation for a class of real Gromov–Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus $0$ real invariants with conjugate pairs of constraints to genus $0$ invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide nontrivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the $3$ -dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo $4$ .
</p>projecteuclid.org/euclid.dmj/1507687598_20171116040048Thu, 16 Nov 2017 04:00 ESTHasse principle for three classes of varieties over global function fieldshttps://projecteuclid.org/euclid.dmj/1505808016<strong>Zhiyu Tian</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 17, 3349--3424.</p><p><strong>Abstract:</strong><br/>
We give a geometric proof that the Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces, smooth cubic hypersurfaces of dimension at least $4$ in characteristic at least $7$ , and smooth complete intersections of two quadrics, which are of dimension at least $3$ , in odd characteristics.
</p>projecteuclid.org/euclid.dmj/1505808016_20171116040048Thu, 16 Nov 2017 04:00 ESTQuantum ergodicity and Benjamini–Schramm convergence of hyperbolic surfaceshttps://projecteuclid.org/euclid.dmj/1507860019<strong>Etienne Le Masson</strong>, <strong>Tuomas Sahlsten</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3425--3460.</p><p><strong>Abstract:</strong><br/>
We present a quantum ergodicity theorem for fixed spectral window and sequences of compact hyperbolic surfaces converging to the hyperbolic plane in the sense of Benjamini and Schramm. This addresses a question posed by Colin de Verdière. Our theorem is inspired by results for eigenfunctions on large regular graphs by Anantharaman and Le Masson. It applies in particular to eigenfunctions on compact arithmetic surfaces in the level aspect, which connects it to a question of Nelson on Maass forms. The proof is based on a wave propagation approach recently considered by Brooks, Le Masson, and Lindenstrauss on discrete graphs. It does not use any microlocal analysis, making it quite different from the usual proof of quantum ergodicity in the large eigenvalue limit. Moreover, we replace the wave propagator with renormalized averaging operators over disks, which simplifies the analysis and allows us to make use of a general ergodic theorem of Nevo. As a consequence of this approach, we require little regularity on the observables.
</p>projecteuclid.org/euclid.dmj/1507860019_20171204040052Mon, 04 Dec 2017 04:00 ESTEquidistribution in $\operatorname{Bun}_{2}(\mathbb{P}^{1})$https://projecteuclid.org/euclid.dmj/1510887861<strong>Vivek Shende</strong>, <strong>Jacob Tsimerman</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3461--3504.</p><p><strong>Abstract:</strong><br/> Fix a finite field. The set of $\operatorname{PGL}_{2}$ bundles over $\mathbb{P}^{1}$ is in bijection with the natural numbers, and carries a natural measure assigning to each bundle the inverse of the number of automorphisms. A branched double cover $\pi:C\to\mathbb{P}^{1}$ determines another measure, given by counting the number of line bundles over $C$ whose image on $\mathbb{P}^{1}$ has a given sheaf of endomorphisms. We show the measures induced by a sequence of such hyperelliptic curves tends to the canonical measure on the space of $\operatorname{PGL}_{2}$ bundles. This is a function field analogue of Duke’s theorem on the equidistribution of Heegner points, and can be proven similarly. Our real interest is the corresponding analogue of the “Mixing Conjecture” of Michel and Venkatesh. This amounts to considering measures on the space of pairs of $\operatorname{PGL}_{2}$ bundles induced by taking a fixed line bundle $\mathcal{L}$ over $C$ , and looking at the distribution of pairs $(\pi_{*}\mathcal{M},\pi_{*}(\mathcal{L}\otimes\mathcal{M}))$ . As in the number field setting, ergodic theory classifies limiting measures for sufficiently special $\mathcal{L}$ . The heart of this work is a geometric attack on the general case. We count points on intersections of translates of loci of special divisors in the Jacobian of a hyperelliptic curve. To prove equidistribution, we would require two results. The first, we prove: in high degree, the cohomologies of these loci match the cohomology of the Jacobian. The second, we establish in characteristic zero and conjecture in characteristic $p$ : the cohomology of these spaces grows at most exponentially in the genus of the curve $C$ . </p>projecteuclid.org/euclid.dmj/1510887861_20171204040052Mon, 04 Dec 2017 04:00 ESTOn the Lagrangian structure of transport equations: The Vlasov–Poisson systemhttps://projecteuclid.org/euclid.dmj/1504836225<strong>Luigi Ambrosio</strong>, <strong>Maria Colombo</strong>, <strong>Alessio Figalli</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3505--3568.</p><p><strong>Abstract:</strong><br/>
The Vlasov–Poisson system is an important nonlinear transport equation, used to describe the evolution of particles under their self-consistent electric or gravitational field. The existence of classical solutions is limited to dimensions $d\leq3$ under strong assumptions on the initial data, whereas weak solutions are known to exist under milder conditions. However, in the setting of weak solutions it is unclear whether the Eulerian description provided by the equation physically corresponds to a Lagrangian evolution of the particles. In this article we develop several general tools concerning the Lagrangian structure of transport equations with nonsmooth vector fields, and we apply these results to show that weak/renormalized solutions of Vlasov–Poisson are Lagrangian and actually that the concepts of renormalized and Lagrangian solutions are equivalent. As a corollary, we prove that finite-energy solutions in dimension $d\leq4$ are transported by a global flow (in particular, they preserve all the natural Casimir invariants), and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data.
</p>projecteuclid.org/euclid.dmj/1504836225_20171204040052Mon, 04 Dec 2017 04:00 ESTThe C∗-algebra of a minimal homeomorphism of zero mean dimensionhttps://projecteuclid.org/euclid.dmj/1511233446<strong>George A. Elliott</strong>, <strong>Zhuang Niu</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 166, Number 18, 3569--3594.</p><p><strong>Abstract:</strong><br/>
Let $X$ be an infinite metrizable compact space, and let $\sigma:X\toX$ be a minimal homeomorphism. Suppose that $(X,\sigma)$ has zero mean topological dimension. The associated C∗-algebra $A=\mathrm{C}(X)\rtimes_{\sigma}\mathbb{Z}$ is shown to absorb the Jiang–Su algebra $\mathcal{Z}$ tensorially; that is, $A\cong A\otimes\mathcal{Z}$ . This implies that $A$ is classifiable when $(X,\sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang–Su algebra.
</p>projecteuclid.org/euclid.dmj/1511233446_20171204040052Mon, 04 Dec 2017 04:00 ESTFull-rank affine invariant submanifoldshttps://projecteuclid.org/euclid.dmj/1512529217<strong>Maryam Mirzakhani</strong>, <strong>Alex Wright</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 1--40.</p><p><strong>Abstract:</strong><br/>
We show that every $\operatorname{GL}(2,R)$ orbit closure of translation surfaces is a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense $\operatorname{GL}(2,R)$ orbit.
</p>projecteuclid.org/euclid.dmj/1512529217_20180104040138Thu, 04 Jan 2018 04:01 ESTQuantitative nonorientability of embedded cycleshttps://projecteuclid.org/euclid.dmj/1510736537<strong>Robert Young</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 41--108.</p><p><strong>Abstract:</strong><br/>
We introduce an invariant linked to some foundational questions in geometric measure theory and provide bounds on this invariant by decomposing an arbitrary cycle into uniformly rectifiable pieces. Our invariant measures the difficulty of cutting a nonorientable closed manifold or mod- $2$ cycle in $\mathbb{R}^{n}$ into orientable pieces, and we use it to answer some simple but long-open questions on filling volumes and mod- $\nu$ currents.
</p>projecteuclid.org/euclid.dmj/1510736537_20180104040138Thu, 04 Jan 2018 04:01 ESTSpectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact casehttps://projecteuclid.org/euclid.dmj/1510304420<strong>Yves Colin de Verdière</strong>, <strong>Luc Hillairet</strong>, <strong>Emmanuel Trélat</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 109--174.</p><p><strong>Abstract:</strong><br/> This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian (sR) Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasimodes, and trace formulae. Even if hypoelliptic operators have been well studied from the point of view of PDEs, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sR setting are quite different from those in the Riemannian one. Let us consider an sR metric on a closed $3$ -dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a quantum ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure. This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace–Beltrami operator on a closed Riemannian manifold with ergodic geodesic flow. To prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all $3$ -dimensional contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization, and the ergodicity assumption, we finally establish a variance estimate, from which QE follows. We also obtain a second result, which is valid without any ergodicity assumption: every quantum limit (QL) can be decomposed in a sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold of the sR Laplacian and is invariant under the lift of the Reeb flow. Moreover, we prove that the first measure is zero for most QLs. </p>projecteuclid.org/euclid.dmj/1510304420_20180104040138Thu, 04 Jan 2018 04:01 ESTOn the marked length spectrum of generic strictly convex billiard tableshttps://projecteuclid.org/euclid.dmj/1512702098<strong>Guan Huang</strong>, <strong>Vadim Kaloshin</strong>, <strong>Alfonso Sorrentino</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 1, 175--209.</p><p><strong>Abstract:</strong><br/>
In this paper we show that for a generic strictly convex domain, one can recover the eigendata corresponding to Aubry–Mather periodic orbits of the induced billiard map from the (maximal) marked length spectrum of the domain.
</p>projecteuclid.org/euclid.dmj/1512702098_20180104040138Thu, 04 Jan 2018 04:01 ESTThe Abelianization of the real Cremona grouphttps://projecteuclid.org/euclid.dmj/1513998141<strong>Susanna Zimmermann</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 2, 211--267.</p><p><strong>Abstract:</strong><br/>
We present the Abelianization of the group of birational transformations of $\mathbb{P}^{2}_{\mathbb{R}}$ .
</p>projecteuclid.org/euclid.dmj/1513998141_20180129040341Mon, 29 Jan 2018 04:03 ESTCurrent fluctuations of the stationary ASEP and six-vertex modelhttps://projecteuclid.org/euclid.dmj/1515812504<strong>Amol Aggarwal</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 2, 269--384.</p><p><strong>Abstract:</strong><br/> Our results in this article are twofold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$ , and show that they are of order $T^{1/3}$ along a characteristic line. Upon scaling by $T^{1/3}$ , we establish that these fluctuations converge to the long-time height fluctuations of the stationary Kardar–Parisi–Zhang (KPZ) equation, that is, to the Baik–Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn from 2005 (when they established the same result for the TASEP) and the work of Balázs and Seppäläinen from 2010 (when they established the $T^{1/3}$ -scaling for the general ASEP). Second, we introduce a class of translation-invariant Gibbs measures that characterizes a one-parameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point ) in the free-energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size $T$ and show that they too are of order $T^{1/3}$ along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995, stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity. Upon scaling the height fluctuations by $T^{1/3}$ , we again recover the Baik–Rains distribution in the large $T$ limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992. </p>projecteuclid.org/euclid.dmj/1515812504_20180129040341Mon, 29 Jan 2018 04:03 ESTTotaro’s question on zero-cycles on torsorshttps://projecteuclid.org/euclid.dmj/1513998140<strong>R. Gordon-Sarney</strong>, <strong>V. Suresh</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 2, 385--395.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a smooth connected linear algebraic group, and let $X$ be a $G$ -torsor. Totaro asked: If $X$ admits a zero-cycle of degree $d\geq1$ , then does $X$ have a closed étale point of degree dividing $d$ ? While the literature contains affirmative answers in some special cases, we give examples to show that the answer is negative in general.
</p>projecteuclid.org/euclid.dmj/1513998140_20180129040341Mon, 29 Jan 2018 04:03 ESTThe colored HOMFLYPT function is $q$ -holonomichttps://projecteuclid.org/euclid.dmj/1510304421<strong>Stavros Garoufalidis</strong>, <strong>Aaron D. Lauda</strong>, <strong>Thang T. Q. Lê</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 397--447.</p><p><strong>Abstract:</strong><br/>
We prove that the HOMFLYPT polynomial of a link colored by partitions with a fixed number of rows is a $q$ -holonomic function. By specializing to the case of knots colored by a partition with a single row, it proves the existence of an $(a,q)$ superpolynomial of knots in $3$ -space, as was conjectured by string theorists. Our proof uses skew-Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincaré–Birkhoff–Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram. The result is a concrete and algorithmic web evaluation algorithm that is manifestly $q$ -holonomic.
</p>projecteuclid.org/euclid.dmj/1510304421_20180209220044Fri, 09 Feb 2018 22:00 ESTCanonical growth conditions associated to ample line bundleshttps://projecteuclid.org/euclid.dmj/1515143006<strong>David Witt Nyström</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 449--495.</p><p><strong>Abstract:</strong><br/>
We propose a new construction which associates to any ample (or big) line bundle $L$ on a projective manifold $X$ a canonical growth condition (i.e., a choice of a plurisubharmonic (psh) function well defined up to a bounded term) on the tangent space $T_{p}X$ of any given point $p$ . We prove that it encodes such classical invariants as the volume and the Seshadri constant. Even stronger, it allows one to recover all the infinitesimal Okounkov bodies of $L$ at $p$ . The construction is inspired by toric geometry and the theory of Okounkov bodies; in the toric case, the growth condition is “equivalent” to the moment polytope. As in the toric case, the growth condition says a lot about the Kähler geometry of the manifold. We prove a theorem about Kähler embeddings of large balls, which generalizes the well-known connection between Seshadri constants and Gromov width established by McDuff and Polterovich.
</p>projecteuclid.org/euclid.dmj/1515143006_20180209220044Fri, 09 Feb 2018 22:00 ESTCarathéodory’s metrics on Teichmüller spaces and $L$ -shaped pillowcaseshttps://projecteuclid.org/euclid.dmj/1516762971<strong>Vladimir Markovic</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 497--535.</p><p><strong>Abstract:</strong><br/>
One of the most important results in Teichmüller theory is Royden’s theorem, which says that the Teichmüller and Kobayashi metrics agree on the Teichmüller space of a given closed Riemann surface. The problem that remained open is whether the Carathéodory metric agrees with the Teichmüller metric as well. In this article, we prove that these two metrics disagree on each $\mathcal{T}_{g}$ , the Teichmüller space of a closed surface of genus $g\ge2$ . The main step is to establish a criterion to decide when the Teichmüller and Carathéodory metrics agree on the Teichmüller disk corresponding to a rational Jenkins–Strebel differential $\varphi$ . First, we construct a holomorphic embedding $\mathcal{E}:\mathbb{H}^{k}\to\mathcal{T}_{g,n}$ corresponding to $\varphi$ . The criterion says that the two metrics agree on this disk if and only if a certain function $\mathbf{\Phi}:\mathcal{E}(\mathbb{H}^{k})\to\mathbb{H}$ can be extended to a holomorphic function $\mathbf{\Phi}:\mathcal{T}_{g,n}\to\mathbb{H}$ . We then show by explicit computation that this is not the case for quadratic differentials arising from $L$ -shaped pillowcases.
</p>projecteuclid.org/euclid.dmj/1516762971_20180209220044Fri, 09 Feb 2018 22:00 ESTGroups quasi-isometric to right-angled Artin groupshttps://projecteuclid.org/euclid.dmj/1515467194<strong>Jingyin Huang</strong>, <strong>Bruce Kleiner</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 3, 537--602.</p><p><strong>Abstract:</strong><br/>
We characterize groups quasi-isometric to a right-angled Artin group (RAAG) $G$ with finite outer automorphism group. In particular, all such groups admit a geometric action on a $\operatorname{CAT}(0)$ cube complex that has an equivariant “fibering” over the Davis building of $G$ . This characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi-isometric to certain RAAGs.
</p>projecteuclid.org/euclid.dmj/1515467194_20180209220044Fri, 09 Feb 2018 22:00 ESTThe Breuil–Mézard conjecture when $l\neq p$https://projecteuclid.org/euclid.dmj/1513998139<strong>Jack Shotton</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 4, 603--678.</p><p><strong>Abstract:</strong><br/> Let $l$ and $p$ be primes, let $F/\mathbb{Q}_{p}$ be a finite extension with absolute Galois group $G_{F}$ , let $\mathbb{F}$ be a finite field of characteristic $l$ , and let \[\overline{\rho}:G_{F}\rightarrow \operatorname{GL}_{n}(\mathbb{F})\] be a continuous representation. Let $R^{\square}(\overline{\rho})$ be the universal framed deformation ring for $\overline{\rho}$ . If $l=p$ , then the Breuil–Mézard conjecture (as recently formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^{\square}(\overline{\rho})$ to the mod $l$ reduction of certain representations of $\operatorname{GL}_{n}(\mathcal{O}_{F})$ . We state an analogue of the Breuil–Mézard conjecture when $l\neq p$ , and we prove it whenever $l\gt 2$ using automorphy lifting theorems. We give a local proof when $l$ is “quasibanal” for $F$ and $\overline{\rho}$ is tamely ramified. We also analyze the reduction modulo $l$ of the types $\sigma(\tau)$ defined by Schneider and Zink. </p>projecteuclid.org/euclid.dmj/1513998139_20180301040030Thu, 01 Mar 2018 04:00 EST$\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of stratahttps://projecteuclid.org/euclid.dmj/1517281370<strong>Paul Apisa</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 4, 679--742.</p><p><strong>Abstract:</strong><br/>
The object of this article is to study $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata of Abelian differentials. The main result is that all higher-rank affine invariant submanifolds in hyperelliptic components are branched covering constructions; that is, every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of Abelian differentials. This result implies the finiteness of algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two. A classification of all $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.
</p>projecteuclid.org/euclid.dmj/1517281370_20180301040030Thu, 01 Mar 2018 04:00 ESTA $p$ -adic Waldspurger formulahttps://projecteuclid.org/euclid.dmj/1518166812<strong>Yifeng Liu</strong>, <strong>Shouwu Zhang</strong>, <strong>Wei Zhang</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 4, 743--833.</p><p><strong>Abstract:</strong><br/>
In this article, we study $p$ -adic torus periods for certain $p$ -adic-valued functions on Shimura curves of classical origin. We prove a $p$ -adic Waldspurger formula for these periods as a generalization of recent work of Bertolini, Darmon, and Prasanna. In pursuing such a formula, we construct a new anti-cyclotomic $p$ -adic $L$ -function of Rankin–Selberg type. At a character of positive weight, the $p$ -adic $L$ -function interpolates the central critical value of the complex Rankin–Selberg $L$ -function. Its value at a finite-order character, which is outside the range of interpolation, essentially computes the corresponding $p$ -adic torus period.
</p>projecteuclid.org/euclid.dmj/1518166812_20180301040030Thu, 01 Mar 2018 04:00 ESTPicard–Lefschetz oscillators for the Drinfeld–Lafforgue–Vinberg degeneration for $\operatorname{SL}_{2}$https://projecteuclid.org/euclid.dmj/1520906430<strong>Simon Schieder</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 835--921.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a reductive group, and let $\operatorname{Bun}_{G}$ denote the moduli stack of $G$ -bundles on a smooth projective curve. We begin the study of the singularities of a canonical compactification of $\operatorname{Bun}_{G}$ due to Drinfeld (unpublished), which we refer to as the Drinfeld–Lafforgue–Vinberg compactification $\overline{\operatorname{Bun}}_{G}$ . For $G=\operatorname{GL}_{2}$ and $G=\operatorname{GL}_{n}$ , certain smooth open substacks of this compactification have already appeared in the work of Drinfeld and Lafforgue on the Langlands correspondence for function fields. The stack $\overline{\operatorname{Bun}}_{G}$ is, however, already singular for $G=\operatorname{SL}_{2}$ ; questions about its singularities arise naturally in the geometric Langlands program, and form the topic of the present article. Drinfeld’s definition of $\overline{\operatorname{Bun}}_{G}$ for a general reductive group $G$ relies on the Vinberg semigroup of $G$ (we will study this case in a forthcoming article). In the present article we focus on the case $G=\operatorname{SL}_{2}$ , where the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of $\operatorname{SL}_{2}$ -bundles. We study the singularities of this one-parameter degeneration via the weight-monodromy theory of its nearby cycles. We give an explicit description of the nearby cycles sheaf together with its monodromy action in terms of certain novel perverse sheaves which we call Picard–Lefschetz oscillators and which govern the singularities of $\overline{\operatorname{Bun}}_{G}$ . We then use this description to determine the intersection cohomology sheaf of $\overline{\operatorname{Bun}}_{G}$ and other invariants of its singularities. Our proofs rely on the construction of certain local models which themselves form one-parameter families of spaces which are factorizable in the sense of Beilinson and Drinfeld. We also briefly discuss the relationship of our results for $G=\operatorname{SL}_{2}$ with the miraculous duality of Drinfeld and Gaitsgory in the geometric Langlands program, as well as two applications of our results to the classical theory on the level of functions: to Drinfeld’s and Wang’s strange invariant bilinear form on the space of automorphic forms, and to the categorification of the Bernstein asymptotics map studied by Bezrukavnikov and Kazhdan as well as by Sakellaridis and Venkatesh.
</p>projecteuclid.org/euclid.dmj/1520906430_20180330040050Fri, 30 Mar 2018 04:00 EDTA geometric characterization of toric varietieshttps://projecteuclid.org/euclid.dmj/1520046166<strong>Morgan V. Brown</strong>, <strong>James McKernan</strong>, <strong>Roberto Svaldi</strong>, <strong>Hong R. Zong</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 923--968.</p><p><strong>Abstract:</strong><br/>
We prove a conjecture of Shokurov which characterizes toric varieties using log pairs.
</p>projecteuclid.org/euclid.dmj/1520046166_20180330040050Fri, 30 Mar 2018 04:00 EDTRegularization under diffusion and anticoncentration of the information contenthttps://projecteuclid.org/euclid.dmj/1515747886<strong>Ronen Eldan</strong>, <strong>James R. Lee</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 969--993.</p><p><strong>Abstract:</strong><br/>
Under the Ornstein–Uhlenbeck semigroup $\{U_{t}\}$ , any nonnegative measurable $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ exhibits a uniform tail bound better than that implied by Markov’s inequality and conservation of mass. For every $\alpha\geq e^{3}$ , and $t\gt 0$ ,
\[\gamma_{n}(\{x\in\mathbb{R}^{n}:U_{t}f(x)\gt \alpha\int f\,d\gamma_{n}\})\leq C(t)\frac{1}{\alpha}\sqrt{\frac{\log\log\alpha}{\log\alpha}},\] where $\gamma_{n}$ is the $n$ -dimensional Gaussian measure and $C(t)$ is a constant depending only on $t$ . This confirms positively the Gaussian limiting case of Talagrand’s convolution conjecture (1989). This is shown to follow from a more general phenomenon. Suppose that $f:\mathbb{R}^{n}\to\mathbb{R}_{+}$ is semi-log-convex in the sense that for some $\beta\gt 0$ , for all $x\in\mathbb{R}^{n}$ , the eigenvalues of $\nabla^{2}\log f(x)$ are at least $-\beta$ . Then $f$ satisfies a tail bound asymptotically better than that implied by Markov’s inequality.
</p>projecteuclid.org/euclid.dmj/1515747886_20180330040050Fri, 30 Mar 2018 04:00 EDTOdd degree number fields with odd class numberhttps://projecteuclid.org/euclid.dmj/1520046167<strong>Wei Ho</strong>, <strong>Arul Shankar</strong>, <strong>Ila Varma</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 5, 995--1047.</p><p><strong>Abstract:</strong><br/>
For every odd integer $n\geq3$ , we prove that there exist infinitely many number fields of degree $n$ and associated Galois group $S_{n}$ whose class number is odd. To do so, we study the class groups of families of number fields of degree $n$ whose rings of integers arise as the coordinate rings of the subschemes of ${\mathbb{P}}^{1}$ cut out by integral binary $n$ -ic forms. By obtaining upper bounds on the mean number of $2$ -torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to $1$ as $n$ tends to $\infty$ ) of such fields have trivial $2$ -torsion subgroups in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen and Lenstra, Cohen and Martinet, Malle, and Dummit and Voight. Additionally, for any order ${\mathcal{O}}_{f}$ of degree $n$ arising from an integral binary $n$ -ic form $f$ , we compare the sizes of $\operatorname{Cl}_{2}({\mathcal{O}}_{f})$ , the $2$ -torsion subgroup of ideal classes in ${\mathcal{O}}_{f}$ , and of ${\mathcal{I}}_{2}({\mathcal{O}}_{f})$ , the $2$ -torsion subgroup of ideals in ${\mathcal{O}}_{f}$ . For the family of orders arising from integral binary $n$ -ic forms and contained in fields with fixed signature $(r_{1},r_{2})$ , we prove that the mean value of the difference $\vert \operatorname{Cl}_{2}({\mathcal{O}}_{f})\vert -{2^{1-r_{1}-r_{2}}}\vert {\mathcal{I}}_{2}({\mathcal{O}}_{f})\vert $ is equal to $1$ , generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of $\vert \operatorname{Cl}_{2}({\mathcal{O}}_{f})\vert -{2^{1-r_{1}-r_{2}}}\vert {\mathcal{I}}_{2}({\mathcal{O}}_{f})\vert $ remains $1$ for certain families obtained by imposing local splitting and maximality conditions.
</p>projecteuclid.org/euclid.dmj/1520046167_20180330040050Fri, 30 Mar 2018 04:00 EDTGroup cubizationhttps://projecteuclid.org/euclid.dmj/1520499610<strong>Damian Osajda</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 6, 1049--1055.</p><p><strong>Abstract:</strong><br/>
We present a procedure of group cubization: it results in a group whose features resemble some of those of a given group and which acts without fixed points on a $\operatorname{CAT}(0)$ cubical complex. As a main application, we establish the lack of Kazhdan’s property (T) for Burnside groups.
</p>projecteuclid.org/euclid.dmj/1520499610_20180413040030Fri, 13 Apr 2018 04:00 EDTGalois and Cartan cohomology of real groupshttps://projecteuclid.org/euclid.dmj/1520928011<strong>Jeffrey Adams</strong>, <strong>Olivier Taïbi</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 6, 1057--1097.</p><p><strong>Abstract:</strong><br/>
Suppose that $G$ is a complex, reductive algebraic group. A real form of $G$ is an antiholomorphic involutive automorphism $\sigma$ , so $G(\mathbb{R})=G(\mathbb{C})^{\sigma}$ is a real Lie group. Write $H^{1}(\sigma,G)$ for the Galois cohomology (pointed) set $H^{1}(\operatorname{Gal}(\mathbb{C}/\mathbb{R}),G)$ . A Cartan involution for $\sigma$ is an involutive holomorphic automorphism $\theta$ of $G$ , commuting with $\sigma$ , so that $\theta\sigma$ is a compact real form of $G$ . Let $H^{1}(\theta,G)$ be the set $H^{1}(\mathbb{Z}_{2},G)$ , where the action of the nontrivial element of $\mathbb{Z}_{2}$ is by $\theta$ . By analogy with the Galois group, we refer to $H^{1}(\theta,G)$ as the Cartan cohomology of $G$ with respect to $\theta$ . Cartan’s classification of real forms of a connected group, in terms of their maximal compact subgroups, amounts to an isomorphism $H^{1}(\sigma,G_{\mathrm{ad}})\simeq H^{1}(\theta,G_{\mathrm{ad}})$ , where $G_{\mathrm{ad}}$ is the adjoint group. Our main result is a generalization of this: there is a canonical isomorphism $H^{1}(\sigma,G)\simeq H^{1}(\theta,G)$ .
We apply this result to give simple proofs of some well-known structural results: the Kostant–Sekiguchi correspondence of nilpotent orbits; Matsuki duality of orbits on the flag variety; conjugacy classes of Cartan subgroups; and structure of the Weyl group. We also use it to compute $H^{1}(\sigma,G)$ for all simple, simply connected groups and to give a cohomological interpretation of strong real forms. For the applications it is important that we do not assume that $G$ is connected.
</p>projecteuclid.org/euclid.dmj/1520928011_20180413040030Fri, 13 Apr 2018 04:00 EDTAlmost sure multifractal spectrum of Schramm–Loewner evolutionhttps://projecteuclid.org/euclid.dmj/1522224103<strong>Ewain Gwynne</strong>, <strong>Jason Miller</strong>, <strong>Xin Sun</strong>. <p><strong>Source: </strong>Duke Mathematical Journal, Volume 167, Number 6, 1099--1237.</p><p><strong>Abstract:</strong><br/>
Suppose that $\eta$ is a Schramm–Loewner evolution ( $\operatorname{SLE}_{\kappa}$ ) in a smoothly bounded simply connected domain $D\subset{\mathbf{C}}$ and that $\phi$ is a conformal map from $\mathbf{D}$ to a connected component of $D\setminus\eta([0,t])$ for some $t\gt 0$ . The multifractal spectrum of $\eta$ is the function $(-1,1)\to[0,\infty)$ which, for each $s\in(-1,1)$ , gives the Hausdorff dimension of the set of points $x\in\partial\mathbf{D}$ such that $|\phi'((1-\epsilon)x)|=\epsilon^{-s+o(1)}$ as $\epsilon\to0$ . We rigorously compute the almost sure multifractal spectrum of $\operatorname{SLE}$ , confirming a prediction due to Duplantier. As corollaries, we confirm a conjecture made by Beliaev and Smirnov for the almost sure bulk integral means spectrum of $\operatorname{SLE}$ , we obtain the optimal Hölder exponent for a conformal map which uniformizes the complement of an $\operatorname{SLE}$ curve, and we obtain a new derivation of the almost sure Hausdorff dimension of the $\operatorname{SLE}$ curve for $\kappa\leq4$ . Our results also hold for the $\operatorname{SLE}_{\kappa}(\underline{\rho})$ processes with general vectors of weight $\underline{\rho}$ .
</p>projecteuclid.org/euclid.dmj/1522224103_20180413040030Fri, 13 Apr 2018 04:00 EDT