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  • QUANTUM COHOMOLOGY AND CLOSED-STRING MIRROR SYMMETRY FOR TORIC VARIETIES
    Q. J. Math. (IF 0.636) Pub Date : 2020-01-11
    Smith J.

    We give a short new computation of the quantum cohomology of an arbitrary smooth (semiprojective) toric variety $X$, by showing directly that the Kodaira–Spencer map of Fukaya–Oh–Ohta–Ono defines an isomorphism onto a suitable Jacobian ring. In contrast to previous results of this kind, $X$ need not be compact. The proof is based on the purely algebraic fact that a class of generalized Jacobian rings associated to $X$ are free as modules over the Novikov ring. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure.

    更新日期:2020-01-13
  • UN SCINDAGE DU MORPHISME DE FROBENIUS SUR L’ALGÈBRE DES DISTRIBUTIONS D’UN GROUPE RÉDUCTIF
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-20
    Gros M, Masaharu K.

    Pour un groupe algébrique semi-simple simplement connexe sur un corps algébriquement clos de caractéristique positive, nous avons précédemment construit un scindage de l’endomorphisme de Frobenius sur son algèbre des distributions. Nous généralisons la construction au cas de des groupes réductifs connexes et en dégageons les corollaires correspondants.For a simply connected semisimple algebraic group over an algebraically closed field of positive characteristic we have already constructed a splitting of the Frobenius endomorphism on its algebra of distributions. We generalize the construction to the case of general connected reductive groups and derive the corresponding corollaries.

    更新日期:2020-01-04
  • SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION
    Q. J. Math. (IF 0.636) Pub Date : 2019-12-27
    Guillou B, May J, Merling M, et al.

    We give an operadic definition of a genuine symmetric monoidal $G$-category, and we prove that its classifying space is a genuine $E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $G$-categories to genuine permutative $G$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $G$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $G$-categories as input to an equivariant infinite loop space machine that gives genuine $\Omega $-$G$-spectra as output.

    更新日期:2020-01-04
  • Varieties of involution monoids with extreme properties
    Q. J. Math. (IF 0.636) Pub Date : 2019-06-22
    Lee E.

    A variety that contains continuum many subvarieties is said to be huge. A sufficient condition is established under which an involution monoid generates a variety that is huge by virtue of its lattice of subvarieties order-embedding the power set lattice of the positive integers. Based on this result, several examples of finite involution monoids with extreme varietal properties are exhibited. These examples—all first of their kinds—include the following: finite involution monoids that generate huge varieties but whose reduct monoids generate Cross varieties; two finite involution monoids sharing a common reduct monoid such that one generates a huge, non-finitely based variety while the other generates a Cross variety; and two finite involution monoids that generate Cross varieties, the join of which is huge.

    更新日期:2020-01-04
  • Differential operators on polar harmonic Maass forms and elliptic duality
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-03
    Bringmann K, Jenkins P, Kane B.

    In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

    更新日期:2020-01-04
  • The Ricci pinching functional on solvmanifolds
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    Lauret J, Will C.

    We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of $F$ are clarified.

    更新日期:2020-01-04
  • Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    Fino A, Rollenske S, Ruppenthal J.

    It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalizes previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

    更新日期:2020-01-04
  • Fillings and fittings of unit cotangent bundles of odd-dimensional spheres
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    Kwon M, Zehmisch K.

    We introduce the concept of fittings to symplectic fillings of the unit cotangent bundle of odd-dimensional spheres. Assuming symplectic asphericity we show that all fittings are diffeomorphic to the respective unit co-disc bundle.

    更新日期:2020-01-04
  • Hopf invariants for sectional category with applications to topological robotics
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-15
    González J, Grant M, Vandembroucq L.

    We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular, we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our results are applied to the study of Farber’s topological complexity for two-cell complexes, as well as to the construction of a counterexample to the analogue for topological complexity of Ganea’s conjecture on Lusternik–Schnirelmann category.

    更新日期:2020-01-04
  • On Local definability of holomorphic functions
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-16
    Jones G, Kirby J, Le Gal O, et al.

    Given a collection $\mathcal {A}$ of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from $\mathcal {A}$. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from $\mathcal {A}$ in the neighbourhood of a generic point. We prove that this description is no longer complete in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions that suggest that at least three new operations need to be added to Wilkie’s description in order to capture local definability in its entirety. The constructions illustrate the interaction between resolution of singularities and definability in the o-minimal setting.

    更新日期:2020-01-04
  • Calculations with graded perverse-coherent sheaves
    Q. J. Math. (IF 0.636) Pub Date : 2019-07-24
    Achar P, Hardesty W.

    In this paper, we carry out several computations involving graded (or ${{\mathbb {G}}_{\textrm {m}}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the ${{\mathbb {G}}_{\textrm {m}}}$-action on certain normalized (or ‘canonical’) simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse-coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the cohomology of tilting modules for the corresponding quantum group, as well as a proof that $\textsf {PCoh}^{{{\mathbb {G}}_{\textrm {m}}}}({\mathcal {N}})$ never admits a positive grading when the characteristic of the field is greater than 3.

    更新日期:2020-01-04
  • On the 2-head of the colored Jones polynomial for pretzel knots
    Q. J. Math. (IF 0.636) Pub Date : 2019-08-27
    Beirne P.

    In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.

    更新日期:2020-01-04
  • COUNTING INTEGERS WITH A SMOOTH TOTIENT
    Q. J. Math. (IF 0.636) Pub Date : 2019-09-10
    Banks W, Friedlander J, Pomerance C, et al.

    In an earlier paper we considered the distribution of integers $n$ for which Euler’s totient function at $n$ has all small prime factors. Here we obtain an improvement that is likely to be best possible.

    更新日期:2020-01-04
  • SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION
    Q. J. Math. (IF 0.636) Pub Date : 2019-09-26
    Heap W, Radziwiłł M, Soundararajan K.

    We establish sharp upper bounds for the $2k$th moment of the Riemann zeta function on the critical line, for all real $0 \leqslant k \leqslant 2$. This improves on earlier work of Ramachandra, Heath-Brown and Bettin–Chandee–Radziwiłł.

    更新日期:2020-01-04
  • Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-04
    Basu S, Lerario A, Natarajan A.

    Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in ${\mathbb{R}}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$ such that each manifold $Z_d$ is diffeomorphic to a component of the zero set on $\Gamma$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $\Gamma$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $\Gamma$ is bounded by a polynomial in $\deg (p)$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$ containing a subset $D$ homeomorphic to a disk, and a family of polynomials $\{p_m\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that $(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$ i.e. the zero set of $p_m$ in $D$ is isotopic to $Z_{d_m}$ in ${\mathbb{R}}^{n-1}$. This says that, up to extracting subsequences, the intersection of $\Gamma$ with a hypersurface of degree $d$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $0 \leq k \leq n-2$ and every sequence of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$, a subsequence $\{a_{d_m}\}_{m\in \mathbb{N}}$ and homogeneous polynomials $\{p_{m}\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $b_k$ denotes the $k$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $\Gamma$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $d$, of the set $\Sigma _{d_m,a, \Gamma }$ of polynomials verifying (0.10.1) is positive, but there exists a constant $c_\Gamma$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $a$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $\Gamma$, for most polynomials a Bézout-type bound holds for the intersection $\Gamma \cap Z(p)$: for every $0\leq k\leq n-2$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$

    更新日期:2020-01-04
  • HYPERSYMPLECTIC MANIFOLDS AND ASSOCIATED GEOMETRIES
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-21
    Thakre V.

    We investigate an obstruction for hypersymplectic manifolds equipped with a free, isometric action of SU$(1,1)$. When the obstruction vanishes, we show that the manifold is a metric cone over a split 3-Sasakian manifold. Furthermore, if the action of SU$(1,1)$ is also proper, then the hypersymplectic manifold fibres over a para-quaternionic Kähler manifold. We conclude the article with some examples for which the obstruction vanishes. In particular, we show that the moduli space to Nahm–Schmid equations admits a fibration over a para-quaternionic Kähler manifold.

    更新日期:2020-01-04
  • BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-21
    Benson D, Kessar R, Linckelmann M.

    Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For ${\mathcal{O}}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantized version of the group algebra of the semidirect product $P\rtimes L$.

    更新日期:2020-01-04
  • MOMENTS OF SPINOR L-FUNCTIONS AND SYMPLECTIC KLOOSTERMAN SUMS
    Q. J. Math. (IF 0.636) Pub Date : 2019-10-21
    Waibel F.

    We compute the second moment of spinor $L$-functions at central points of Siegel modular forms on congruence subgroups of large prime level $N$ and give applications to non-vanishing.

    更新日期:2020-01-04
  • DIOPHANTINE APPROXIMATION WITH GAUSSIAN PRIMES
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-18
    Harman G.

    In this paper we prove that the exact analogue of the author’s work with real irrationals and rational primes (G. Harman, On the distribution of $\alpha p$ modulo one II, Proc. London Math. Soc. (3) 72, 1996, 241–260) holds for approximating $\alpha \in \mathbb{C}\setminus \mathbb{Q}[i]$ with Gaussian primes. To be precise, we show that for such $\alpha $ and arbitrary complex $\beta $ there are infinitely many solutions in Gaussian primes $p$ to $$\begin{equation*} ||\alpha p + \beta|| <| p|^{-7/22}, \end{equation*}$$ where $||\cdot ||$ denotes distance to a nearest member of $\mathbb{Z}[i]$. We shall, in fact, prove a slightly more general result with the Gaussian primes in sectors, and along the way improve a recent result due to Baier (S. Baier, Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints, Eur. J. Math. 3, 2017, 614–649).

    更新日期:2020-01-04
  • THE REAL GRADED BRAUER GROUP
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-18
    Karoubi M, Weibel C.

    We introduce a version of the Brauer–Wall group for Real vector bundles of algebras (in the sense of Atiyah) and compare it to the topological analogue of the Witt group. For varieties over the reals, these invariants capture the topological parts of the Brauer–Wall and Witt groups.

    更新日期:2020-01-04
  • A Unified Factorization Theorem for Lipschitz Summing Operators
    Q. J. Math. (IF 0.636) Pub Date : 2019-11-20
    Botelho G, Maia M, Pellegrino D, et al.

    We prove a general factorization theorem for Lipschitz summing operators in the context of metric spaces that recovers several linear and nonlinear factorization theorems that have been proved recently in different environments. New applications are also given.

    更新日期:2020-01-04
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