Mostrando recursos 1 - 20 de 38

  1. Hasse principle for three classes of varieties over global function fields

    Tian, Zhiyu
    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.

  2. Enumeration of real curves in $\mathbb{C}\mathbb{P}^{2n-1}$ and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

    Georgieva, Penka; Zinger, Aleksey
    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...

  3. Automatic sequences fulfill the Sarnak conjecture

    Müllner, Clemens
    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 $0$ .

  4. K-semistability is equivariant volume minimization

    Li, Chi
    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...

  5. The Coolidge–Nagata conjecture

    Koras, Mariusz; Palka, Karol
    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.

  6. A metric interpretation of reflexivity for Banach spaces

    Motakis, P.; Schlumprecht, T.
    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...

  7. A product for permutation groups and topological groups

    Smith, Simon M.
    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...

  8. A product for permutation groups and topological groups

    Smith, Simon M.
    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...

  9. Rank, combinatorial cost, and homology torsion growth in higher rank lattices

    Abert, Miklos; Gelander, Tsachik; Nikolov, Nikolay
    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...

  10. Rank, combinatorial cost, and homology torsion growth in higher rank lattices

    Abert, Miklos; Gelander, Tsachik; Nikolov, Nikolay
    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...

  11. Moduli of curves as moduli of $A_{\infty}$ -structures

    Polishchuk, Alexander
    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...

  12. Moduli of curves as moduli of $A_{\infty}$ -structures

    Polishchuk, Alexander
    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...

  13. On the geometry of thin exceptional sets in Manin’s conjecture

    Lehmann, Brian; Tanimoto, Sho
    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.

  14. On the geometry of thin exceptional sets in Manin’s conjecture

    Lehmann, Brian; Tanimoto, Sho
    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.

  15. Errata 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), 2447–2519


  16. Errata 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), 2447–2519


  17. A tropical approach to a generalized Hodge conjecture for positive currents

    Babaee, Farhad; Huh, June
    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...

  18. A tropical approach to a generalized Hodge conjecture for positive currents

    Babaee, Farhad; Huh, June
    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...

  19. Sobolev trace inequalities of order four

    Ache, Antonio G.; Chang, Sun-Yung Alice
    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.

  20. Sobolev trace inequalities of order four

    Ache, Antonio G.; Chang, Sun-Yung Alice
    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.

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