Mostrando recursos 1 - 20 de 43

  1. On the $\operatorname{Pin}(2)$ -Equivariant Monopole Floer Homology of Plumbed 3-Manifolds

    Dai, Irving
    We compute the $\operatorname{Pin}(2)$ -equivariant monopole Floer homology for the class of plumbed 3-manifolds considered by Ozsváth and Szabó [18]. We show that for these manifolds, the $\operatorname{Pin}(2)$ -equivariant monopole Floer homology can be calculated in terms of the Heegaard Floer/monopole Floer lattice complex defined by Némethi [15]. Moreover, we prove that in such cases the ranks of the usual monopole Floer homology groups suffice to determine both the Manolescu correction terms and the $\operatorname{Pin}(2)$ -homology as an Abelian group. As an application, we show that $\beta(-Y,s)=\bar{\mu}(Y,s)$ for all plumbed 3-manifolds with at most one “bad” vertex, proving (an analogue...

  2. Legendrian Lens Space Surgeries

    Geiges, Hansjörg; Onaran, Sinem
    We show that every tight contact structure on any of the lens spaces $L(ns^{2}-s+1,s^{2})$ with $n\geq 2$ and $s\geq 1$ can be obtained by a single Legendrian surgery along a suitable Legendrian realisation of the negative torus knot $T(s,-(sn-1))$ in the tight or an overtwisted contact structure on the $3$ -sphere.

  3. Diophantine Approximation Constants for Varieties over Function Fields

    Grieve, Nathan
    By analogy with the program of McKinnon and Roth [10], we define and study approximation constants for points of a projective variety $X$ defined over $\mathbf {K}$ , the function field of an irreducible and nonsingular in codimension $1$ projective variety defined over an algebraically closed field of characteristic zero. In this setting, we use Wang’s theorem, which is an effective version of Schmidt’s subspace theorem, to give a sufficient condition for such approximation constants to be computed on a proper $\mathbf{K}$ -subvariety of $X$ . We also indicate how our approximation constants are related to volume functions and Seshadri...

  4. Givental-Type Reconstruction at a Nonsemisimple Point

    Basalaev, Alexey; Priddis, Nathan
    We consider the orbifold curve that is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov–Witten theory of the orbifold curve via the product of the Gromov–Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental’s action giving the CY/LG correspondence between the Gromov–Witten theory of the orbifold curve $\mathcal{E}/\mathbb{Z}_{4}$ and FJRW theory of the pair defined by the polynomial $x^{4}+y^{4}+z^{2}$ and the maximal group of diagonal symmetries....

  5. Characterization of Aleksandrov Spaces of Curvature Bounded Above by Means of the Metric Cauchy–Schwarz Inequality

    Berg, I. D.; Nikolaev, Igor G.
    We consider the previously introduced notion of the $K$ -quadrilateral cosine, which is the cosine under parallel transport in model $K$ -space, and which is denoted by $\operatorname{cosq}_{K}$ . In $K$ -space, $\vert \operatorname{cosq}_{K}\vert \leq 1$ is equivalent to the Cauchy–Schwarz inequality for tangent vectors under parallel transport. Our principal result states that a geodesically connected metric space (of diameter not greater than $\pi /(2\sqrt{K})$ if $K\gt 0$ ) is an $\Re_{K}$ domain (otherwise known as a $\operatorname{CAT}(K)$ space) if and only if always $\operatorname{cosq}_{K}\leq 1$ or always $\operatorname{cosq}_{K}\geq -1$ . (We prove that in such spaces always $\operatorname{cosq}_{K}\leq 1$...

  6. Multiple Lerch Zeta Functions and an Idea of Ramanujan

    Gun, Sanoli; Saha, Biswajyoti
    In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalizing an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.

  7. Sparse Domination Theorem for Multilinear Singular Integral Operators with $L^{r}$ -Hörmander Condition

    Li, Kangwei
    In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$ -Hörmander condition, then $T$ can be dominated by multilinear sparse operators.

  8. Splitting Criteria for Vector Bundles Induced by Restrictions to Divisors

    Halic, Mihai
    In this article we obtain criteria for the splitting and triviality of vector bundles by restricting them to partially ample divisors. This allows us to study the problem of splitting on the total space of fibre bundles. The statements are illustrated with examples. ¶ For products of minuscule homogeneous varieties, we show that the splitting of vector bundles can be tested by restricting them to subproducts of Schubert $2$ -planes. By using known cohomological criteria for multiprojective spaces, we deduce necessary and sufficient conditions for the splitting of vector bundles on products of minuscule varieties. ¶ The triviality criteria are particularly suited...

  9. Nielsen Realization by Gluing: Limit Groups and Free Products

    Hensel, Sebastian; Kielak, Dawid
    We generalize the Karrass–Pietrowski–Solitar and the Nielsen realization theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel and Mosher and on the outer space of a free product of Guirardel and Levitt, and also a relative version of the Nielsen realization theorem, which, in the case of free groups, answers a question of Karen Vogtmann. We also prove Nielsen realization for limit groups and, as a byproduct, obtain a new proof that limit...

  10. A Note on Brill–Noether Existence for Graphs of Low Genus

    Atanasov, Stanislav; Ranganathan, Dhruv
    In an influential 2008 paper, Baker proposed a number of conjectures relating the Brill–Noether theory of algebraic curves with a divisor theory on finite graphs. In this note, we examine Baker’s Brill–Noether existence conjecture for special divisors. For $g\leq5$ and $\rho(g,r,d)$ nonnegative, every graph of genus $g$ is shown to admit a divisor of rank $r$ and degree at most $d$ . As further evidence, the conjecture is shown to hold in rank $1$ for a number families of highly connected combinatorial types of graphs. In the relevant genera, our arguments give the first combinatorial proof of the Brill–Noether existence...

  11. Almost Gorenstein Rees Algebras of $p_{g}$ -Ideals, Good Ideals, and Powers of the Maximal Ideals

    Goto, Shiro; Matsuoka, Naoyuki; Taniguchi, Naoki; Yoshida, Ken-ichi
    Let $(A,\mathfrak{m})$ be a Cohen–Macaulay local ring, and let $I$ be an ideal of $A$ . We prove that the Rees algebra $\mathcal{R}(I)$ is an almost Gorenstein ring in the following cases: ¶ (1) $(A,\mathfrak{m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K\congA/\mathfrak{m}$ , and $I$ is a $p_{g}$ -ideal; ¶ (2) $(A,\mathfrak{m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity, and $I=\mathfrak{m}^{\ell}$ for all $\ell\ge1$ ; ¶ (3) $(A,\mathfrak{m})$ is a regular local ring of dimension $d\ge2$ , and $I=\mathfrak{m}^{d-1}$ . Conversely, if $\mathcal{R}(\mathfrak{m}^{\ell})$ is an almost Gorenstein graded ring for some $\ell\ge2$ and $d\ge3$...

  12. Symmetric Automorphisms of Free Groups, BNSR-Invariants, and Finiteness Properties

    Zaremsky, Matthew C. B.
    The BNSR-invariants of a group $G$ are a sequence $\Sigma^{1}(G)\supseteq \Sigma^{2}(G)\supseteq \cdots $ of geometric invariants that reveal important information about finiteness properties of certain subgroups of $G$ . We consider the symmetric automorphism group $\operatorname{\Sigma Aut}_{n}$ and pure symmetric automorphism group $\operatorname{P\!\operatorname{\Sigma Aut}}_{n}$ of the free group $F_{n}$ and inspect their BNSR-invariants. We prove that for $n\ge 2$ , all the “positive” and “negative” character classes of $\operatorname{P\!\operatorname{\Sigma Aut}}_{n}$ lie in $\Sigma^{n-2}(\operatorname{P\!\operatorname{\Sigma Aut}}_{n})\setminus \Sigma^{n-1}(\operatorname{P\!\operatorname{\Sigma Aut}}_{n})$ . We use this to prove that for $n\ge 2$ , $\Sigma^{n-2}(\operatorname{\Sigma Aut}_{n})$ equals the full character sphere $S^{0}$ of $\operatorname{\Sigma Aut}_{n}$ but $\Sigma^{n-1}(\operatorname{\Sigma...

  13. Bernstein–Sato Polynomials on Normal Toric Varieties

    Hsiao, Jen-Chieh; Matusevich, Laura Felicia
    We generalize the Bernstein–Sato polynomials of Budur, Mustaţǎ, and Saito to ideals in normal semigroup rings. In the case of monomial ideals, we also relate the roots of the Bernstein–Sato polynomial to the jumping coefficients of the corresponding multiplier ideals. To prove the latter result, we obtain a new combinatorial description for the multiplier ideals of a monomial ideal in a normal semigroup ring.

  14. Nagata’s Compactification Theorem for Normal Toric Varieties over a Valuation Ring of Rank One

    Soto, Alejandro
    Using invariant Zariski–Riemann spaces, we prove that every normal toric variety over a valuation ring of rank one can be embedded as an open dense subset into a proper toric variety equivariantly. This extends a well-known theorem of Sumihiro for toric varieties over a field to this more general setting.

  15. Smooth Rational Curves on Singular Rational Surfaces

    Zhuang, Ziquan
    We classify all complex surfaces with quotient singularities that do not contain any smooth rational curves under the assumption that the canonical divisor of the surface is not pseudo-effective. As a corollary, we show that if $X$ is a log del Pezzo surface such that, for every closed point $p\in X$ , there is a smooth curve (locally analytically) passing through $p$ , then $X$ contains at least one smooth rational curve.

  16. Correction Terms and the Nonorientable Slice Genus

    Golla, Marco; Marengon, Marco
    By considering negative surgeries on a knot $K$ in $S^{3}$ , we derive a lower bound on the nonorientable slice genus $\gamma_{4}(K)$ in terms of the signature $\sigma(K)$ and the concordance invariants $V_{i}(\overline {K})$ ; this bound strengthens a previous bound given by Batson and coincides with Ozsváth–Stipsicz–Szabó’s bound in terms of their $\upsilon$ invariant for L-space knots and quasi-alternating knots. A curious feature of our bound is superadditivity, implying, for instance, that the bound on the stable nonorientable slice genus is sometimes better than that on $\gamma_{4}(K)$ .

  17. Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

    Tshishiku, Bena
    This paper is about the cohomology of certain finite-index subgroups of mapping class groups and its relation to the cohomology of arithmetic groups. For $G=\mathbb{Z}/m\mathbb{Z}$ and for a regular $G$ -cover $S\rightarrow\bar{S}$ (possibly branched), a finite-index subgroup $\Gamma\lt \operatorname{Mod}(\bar{S})$ acts on $H_{1}(S;\mathbb{Z})$ commuting with the deck group action, thus inducing a homomorphism $\Gamma\rightarrow\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z})$ to an arithmetic group. The induced map $H^{*}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})\rightarrow H^{*}(\Gamma;\mathbb{Q})$ can be understood using index theory. To this end, we describe a families version of the $G$ -index theorem for the signature operator and apply this to (i) compute $H^{2}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})\rightarrow H^{2}(\Gamma;\mathbb{Q})$ , (ii) rederive Hirzebruch’s formula for signature...

  18. The Chow Ring of the Stack of Smooth Plane Cubics

    Fulghesu, Damiano; Vistoli, Angelo
    In this paper, we give an explicit presentation of the integral Chow ring of a stack of smooth plane cubics. We also determine some relations in the general case of hypersurfaces of any dimension and degree.

  19. Index


  20. Lifting Homeomorphisms and Cyclic Branched Covers of Spheres

    Ghaswala, Tyrone; Winarski, Rebecca R.
    We characterize the cyclic branched covers of the 2-sphere where every homeomorphism of the sphere lifts to a homeomorphism of the covering surface. This answers the question that appeared in an early version of the erratum of Birman and Hilden [2].

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