The Michigan Mathematical Journal Articles (Project Euclid)
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The latest articles from The Michigan 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 EDTThu, 31 Mar 2011 11:46 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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http://projecteuclid.org/euclid.mmj/1272376024
<p><strong>Source: </strong>Michigan Math. J., Volume 59, Number 1, i--ii.</p>projecteuclid.org/euclid.mmj/1272376024_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTCharacterizations of Some Properties on Weighted Modulation and Wiener Amalgam Spaceshttps://projecteuclid.org/euclid.mmj/1552442712<strong>Weichao Guo</strong>, <strong>Jiecheng Chen</strong>, <strong>Dashan Fan</strong>, <strong>Guoping Zhao</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 32 pages.</p><p><strong>Abstract:</strong><br/>
In this paper, we characterize some properties on weighted modulation and Wiener amalgam spaces by the corresponding properties on weighted Lebesgue spaces. As applications, we obtain sharp conditions for product inequalities, convolution inequalities, and embedding on weighted modulation and Wiener amalgam spaces. By a unified approach different from others we give a complete answer to the question of finding sharp conditions of certain relations on weighted modulation and Wiener amalgam spaces.
</p>projecteuclid.org/euclid.mmj/1552442712_20190312220535Tue, 12 Mar 2019 22:05 EDTRandom Manifolds Have No Totally Geodesic Submanifoldshttps://projecteuclid.org/euclid.mmj/1555034652<strong>Thomas Murphy</strong>, <strong>Frederick Wilhelm</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 2, 323--335.</p><p><strong>Abstract:</strong><br/>
For $n\geq 4$ , we show that generic closed Riemannian $n$ -manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. Although the result is widely believed to be true, we are not aware of any proof in the literature.
</p>projecteuclid.org/euclid.mmj/1555034652_20190611220046Tue, 11 Jun 2019 22:00 EDTOn Multiplicative Dependence of Values of Rational Functions and a Generalization of the Northcott Theoremhttps://projecteuclid.org/euclid.mmj/1556589745<strong>Alina Ostafe</strong>, <strong>Min Sha</strong>, <strong>Igor E. Shparlinski</strong>, <strong>Umberto Zannier</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 2, 385--407.</p><p><strong>Abstract:</strong><br/>
In this paper, we study multiplicative dependence of values of polynomials or rational functions over a number field. As an application, we obtain new results on multiplicative dependence in the orbits of a univariate polynomial dynamical system. We also obtain a generalization of the Northcott theorem replacing the finiteness of preperiodic points from a given number field by the finiteness of algebraic integers having two multiplicatively dependent elements in their orbits.
</p>projecteuclid.org/euclid.mmj/1556589745_20190611220046Tue, 11 Jun 2019 22:00 EDTInfinitely Generated Symbolic Rees Rings of Space Monomial Curves Having Negative Curveshttps://projecteuclid.org/euclid.mmj/1557475399<strong>Kazuhiko Kurano</strong>, <strong>Koji Nishida</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 2, 409--445.</p><p><strong>Abstract:</strong><br/>
In this paper, we study finite generation of symbolic Rees rings of the defining ideal ${\mathfrak{p}}$ of the space monomial curve $(t^{a},t^{b},t^{c})$ for pairwise coprime integers $a$ , $b$ , $c$ . Suppose that the base field is of characteristic $0$ , and the ideal ${\mathfrak{p}}$ is minimally generated by three polynomials. In Theorem 1.1, under the assumption that the homogeneous element $\xi $ of the minimal degree in ${\mathfrak{p}}$ is a negative curve, we determine the minimal degree of an element $\eta $ such that the pair $\{\xi ,\eta \}$ satisfies Huneke’s criterion in the case where the symbolic Rees ring is Noetherian. By this result we can decide whether the symbolic Rees ring $\mathcal{R}_{s}({\mathfrak{p}})$ is Notherian using computers. We give a necessary and sufficient condition for finite generation of the symbolic Rees ring of ${\mathfrak{p}}$ in Proposition 4.10 under some assumptions. We give an example of an infinitely generated symbolic Rees ring of ${\mathfrak{p}}$ in which the homogeneous element of the minimal degree in ${\mathfrak{p}}^{(2)}$ is a negative curve in Example 5.7. We give a simple proof to (generalized) Huneke’s criterion.
</p>projecteuclid.org/euclid.mmj/1557475399_20190611220046Tue, 11 Jun 2019 22:00 EDTOn Separable Higher Gauss Mapshttps://projecteuclid.org/euclid.mmj/1555574416<strong>Katsuhisa Furukawa</strong>, <strong>Atsushi Ito</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 483--503.</p><p><strong>Abstract:</strong><br/>
We study the $m$ th Gauss map in the sense of F. L. Zak of a projective variety $X\subset\mathbb{P}^{N}$ over an algebraically closed field in any characteristic. For all integers $m$ with $n:=\dim(X)\leqslant m\lt N$ , we show that the contact locus on $X$ of a general tangent $m$ -plane is a linear variety if the $m$ th Gauss map is separable. We also show that for smooth $X$ with $n\lt N-2$ , the $(n+1)$ th Gauss map is birational if it is separable, unless $X$ is the Segre embedding $\mathbb{P}^{1}\times\mathbb{P}^{n}\subset\mathbb{P}^{2n-1}$ . This is related to Ein’s classification of varieties with small dual varieties in characteristic zero.
</p>projecteuclid.org/euclid.mmj/1555574416_20190807040041Wed, 07 Aug 2019 04:00 EDTNoncommutative Holomorphic Semicocycleshttps://projecteuclid.org/euclid.mmj/1557302432<strong>Mark Elin</strong>, <strong>Fiana Jacobzon</strong>, <strong>Guy Katriel</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 505--526.</p><p><strong>Abstract:</strong><br/>
In this paper, we study holomorphic semicocycles over semigroups in the unit disk, which take values in an arbitrary unital Banach algebra. We prove that every such semicocycle is the solution to a corresponding evolution problem. We then investigate the linearization problem: which semicocycles are cohomologous to constant semicocycles? In contrast with the case of commutative semicocycles, in the noncommutative case nonlinearizable semicocycles are shown to exist. We derive simple conditions for linearizability and show that they are sharp.
</p>projecteuclid.org/euclid.mmj/1557302432_20190807040041Wed, 07 Aug 2019 04:00 EDTMixed Weak Estimates of Sawyer Type for Commutators of Generalized Singular Integrals and Related Operatorshttps://projecteuclid.org/euclid.mmj/1559894545<strong>Fabio Berra</strong>, <strong>Marilina Carena</strong>, <strong>Gladis Pradolini</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 527--564.</p><p><strong>Abstract:</strong><br/>
We study mixed weak-type inequalities for the commutator $[b,T]$ , where $b$ is a BMO function, and $T$ is a Calderón–Zygmund operator. More precisely, we prove that, for every $t\gt 0$ ,
\[uv(\{x\in\mathbb{R}^{n}:\vert \frac{[b,T](fv)(x)}{v(x)}\vert \gt t\})\leq C\int_{\mathbb{R}^{n}}\Phi (\frac{|f(x)|}{t})u(x)v(x)\,dx,\] where $\Phi(t)=t(1+\log^{+}{t})$ , $u\in A_{1}$ , and $v\in A_{\infty}(u)$ . Our technique involves the classical Calderón–Zygmund decomposition, which allows us to give a direct proof without taking into account the associated maximal operator. We use this result to prove an analogous inequality for higher-order commutators.
For a given Young function $\phi $ we also consider singular integral operators $T$ whose kernels satisfy a $L^{\phi }$ -Hörmander property, and we find sufficient conditions on $\phi $ such that a mixed weak estimate holds for $T$ and also for its higher order commutators $T^{m}_{b}$ .
We also obtain a mixed estimation for a wide class of maximal operators associated to certain Young functions of $L\log L$ type which are in intimate relation with the commutators. This last estimate involves an arbitrary weight $u$ and a radial function $v$ which is not even locally integrable.
</p>projecteuclid.org/euclid.mmj/1559894545_20190807040041Wed, 07 Aug 2019 04:00 EDTOn Certain Complex Projective Manifolds with Hodge Numbers $h^{10}=4$ and $h^{20}=5$https://projecteuclid.org/euclid.mmj/1562032917<strong>Chad Schoen</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 565--596.</p>projecteuclid.org/euclid.mmj/1562032917_20190807040041Wed, 07 Aug 2019 04:00 EDTEstimation of Deviation for Random Covariance Matriceshttps://projecteuclid.org/euclid.mmj/1559894544<strong>Tien-Cuong Dinh</strong>, <strong>Duc-Viet Vu</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 597--620.</p><p><strong>Abstract:</strong><br/>
We give a deviation estimate for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1, and controlled fourth moments. We also give some new properties of Laguerre polynomials.
</p>projecteuclid.org/euclid.mmj/1559894544_20190807040041Wed, 07 Aug 2019 04:00 EDTFurther Evaluation of Wahl Vanishing Theorems for Surface Singularities in Characteristic $p$https://projecteuclid.org/euclid.mmj/1560391418<strong>Masayuki Hirokado</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 621--636.</p><p><strong>Abstract:</strong><br/>
Let $(\operatorname{Spec}R,\mathfrak{m})$ be a rational double point defined over an algebraically closed field $k$ of characteristic $p\geq 0$ . We evaluate further the dimensions of the local cohomology groups, which were treated by Wahl in 1975 as vanishing theorem C (resp., D) under the assumption that $p$ is a very good prime (resp., good prime) with respect to $(\operatorname{Spec}R,\mathfrak{m})$ . We use Artin’s classification of rational double points and completely determine the dimensions $\dim _{k}H_{E}^{1}(S_{X})$ and $\dim _{k}H_{E}^{1}(S_{X}\otimes\mathcal{O}_{X}(E))$ , supplementing Wahl’s theorems. In the proof, we concretely construct derivations that do not lift to the minimal resolution $X\to \operatorname{Spec}R$ and an equisingular family that injects into a versal deformation of the rational double point $(\operatorname{Spec}R,\mathfrak{m})$ .
</p>projecteuclid.org/euclid.mmj/1560391418_20190807040041Wed, 07 Aug 2019 04:00 EDTComplementary Legs and Rational Ballshttps://projecteuclid.org/euclid.mmj/1561708817<strong>Ana G. Lecuona</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 637--649.</p><p><strong>Abstract:</strong><br/>
In this note, we study the Seifert rational homology spheres with two complementary legs, that is, with a pair of invariants whose fractions add up to one. We give a complete classification of the Seifert manifolds with three exceptional fibers and two complementary legs that bound rational homology balls. The result translates into a statement on the sliceness of some Montesinos knots.
</p>projecteuclid.org/euclid.mmj/1561708817_20190807040041Wed, 07 Aug 2019 04:00 EDTThe Picard Group of the Universal Abelian Variety and the Franchetta Conjecture for Abelian Varietieshttps://projecteuclid.org/euclid.mmj/1564106669<strong>Roberto Fringuelli</strong>, <strong>Roberto Pirisi</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 3, 651--671.</p><p><strong>Abstract:</strong><br/>
We compute the Picard group of the universal Abelian variety over the moduli stack $\mathscr{A}_{g,n}$ of principally polarized Abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$ -structure. We then prove that over $\mathbb{C}$ the statement of the Franchetta conjecture holds in a suitable form for $\mathscr{A}_{g,n}$ .
</p>projecteuclid.org/euclid.mmj/1564106669_20190807040041Wed, 07 Aug 2019 04:00 EDTGeneralizations of Intersection Homology and Perverse Sheaves with Duality over the Integershttps://projecteuclid.org/euclid.mmj/1564711315<strong>Greg Friedman</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 675--726.</p><p><strong>Abstract:</strong><br/>
We provide a generalization of the Deligne sheaf construction of intersection homology theory and a corresponding generalization of Poincaré duality on pseudomanifolds such that the Goresky–MacPherson, Goresky–Siegel, and Cappell–Shaneson duality theorems all arise as particular cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky–Siegel IP spaces on which upper-middle perversity intersection homology is self-dual. We also examine torsion-sensitive t-structures and categories of perverse sheaves that contain our torsion-sensitive Deligne sheaves as intermediate extensions.
</p>projecteuclid.org/euclid.mmj/1564711315_20191111040134Mon, 11 Nov 2019 04:01 ESTTopology of Kähler Manifolds with Weakly Pseudoconvex Boundaryhttps://projecteuclid.org/euclid.mmj/1563847454<strong>Brian Weber</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 727--742.</p><p><strong>Abstract:</strong><br/>
We study Kähler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge 2$ boundary components (possibly $l=\infty $ ), then it has the first Betti number at least $l-1$ , and the Levi form of any boundary component is zero. If $K$ has $l\ge 1$ pseudoconvex boundary components and at least one nonparabolic end, then the first Betti number of $K$ is at least $l$ . In either case, any boundary component has a nonvanishing first Betti number. If $K$ has one pseudoconvex boundary component with vanishing first Betti number, then the first Betti number of $K$ is also zero. Especially significant are applications to Kähler ALE manifolds and to Kähler 4-manifolds. This significantly extends prior results in this direction (e.g., those of Kohn and Rossi) and uses substantially simpler methods.
</p>projecteuclid.org/euclid.mmj/1563847454_20191111040134Mon, 11 Nov 2019 04:01 ESTThe Alexander Method for Infinite-Type Surfaceshttps://projecteuclid.org/euclid.mmj/1561773633<strong>Jesús Hernández Hernández</strong>, <strong>Israel Morales</strong>, <strong>Ferrán Valdez</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 743--753.</p><p><strong>Abstract:</strong><br/>
We prove that for any infinite-type orientable surface $S$ , there exists a collection of essential curves $\Gamma$ in $S$ such that any homeomorphism that preserves the isotopy classes of the elements of $\Gamma$ is isotopic to the identity. The collection $\Gamma$ is countable and has an infinite complement in $\mathcal{C}(S)$ , the curve complex of $S$ . As a consequence, we obtain that the natural action of the extended mapping class group of $S$ on $\mathcal{C}(S)$ is faithful.
</p>projecteuclid.org/euclid.mmj/1561773633_20191111040134Mon, 11 Nov 2019 04:01 ESTA Note on Rational Curves on General Fano Hypersurfaceshttps://projecteuclid.org/euclid.mmj/1567735281<strong>Dennis Tseng</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 755--774.</p><p><strong>Abstract:</strong><br/>
We show that the Kontsevich space of rational curves of degree at most roughly $\frac{2-\sqrt{2}}{2}n$ on a general hypersurface $X\subset \mathbb{P}^{n}$ of degree $n-1$ is equidimensional of expected dimension and has two components: one consisting generically of smooth embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows that the Gromov–Witten invariants in these cases are enumerative.
</p>projecteuclid.org/euclid.mmj/1567735281_20191111040134Mon, 11 Nov 2019 04:01 ESTA Bennequin-Type Inequality and Combinatorial Boundshttps://projecteuclid.org/euclid.mmj/1565402474<strong>Carlo Collari</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 775--799.</p><p><strong>Abstract:</strong><br/>
In this paper, we provide a new Bennequin-type inequality for the Rasmussen–Beliakova–Wehrli invariant, featuring the numerical transverse braid invariants (the $c$ -invariants) introduced by the author. From the Bennequin type-inequality and a combinatorial bound on the value of the $c$ -invariants we deduce a new computable bound on the Rasmussen invariant.
</p>projecteuclid.org/euclid.mmj/1565402474_20191111040134Mon, 11 Nov 2019 04:01 ESTHyperplane Arrangements and Tensor Product Invariantshttps://projecteuclid.org/euclid.mmj/1565251217<strong>P. Belkale</strong>, <strong>P. Brosnan</strong>, <strong>S. Mukhopadhyay</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 801--829.</p><p><strong>Abstract:</strong><br/>
In the first part of this paper, we consider, in the context of an arbitrary weighted hyperplane arrangement, the map from compactly supported cohomology to the usual cohomology of a local system. We obtain a formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map.
In the second part, we apply these results to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels) the space of invariants acquires a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants and characterize the subspace of conformal blocks in Hodge theoretic terms.
</p>projecteuclid.org/euclid.mmj/1565251217_20191111040134Mon, 11 Nov 2019 04:01 ESTThe Chow Form of a Reciprocal Linear Spacehttps://projecteuclid.org/euclid.mmj/1571731287<strong>Mario Kummer</strong>, <strong>Cynthia Vinzant</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 831--858.</p><p><strong>Abstract:</strong><br/>
A reciprocal linear space is the image of a linear space under coordinatewise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and appear as part of the defining equations of the central path of a linear program. Their structure is controlled by an underlying matroid. This provides a large family of hyperbolic varieties, recently introduced by Shamovich and Vinnikov. Here we give a definite determinantal representation to the Chow form of a reciprocal linear space. One consequence is the existence of symmetric rank-one Ulrich sheaves on reciprocal linear spaces. Another is a representation of the entropic discriminant as a sum of squares. For generic linear spaces, the determinantal formulas obtained are closely related to the Laplacian of the complete graph and generalizations to simplicial matroids. This raises interesting questions about the combinatorics of hyperbolic varieties and connections with the positive Grassmannian.
</p>projecteuclid.org/euclid.mmj/1571731287_20191111040134Mon, 11 Nov 2019 04:01 ESTEquivariant Khovanov Homology of Periodic Linkshttps://projecteuclid.org/euclid.mmj/1565251218<strong>Wojciech Politarczyk</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 859--889.</p><p><strong>Abstract:</strong><br/>
The purpose of this paper is to construct and study equivariant Khovanov homology, a version of Khovanov homology theory for periodic links. Since our construction works regardless of the characteristic of the coefficient ring, it generalizes a previous construction by Chbili. We establish invariance under equivariant isotopies of links and study algebraic properties of integral and rational version of the homology theory. Moreover, we construct a skein spectral sequence converging to equivariant Khovanov homology and use this spectral sequence to compute, as an example, equivariant Khovanov homology of torus links $T(n,2)$ .
</p>projecteuclid.org/euclid.mmj/1565251218_20191111040134Mon, 11 Nov 2019 04:01 ESTIndexhttps://projecteuclid.org/euclid.mmj/1573462881<p><strong>Source: </strong>The Michigan Mathematical Journal, Volume 68, Number 4, 893--896.</p>projecteuclid.org/euclid.mmj/1573462881_20191111040134Mon, 11 Nov 2019 04:01 ESTInfinitely Many Counterexamples to a Conjecture of Nortonhttps://projecteuclid.org/euclid.mmj/1579683616<strong>Hemar Godinho</strong>, <strong>Michael P. Knapp</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
For any positive integer $k$ , we define $\Gamma ^{*}(k)$ to be the smallest number $s$ such that every diagonal form $a_{1}x_{1}^{k}+a_{2}x_{2}^{k}+\cdots +a_{s}x_{s}^{k}$ in $s$ variables with integer coefficients must have a nontrivial zero in every $p$ -adic field $\mathbb{Q}_{p}$ . An old conjecture of Norton is that we should have $\Gamma ^{*}(k)\equiv 1(\text{mod }{k})$ for all $k$ . For many years, $\Gamma ^{*}(8)=39$ was the only known counterexample to this conjecture, and in recent years two more counterexamples have been found. In this article, we produce infinitely many counterexamples to Norton’s conjecture.
</p>projecteuclid.org/euclid.mmj/1579683616_20200122040029Wed, 22 Jan 2020 04:00 ESTQuadric Complexeshttps://projecteuclid.org/euclid.mmj/1576832418<strong>Nima Hoda</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 31 pages.</p><p><strong>Abstract:</strong><br/>
Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize $2$ -dimensional $\operatorname{CAT}(0)$ cube complexes and are a square analog of systolic complexes. We introduce and study the basic properties of these complexes. Using a form of dismantlability for the $1$ -skeleta of finite quadric complexes, we show that every finite group acting on a quadric complex stabilizes a complete bipartite subgraph of its $1$ -skeleton. Finally, we prove that $\mathrm{C}(4)\mbox{-}\mathrm{T}(4)$ small cancelation groups act on quadric complexes.
</p>projecteuclid.org/euclid.mmj/1576832418_20200122040029Wed, 22 Jan 2020 04:00 ESTExtremal and Stationary Discs for the Kobayashi $k$ -Metrichttps://projecteuclid.org/euclid.mmj/1576033217<strong>F. Bertrand</strong>, <strong>G. Della Sala</strong>, <strong>J.-C. Joo</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 13 pages.</p><p><strong>Abstract:</strong><br/>
We generalize Lempert’s and Poletsky’s works [11, 14] on the description of extremal discs for the Kobayashi metric to a higher order setting with $k$ -stationarity condition introduced in [1].
</p>projecteuclid.org/euclid.mmj/1576033217_20200122040029Wed, 22 Jan 2020 04:00 ESTFano Threefolds as Equivariant Compactifications of the Vector Grouphttps://projecteuclid.org/euclid.mmj/1576033218<strong>Zhizhong Huang</strong>, <strong>Pedro Montero</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
In this article, we determine all equivariant compactifications of the three-dimensional vector group $\mathbf{G}_{a}^{3}$ that are smooth Fano threefolds with Picard number greater than or equal to two.
</p>projecteuclid.org/euclid.mmj/1576033218_20200122040029Wed, 22 Jan 2020 04:00 ESTGeodesic Gaussian Integer Continued Fractionshttps://projecteuclid.org/euclid.mmj/1576033219<strong>Meira Hockman</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 26 pages.</p><p><strong>Abstract:</strong><br/>
This paper sets out to extend the results in the paper Geodesic Continued Fractions to continued fractions with Gaussian integer coefficients. The Farey graph $\mathcal{F}$ , whose vertices are reduced Gaussian rationals in $\mathbb{Q}_{\infty }(i)$ and whose edges join Farey neighbors, is introduced. The graph is modeled by the concrete realization in $\mathbb{H}^{3}$ where Farey neighbors are joined by hyperbolic geodesics (Farey geodesics) as seen in the Farey tessellation of $\mathbb{H}^{3}$ by Farey octahedrons. A natural distance $\varrho $ on $\mathbb{Q}_{\infty }(i)$ is also recalled, where $\varrho (\infty ,w)=n$ is the least number of edges in $\mathcal{F}$ from $\infty $ to $w\in \mathbb{Q}(i)$ , where $n$ is called the generation of w and a relevant path in $\mathcal{F}$ is called a geodesic expansion for $w$ . The Farey neighborhood of a reduced Gaussian rational is introduced and partitioned into neighbors of generation $n-1$ , $n$ , and $n+1$ . Subsequently, it is seen that there can be at most four Farey neighbors of generation $n-1$ in the neighborhood. An ancestral path is introduced, and a bound on the number of geodesic paths to any $w$ is established. Central to the paper are conditions for a path to be a geodesic path. The paper also addresses conditions for the existence of an infinite geodesic Gaussian integer continued fraction and suggestions of extending the paper to continued fraction with integer quaternion entries.
</p>projecteuclid.org/euclid.mmj/1576033219_20200122040029Wed, 22 Jan 2020 04:00 ESTVirtually Abelian Subgroups of $\operatorname{IA}_{n}(\mathbb{Z}/3)$ Are Abelianhttps://projecteuclid.org/euclid.mmj/1574845271<strong>Michael Handel</strong>, <strong>Lee Mosher</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 21 pages.</p><p><strong>Abstract:</strong><br/>
When studying subgroups of $\mathsf{Out}(F_{n})$ , one often replaces a given subgroup $\mathcal{H}$ with one of its finite index subgroups $\mathcal{H}_{0}$ so that virtual properties of $\mathcal{H}$ become actual properties of $\mathcal{H}_{0}$ . In many cases, the finite index subgroup is $\mathcal{H}_{0}=\mathcal{H}\cap \operatorname{IA}_{n}(\mathbb{Z}/3)$ . For which properties is this a good choice? Our main theorem states that being abelian is such a property. Namely, every virtually abelian subgroup of $\operatorname{IA}_{n}(\mathbb{Z}/3)$ is abelian.
</p>projecteuclid.org/euclid.mmj/1574845271_20200122040029Wed, 22 Jan 2020 04:00 ESTAlgebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebrashttps://projecteuclid.org/euclid.mmj/1574845272<strong>Pavlos Motakis</strong>, <strong>Daniele Puglisi</strong>, <strong>Andreas Tolias</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 56 pages.</p><p><strong>Abstract:</strong><br/>
For every Banach space $X$ with a Schauder basis, consider the Banach algebra $\mathbb{R}I\oplus \mathcal{K}_{\mathrm{diag}}(X)$ of all diagonal operators that are of the form $\lambda I+K$ . We prove that $\mathbb{R}I\oplus \mathcal{K}_{\mathrm{diag}}(X)$ is a Calkin algebra, that is, there exists a Banach space $\mathcal{Y}_{X}$ such that the Calkin algebra of $\mathcal{Y}_{X}$ is isomorphic as a Banach algebra to $\mathbb{R}I\oplus \mathcal{K}_{\mathrm{diag}}(X)$ . Among other applications of this theorem, we obtain that certain hereditarily indecomposable spaces and the James spaces $J_{p}$ and their duals endowed with natural multiplications are Calkin algebras; that all nonreflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras; and that sums of reflexive spaces with unconditional bases with certain James–Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.
</p>projecteuclid.org/euclid.mmj/1574845272_20200122040029Wed, 22 Jan 2020 04:00 ESTMultivariable Signatures, Genus Bounds, and $0.5$ -Solvable Cobordismshttps://projecteuclid.org/euclid.mmj/1574845273<strong>Anthony Conway</strong>, <strong>Matthias Nagel</strong>, <strong>Enrico Toffoli</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 47 pages.</p><p><strong>Abstract:</strong><br/>
We refine prior bounds on how the multivariable signature and the nullity of a link change under link cobordisms. The formula generalizes a series of results about the $4$ -genus having their origins in the Murasugi–Tristram inequality, and at the same time extends previously known results about concordance invariance of the signature to a bigger set of allowed variables. Finally, we show that the multivariable signature and nullity are also invariant under $0.5$ -solvable cobordism.
</p>projecteuclid.org/euclid.mmj/1574845273_20200122040029Wed, 22 Jan 2020 04:00 ESTBounds on Homological Invariants of VI-Moduleshttps://projecteuclid.org/euclid.mmj/1574326878<strong>Wee Liang Gan</strong>, <strong>Liping Li</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
We give bounds for various homological invariants (including Castelnuovo–Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the nondescribing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.
</p>projecteuclid.org/euclid.mmj/1574326878_20200122040029Wed, 22 Jan 2020 04:00 ESTFujita’s Freeness Conjecture for $T$ -Varieties of Complexity Onehttps://projecteuclid.org/euclid.mmj/1574326879<strong>Klaus Altmann</strong>, <strong>Nathan Ilten</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
We prove Fujita’s freeness conjecture for Gorenstein complexity-one $T$ -varieties with rational singularities.
</p>projecteuclid.org/euclid.mmj/1574326879_20200122040029Wed, 22 Jan 2020 04:00 ESTHyperbolicity Notions for Varieties Defined over a Non-Archimedean Fieldhttps://projecteuclid.org/euclid.mmj/1574326880<strong>R. Rodríguez Vázquez</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 38 pages.</p><p><strong>Abstract:</strong><br/>
Firstly, we pursue the work of W. Cherry on the analogue of the Kobayashi semidistance $d_{\mathrm{CK}}$ , which he introduced for analytic spaces defined over a non-Archimedean metrized field $k$ . We prove various characterizations of smooth projective varieties for which $d_{\mathrm{CK}}$ is an actual distance.
Secondly, we explore several notions of hyperbolicity for a smooth algebraic curve $X$ defined over $k$ . We prove a non-Archimedean analogue of the equivalence between having a negative Euler characteristic and the normality of certain families of analytic maps taking values in $X$ .
</p>projecteuclid.org/euclid.mmj/1574326880_20200122040029Wed, 22 Jan 2020 04:00 ESTAbsolutely Convergent Fourier Series of Functions over Homogeneous Spaces of Compact Groupshttps://projecteuclid.org/euclid.mmj/1574326881<strong>Arash Ghaani Farashahi</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 22 pages.</p><p><strong>Abstract:</strong><br/>
This paper presents a systematic study for classical aspects of functions with absolutely convergent Fourier series over homogeneous spaces of compact groups. Let $G$ be a compact group, $H$ be a closed subgroup of $G$ , and $\mu$ be the normalized $G$ -invariant measure over the left coset space $G/H$ associated with Weil’s formula with respect to the probability measures of $G$ and $H$ . We introduce the abstract notion of functions with absolutely convergent Fourier series in the Banach function space $L^{1}(G/H,\mu)$ . We then present some analytic characterizations for the linear space consisting of functions with absolutely convergent Fourier series over the compact homogeneous space $G/H$ .
</p>projecteuclid.org/euclid.mmj/1574326881_20200122040029Wed, 22 Jan 2020 04:00 ESTA Dirichlet Problem in Noncommutative Potential Theoryhttps://projecteuclid.org/euclid.mmj/1574326882<strong>Kuang-Ru Wu</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.
</p>projecteuclid.org/euclid.mmj/1574326882_20200122040029Wed, 22 Jan 2020 04:00 ESTLogarithmic Comparison with Smooth Boundary Divisor in Mixed Hodge Moduleshttps://projecteuclid.org/euclid.mmj/1574326883<strong>Chuanhao Wei</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 23 pages.</p>projecteuclid.org/euclid.mmj/1574326883_20200122040029Wed, 22 Jan 2020 04:00 ESTAut-Invariant Word Norm on Right-Angled Artin and Right-Angled Coxeter Groupshttps://projecteuclid.org/euclid.mmj/1573873440<strong>Michał Marcinkowski</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 11 pages.</p><p><strong>Abstract:</strong><br/>
We prove that the $\mathrm{Aut}$ -invariant word norm on right-angled Artin and right-angled Coxeter groups is unbounded (except in few special cases). To prove unboundedness, we exhibit certain characteristic subgroups. This allows us to find unbounded quasi-morphisms which are Lipschitz with respect to the $\mathrm{Aut}$ -invariant word norm.
</p>projecteuclid.org/euclid.mmj/1573873440_20200122040029Wed, 22 Jan 2020 04:00 ESTOn Gromov–Witten Theory of Projective Bundleshttps://projecteuclid.org/euclid.mmj/1573700736<strong>Honglu Fan</strong>, <strong>Yuan-Pin Lee</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 26 pages.</p>projecteuclid.org/euclid.mmj/1573700736_20200122040029Wed, 22 Jan 2020 04:00 ESTA Refinement of the Burgess Bound for Character Sumshttps://projecteuclid.org/euclid.mmj/1573700737<strong>Bryce Kerr</strong>, <strong>Igor E. Shparlinski</strong>, <strong>Kam Hung Yau</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
In this paper we give a refinement of the Burgess bound for multiplicative character sums modulo a prime number $q$ . This continues a series of previous logarithmic improvements, which are mostly due to Friedlander, Iwaniec, and Kowalski. In particular, for any nontrivial multiplicative character $\chi $ modulo a prime $q$ and any integer $r\geqslant 2$ , we show that \begin{equation*}\sum_{M\lt n\leqslant M+N}\chi (n)=O(N^{1-1/r}q^{(r+1)/4r^{2}}(\log q)^{1/4r}),\end{equation*} which sharpens the previous results by a factor $(\log q)^{1/4r}$ . Our improvement comes from averaging over numbers with no small prime factors rather than over an interval as in the previous approaches.
</p>projecteuclid.org/euclid.mmj/1573700737_20200122040029Wed, 22 Jan 2020 04:00 EST$L^{2}$ Estimates and Vanishing Theorems for Holomorphic Vector Bundles Equipped with Singular Hermitian Metricshttps://projecteuclid.org/euclid.mmj/1573700740<strong>Takahiro Inayama</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 18 pages.</p><p><strong>Abstract:</strong><br/>
We investigate singular hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^{2}$ estimates and vanishing theorems usually require an assumption that the vector bundles are Nakano positive. However, there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^{2}$ estimates and vanishing theorems by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.
</p>projecteuclid.org/euclid.mmj/1573700740_20200122040029Wed, 22 Jan 2020 04:00 ESTChern–Weil Theory for Line Bundles with the Family Arakelov Metrichttps://projecteuclid.org/euclid.mmj/1564711314<strong>Michiel Jespers</strong>, <strong>Robin de Jong</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 38 pages.</p><p><strong>Abstract:</strong><br/>
We prove a result of Chern–Weil type for canonically metrized line bundles on one-parameter families of smooth complex curves. Our result generalizes a result due to J. I. Burgos Gil, J. Kramer, and U. Kühn that deals with a line bundle of Jacobi forms on the universal elliptic curve over the modular curve with full level structure, equipped with the Petersson metric. Our main tool, as in the work by Burgos Gil, Kramer, and Kühn, is the notion of a b-divisor.
</p>projecteuclid.org/euclid.mmj/1564711314_20200122040029Wed, 22 Jan 2020 04:00 ESTOn the Hilbert Function of General Fat Points in $\mathbb{P}^{1}\times \mathbb{P}^{1}$https://projecteuclid.org/euclid.mmj/1580180455<strong>Enrico Carlini</strong>, <strong>Maria Virginia Catalisano</strong>, <strong>Alessandro Oneto</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 32 pages.</p><p><strong>Abstract:</strong><br/>
We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^{1}\times \mathbb{P}^{1}$ . Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A. V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.
</p>projecteuclid.org/euclid.mmj/1580180455_20200127220118Mon, 27 Jan 2020 22:01 ESTToeplitz Algebras of Correspondences and Endomorphisms of Sums of Type I Factorshttps://projecteuclid.org/euclid.mmj/1580180456<strong>Philip M. Gipson</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 12 pages.</p><p><strong>Abstract:</strong><br/>
It is a well-known fact that endomorphisms of $B(H)$ are intimately connected with families of mutually orthogonal isometries, that is, with representations of the so-called Toeplitz $C^{*}$ -algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^{*}$ -algebras associated with graphs and endomorphisms of certain von Neumann subalgebras of $B(H)$ . Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.
</p>projecteuclid.org/euclid.mmj/1580180456_20200127220118Mon, 27 Jan 2020 22:01 ESTOn Existence of Euclidean Ideal Classes in Real Cubic and Quadratic Fields with Cyclic Class Grouphttps://projecteuclid.org/euclid.mmj/1580180457<strong>Sanoli Gun</strong>, <strong>Jyothsnaa Sivaraman</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 20 pages.</p><p><strong>Abstract:</strong><br/>
Lenstra introduced the notion of Euclidean ideal classes for number fields to study cyclicity of their class groups. In particular, he showed that the class group of a number field with unit rank greater than or equal to one is cyclic if and only if it has a Euclidean ideal class. The only if part in the above result is conditional on the extended Riemann hypothesis. Graves and Murty showed that one does not require the extended Riemann hypothesis if the unit rank of the number field is greater than or equal to four and its Hilbert class field is abelian over rationals. In this article, we study real cubic and quadratic fields with cyclic class groups and show that they have a Euclidean ideal class under certain conditions.
</p>projecteuclid.org/euclid.mmj/1580180457_20200127220118Mon, 27 Jan 2020 22:01 ESTGromov–Witten Invariants Under Blow-Ups Along $(-1,-1)$ -Curveshttps://projecteuclid.org/euclid.mmj/1580439626<strong>Hua-Zhong Ke</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 17 pages.</p><p><strong>Abstract:</strong><br/>
For blow-ups of threefolds along $(-1,-1)$ -curves, we use the degeneration formula and the absolute/relative correspondence to obtain some closed blow-up formulae for Gromov–Witten invariants and generalized BPS numbers.
</p>projecteuclid.org/euclid.mmj/1580439626_20200130220100Thu, 30 Jan 2020 22:01 ESTOn the Bielliptic and Bihyperelliptic Locihttps://projecteuclid.org/euclid.mmj/1580439627<strong>Paola Frediani</strong>, <strong>Paola Porru</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 30 pages.</p><p><strong>Abstract:</strong><br/>
We study some particular loci inside the moduli space $\mathcal{M}_{g}$ , namely the bielliptic locus (i.e. the locus of curves admitting a $2:1$ cover over an elliptic curve $E$ ) and the bihyperelliptic locus (i.e. the locus of curves admitting a $2:1$ cover over a hyperelliptic curve $C'$ , $g(C')\geq 2$ ). We show that the bielliptic locus is not a totally geodesic subvariety of $\mathcal{A}_{g}$ if $g\geq 4$ (whereas it is for $g=3$ , see [18]) and that the bihyperelliptic locus is not totally geodesic in $\mathcal{A}_{g}$ if $g\geq 3g'$ . We also give a lower bound for the rank of the second Gaussian map at the generic point of the bielliptic locus and an upper bound for this rank for every bielliptic curve.
</p>projecteuclid.org/euclid.mmj/1580439627_20200130220100Thu, 30 Jan 2020 22:01 ESTEquations Defining Certain Graphshttps://projecteuclid.org/euclid.mmj/1580439628<strong>Youngsu Kim</strong>, <strong>Vivek Mukundan</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 36 pages.</p><p><strong>Abstract:</strong><br/>
Consider the rational map $\phi:\mathbb{P}^{n-1}_{{\mathbf{k}}}\mathop{\dashrightarrow}\limits^{[f_{0}:\cdots:f_{n}]}\mathbb{P}^{n}_{{\mathbf{k}}}$ defined by homogeneous polynomials $f_{0},\dots,f_{n}$ of the same degree $d$ in a polynomial ring $R={\mathbf{k}}[x_{1},\dots,x_{n}]$ over a field ${\mathbf{k}}$ . Suppose that $I=(f_{0},\dots,f_{n})$ is a height two perfect ideal satisfying $\mu(I_{p})\leq\dim R_{p}$ for $p\in\operatorname{Spec}(R)\setminusV(x_{1},\dots,x_{n})$ . We study the equations defining the graph of $\phi$ whose coordinate ring is the Rees algebra $R[\mathit{It}]$ . We provide new methods to construct these equations using the work of Buchsbaum and Eisenbud. Furthermore, for certain classes of ideals satisfying the conditions above, our methods lead to explicit equations defining Rees algebras of the ideals in these classes. These classes of examples are interesting in that there are no known methods to compute the defining ideal of the Rees algebra of such ideals. Our new methods also give effective criteria to check that $\phi$ is birational onto its image.
</p>projecteuclid.org/euclid.mmj/1580439628_20200130220100Thu, 30 Jan 2020 22:01 ESTCohomology and the Bowditch Boundaryhttps://projecteuclid.org/euclid.mmj/1580461363<strong>Jason F. Manning</strong>, <strong>Oliver H. Wang</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 37 pages.</p><p><strong>Abstract:</strong><br/>
We give a group cohomological description of the Čech cohomology of the Bowditch boundary of a relatively hyperbolic group pair, generalizing a result of Bestvina–Mess about hyperbolic groups. In the case of a relatively hyperbolic Poincaré duality group pair, we show that the Bowditch boundary is a homology manifold. For a three-dimensional Poincaré duality pair, we recover the theorem of Tshishiku–Walsh stating that the boundary is homeomorphic to a two-sphere.
</p>projecteuclid.org/euclid.mmj/1580461363_20200131040324Fri, 31 Jan 2020 04:03 ESTArithmetically Nef Line Bundleshttps://projecteuclid.org/euclid.mmj/1581735954<strong>Dennis Keeler</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 14 pages.</p><p><strong>Abstract:</strong><br/>
Let $L$ be a line bundle on a scheme $X$ , proper over a field. The property of $L$ being nef can sometimes be “thickened”, allowing reductions to positive characteristic. We call such line bundles arithmetically nef . It is known that a line bundle $L$ may be nef, but not arithmetically nef. We show that $L$ is arithmetically nef if and only if its restriction to its stable base locus is arithmetically nef. Consequently, if $L$ is nef and its stable base locus has dimension $1$ or less, then $L$ is arithmetically nef.
</p>projecteuclid.org/euclid.mmj/1581735954_20200214220618Fri, 14 Feb 2020 22:06 ESTVGIT Presentation of the Second Flip of $\overline{M}_{2,1}$https://projecteuclid.org/euclid.mmj/1582081218<strong>Maksym Fedorchuk</strong>, <strong>Matthew Grimes</strong>. <p><strong>Source: </strong>The Michigan Mathematical Journal, Advance publication, 28 pages.</p><p><strong>Abstract:</strong><br/>
We perform a variation of geometric invariant theory stability analysis for 2nd Hilbert points of bi-log-canonically embedded pointed curves of genus $2$ . As a result, we give a GIT construction of the log canonical models $\overline{M}_{2,1}(\alpha )$ for $\alpha =2/3\pm \epsilon $ and obtain a VGIT presentation of the second flip in the Hassett–Keel program for the moduli space of pointed genus $2$ curves.
</p>projecteuclid.org/euclid.mmj/1582081218_20200218220051Tue, 18 Feb 2020 22:00 EST