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2. Addendum to the paper ??On partial analyticity of CR mappings??
- Author
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Bernard Coupet, Sergey Pinchuk, and Alexandre Sukhov
- Subjects
Pure mathematics ,General Mathematics ,Addendum ,Algorithm ,Mathematics - Published
- 2004
3. A note on a paper by W.J. Ricker and H.H. Schaefer
- Author
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Ben de Pagter
- Subjects
Discrete mathematics ,General Mathematics ,Mathematics - Published
- 1990
4. Corrigendum for the paper 'Invariant tori for nearly integrable Hamiltonian systems with degeneracy'
- Author
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Jiangong You and Junxiang Xu
- Subjects
Null set ,Pure mathematics ,Integrable system ,Kolmogorov–Arnold–Moser theorem ,General Mathematics ,Mathematical analysis ,Torus ,Superintegrable Hamiltonian system ,Invariant (physics) ,Degeneracy (mathematics) ,Mathematics ,Hamiltonian system - Abstract
In the paper [1], the authors obtain a KAM theorem for nearly integrable hamiltonian systems under the Russmann’s non-degeneracy condition, which is known to be sharpest one for small divisor conditions. However, the Remark 1.3 is wrong because we have ignored the null set − ∗, which may contain zeros of ω of high order such that (1.4) does not hold for all p ∈ . The Remark 1.3 might mislead the readers that the condition (1.5) of Theorem B is equivalent to the Russmann’s non-degeneracy condition. Actually, the Russmann’s non-degeneracy condition is equivalent to the condition (1.4) of Theorem A as proved in [1]. Under the Russmann’s non-degeneracy condition (1.4), as proved in the Remark 3.1 the condition (1.5) holds if we replace n − 1 by a sufficiently large number N depending on h, and then the conclusion of Theorem B remains valid if in the measure estimate n − 1 is replaced by N .
- Published
- 2007
5. Addendum to the paper 'Characteristic cycles of perverse sheaves and Milnor fibers'
- Author
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Philibert Nang and Kiyoshi Takeuchi
- Subjects
Algebra ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,General Mathematics ,FOS: Mathematics ,14B05, 32C38, 32S40, 35A27 ,Algebraic Topology (math.AT) ,Addendum ,Mathematics - Algebraic Topology ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We note that Theorem 5.4 of the paper ``Characteristic cycles of perverse sheaves and Milnor fibers" (by P. Nang and K. Takeuchi, published online in Math. Zeitschrift, October 15, 2004) holds for "any" non-zero complex number $a$., Comment: 1page
- Published
- 2005
6. Special Ulrich bundles on regular Weierstrass fibrations
- Author
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Joan Pons-Llopis and Rosa M. Miró-Roig
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Short paper ,Elliptic surfaces ,Ulrich bundles ,01 natural sciences ,Mathematics::Algebraic Geometry ,Simple (abstract algebra) ,0103 physical sciences ,Weierstrass fibrations ,Rank (graph theory) ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.
- Published
- 2019
7. A remark on the preceding paper of Fu?ik and Krbec
- Author
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Peter Hess
- Subjects
General Mathematics ,Calculus ,Mathematics - Published
- 1977
8. Relations of a paper of Ky Fan to a theorem of Krein-Milman type
- Author
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Marvin W. Grossman
- Subjects
Pure mathematics ,General Mathematics ,Calculus ,Type (model theory) ,Mathematics - Published
- 1965
9. Note on a paper by L. Neder
- Author
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W. L. Ferra
- Subjects
General Mathematics ,Mathematics education ,Mathematics - Published
- 1930
10. A note on the paper ?Hermitian operators onC (X, E) and the Banach-Stone theorem? by R. J. Fleming and J.E. Jamison
- Author
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Peter Greim and Ehrhard Behrends
- Subjects
Pure mathematics ,Banach–Stone theorem ,General Mathematics ,Hermitian matrix ,Self-adjoint operator ,Mathematics - Published
- 1980
11. On the filling radius of positively curved Alexandrov spaces
- Author
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Takumi Yokota
- Subjects
General Mathematics ,Round sphere ,Mathematical analysis ,Short paper ,Filling radius ,Mathematics::Metric Geometry ,Radius of curvature ,Geometry ,Mathematics::Differential Geometry ,Mathematics - Abstract
It was shown by F. Wilhelm that Gromov’s filling radius of any positively curved closed Riemannian manifolds are less than that of the round sphere unless they are isometric to each other. In this short paper, we adapt his proof to see that the same is true for any positively curved closed Alexandrov spaces as well.
- Published
- 2012
12. Order 3 symplectic automorphisms on K3 surfaces
- Author
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Alice Garbagnati and Yulieth Prieto Montañez
- Subjects
Pure mathematics ,Endomorphism ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Order (ring theory) ,Automorphism ,01 natural sciences ,Cohomology ,14J28, 14J50 ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Quotient ,Mathematics ,Symplectic geometry - Abstract
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $\Lambda_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $\pi_*$ and $\pi^*$ induced in cohomology by the rational quotient map $\pi:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $\sigma$ and $Y$ is the minimal resolution of the quotient $X/\sigma$; we deduce the relation between the N\'eron--Severi group of $X$ and the one of $Y$. Applying these results we describe explicit geometric examples and generalize the Shioda--Inose structures, relating Abelian surfaces admitting order 3 endomorphisms with certain specific K3 surfaces admitting particular order 3 symplectic automorphisms., Comment: 28 pages. Version 2: this is the published version of the paper. The last section of the previous version (v1) was erased (the results are only stated) and it is now contained in arXiv:2209.10141
- Published
- 2021
13. Maximal families of nodal varieties with defect
- Author
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REMKE NANNE KLOOSTERMAN
- Subjects
Surface (mathematics) ,Double cover ,Degree (graph theory) ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Hypersurface ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,NODAL ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large., v2: A proof for the Ciliberto-Di Gennaro conjecture is added (Section 5); Some minor corrections in the other sections. v3: some minor corrections in the abstract v4: The proof for the Ciliberto-Di Gennaro conjecture has been modified; The paper is split into two parts, the complete intersection case will be discussed in a different paper
- Published
- 2021
14. Approximations in $$L^1$$ with convergent Fourier series
- Author
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Michael Ruzhansky, Zhirayr Avetisyan, and M. G. Grigoryan
- Subjects
Measurable function ,General Mathematics ,010102 general mathematics ,Hausdorff space ,Second-countable space ,Space (mathematics) ,01 natural sciences ,Functional Analysis (math.FA) ,Separable space ,Mathematics - Functional Analysis ,010101 applied mathematics ,Combinatorics ,Mathematics and Statistics ,Bounded function ,41A99, 43A15, 43A50, 43A85, 46E30 ,Homogeneous space ,FOS: Mathematics ,Orthonormal basis ,0101 mathematics ,Mathematics - Abstract
For a separable finite diffuse measure space $${\mathcal {M}}$$ M and an orthonormal basis $$\{\varphi _n\}$$ { φ n } of $$L^2({\mathcal {M}})$$ L 2 ( M ) consisting of bounded functions $$\varphi _n\in L^\infty ({\mathcal {M}})$$ φ n ∈ L ∞ ( M ) , we find a measurable subset $$E\subset {\mathcal {M}}$$ E ⊂ M of arbitrarily small complement $$|{\mathcal {M}}{\setminus } E| | M \ E | < ϵ , such that every measurable function $$f\in L^1({\mathcal {M}})$$ f ∈ L 1 ( M ) has an approximant $$g\in L^1({\mathcal {M}})$$ g ∈ L 1 ( M ) with $$g=f$$ g = f on E and the Fourier series of g converges to g, and a few further properties. The subset E is universal in the sense that it does not depend on the function f to be approximated. Further in the paper this result is adapted to the case of $${\mathcal {M}}=G/H$$ M = G / H being a homogeneous space of an infinite compact second countable Hausdorff group. As a useful illustration the case of n-spheres with spherical harmonics is discussed. The construction of the subset E and approximant g is sketched briefly at the end of the paper.
- Published
- 2021
15. High perturbations of quasilinear problems with double criticality
- Author
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Prashanta Garain, Vicenţiu D. Rădulescu, Claudianor O. Alves, Universidade Federal de Campina Grande, Department of Mathematics and Systems Analysis, AGH University of Science and Technology, Aalto-yliopisto, and Aalto University
- Subjects
General Mathematics ,010102 general mathematics ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,Qualitative analysis ,Variational methods ,Domain (ring theory) ,Musielak–Sobolev space ,Nabla symbol ,0101 mathematics ,Quasilinear problems ,Mathematics - Abstract
This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ - Δ Φ u = f ( x , u ) in Ω , u = 0 on ∂ Ω , where $$\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\nabla u|)\nabla u)$$ Δ Φ u = div ( φ ( x , | ∇ u | ) ∇ u ) and $$\Phi (x,t)=\int _{0}^{|t|}\varphi (x,s)s\,ds$$ Φ ( x , t ) = ∫ 0 | t | φ ( x , s ) s d s is a generalized N-function. We assume that $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a smooth bounded domain that contains two open regions $$\Omega _N,\Omega _p$$ Ω N , Ω p with $${\overline{\Omega }}_N \cap {\overline{\Omega }}_p=\emptyset $$ Ω ¯ N ∩ Ω ¯ p = ∅ . The features of this paper are that $$-\Delta _{\Phi }u$$ - Δ Φ u behaves like $$-\Delta _N u $$ - Δ N u on $$\Omega _N$$ Ω N and $$-\Delta _p u $$ - Δ p u on $$\Omega _p$$ Ω p , and that the growth of $$f:\Omega \times {\mathbb {R}} \rightarrow {\mathbb {R}}$$ f : Ω × R → R is like that of $$e^{\alpha |t|^{\frac{N}{N-1}}}$$ e α | t | N N - 1 on $$\Omega _N$$ Ω N and as $$|t|^{p^{*}-2}t$$ | t | p ∗ - 2 t on $$\Omega _p$$ Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.
- Published
- 2021
16. Graded Bourbaki ideals of graded modules
- Author
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Jürgen Herzog, Dumitru I. Stamate, and Shinya Kumashiro
- Subjects
Noetherian ,Pure mathematics ,Sequence ,Class (set theory) ,Ideal (set theory) ,Mathematics::Commutative Algebra ,Mathematics::General Mathematics ,General Mathematics ,Mathematics::History and Overview ,010102 general mathematics ,Structure (category theory) ,Mathematics::General Topology ,Field (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematik ,0103 physical sciences ,FOS: Mathematics ,Homomorphism ,13A02, 13A30, 13D02, 13H10 ,010307 mathematical physics ,0101 mathematics ,Rees algebra ,Mathematics - Abstract
In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle and the Rees algebra of its Bourbaki ideal., Comment: 29 pages
- Published
- 2021
17. Low dimensional orders of finite representation type
- Author
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Daniel Chan and Colin Ingalls
- Subjects
Ring (mathematics) ,Plane curve ,Root of unity ,General Mathematics ,010102 general mathematics ,14E16 ,Local ring ,Order (ring theory) ,Mathematics - Rings and Algebras ,Type (model theory) ,01 natural sciences ,Noncommutative geometry ,Combinatorics ,Minimal model program ,Mathematics - Algebraic Geometry ,Rings and Algebras (math.RA) ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $$G \subset {{{\,\mathrm{GL}\,}}_2}$$ , explicitly computing $$H^2(G,k^*)$$ , and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $$B = k_{\zeta } \llbracket x,y \rrbracket $$ be the skew power series ring where $$\zeta $$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $$A = B/(f)$$ where $$f \in Z(B)$$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.
- Published
- 2020
18. Remarks on the geodesic-Einstein metrics of a relative ample line bundle
- Author
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Xueyuan Wan and Xu Wang
- Subjects
Ample line bundle ,Pure mathematics ,Geodesic ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Fibration ,Type (model theory) ,01 natural sciences ,Mathematics::Algebraic Geometry ,Flow (mathematics) ,Bounded function ,Bundle ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this paper, we introduce the associated geodesic-Einstein flow for a relative ample line bundle L over the total space $$\mathcal {X}$$ of a holomorphic fibration and obtain a few properties of that flow. In particular, we prove that the pair $$(\mathcal {X}, L)$$ is nonlinear semistable if the associated Donaldson type functional is bounded from below and the geodesic-Einstein flow has long-time existence property. We also define the associated S-classes and C-classes for $$(\mathcal {X}, L)$$ and obtain two inequalities between them when L admits a geodesic-Einstein metric. Finally, in the appendix of this paper, we prove that a relative ample line bundle is geodesic-Einstein if and only if an associated infinite rank bundle is Hermitian–Einstein.
- Published
- 2020
19. On the local density formula and the Gross–Keating invariant with an Appendix ‘The local density of a binary quadratic form’ by T. Ikeda and H. Katsurada
- Author
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Cho Sungmun
- Subjects
Pure mathematics ,Mathematics - Number Theory ,Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Local factor ,01 natural sciences ,Quadratic form ,0103 physical sciences ,FOS: Mathematics ,11E08, 11E95, 14L15, 20G25 ,Binary quadratic form ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Local field ,Fourier series ,Mathematics - Abstract
T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating invariant of a quadratic form in their recent papers [IK1] and [IK2]. In particular, they prove that the local factor of the Fourier coefficients of the Siegel-Eisenstein series is completely determined by the Gross-Keating invariant with extra datum, called the extended GK datum, in [IK2]. On the other hand, such local factor is a special case of the local densities for a pair of two quadratic forms. Thus we propose a general question if the local density can be determined by certain series of the Gross-Keating invariants and the extended GK datums. In this paper, we prove that the answer to this question is affirmative, for the local density of a single quadratic form defined over an unramified finite extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute the local density formula of a single binary quadratic form defined over any finite extension of $\mathbb{Z}_2$., 32 pages
- Published
- 2020
20. Archimedean non-vanishing, cohomological test vectors, and standard L-functions of $${\mathrm {GL}}_{2n}$$: real case
- Author
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Cheng Chen, Fangyang Tian, Dihua Jiang, and Bingchen Lin
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Linear model ,Structure (category theory) ,22E45 (Primary), 11F67 (Secondary) ,Type (model theory) ,Lambda ,Infinity ,01 natural sciences ,Invariant theory ,Linear form ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics ,media_common - Abstract
The standard $L$-functions of $\mathrm{GL}_{2n}$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existance or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Binyong Sun, by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $\Lambda_{s,\chi}$, which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard $L$-function $L(s,\pi\otimes\chi)$ as a meromorphic function of $s\in \mathbb{C}$. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector, and hence recovers a non-vanishing result of Binyong Sun via a completely different method. Our main result indicates a complete solution to (2), which will be presented in a paper of Dihua Jiang, Binyong Sun and Fangyang Tian with full details and with applications to the global period relations for the twisted standard $L$-functions at critical places., Comment: 39 pages. The current version of this paper is significantly shorter than the previous one, as the first author pointed out a conceptual intepretation of construction of cohomological test vector in the old version of this paper. Section 4 is completely rewritten. Also fix some inaccuracies
- Published
- 2019
21. A sparse approach to mixed weak type inequalities
- Author
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Marcela Caldarelli and Israel P. Rivera-Ríos
- Subjects
Pure mathematics ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Novelty ,Singular integral ,Weak type ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,GEOM ,media_common ,Mathematics - Abstract
In this paper we provide some quantitative mixed weak-type estimates assuming conditions that imply that $$uv\in A_{\infty }$$ for Calderon–Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced in Domingo-Salazar et al. (Bull Lond Math Soc 48(1):63–73, 2016) and extended in Lerner et al. (Adv Math 319:153–181, 2017) and Li et al. (J Geom Anal, 2018).
- Published
- 2019
22. Surgery for partially hyperbolic dynamical systems II. Blow-up of a complex curve
- Author
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Federico Rodriguez Hertz and Andrey Gogolev
- Subjects
Flexibility (engineering) ,medicine.medical_specialty ,Dynamical systems theory ,General Mathematics ,010102 general mathematics ,Hyperbolic manifold ,Dynamical Systems (math.DS) ,Mathematics::Geometric Topology ,01 natural sciences ,Surgery ,0103 physical sciences ,FOS: Mathematics ,Geodesic flow ,medicine ,Totally geodesic ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
In this paper we use the blow-up surgery introduced in [G] to produce new higher dimensional partially hyperbolic flows. The main contribution of the paper is the slow-down construction which accompanies the blow-up construction. This new ingredient allows to dispose of a rather strong domination assumption which was crucial for results in [G]. Consequently we gain more flexibility which allows to construct new volume-preserving partially hyperbolic flows as well as new examples which are not fiberwise Anosov. The latter are produced by starting with the geodesic flow on complex hyperbolic manifold which admits a totally geodesic complex curve. Then by performing the slow-down first and the blow-up second we obtain a new (volume-preserving) partially hyperbolic flows., Comment: 16 pages. This is part II. Part I being arXiv:1609.05925
- Published
- 2021
23. On the averages of generalized Hasse–Witt invariants of pointed stable curves in positive characteristic
- Author
-
Yu Yang
- Subjects
Fundamental group ,Stable curve ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,Anabelian geometry ,0103 physical sciences ,010307 mathematical physics ,Isomorphism class ,0101 mathematics ,Abelian group ,Algebraically closed field ,Invariant (mathematics) ,Mathematics - Abstract
In the present paper, we study fundamental groups of curves in positive characteristic. Let $$X^{\bullet }$$ be a pointed stable curve of type $$(g_{X}, n_{X})$$ over an algebraically closed field of characteristic $$p>0$$, $$\Gamma _{X^{\bullet }}$$ the dual semi-graph of $$X^{\bullet }$$, and $$\Pi _{X^{\bullet }}$$ the admissible fundamental group of $$X^{\bullet }$$. In the present paper, we study a kind of group-theoretical invariant $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ associated to the isomorphism class of $$\Pi _{X^{\bullet }}$$ called the limit of p-averages of $$\Pi _{X^{\bullet }}$$, which plays a central role in the theory of anabelian geometry of curves over algebraically closed fields of positive characteristic. Without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, we give a lower bound and a upper bound of $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$. In particular, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ under a certain assumption concerning $$\Gamma _{X^{\bullet }}$$ which generalizes a formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ obtained by Tamagawa. Moreover, if $$X^{\bullet }$$ is a component-generic pointed stable curve, we prove an explicit formula for $$\text {Avr}_{p}(\Pi _{X^{\bullet }})$$ without any assumptions concerning $$\Gamma _{X^{\bullet }}$$, which can be regarded as an averaged analogue of the results of Nakajima, Zhang, and Ozman–Pries concerning p-rank of abelian etale coverings of projective generic curves for admissible coverings of component-generic pointed stable curves.
- Published
- 2019
24. The prime end capacity of inaccessible prime ends, resolutivity, and the Kellogg property
- Author
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Nageswari Shanmugalingam and Tomasz Adamowicz
- Subjects
Mathematics::Number Theory ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Boundary (topology) ,Metric Geometry (math.MG) ,Lipschitz continuity ,01 natural sciences ,Prime (order theory) ,Domain (mathematical analysis) ,Combinatorics ,Metric space ,Mathematics - Analysis of PDEs ,Prime end ,Mathematics - Metric Geometry ,Bounded function ,31E05, 31B15, 31B25, 31C15, 30L99 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Prime end boundaries $\partial_P\Omega$ of domains $\Omega$ are studied in the setting of complete doubling metric measure spaces supporting a $p$-Poincar\'e inequality. Notions of rectifiably (in)accessible- and (in)finitely far away prime ends are introduced and employed in classification of prime ends. We show that, for a given domain, the prime end capacity of the collection of all rectifiably inaccessible prime ends together will all non-singleton prime ends is zero. We show the resolutivity of continouous functions on $\partial_P\Omega$ which are Lipschitz continuous with respect to the Mazurkiewicz metric when restricted to the collection $\partial_{SP}\Omega$ of all accessible prime ends. Furthermore, bounded perturbations of such functions in $\partial_P\Omega\setminus\partial_{SP}\Omega$ yield the same Perron solution. In the final part of the paper, we demonstrate the (resolutive) Kellogg property with respect to the prime end boundary of bounded domains in the metric space. Notions given in this paper are illustrated by a number of examples., Comment: 23 pages, 3 figures
- Published
- 2019
25. Signature characters of invariant Hermitian forms on irreducible Verma modules and Hall–Littlewood polynomials
- Author
-
Wai Ling Yee
- Subjects
Pure mathematics ,Verma module ,General Mathematics ,010102 general mathematics ,Positive-definite matrix ,01 natural sciences ,Unitary state ,Hermitian matrix ,Hall–Littlewood polynomials ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Representation Theory ,Alcove ,Affine Hecke algebra ,Mathematics - Abstract
The Unitary Dual Problem is one of mathematics’ most important open problems: classify the irreducible unitary representations of a group. The general approach has been to classify all representations admitting non-degenerate invariant Hermitian forms, compute the signatures of those forms, and then determine which forms are positive definite. Signature character algorithms and formulas arising from deforming representations and analysing changes at reducibility points, as in Adams et al. (Unitary representations of real reductive groups (ArXiv e-prints), 2012) and Yee (Represent Theory 9:638–677, 2005), produce very complicated formulas or algorithms from the resulting recursion. This paper shows that in the case of irreducible Verma modules all of the complexity can be encapsulated by the affine Hecke algebra: for compact real forms and for alcoves corresponding to translations of the fundamental alcove by a regular weight, signature characters of irreducible Verma modules are in fact “negatives” of Hall–Littlewood polynomial summands evaluated at $$q=-\,1$$ times a version of the Weyl denominator, establishing a simple signature character formula and drawing an important connection between signature characters and the affine Hecke algebra. Signature characters of irreducible highest weight modules are shown to be related to Kazhdan-Lusztig basis elements. This paper also handles noncompact real forms. The current state of the art for the unitary dual is a computer algorithm for determining if a given representation is unitary. These results suggest the potential to move the state of the art to a closed form classification for the entire unitary dual.
- Published
- 2018
26. On central leaves of Hodge-type Shimura varieties with parahoric level structure
- Author
-
Wansu Kim
- Subjects
Pure mathematics ,Reduction (recursion theory) ,Mathematics - Number Theory ,Group (mathematics) ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Neighbourhood (graph theory) ,Structure (category theory) ,Type (model theory) ,Space (mathematics) ,14L05, 14G35 ,01 natural sciences ,Mathematics - Algebraic Geometry ,Product (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Kisin and Pappas constructed integral models of Hodge-type Shimura varieties with parahoric level structure at $p>2$, such that the formal neighbourhood of a mod~$p$ point can be interpreted as a deformation space of $p$-divisible group with some Tate cycles (generalising Faltings' construction). In this paper, we study the central leaf and the closed Newton stratum in the formal neighbourhoods of mod~$p$ points of Kisin-Pappas integral models with parahoric level structure; namely, we obtain the dimension of central leaves and the almost product structure of Newton strata. In the case of hyperspecial level strucure (i.e., in the good reduction case), our main results were already obtained by Hamacher, and the result of this paper holds for ramified groups as well., 33 pages; section 2.5 added to fill in the gap in the earlier version
- Published
- 2018
27. Special Lagrangian and deformed Hermitian Yang–Mills on tropical manifold
- Author
-
Hikaru Yamamoto
- Subjects
General Mathematics ,010102 general mathematics ,Fibration ,Yang–Mills existence and mass gap ,Torus ,String theory ,Submanifold ,01 natural sciences ,Hermitian matrix ,Manifold ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mirror symmetry ,Mathematics::Symplectic Geometry ,Mathematical physics ,Mathematics - Abstract
From string theory, the notion of deformed Hermitian Yang–Mills connections has been introduced by Marino et al. (J High Energy Phys Paper 5, 2000). After that, Leung et al. (Adv Theor Math Phys 4(6):1319–1341, 2000) proved that it naturally appears as mirror objects of special Lagrangian submanifolds via Fourier–Mukai transform between dual torus fibrations. In their paper, some conditions are imposed for simplicity. In this paper, data to glue their construction on tropical manifolds are proposed and a generalization of the correspondence is proved without the assumption that the Lagrangian submanifold is a section of the torus fibration.
- Published
- 2018
28. A sharp lower bound for the geometric genus and Zariski multiplicity question
- Author
-
Huaiqing Zuo and Stephen S.-T. Yau
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Geometric genus ,Multiplicity (mathematics) ,Mathematics::Geometric Topology ,01 natural sciences ,Upper and lower bounds ,Milnor number ,Mathematics::Algebraic Geometry ,Hypersurface ,Singularity ,0103 physical sciences ,Gravitational singularity ,0101 mathematics ,Mathematics - Abstract
It is well known that the geometric genus and multiplicity are two important invariants for isolated singularities. In this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities. In 1971, Zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type. This problem remains unsettled. In this paper we answer positively Zariski’s multiplicity question for isolated hypersurface singularity if Milnor number or geometric genus is small.
- Published
- 2017
29. Prüfer algebraic spaces
- Author
-
Ilya Tyomkin and Michael Temkin
- Subjects
Mathematics::Commutative Algebra ,General Mathematics ,Topological tensor product ,010102 general mathematics ,Birational geometry ,01 natural sciences ,Algebra ,Algebraic cycle ,Mathematics::Algebraic Geometry ,0103 physical sciences ,Real algebraic geometry ,Interpolation space ,010307 mathematical physics ,Compactification (mathematics) ,0101 mathematics ,Algebraic number ,Algebraic geometry and analytic geometry ,Mathematics - Abstract
This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Prufer spaces and Prufer pairs of algebraic spaces that generalize spectra of Prufer rings. As a particular case of Prufer spaces we introduce valuation algebraic spaces, and use them to establish valuative criteria of separatedness and properness that sharpen the standard criteria. In a sequel paper, we introduce a version of Riemann–Zariski spaces, and prove Nagata’s compactification theorem for algebraic spaces.
- Published
- 2016
30. Algebraic cycles representing cohomology operations
- Author
-
Marie-Louise Michelsohn
- Subjects
General Mathematics ,Codimension ,Homology (mathematics) ,Mathematics::Algebraic Topology ,Cohomology ,14C25, 55S10, 55S15 ,Algebraic cycle ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::K-Theory and Homology ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we will show that certain universal homology classes which are fundamental in topology are algebraic. To be specific, the products of Eilenberg–MacLane spaces \({\mathcal K}_{2q}\equiv K(\mathbf Z,{2}) \times K(\mathbf Z,{4}) \times \cdots \times K(\mathbf Z,{2q}) \) have models which are limits of complex projective varieties. Precisely, we have \({\mathcal K}_{2q}= \varinjlim \nolimits _{d\rightarrow \infty }\mathcal C^{q}_{d}(\mathbf P^{n})\) where \(\mathcal C^{q}_{d}(\mathbf P^{n})\) denotes the Chow variety of effective cycles of codimension q and degree d on \(\mathbf P_{\mathbf C}^{n}\). It is natural to ask which elements in the homology of \({\mathcal K}_{2q}\) are represented by algebraic cycles in these approximations. In this paper we find such representations for the even dimensional classes which are known as Steenrod squares (as well as their Pontrjagin and join products). These classes are dual to the cohomology classes which correspond to the basic cohomology operations also known as the Steenrod squares.
- Published
- 2016
31. On log local Cartier transform of higher level in characteristic p
- Author
-
Sachio Ohkawa
- Subjects
Discrete mathematics ,Smooth morphism ,General Mathematics ,Modulo ,010102 general mathematics ,Scalar (mathematics) ,13N10, 16H05, 16S32 ,Differential operator ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Mathematics - Algebraic Geometry ,Azumaya algebra ,FOS: Mathematics ,Higgs boson ,Sheaf ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In our previous paper, given an integral log smooth morphism $X\to S$ of fine log schemes of characteristic $p>0$, we studied the Azumaya nature of the sheaf of log differential operators of higher level and constructed a splitting module of it under an existence of a certain lifting modulo $p^{2}$. In this paper, under a certain liftability assumption which is stronger than our previous paper, we construct another splitting module of our Azumaya algebra over a scalar extension, which is smaller than our previous paper. As an application, we construct an equivalence, which we call the log local Cartier transform of higher level, between certain $\cal D$-modules and certain Higgs modules. We also discuss about the compatibility of the log Frobenius descent and the log local Cartier transform and the relation between the splitting module constructed in this paper and that constructed in the previous paper. Our result can be considered as a generalization of the result of Ogus-Vologodsky, Gros-Le Stum-Quir��s to the case of log schemes and that of Schepler to the case of higher level.
- Published
- 2016
32. A study of variations of pseudoconvex domains via Kähler-Einstein metrics
- Author
-
Young-Jun Choi
- Subjects
Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,Mathematical analysis ,Dimension (graph theory) ,Triviality ,Domain (mathematical analysis) ,symbols.namesake ,Kähler–Einstein metric ,Pseudoconvexity ,Bounded function ,symbols ,Einstein ,Pseudoconvex function ,Mathematics - Abstract
This paper is a sequel to Choi (Math Ann 362(1–2):121–146, 2015) in Math. Ann. In that paper we studied the subharmonicity of Kahler–Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to 3. In this paper, we study the variations Kahler–Einstein metrics on bounded strongly pseudoconvex domains of dimension 2. In addition, we discuss the previous result with general bounded pseudoconvex domain and local triviality of a family of bounded strongly pseudoconvex domains.
- Published
- 2015
33. On the functional equation of the normalized Shintani L-function of several variables
- Author
-
Minoru Hirose and Nobuo Sato
- Subjects
Integral representation ,Mathematics - Number Theory ,Generalization ,Mathematics::Number Theory ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,Special values ,Riemann zeta function ,Algebra ,symbols.namesake ,11M32 (Primary) 11M35 (Secondary) ,Functional equation ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,L-function ,Mathematics::Representation Theory ,Mathematics - Abstract
In this paper, we introduce the normalized Shintani L-function of several variables by an integral representation and prove its functional equation. The Shintani L-function is a generalization to several variables of the Hurwitz-Lerch zeta function and the functional equation given in this paper is a generalization of the functional equation of Hurwitz-Lerch zeta function. In addition to the functional equation, we give special values of the normalized Shintani L-function at non-positive integers and some positive integers.
- Published
- 2015
34. Multiplicative forms and Spencer operators
- Author
-
Crainic, Marius, Salazar Pinzon, Maria, Struchiner, Ivan, Fundamental mathematics, Sub Fundamental Mathematics, Sub Algebra,Geometry&Mathem. Logic begr., Fundamental mathematics, Sub Fundamental Mathematics, and Sub Algebra,Geometry&Mathem. Logic begr.
- Subjects
Mathematics - Differential Geometry ,Functor ,Multiplicative forms ,General Mathematics ,Modulo ,Multiplicative function ,Pfaffian ,Poisson geometry ,Algebra ,Operator (computer programming) ,Differential Geometry (math.DG) ,Symplectic groupoids ,Linearization ,Lie groupoids ,FOS: Mathematics ,58H05 (primary) ,Point (geometry) ,Mathematics::Differential Geometry ,ANÁLISE GLOBAL ,Mathematics - Abstract
Motivated by our attempt to recast Cartan's work on Lie pseudogroups in a more global and modern language, we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem. From this point of view, the novelty of this paper is that we study forms with coefficients. However, the main contribution of this paper is conceptual: the finding of the relationship between multiplicative forms and Cartan's work, which provides a completely new approach to integrability theorems for multiplicative forms. Back to Cartan, the multiplicative point of view shows that, modulo Lie's functor, the Cartan Pfaffian system (itself a multiplicative form with coefficients!) is the same thing as the classical Spencer operator.
- Published
- 2014
35. On a result of Moeglin and Waldspurger in residual characteristic 2
- Author
-
Sandeep Varma
- Subjects
Admissible representation ,Pure mathematics ,Character (mathematics) ,General Mathematics ,Mathematical analysis ,Degenerate energy levels ,Nilpotent orbit ,Field (mathematics) ,Reductive group ,Identity element ,Residual ,Mathematics - Abstract
Let \(F\) be a \(p\)-adic field, \(\mathbf G\) a connected reductive group over \(F\), and \(\pi \) an irreducible admissible representation of \(\mathbf G(F)\). A result of Moeglin and Waldspurger states that, if the residual characteristic of \(F\) is different from \(2\), then the ‘leading’ coefficients in the character expansion of \(\pi \) at the identity element of \(\mathbf G(F)\) give the dimensions of certain spaces of degenerate Whittaker forms. In this paper, we extend their result to residual characteristic 2. The outline of the proof is the same as in the original paper of Moeglin and Waldspurger, but certain constructions are modified to accommodate the case of even residual characteristic.
- Published
- 2014
36. First eigenvalue of Laplace operator on locally symmetric space
- Author
-
Yufa Huang
- Subjects
Combinatorics ,Admissible representation ,Square-integrable function ,General Mathematics ,Quantum mechanics ,Symmetric space ,Schur's lemma ,Quadratic field ,Spectral gap ,Laplace operator ,Maximal compact subgroup ,Mathematics - Abstract
This paper is concerned with the lower bound for the first positive eigenvalue \(\lambda _{1}(\Gamma )\) of the Laplace operator \(\bigtriangleup \) on the space of square integrable functions on some locally symmetric space \(\Gamma \backslash G / K\), where \(G=SO_{2n+1}(\mathbb {C})\), \(K\) is a maximal compact subgroup of \(G\) and \(\Gamma \) is a lattice in \(G\) which arises from some imaginary quadratic field. It is well-known that the Casimir operator \(\fancyscript{C}\) acts by a scalar \(-\lambda _{\pi }\) on every irreducible admissible representation \(\pi \) of \(G\) by Diximir’s Schur Lemma. In this paper, we prove that when \(n\ge 5\), there exists a spectral gap between \(\lambda _{1} (\Gamma )\) and the infimum \(\lambda _{1}(G)\) of \(\lambda _{\pi }\) when \(\pi \) passes through all non-trivial irreducible spherical unitary representations of \(G\).
- Published
- 2014
37. Hurwitz-type bound, knot surgery, and smooth $${\mathbb{S }}^1$$ S 1 -four-manifolds
- Author
-
Weimin Chen
- Subjects
medicine.medical_specialty ,Fundamental group ,General Mathematics ,010102 general mathematics ,Homology (mathematics) ,Mathematics::Geometric Topology ,01 natural sciences ,Manifold ,Surgery ,0103 physical sciences ,medicine ,Equivariant map ,Intersection form ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Mathematics::Symplectic Geometry ,Mathematics ,Knot (mathematics) - Abstract
In this paper we prove several related results concerning smooth \(\mathbb{Z }_p\) or \(\mathbb{S }^1\) actions on \(4\)-manifolds. We show that there exists an infinite sequence of smooth \(4\)-manifolds \(X_n\), \(n\ge 2\), which have the same integral homology and intersection form and the same Seiberg-Witten invariant, such that each \(X_n\) supports no smooth \(\mathbb{S }^1\)-actions but admits a smooth \(\mathbb{Z }_n\)-action. In order to construct such manifolds, we devise a method for annihilating smooth \(\mathbb{S }^1\)-actions on \(4\)-manifolds using Fintushel-Stern knot surgery, and apply it to the Kodaira-Thurston manifold in an equivariant setting. Finally, the method for annihilating smooth \(\mathbb{S }^1\)-actions relies on a new obstruction we derived in this paper for existence of smooth \(\mathbb{S }^1\)-actions on a \(4\)-manifold: the fundamental group of a smooth \(\mathbb{S }^1\)-four-manifold with nonzero Seiberg-Witten invariant must have infinite center. We also include a discussion on various analogous or related results in the literature, including locally linear actions or smooth actions in dimensions other than four.
- Published
- 2013
38. On locally analytic Beilinson–Bernstein localization and the canonical dimension
- Author
-
Tobias Schmidt
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Localization theorem ,Dimension (graph theory) ,Congruence (manifolds) ,Field (mathematics) ,Reductive group ,Type (model theory) ,Prime (order theory) ,Mathematics - Abstract
Let $$\mathbf{G}$$ be a connected split reductive group over a $$p$$ -adic field. In the first part of the paper we prove, under certain assumptions on $$\mathbf{G}$$ and the prime $$p$$ , a localization theorem of Beilinson–Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of $$\mathbf{G}$$ . In doing so we take up and extend some recent methods and results of Ardakov–Wadsley on completed universal enveloping algebras (Ardakov and Wadsley, Ann. Math., 2013) to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith’s theorem on the canonical dimension.
- Published
- 2013
39. An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index
- Author
-
Hochs, Peter, Wang, Bai-Ling, and Wang, Hang
- Subjects
Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,Mathematics::K-Theory and Homology ,General Mathematics ,Mathematics - K-Theory and Homology ,Mathematics - Operator Algebras ,FOS: Mathematics ,K-Theory and Homology (math.KT) ,Operator Algebras (math.OA) ,Mathematics - Abstract
Consider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. Then an equivariant Dirac-type operator $D$ on $M$ under a suitable boundary condition has an equivariant index $\operatorname{index}_G(D)$ in the $K$-theory of the reduced group $C^*$-algebra $C^*_rG$ of $G$. This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index $\operatorname{index}_g(D)$ was defined for an element $g \in G$, in terms of a parametrix of $D$ and a trace associated to $g$. An Atiyah-Patodi-Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, $\tau_g(\operatorname{index}_G(D)) = \operatorname{index}_g(D)$, for a trace $\tau_g$ defined by the orbital integral over the conjugacy class of $g$. This implies that the index theorem from part I yields information about the $K$-theoretic index $\operatorname{index}_G(D)$. It also shows that $\operatorname{index}_g(D)$ is a homotopy-invariant quantity., Comment: 44 pages. The first version of the preprint 1904.11146 was split into two parts, this is the second part
- Published
- 2022
40. On Oka’s extra-zero problem and examples
- Author
-
Junjiro Noguchi, Makoto Abe, and Sachiko Hamano
- Subjects
Discrete mathematics ,Pure mathematics ,General Mathematics ,Dimension (graph theory) ,Cousin ,Zero (complex analysis) ,Disjoint sets ,Special case ,Mathematics - Abstract
After the solution of Cousin II problem by Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and some new questions.
- Published
- 2012
41. Eichler integrals, period relations and Jacobi forms
- Author
-
YoungJu Choie and Subong Lim
- Subjects
Algebra ,Jacobi identity ,Ramanujan theta function ,symbols.namesake ,Jacobi operator ,General Mathematics ,symbols ,Holomorphic function ,Theta function ,Jacobi group ,Jacobi integral ,Mathematics ,Jacobi elliptic functions - Abstract
This paper contains three main results: the first one is to derive two “period relations” and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of “Jacobi integrals” on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincare series explicitly. This is to say that every holomorphic function with “period relations” is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607–632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the “Mordell integrals” associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et al. in Commun Math Phys 255(2):469–512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.
- Published
- 2011
42. Algebraic groups over the field with one element
- Author
-
Oliver Lorscheid
- Subjects
Discrete mathematics ,Weyl group ,Mathematics::Commutative Algebra ,Group (mathematics) ,General Mathematics ,Toric variety ,Field (mathematics) ,Group Theory (math.GR) ,Field with one element ,Reductive group ,Mathematics - Algebraic Geometry ,symbols.namesake ,Group scheme ,Algebraic group ,FOS: Mathematics ,symbols ,Computer Science::Symbolic Computation ,Mathematics - Group Theory ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we provide a first realization of an idea of Jacques Tits from a 1956 paper, which first mentioned that there should be a field of characteristique une, which is now called $${\mathbb{F}_1}$$ , the field with one element. This idea was that every split reductive group scheme over $${\mathbb{Z}}$$ should descend to $${\mathbb{F}_1}$$ , and its group of $${\mathbb{F}_1}$$ -rational points should be its Weyl group. We connect the notion of a torified scheme to the notion of $${\mathbb{F}_1}$$ -schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive group schemes as $${\mathbb{F}_1}$$ -schemes. We endow the class of $${\mathbb{F}_1}$$ -schemes with two classes of morphisms, one leading to a satisfying notion of $${\mathbb{F}_1}$$ -rational points, the other leading to the notion of an algebraic group over $${\mathbb{F}_1}$$ such that every split reductive group is defined as an algebraic group over $${\mathbb{F}_1}$$ . Furthermore, we show that certain combinatorics that are expected from parabolic subgroups of GL(n) and Grassmann varieties are realized in this theory.
- Published
- 2011
43. On the Picard principle for Δ + μ
- Author
-
Ivan Netuka and Wolfhard Hansen
- Subjects
Unit sphere ,Combinatorics ,General Mathematics ,Quantum mechanics ,Mathematics - Abstract
Given a (local) Kato measure μ on \({{\mathbb{R}^d} \setminus \{0\},\,d \ge 2}\), let \({{\mathcal H}_0^{\Delta+\mu}(U)}\) be the convex cone of all continuous real solutions u ≥ 0 to the equation Δu + uμ = 0 on the punctured unit ball U satisfying \({\lim_{|x|\to 1} u(x)=0}\). It is shown that \({{\mathcal H}_0^{\Delta+\mu}(U)\ne \{0\}}\) if and only if the operator \({f\mapsto \int_U G(\cdot,y)f(y)\,d\mu(y)}\), where G denotes the Green function on U, is bounded on \({\mathcal L^2(U,\mu)}\) and has a norm which is at most one. Moreover, extremal rays in \({{\mathcal H}_0^{\Delta+\mu}(U)}\) are characterized and it is proven that Δ + μ satisfies the Picard principle on U, that is, that \({{\mathcal H}_0^{\Delta+\mu}(U)}\) consists of one ray, provided there exists a suitable sequence of shells in U such that, on these shells, μ is either small or not too far from being radial. Further, it is shown that the verification of the Picard principle can be localized. Several results on L2-(sub)eigenfunctions and 3G-inequalities which are used in the paper, but may be of independent interest, are proved at the end of the paper.
- Published
- 2011
44. Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature
- Author
-
Shaoqiang Deng and Zhiguang Hu
- Subjects
Large class ,Pure mathematics ,General Mathematics ,Isotropy ,Mathematical analysis ,Rigidity (psychology) ,Space (mathematics) ,Curvature ,Homogeneous ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,Mathematics ,Scalar curvature ,Flag (geometry) - Abstract
In this paper, we will give a complete classification of homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. This results in a large class of Finsler spaces with non-constant positive flag curvature. At the final part of the paper, we prove a rigidity result asserting that a homogeneous Randers space with almost isotropic S-curvature and negative Ricci scalar must be Riemannian.
- Published
- 2011
45. Strict and nonstrict positivity of direct image bundles
- Author
-
Bo Berndtsson
- Subjects
Vector-valued differential form ,Pure mathematics ,Mathematics::Complex Variables ,General Mathematics ,Mathematical analysis ,Connection (vector bundle) ,Vector bundle ,Frame bundle ,Principal bundle ,Tautological line bundle ,Mathematics::Algebraic Geometry ,Line bundle ,Normal bundle ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
This paper is a sequel to (Berndtsson in Ann Math 169:531-560, 2009). In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kahler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we address the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. We prove that this is so provided the twisting line bundle is strictly positive along fibers, but not in general.
- Published
- 2010
46. Enriques surfaces and Jacobian elliptic K3 surfaces
- Author
-
Matthias Schütt and Klaus Hulek
- Subjects
General Mathematics ,Enriques surface ,Fibration ,Automorphism ,K3 surface ,Algebra ,symbols.namesake ,Mathematics::Algebraic Geometry ,Quadratic equation ,Jacobian matrix and determinant ,symbols ,Point (geometry) ,Brauer group ,Mathematics - Abstract
This paper proposes a new geometric construction of Enriques surfaces. Its starting point are K3 surfaces with Jacobian elliptic fibration which arise from rational elliptic surfaces by a quadratic base change. The Enriques surfaces obtained in this way are characterised by elliptic fibrations with a rational curve as bisection which splits into two sections on the covering K3 surface. The construction has applications to the study of Enriques surfaces with specific automorphisms. It also allows us to answer a question of Beauville about Enriques surfaces whose Brauer groups show an exceptional behaviour. In a forthcoming paper, we will study arithmetic consequences of our construction.
- Published
- 2010
47. Rectifiability of flat chains in Banach spaces with coefficients in Z p
- Author
-
Stefan Wenger, Luigi Ambrosio, Ambrosio, Luigi, and Wenger, S.
- Subjects
Class (set theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,rectifiability ,Banach space ,01 natural sciences ,Measure (mathematics) ,Ambient space ,Metric space ,Mathematics - Classical Analysis and ODEs ,flat chains ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Mathematics::Metric Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Quotient - Abstract
Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and with respect to weaker mass bounds. A crucial tool in many proofs is the isoperimetric inequality proved by Ambrosio-Katz with universal constants.
- Published
- 2010
48. The HcscK equations in symplectic coordinates
- Author
-
Carlo Scarpa and Jacopo Stoppa
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Kaehler metrics, scalar curvature, moment maps ,General Mathematics ,Space (mathematics) ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0502 economics and business ,FOS: Mathematics ,scalar curvature ,Tensor ,Uniqueness ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Moment map ,050205 econometrics ,Mathematics ,010102 general mathematics ,05 social sciences ,Manifold ,Kaehler metrics ,moment maps ,Differential Geometry (math.DG) ,Settore MAT/03 - Geometria ,Mathematics::Differential Geometry ,Symplectic geometry ,Scalar curvature - Abstract
The Donaldson-Fujiki K\"ahler reduction of the space of compatible almost complex structures, leading to the interpretation of the scalar curvature of K\"ahler metrics as a moment map, can be lifted canonically to a hyperk\"ahler reduction. Donaldson proposed to consider the corresponding vanishing moment map conditions as (fully nonlinear) analogues of Hitchin's equations, for which the underlying bundle is replaced by a polarised manifold. However this construction is well understood only in the case of complex curves. In this paper we study Donaldson's hyperk\"ahler reduction on abelian varieties and toric manifolds. We obtain a decoupling result, a variational characterisation, a relation to $K$-stability in the toric case, and prove existence and uniqueness under suitable assumptions on the ``Higgs tensor''. We also discuss some aspects of the analogy with Higgs bundles., Comment: 43 pages. Accepted version. Upgraded the previous results to consider uniform toric K-stability
- Published
- 2021
49. Deformations of Dolbeault cohomology classes
- Author
-
Wei Xia
- Subjects
Mathematics - Differential Geometry ,Power series ,Pure mathematics ,Parallelizable manifold ,Mathematics - Complex Variables ,Mathematics::Complex Variables ,General Mathematics ,Deformation theory ,Holomorphic function ,Dolbeault cohomology ,Extension (predicate logic) ,Mathematics::Algebraic Topology ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,Simple (abstract algebra) ,Tensor (intrinsic definition) ,FOS: Mathematics ,32G05, 32L10, 55N30, 32G99 ,Mathematics::Differential Geometry ,Complex Variables (math.CV) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this paper, we establish a deformation theory for Dolbeault cohomology classes valued in holomorphic tensor bundles. We prove the extension equation which will play the role of Maurer-Cartan equation. Following the classical theory of Kodaira-Spencer-Kuranishi, we construct a canonical complete family of deformations by using the power series method. We also prove a simple relation between the existence of deformations and the varying of the dimensions of Dolbeault cohomology. The deformations of $(p,q)$-forms is shown to be unobstructed under some mild conditions. By analyzing Nakamura's example of complex parallelizable manifolds, we will see that the deformation theory developed in this work provides precise explanations to the jumping phenomenon of Dolbeault cohomology., 46 pages, published version (https://doi.org/10.1007/s00209-021-02900-w)
- Published
- 2021
50. An analogue of a result of Tits for transvection groups
- Author
-
Pratyusha Chattopadhyay
- Subjects
Pure mathematics ,Symplectic group ,Group (mathematics) ,General Mathematics ,Elementary proof ,Context (language use) ,Special case ,Square (algebra) ,Mathematics ,Transvection - Abstract
In (L’Enseignement Math 61(2):151–159, 2015) Nica presented an elementary proof of a result which says that the relative elementary linear group with respect to square of an ideal of a ring is a subset of the true relative elementary linear group. The original result was proved by Tits (C R Acad Sci Paris Ser A 283:693–695, 1976) in the much general context of Chevalley groups. In this paper we prove analogues of this result of Tits for transvection groups. We also obtain an elementary proof of a special case of Tits’s result, namely the case of elementary symplectic group, using commutator identities for generators of this group.
- Published
- 2021
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