68 results
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2. Essential regularity of the model space for the Weil–Petersson metric
- Author
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Georgios Daskalopoulos and Chikako Mese
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Metric (mathematics) ,Mathematical analysis ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,Space (mathematics) ,01 natural sciences ,Mathematics - Abstract
This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion 𝒯 ¯ {\overline{\mathcal{T}}} of Teichmüller space. The boundary of 𝒯 ¯ {\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space 𝐇 ¯ {\overline{\bf H}} called the model space. The significance of 𝐇 ¯ {\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of 𝒯 ¯ {\overline{\mathcal{T}}} . The main result of the paper is that certain subsets of 𝐇 ¯ {\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on 𝐇 ¯ {\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.
- Published
- 2016
3. The Asaeda–Haagerup fusion categories
- Author
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Noah Snyder, Pinhas Grossman, and Masaki Izumi
- Subjects
Class (set theory) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Center (group theory) ,01 natural sciences ,Subfactor ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Morita equivalence ,Symmetry (geometry) ,Orbifold ,Quotient ,Mathematics - Abstract
The classification of subfactors of small index revealed several new subfactors. The first subfactor above index 4, the Haagerup subfactor, is increasingly well understood and appears to lie in a (discrete) infinite family of subfactors where the ℤ \mathbb{Z} /3 ℤ \mathbb{Z} symmetry is replaced by other finite abelian groups. The goal of this paper is to give a similarly good description of the Asaeda–Haagerup subfactor which emerged from our study of its Brauer–Picard groupoid. More specifically, we construct a new subfactor 𝒮 {\mathcal{S}} which is a ℤ \mathbb{Z} /4 ℤ \mathbb{Z} × \times ℤ \mathbb{Z} /2 ℤ \mathbb{Z} analogue of the Haagerup subfactor and we show that the even parts of the Asaeda–Haagerup subfactor are higher Morita equivalent to an orbifold quotient of 𝒮 {\mathcal{S}} . This gives a new construction of the Asaeda–Haagerup subfactor which is much more symmetric and easier to work with than the original construction. As a consequence, we can settle many open questions about the Asaeda–Haagerup subfactor: calculating its Drinfeld center, classifying all extensions of the Asaeda–Haagerup fusion categories, finding the full higher Morita equivalence class of the Asaeda–Haagerup fusion categories, and finding intermediate subfactor lattices for subfactors coming from the Asaeda–Haagerup categories. The details of the applications will be given in subsequent papers.
- Published
- 2016
4. Fermat curves and a refinement of the reciprocity law on cyclotomic units
- Author
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Tomokazu Kashio
- Subjects
Fermat's Last Theorem ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Reciprocity law ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s = 0 s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.
- Published
- 2016
5. Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids
- Author
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Maxence Mayrand
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Holomorphic function ,Kähler manifold ,01 natural sciences ,Section (fiber bundle) ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic manifold ,Mathematics::Complex Variables ,Applied Mathematics ,010102 general mathematics ,Zero (complex analysis) ,53D17, 53C26, 53C28, 32G05 ,Submanifold ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,Symplectic Geometry (math.SG) ,Cotangent bundle ,Mathematics::Differential Geometry ,010307 mathematical physics ,Symplectic geometry - Abstract
The first part of this paper is a generalization of the Feix-Kaledin theorem on the existence of a hyperkahler metric on a neighbourhood of the zero section of the cotangent bundle of a Kahler manifold. We show that the problem of constructing a hyperkahler structure on a neighbourhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix-Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kahler type has a hyperkahler structure on a neighbourhood of its identity section. More generally, we reduce the existence of a hyperkahler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin's unobstructedness theorem., 20 pages. To appear in Journal fur die reine und angewandte Mathematik (Crelle's Journal)
- Published
- 2021
6. Modular symbols for Teichmüller curves
- Author
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Curtis T. McMullen
- Subjects
Algebra ,business.industry ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0502 economics and business ,05 social sciences ,0101 mathematics ,Modular design ,business ,01 natural sciences ,050203 business & management ,Mathematics - Abstract
This paper introduces a space of nonabelian modular symbols 𝒮 ( V ) {{\mathcal{S}}(V)} attached to any hyperbolic Riemann surface V, and applies it to obtain new results on polygonal billiards and holomorphic 1-forms. In particular, it shows the scarring behavior of periodic trajectories for billiards in a regular polygon is governed by a countable set of measures homeomorphic to ω ω + 1 {\omega^{\omega}+1} .
- Published
- 2021
7. On the geometric André–Oort conjecture for variations of Hodge structures
- Author
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Jiaming Chen
- Subjects
André–Oort conjecture ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Let 𝕍 {{\mathbb{V}}} be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety S. In this paper, we show that the union of the non-factor special subvarieties for ( S , 𝕍 ) {(S,{\mathbb{V}})} , which are of Shimura type with dominant period maps, is a finite union of special subvarieties of S. This generalizes previous results of Clozel and Ullmo (2005) and Ullmo (2007) on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part of a conjecture of Klingler on the André–Oort conjecture for variations of Hodge structures.
- Published
- 2021
8. Area minimizing surfaces of bounded genus in metric spaces
- Author
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Stefan Wenger and Martin Fitzi
- Subjects
Mathematics - Differential Geometry ,Surface (mathematics) ,Pure mathematics ,General Mathematics ,Boundary (topology) ,01 natural sciences ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mathematics - Metric Geometry ,Genus (mathematics) ,FOS: Mathematics ,0101 mathematics ,49Q05, 53C23 ,Mathematics ,Euclidean space ,Applied Mathematics ,010102 general mathematics ,Metric Geometry (math.MG) ,Jordan curve theorem ,010101 applied mathematics ,Metric space ,Differential Geometry (math.DG) ,Bounded function ,symbols ,Isoperimetric inequality ,Analysis of PDEs (math.AP) - Abstract
The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.
- Published
- 2021
9. On birational boundedness of foliated surfaces
- Author
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Christopher D. Hacon and Adrian Langer
- Subjects
Polynomial (hyperelastic model) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Function (mathematics) ,Type (model theory) ,01 natural sciences ,Combinatorics ,Mathematics - Algebraic Geometry ,Integer ,0103 physical sciences ,FOS: Mathematics ,Foliation (geology) ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer {N>0} such that if {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all {m\in\mathbb{Z}_{\geq 0}} , then {|mK_{\mathcal{F}}|} defines a birational map for all {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.
- Published
- 2021
10. Half-space theorems for the Allen–Cahn equation and related problems
- Author
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Yong Liu, Kelei Wang, Juncheng Wei, François Hamel, and Pieralberto Sicbaldi
- Subjects
0209 industrial biotechnology ,Pure mathematics ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Rigidity (psychology) ,02 engineering and technology ,Half-space ,01 natural sciences ,020901 industrial engineering & automation ,Bounded function ,0101 mathematics ,Allen–Cahn equation ,Mathematics - Abstract
In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in {\mathbb{R}^{n}} , whose level set {\{u=0\}} is contained in a half-space. If {n\leq 3} , we prove that the solution must be one-dimensional. In dimension {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.
- Published
- 2021
11. Eichler cohomology and zeros of polynomials associated to derivatives of L-functions
- Author
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Larry Rolen and Nikolaos Diamantis
- Subjects
Cusp (singularity) ,Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular form ,State (functional analysis) ,01 natural sciences ,Cohomology ,010101 applied mathematics ,symbols.namesake ,Eisenstein series ,FOS: Mathematics ,symbols ,Number Theory (math.NT) ,0101 mathematics ,Special case ,Mathematics - Abstract
In recent years, a number of papers have been devoted to the study of roots of period polynomials of modular forms. Here, we study cohomological analogues of the Eichler-Shimura period polynomials corresponding to higher $L$-derivatives. We state general conjectures about the locations of the roots of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. We prove a special case of this conjecture in the case of Eisenstein series., 21 pages
- Published
- 2021
12. On the regular-convexity of Ricci shrinker limit spaces
- Author
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Bing Wang, Shaosai Huang, and Yu Li
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,Structure (category theory) ,Space (mathematics) ,01 natural sciences ,Upper and lower bounds ,Convexity ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,Limit (mathematics) ,0101 mathematics ,Smoothing ,Ricci curvature ,Mathematics - Abstract
In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions., Comment: revised version, to appear in J. Reine Angew. Math
- Published
- 2020
13. Endoscopic character identities for metaplectic groups
- Author
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Caihua Luo
- Subjects
Pure mathematics ,Formalism (philosophy) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Character (mathematics) ,Metaplectic group ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper, we prove the conjectural endoscopic character identities for tempered representations of metaplectic group $Mp_{2n}$ based on the formalism of endoscopy theory by J. Adams, D. Renard and W.W. Li., This is part of my thesis. Submitted
- Published
- 2020
14. Counterexamples to the tilting and (p,r)-filtration conjectures
- Author
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Paul Sobaje, Daniel K. Nakano, Christopher P. Bendel, and Cornelius Pillen
- Subjects
Pure mathematics ,Conjecture ,Weyl module ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Simple (abstract algebra) ,Algebraic group ,0103 physical sciences ,Filtration (mathematics) ,010307 mathematical physics ,0101 mathematics ,Indecomposable module ,Kernel (category theory) ,Mathematics ,Counterexample - Abstract
In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the ( p , r ) {(p,r)} -Filtration Conjecture.
- Published
- 2019
15. Almost isotropic Kähler manifolds
- Author
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Ralf Spatzier, Krishnan Shankar, and Benjamin Schmidt
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Isotropy ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematical physics ,Mathematics - Abstract
Let M be a complete Riemannian manifold and suppose p ∈ M {p\in M} . For each unit vector v ∈ T p M {v\in T_{p}M} , the Jacobi operator, 𝒥 v : v ⊥ → v ⊥ {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, 𝒥 v ( w ) = R ( w , v ) v {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant κ p ∈ ℝ {\kappa_{p}\in{\mathbb{R}}} such that 𝒥 v = κ p Id v ⊥ {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector v ∈ T p M {v\in T_{p}M} . If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each p ∈ M {p\in M} , there exists a constant κ p ∈ ℝ {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators 𝒥 v {\mathcal{J}_{v}} satisfy rank ( 𝒥 v - κ p Id v ⊥ ) ≤ 1 {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector v ∈ T p M {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension d = 2 n ⩾ 4 {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to ℂ n - 1 {{\mathbb{C}}^{n-1}} .
- Published
- 2019
16. Gap theorem on Kähler manifolds with nonnegative orthogonal bisectional curvature
- Author
-
Yanyan Niu and Lei Ni
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics::Differential Geometry ,010307 mathematical physics ,Gap theorem ,0101 mathematics ,Curvature ,01 natural sciences ,Mathematics - Abstract
In this paper we prove a gap theorem for Kähler manifolds with nonnegative orthogonal bisectional curvature and nonnegative Ricci curvature, which generalizes an earlier result of the first author [L. Ni, An optimal gap theorem, Invent. Math. 189 2012, 3, 737–761]. We also prove a Liouville theorem for plurisubharmonic functions on such a manifold, which generalizes a previous result of L.-F. Tam and the first author [L. Ni and L.-F. Tam, Plurisubharmonic functions and the structure of complete Kähler manifolds with nonnegative curvature, J. Differential Geom. 64 2003, 3, 457–524] and complements a recent result of Liu [G. Liu, Three-circle theorem and dimension estimate for holomorphic functions on Kähler manifolds, Duke Math. J. 165 2016, 15, 2899–2919].
- Published
- 2019
17. Dehn functions and Hölder extensions in asymptotic cones
- Author
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Robert Young, Stefan Wenger, and Alexander Lytchak
- Subjects
Mathematics::Group Theory ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics::Geometric Topology ,01 natural sciences ,Mathematics - Abstract
The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.
- Published
- 2019
18. Identifiability of homogeneous polynomials and Cremona transformations
- Author
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Massimiliano Mella and Francesco Galuppi
- Subjects
Pure mathematics ,Degree (graph theory) ,polynomials, idetifiability, Waring decomposition ,polynomials ,14J70 (Primary), 14N05, 14E05 (Secondary) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Special class ,01 natural sciences ,NO ,idetifiability ,Mathematics - Algebraic Geometry ,PE1_2 ,Waring decomposition ,Homogeneous ,Homogeneous polynomial ,FOS: Mathematics ,Identifiability ,General polynomial ,0101 mathematics ,PE1_5 ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
A homogeneous polynomial of degree $d$ in $n+1$ variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of $d$ and $n$ for which a general polynomial of degree $d$ in $n+1$ variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces., 25 pages. Proof of Proposition 17 and Lemma 18 fixed, some typos corrected
- Published
- 2017
19. The structure of spaces with Bakry–Émery Ricci curvature bounded below
- Author
-
Feng Wang and Xiaohua Zhu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,01 natural sciences ,Bounded function ,0103 physical sciences ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics - Abstract
We explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space as in a paper of Cheeger, Colding and Tian. Our results will be applied to study the limit spaces for a sequence of Kähler metrics arising from solutions of certain complex Monge–Ampère equations for the existence problem of Kähler–Ricci solitons on a Fano manifold via the continuity method.
- Published
- 2017
20. 𝔸-curves on log smooth varieties
- Author
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Qile Chen and Yi Zhu
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study 𝔸 1 {\mathbb{A}^{1}} -connected varieties from log geometry point of view, and prove a criterion for 𝔸 1 {\mathbb{A}^{1}} -connectedness. As applications, we provide many interesting examples of 𝔸 1 {\mathbb{A}^{1}} -connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces.
- Published
- 2017
21. Gravitational instantons with faster than quadratic curvature decay (II)
- Author
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Gao Chen and Xiuxiong Chen
- Subjects
Mathematics - Differential Geometry ,Instanton ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Curvature ,01 natural sciences ,Gravitation ,Quadratic equation ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
This is our second paper in a series to study gravitational instantons, i.e. complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay. We prove two main theorems: 1.The asymptotic rate of gravitational instantons to the standard models can be improved automatically. 2.Any ALF-D_k gravitational instanton must be the Cherkis-Hitchin-Ivanov-Kapustin-Lindstr\"om-Ro\v{c}ek metric., Comment: We add a corollary and the applications, correct the asymptotic rate of the multi-Taub-NUT metric
- Published
- 2017
22. On the global Gan–Gross–Prasad conjecture for unitary groups: Approximating smooth transfer of Jacquet–Rallis
- Author
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Hang Xue
- Subjects
Pure mathematics ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Unitary state ,Combinatorics ,Transfer (group theory) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Prasad ,Mathematics - Abstract
Zhang proved the global Gan–Gross–Prasad conjecture for U ( n + 1 ) × U ( n ) \operatorname{U}(n+1)\times\operatorname{U}(n) under some local conditions [19]. One of the conditions is that the unitary groups are split at the archimedean places. We remove this assumption at the archimedean places in this paper.
- Published
- 2017
23. Embedded minimal surfaces of finite topology
- Author
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Joaquín Pérez and William H. Meeks
- Subjects
Mathematics - Differential Geometry ,0209 industrial biotechnology ,Finite topological space ,Pure mathematics ,Minimal surface ,Mathematics::Complex Variables ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,Conformal map ,02 engineering and technology ,Type (model theory) ,53A10, 49Q05, 53C42 ,01 natural sciences ,020901 industrial engineering & automation ,Differential Geometry (math.DG) ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,Finite set ,Mathematics ,Meromorphic function - Abstract
In this paper we prove that a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface $\overline{M}$ with boundary punctured in a finite number of interior points and that $M$ can be represented in terms of meromorphic data on its conformal completion $\overline{M}$. In particular, we demonstrate that $M$ is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of $M$., Comment: 33 pages, 6 figures. Comments are welcome
- Published
- 2017
24. Supercuspidal representations and preservation principle of theta correspondence
- Author
-
Shu-Yen Pan
- Subjects
Mathematics::Number Theory ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Calculus ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,01 natural sciences ,Mathematics - Abstract
The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p-adic classical groups. In this paper, we give an explicit characterization of the chain starting from an irreducible supercuspidal representations of a unitary group of one variable or an orthogonal group of two variables. In particular, we define the Lusztig-like correspondence of generic cuspidal data for p-adic groups and establish its relation with local theta correspondence of supercuspidal representations for p-adic dual pairs.
- Published
- 2016
25. Limit lamination theorems for H-surfaces
- Author
-
Giuseppe Tinaglia and William H. Meeks
- Subjects
Mathematics - Differential Geometry ,0209 industrial biotechnology ,Pure mathematics ,Minimal surface ,Mean curvature ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Structure (category theory) ,53A10 ,02 engineering and technology ,Lamination (topology) ,Infinity ,01 natural sciences ,020901 industrial engineering & automation ,Differential Geometry (math.DG) ,Genus (mathematics) ,FOS: Mathematics ,Mathematics::Differential Geometry ,Limit (mathematics) ,0101 mathematics ,Constant (mathematics) ,Mathematics ,media_common - Abstract
In this paper we prove some general results on constant mean curvature lamination limits of certain sequences of compact surfaces $M_n$ embedded in $\mathbb R^3$ with constant mean curvature $H_n$ and fixed finite genus, when the boundaries of these surfaces tend to infinity. Two of these theorems generalize to the non-zero constant mean curvature case, similar structure theorems by Colding and Minicozzi in~[6,8] for limits of sequences of minimal surfaces of fixed finite genus., Comment: Details added at referee's request. To appear in Crelle
- Published
- 2016
26. Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants
- Author
-
Yukinobu Toda and Dulip Piyaratne
- Subjects
Pure mathematics ,Conjecture ,Generalization ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Moduli ,Stability conditions ,Mathematics::Algebraic Geometry ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Algebraic number ,Quotient ,Mathematics - Abstract
In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.
- Published
- 2016
27. Geometric structures of collapsing Riemannian manifolds II
- Author
-
Aaron Naber and Gang Tian
- Subjects
010101 applied mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper we study collapsing sequences M i → GH X M_{i}\xrightarrow{\rm GH}X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M i {M_{i}} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generalization of Gromov’s Almost Flat Theorem and prove new Ricci pinching theorems which extend those known in the noncollapsed setting. There are also interesting topological consequences to the structure.
- Published
- 2016
28. Behavior of canonical divisors under purely inseparable base changes
- Author
-
Hiromu Tanaka
- Subjects
Divisor ,Fiber (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Base (group theory) ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Chain (algebraic topology) ,Cone (topology) ,FOS: Mathematics ,0101 mathematics ,Variety (universal algebra) ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Let $k$ be an imperfect field. Let $X$ be a regular variety over $k$ and set $Y$ to be the normalization of $(X \times_k k^{1/p^{\infty}})_{{\rm red}}$. In this paper, we show that $K_Y+C=f^*K_X$ for some effective divisor $C$ on $Y$. We obtain the following three applications. First, we show that a $K_X$-trivial fiber space with non-normal fibers is uniruled. Second, we prove that general fibers of Mori fiber spaces are rationally chain connected. Third, we obtain a weakening of the cone theorem for surfaces and threefolds defined over an imperfect field., Comment: 33 pages; v2: minor revisions; v3: minor changes, v4: the numberings of the theorems are changed
- Published
- 2016
29. Equivariant basic cohomology of Riemannian foliations
- Author
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Dirk Töben and Oliver Goerisches
- Subjects
Pure mathematics ,Betti number ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lie group ,Fixed point ,Mathematics::Algebraic Topology ,01 natural sciences ,Manifold ,Cohomology ,Transfer (group theory) ,0103 physical sciences ,Equivariant cohomology ,Equivariant map ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The basic cohomology of a complete Riemannian foliation with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing foliations): We show that the total basic Betti number of the union C of the closed leaves is smaller than or equal to the total basic Betti number of the foliated manifold, and we give sufficient conditions for equality. If there is a basic Morse–Bott function with critical set equal to C, we can compute the basic cohomology explicitly. Another case in which the basic cohomology can be determined is if the space of leaf closures is a simple, convex polytope. Our results are based on Molino’s observation that the existence of non-closed leaves yields a distinguished transverse action on the foliated manifold with fixed point set C. We introduce equivariant basic cohomology of transverse actions in analogy to equivariant cohomology of Lie group actions enabling us to transfer many results from the theory of Lie group actions to Riemannian foliations. The prominent role of the fixed point set in the theory of torus actions explains the relevance of the set C in the basic setting.
- Published
- 2016
30. Algebraic flows on abelian varieties
- Author
-
Emmanuel Ullmo and Andrei Yafaev
- Subjects
Abelian variety ,Pure mathematics ,Subvariety ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Closure (topology) ,01 natural sciences ,Flow (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,010307 mathematical physics ,Algebraic curve ,0101 mathematics ,Abelian group ,Algebraic number ,Topological closure ,Mathematics - Abstract
Let A be an abelian variety. The abelian Ax–Lindemann theorem shows that the Zariski closure of an algebraic flow in A is a translate of an abelian subvariety of A. The paper discusses some conjectures on the usual topological closure of an algebraic flow in A. The main result is a proof of these conjectures when the algebraic flow is given by an algebraic curve.
- Published
- 2016
31. Hausdorff theory of dual approximation on planar curves
- Author
-
Jing-Jing Huang
- Subjects
Pure mathematics ,Class (set theory) ,Mathematics - Number Theory ,11J83 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hausdorff space ,16. Peace & justice ,01 natural sciences ,Manifold ,Dual (category theory) ,0103 physical sciences ,Convergence (routing) ,FOS: Mathematics ,Complete theory ,Hausdorff measure ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Divergence (statistics) ,Mathematics - Abstract
Ten years ago, Beresnevich-Dickinson-Velani initiated a project that develops the general Hausdorff measure theory of dual approximation on non-degenerate manifolds. In particular, they established the divergence part of the theory based on their general ubiquity framework. However, the convergence counterpart of the project remains wide open and represents a major challenging question in the subject. Until recently, it was not even known for any single non-degenerate manifold. In this paper, we settle this problem for all curves in $\mathbb{R}^2$, which represents the first complete theory of its kind for a general class of manifolds., 17 pages, to appear in Crelle's Journal
- Published
- 2015
32. Analysis of gauged Witten equation
- Author
-
Gang Tian and Guangbo Xu
- Subjects
Series (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Superpotential ,Type (model theory) ,01 natural sciences ,Moduli space ,High Energy Physics::Theory ,symbols.namesake ,Mathematics::Algebraic Geometry ,Compact space ,Lagrange multiplier ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematical physics ,Mathematics - Abstract
The gauged Witten equation was essentially introduced by Witten in his formulation of the gauged linear σ-model (GLSM), which explains the so-called Landau–Ginzburg/Calabi–Yau correspondence. This is the first paper in a series towards a mathematical construction of GLSM, in which we set up a proper framework for studying the gauged Witten equation and its perturbations. We also prove several analytical properties of solutions and moduli spaces of the perturbed gauged Witten equation. We prove that solutions have nice asymptotic behavior on cylindrical ends of the domain. Under a good perturbation scheme, the energies of solutions are shown to be uniformly bounded by a constant depending only on the topological type. We prove that the linearization of the perturbed gauged Witten equation is Fredholm, and we calculate its Fredholm index. Finally, we define a notion of stable solutions and prove a compactness theorem for the moduli space of solutions over a fixed domain curve.
- Published
- 2015
33. The completion problem for equivariant K-theory
- Author
-
Amalendu Krishna
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Equivariant K-theory ,Topology ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the Atiyah–Segal completion problem for the equivariant algebraic K-theory. We show that this completion problem has a positive solution for the action of connected groups on smooth projective schemes. In contrast, we show that this problem has a negative solution for non-projective smooth schemes, even if the action has only finite stabilizers.
- Published
- 2015
34. Langlands program for p-adic coefficients and the petits camarades conjecture
- Author
-
Tomoyuki Abe
- Subjects
Pure mathematics ,Conjecture ,Rank (linear algebra) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Langlands program ,Finite field ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Mathematics - Abstract
In this paper, we prove that, if Deligne’s “petits camarades conjecture” holds, then a Langlands type correspondence holds also for p-adic coefficients on a smooth curve over a finite field. As an application, we prove that any overconvergent F-isocrystal of rank less than or equal to 2 on a smooth curve is ι-mixed.
- Published
- 2015
35. Separable monoids in Dqc (X)
- Author
-
Amnon Neeman
- Subjects
Monoid ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Separable space - Abstract
Suppose ( 𝒯 , ⊗ , 𝟙 ) {({\mathscr{T}},\otimes,\mathds{1})} is a tensor triangulated category. In a number of recent articles Balmer defines and explores the notion of “separable tt-rings” in 𝒯 {{\mathscr{T}}} (in this paper we will call them “separable monoids”). The main result of this article is that, if 𝒯 {{\mathscr{T}}} is the derived quasicoherent category of a noetherian scheme X, then the only separable monoids are the pushforwards by étale maps of smashing Bousfield localizations of the structure sheaf.
- Published
- 2015
36. Effective bounds of linear series on algebraic varieties and arithmetic varieties
- Author
-
Xinyi Yuan and Tong Zhang
- Subjects
General Mathematics ,Linear series ,Fibered knot ,01 natural sciences ,Mathematics - Algebraic Geometry ,symbols.namesake ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,Arithmetic ,Algebraic Geometry (math.AG) ,Mathematics ,Mathematics::Commutative Algebra ,Mathematics - Number Theory ,Applied Mathematics ,010102 general mathematics ,Algebraic variety ,14G40, 11G50 ,Hermitian matrix ,Line (geometry) ,symbols ,010307 mathematical physics ,Preprint ,Noether's theorem - Abstract
In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert–Samuel formula and the arithmetic Hilbert–Samuel formula. The treatments are high-dimensional generalizations of [Duke. Math. J. 162 (2013), 1723–1770] and [`Relative Noether inequality on fibered surfaces', preprint 2013]. Similar results are obtained independently by Huayi Chen [`Majorations explicites de fonctions de Hilbert–Samuel géométrique et arithmétique', preprint 2014] with less explicit error terms.
- Published
- 2015
37. Gromov–Witten theory and cycle-valued modular forms
- Author
-
Todor Milanov, Yefeng Shen, and Yongbin Ruan
- Subjects
Discrete mathematics ,Modularity (networks) ,Generalization ,Applied Mathematics ,General Mathematics ,Modulo ,010102 general mathematics ,Modular form ,01 natural sciences ,Cohomology ,Simple (abstract algebra) ,0103 physical sciences ,Gravitational singularity ,010307 mathematical physics ,0101 mathematics ,Orbifold ,Mathematics - Abstract
In this paper, we review Teleman’s work on lifting Givental’s quantization of ℒ + ( 2 ) GL ( H ) {\mathcal{L}^{(2)}_{+}{\rm GL}(H)} action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight ( 3 , 3 , 3 ) (3,3,3) , ( 2 , 4 , 4 ) (2,4,4) , ( 2 , 3 , 6 ) (2,3,6) . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.
- Published
- 2015
38. Pro unitality and pro excision in algebraic K-theory and cyclic homology
- Author
-
Matthew Morrow, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), and Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Noetherian ,Pure mathematics ,19D55 (primary), 16E40, 13D03 (secondary) ,General Mathematics ,Cyclic homology ,Mathematics::Algebraic Topology ,01 natural sciences ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Algebraic number ,Excision theorem ,Commutative property ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Mathematics::Commutative Algebra ,Hochschild homology ,Applied Mathematics ,Unital ,010102 general mathematics ,K-Theory and Homology (math.KT) ,Algebraic K-theory ,Mathematics - K-Theory and Homology ,[MATH.MATH-KT]Mathematics [math]/K-Theory and Homology [math.KT] ,010307 mathematical physics - Abstract
We study pro excision in algebraic K-theory, following Suslin--Wodzicki, Cuntz--Quillen, Corti\~nas, and Geisser--Hesselholt, as well as Artin--Rees and continuity properties of Andr\'e--Quillen, Hochschild, and cyclic homology. Our key tool is to first establish the equivalence of various pro Tor vanishing conditions which appear in the literature. Using this we prove that all ideals of commutative, Noetherian rings are pro unital in a certain sense, and show that such ideals satisfy pro excision in $K$-theory as well as in cyclic and topological cyclic homology. In addition, our techniques yield a strong form of the pro Hochschild--Kostant--Rosenberg theorem, an extension to general base rings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and a generalisation of the Fe\u{\i}gin--Tsygan theorem., Comment: Version 2: The paper now includes generalisations of the Cuntz--Quillen and Fe\u{i}gin--Tsygan theorems on periodic cyclic homology to general base rings of characteristic zero. Various minor improvements and corrections. To appear in Journal fur die reine und angewandte Mathematik
- Published
- 2015
39. The equivariant Tamagawa number conjecture and the extended abelian Stark conjecture
- Author
-
Daniel Vallières
- Subjects
Discrete mathematics ,Pure mathematics ,Conjecture ,Elliott–Halberstam conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Equivariant map ,Beal's conjecture ,0101 mathematics ,Abelian group ,Mathematics ,Arithmetic of abelian varieties - Abstract
The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field ℚ {\mathbb{Q}} , since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14].
- Published
- 2015
40. Partial regularity for mass-minimizing currents in Hilbert spaces
- Author
-
Camillo De Lellis, Thomas Schmidt, Luigi Ambrosio, Ambrosio, Luigi, De Lellis, Camillo, Schmidt, Thomas, and University of Zurich
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,Open set ,Codimension ,Space (mathematics) ,Lipschitz continuity ,01 natural sciences ,Ambient space ,010101 applied mathematics ,10123 Institute of Mathematics ,symbols.namesake ,Metric space ,510 Mathematics ,2604 Applied Mathematics ,Dimension (vector space) ,Settore MAT/05 - Analisi Matematica ,symbols ,0101 mathematics ,2600 General Mathematics ,Mathematics - Abstract
Recently, the theory of currents and the existence theory for Plateau’s problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1–80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121–1142], [Adv. Calc. Var. 7 (2014), 227–240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n-dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415–434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.
- Published
- 2015
41. Remarks on commutators in finite groups
- Author
-
Marian Deaconescu and Gary L. Walls
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The main result of this paper is a formula, in terms of characters, for the number of elements of a normal subgroup H of a finite group G which are not commutators in G.
- Published
- 2015
42. Topological invariance of the homological index
- Author
-
Alan L. Carey and Jens Kaad
- Subjects
Pure mathematics ,Class (set theory) ,Index (economics) ,Applied Mathematics ,General Mathematics ,Homotopy ,010102 general mathematics ,Cyclic homology ,01 natural sciences ,Cohomology ,Mathematics::K-Theory and Homology ,Pairing ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Trace class ,Mathematics - Abstract
R. W. Carey and J. Pincus in [6] proposed an index theory for non-Fredholm bounded operators T on a separable Hilbert space ℋ {\mathcal{H}} such that T T * - T * T {TT^{*}-T^{*}T} is in the trace class. We showed in [3] using Dirac-type operators acting on sections of bundles over ℝ 2 n {\mathbb{R}^{2n}} that we could construct bounded operators T satisfying the more general condition that the operator ( 1 - T T * ) n - ( 1 - T * T ) n {(1-TT^{*})^{n}-(1-T^{*}T)^{n}} is in the trace class. We proposed there a ‘homological index’ for these Dirac-type operators given by Tr ( ( 1 - T T * ) n - ( 1 - T * T ) n ) {{\rm Tr}((1-TT^{*})^{n}-(1-T^{*}T)^{n})} . In this paper we show that the index introduced in [3] represents the result of a paring between a cyclic homology theory for the algebra generated by T and T * {T^{*}} and its dual cohomology theory. This leads us to establish the homotopy invariance of our homological index (in the sense of cyclic theory). We are then able to define in a very general fashion a homological index for certain unbounded operators and prove invariance of this index under a class of unbounded perturbations.
- Published
- 2015
43. Towards the Andre–Oort conjecture for mixed Shimura varieties: The Ax–Lindemann theorem and lower bounds for Galois orbits of special points
- Author
-
Ziyang Gao
- Subjects
Shimura variety ,Pure mathematics ,Conjecture ,Subvariety ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,André–Oort conjecture ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics - Abstract
We prove in this paper the Ax–Lindemann–Weierstraß theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular, we reprove a result of Silverberg [57] in a different approach. Then combining these results we prove the André–Oort conjecture unconditionally for any mixed Shimura variety whose pure part is a subvariety of 𝒜 6 n {\mathcal{A}_{6}^{n}} and under the Generalized Riemann Hypothesis for all mixed Shimura varieties of abelian type.
- Published
- 2015
44. Fourier multipliers for weighted L2{L^{2}} spaces with Lévy–Khinchin–Schoenberg weights
- Author
-
Igor E. Verbitsky and Nikolai Nikolski
- Subjects
symbols.namesake ,Pure mathematics ,Fourier transform ,010201 computation theory & mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,symbols ,0102 computer and information sciences ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we present a class of weight functions w on the circle 𝕋 {\mathbb{T}} , called Lévy–Khinchin–Schoenberg (LKS) weights, for which we are able to completely characterize (in terms of a capacitary inequality) all Fourier multipliers for the weighted space L 2 ( 𝕋 , w ) {L^{2}(\mathbb{T},w)} . We show that the multiplier algebra is nontrivial if and only if 1 / w ∈ L 1 ( 𝕋 ) {1/w\in L^{1}(\mathbb{T})} , and in this case multipliers satisfy the Spectral Localization Property (no “hidden spectrum”). On the other hand, the Muckenhoupt ( A 2 ) {(A_{2})} condition responsible for the basis property of exponentials ( e i k x ) {(e^{ikx})} is more or less independent of the Spectral Localization Property and LKS requirements. Some more complicated compositions of LKS weights are considered as well.
- Published
- 2015
45. On Lagrange’s four squares theorem with almost prime variables
- Author
-
Lilu Zhao and Kai-Man Tsang
- Subjects
Discrete mathematics ,Wilson's theorem ,Almost prime ,Proofs of Fermat's theorem on sums of two squares ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Fermat's theorem on sums of two squares ,01 natural sciences ,Euler's four-square identity ,symbols.namesake ,Mean value theorem (divided differences) ,0103 physical sciences ,Lagrange inversion theorem ,symbols ,010307 mathematical physics ,Lagrange's four-square theorem ,0101 mathematics ,Mathematics - Abstract
In 1994, Brüdern and Fouvry [1] initiated the investigation of Lagrange’s four squares theorem with almost prime variables. In this paper, we prove that every sufficiently large integer, congruent to 4 modulo 24, can be represented as a sum of four squares of integers, each of which has at most four prime factors. Instead of the four-dimensional vector sieve developed by Brüdern and Fouvry [1], we establish this result by combining the three-dimensional sieve and the switching principle.
- Published
- 2014
46. The volume of Kähler–Einstein Fano varieties and convex bodies
- Author
-
Robert J. Berman and Bo Berndtsson
- Subjects
Pure mathematics ,Convex geometry ,Applied Mathematics ,General Mathematics ,Complex projective space ,010102 general mathematics ,Holomorphic function ,Volume conjecture ,Polytope ,Fano plane ,Fixed point ,01 natural sciences ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic geometry - Abstract
We show that the complex projective space ℙ n ${\mathbb{P}^{n}}$ has maximal degree (volume) among all n-dimensional Kähler–Einstein Fano manifolds admitting a non-trivial holomorphic ℂ * ${\mathbb{C}^{*}}$ -action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart’s volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also discussed. The proof uses Moser–Trudinger type inequalities for Stein domains and also leads to criticality results for mean field type equations in ℂ n ${\mathbb{C}^{n}}$ of independent interest. The paper supersedes our previous preprint [5] concerning the case of toric Fano manifolds.
- Published
- 2014
47. Multiple Dedekind zeta functions
- Author
-
Ivan Horozov
- Subjects
Pure mathematics ,11G55, 11M32 ,Integral representation ,Mathematics - Number Theory ,Mathematics::General Mathematics ,Mathematics::Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Type (model theory) ,01 natural sciences ,Mathematics - Algebraic Geometry ,010104 statistics & probability ,Continuation ,Iterated integrals ,FOS: Mathematics ,Dedekind cut ,Number Theory (math.NT) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Dedekind zeta function ,Mathematics ,Meromorphic function - Abstract
In this paper we define multiple Dedekind zeta values (MDZV), using a new type of iterated integrals, called iterated integrals on a membrane. One should consider MDZV as a number theoretic generalization of Euler's multiple zeta values. Over imaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series (Gangl, Kaneko and Zagier). We give an analogue of multiple Eisenstein series over real quadratic field and an alternative definition of values of multiple Eisenstein-Kronecker series (Goncharov). Each of them is a special case of multiple Dedekind zeta values. MDZV are interpolated into functions that we call multiple Dedekind zeta functions (MDZF). We show that MDZF have integral representation, can be written as infinite sum, and have analytic continuation. We compute explicitly the value of a multiple residue of certain MDZF over a quadratic number field at the point (1,1,1,1). Based on such computations, we state two conjectures about MDZV., Comment: This version has substantial improvements in the content and the style. There are more details about the analytic continuation together with new examples of multiple residues. 43 pages
- Published
- 2014
48. The spt-crank for ordinary partitions
- Author
-
William Y. C. Chen, Wenston J. T. Zang, and Kathy Q. Ji
- Subjects
Crank ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,Combinatorial interpretation ,010102 general mathematics ,Congruence relation ,01 natural sciences ,05A17, 05A19, 11P81, 11P83 ,010101 applied mathematics ,Combinatorics ,FOS: Mathematics ,Bijection ,Mathematics - Combinatorics ,Partition (number theory) ,Combinatorics (math.CO) ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
The spt-function $spt(n)$ was introduced by Andrews as the weighted counting of partitions of $n$ with respect to the number of occurrences of the smallest part. Andrews, Garvan and Liang defined the spt-crank of an $S$-partition which leads to combinatorial interpretations of the congruences of $spt(n)$ mod 5 and 7. Let $N_S(m,n)$ denote the net number of $S$-partitions of $n$ with spt-crank $m$. Andrews, Garvan and Liang showed that $N_S(m,n)$ is nonnegative for all integers $m$ and positive integers $n$, and they asked the question of finding a combinatorial interpretation of $N_S(m,n)$. In this paper, we introduce the structure of doubly marked partitions and define the spt-crank of a doubly marked partition. We show that $N_S(m,n)$ can be interpreted as the number of doubly marked partitions of $n$ with spt-crank $m$. Moreover, we establish a bijection between marked partitions of $n$ and doubly marked partitions of $n$. A marked partition is defined by Andrews, Dyson and Rhoades as a partition with exactly one of the smallest parts marked. They consider it a challenge to find a definition of the spt-crank of a marked partition so that the set of marked partitions of $5n+4$ and $7n+5$ can be divided into five and seven equinumerous classes. The definition of spt-crank for doubly marked partitions and the bijection between the marked partitions and doubly marked partitions leads to a solution to the problem of Andrews, Dyson and Rhoades., 22 pages, 6 figures
- Published
- 2014
49. The algebra and model theory of tame valued fields
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Algebraic function field ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Normal extension ,Field (mathematics) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Algebraic closure ,Algebra ,Residue field ,Algebraic theory ,0103 physical sciences ,FOS: Mathematics ,12J10, 12J15 ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Valuation (algebra) ,Mathematics - Abstract
A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.
- Published
- 2014
50. Matrix factorizations and cohomological field theories
- Author
-
Arkady Vaintrob and Alexander Polishchuk
- Subjects
Pure mathematics ,General Mathematics ,Field (mathematics) ,Commutative Algebra (math.AC) ,Mathematics::Algebraic Topology ,01 natural sciences ,Moduli ,Coherent sheaf ,Mathematics - Algebraic Geometry ,Matrix (mathematics) ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics ,Functor ,Hochschild homology ,Group (mathematics) ,Applied Mathematics ,010102 general mathematics ,Mathematics - Commutative Algebra ,16. Peace & justice ,Equivariant map ,010307 mathematical physics - Abstract
We give a purely algebraic construction of a cohomological field theory associated with a quasihomogeneous isolated hypersurface singularity W and a subgroup G of the diagonal group of symmetries of W. This theory can be viewed as an analogue of the Gromov-Witten theory for an orbifoldized Landau-Ginzburg model for W/G. The main geometric ingredient for our construction is provided by the moduli of curves with W-structures introduced by Fan, Jarvis and Ruan. We construct certain matrix factorizations on the products of these moduli stacks with affine spaces which play a role similar to that of the virtual fundamental classes in the Gromov-Witten theory. These matrix factorizations are used to produce functors from the categories of equivariant matrix factorizations to the derived categories of coherent sheaves on the Deligne-Mumford moduli stacks of stable curves. The structure maps of our cohomological field theory are then obtained by passing to the induced maps on Hochschild homology. We prove that for simple singularities a specialization of our theory gives the cohomological field theory constructed by Fan, Jarvis and Ruan using analytic tools., v2:many corrections; added a Theorem on vanishing of the Chern character of a Koszul matrix factorization in section 5.6; v3: further corrections; v4: the section 5.9 on the change of group is deleted (it contained a mistake that did not affect the rest of the paper)
- Published
- 2014
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