D. Croydon, Tsuyoshi Kato, M. Sasada, S. Tsujimoto
The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than 1 2 frac {1}{2} , and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by ‘reflection in the past maximum’, which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, we give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter p ↑ 1 2 puparrow frac 12 (which may be considered the ‘high density’ regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.
盒式球系统(BBS)由Takahashi和Satsuma于1990年提出，是一种具有孤子行为的元胞自动机。在本文中，我们从一个随机的双面无限粒子结构开始研究BBS。对于这样的模型，Ferrari等人最近证明了密度严格小于12 frac{1}{2}的伯努利积测度分布的不变量，并给出了更一般的不变量测度的孤子分解。我们使用由“过去极大值反射”给出的粒子构型的最近邻路径编码的变换来研究BBS动力学，这是Pitman著名的将布朗运动和三维贝塞尔过程联系起来的方法。我们用它来描述一组配置，其中动力学在任何时候都是定义良好的和可逆的(即可以反转)。为理解确定性动力学而开发的技术随后被应用于研究BBS从随机初始配置的演变。具体地说，我们给出了在BBS动力学下随机初始条件在分布上不变的简单充分条件，并在几个自然例子中进行了验证，同时研究了相关变换的遍历性。此外，我们还分析了通常用于相互作用粒子系统的BBS的各种概率性质，例如粒子和标记粒子的积分电流的渐近行为。最后，对于参数p ^ 1 ^ 2 p uprow frac 12的伯努利积测度(可以被认为是“高密度”状态)，我们考虑的路径编码有一个自然的尺度限制，这促使我们引入新的连续版本的BBS，我们相信它将作为一个动态系统具有独立的兴趣。
{"title":"Dynamics of the Box-Ball System with Random Initial Conditions via Pitman’s Transformation","authors":"D. Croydon, Tsuyoshi Kato, M. Sasada, S. Tsujimoto","doi":"10.1090/memo/1398","DOIUrl":"https://doi.org/10.1090/memo/1398","url":null,"abstract":"The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. In this article, we study the BBS when started from a random two-sided infinite particle configuration. For such a model, Ferrari et al. recently showed the invariance in distribution of Bernoulli product measures with density strictly less than \u0000\u0000 \u0000 \u0000 1\u0000 2\u0000 \u0000 frac {1}{2}\u0000 \u0000\u0000, and gave a soliton decomposition for invariant measures more generally. We study the BBS dynamics using the transformation of a nearest neighbour path encoding of the particle configuration given by ‘reflection in the past maximum’, which was famously shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible (i.e. can be inverted) for all times. The techniques developed to understand the deterministic dynamics are subsequently applied to study the evolution of the BBS from a random initial configuration. Specifically, we give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. Finally, for Bernoulli product measures with parameter \u0000\u0000 \u0000 \u0000 p\u0000 ↑\u0000 \u0000 1\u0000 2\u0000 \u0000 \u0000 puparrow frac 12\u0000 \u0000\u0000 (which may be considered the ‘high density’ regime), the path encoding we consider has a natural scaling limit, which motivates the introduction of a new continuous version of the BBS that we believe will be of independent interest as a dynamical system.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47334698","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.
{"title":"Positive Gaussian Kernels also Have Gaussian Minimizers","authors":"F. Barthe, P. Wolff","doi":"10.1090/memo/1359","DOIUrl":"https://doi.org/10.1090/memo/1359","url":null,"abstract":"We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45931619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Introduction • Acknowledgements • Preliminaries • Some general results • The cases $c_1=4$ and $c_1 = 5$ on ${mathbb P}^2$ • The case $c_1 = 4$, $c_2 = 5, 6$ on ${mathbb P}^3$ • The case $c_1 = 4$, $c_2 = 7$ on ${mathbb P}^3$ • The case $c_1 = 4$, $c_2 = 8$ on ${mathbb P}^3$ • The case $c_1 = 4$, $5 leq c_2 leq 8$ on ${mathbb P}^n$, $n geq 4$ • Appendix A. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 2$ on ${mathbb P}^3$ • Appendix B. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 4$ on ${mathbb P}^3$ • Bibliography •
{"title":"Globally Generated Vector Bundles with Small _{tiny}1 on Projective Spaces","authors":"C. Anghel, I. Coandă, N. Manolache","doi":"10.1090/memo/1209","DOIUrl":"https://doi.org/10.1090/memo/1209","url":null,"abstract":"Introduction • Acknowledgements • Preliminaries • Some general results • The cases $c_1=4$ and $c_1 = 5$ on ${mathbb P}^2$ • The case $c_1 = 4$, $c_2 = 5, 6$ on ${mathbb P}^3$ • The case $c_1 = 4$, $c_2 = 7$ on ${mathbb P}^3$ • The case $c_1 = 4$, $c_2 = 8$ on ${mathbb P}^3$ • The case $c_1 = 4$, $5 leq c_2 leq 8$ on ${mathbb P}^n$, $n geq 4$ • Appendix A. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 2$ on ${mathbb P}^3$ • Appendix B. The case $c_1 = 4$, $c_2 = 8$, $c_3 = 4$ on ${mathbb P}^3$ • Bibliography •","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"253 1","pages":"0-0"},"PeriodicalIF":1.9,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43046158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defined a priori but must be chosen randomly. Spectral properties of an operator should hence be independent of this left multiplication. We show that results derived from functional calculi for intrinsic slice functions can be formulated without the assumption of a left multiplication. We develop the S-functional calculus in this setting and a new approach to spectral integration. This approach has a clear interpretation in terms of the right linear structure on the space and allows to formulate the spectral theorem without using any randomly chosen structure. Our techniques only apply to intrinsic slice functions, but only these functions are compatible with the basic intuition of a functional calculus that $f(T)$ should be defined by letting $f$ act on the spectral values of $T$. Using these tools, we develop a theory of quaternionic spectral operators. In particular, we show the existence of a canonical decomposition of such operator and discuss its behavior under the S-functional calculus. Finally, we show a relation with complex operator theory: if we embed the complex numbers into the quaternions, then complex and quaternionic operator theory are consistent. The symmetry of intrinsic slice functions guarantees that this compatibility is true for any imbedding of the complex numbers.
{"title":"Operator Theory on One-Sided Quaternionic\u0000 Linear Spaces: Intrinsic S-Functional Calculus and\u0000 Spectral Operators","authors":"J. Gantner","doi":"10.1090/memo/1297","DOIUrl":"https://doi.org/10.1090/memo/1297","url":null,"abstract":"Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. \u0000The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defined a priori but must be chosen randomly. Spectral properties of an operator should hence be independent of this left multiplication. \u0000We show that results derived from functional calculi for intrinsic slice functions can be formulated without the assumption of a left multiplication. We develop the S-functional calculus in this setting and a new approach to spectral integration. This approach has a clear interpretation in terms of the right linear structure on the space and allows to formulate the spectral theorem without using any randomly chosen structure. Our techniques only apply to intrinsic slice functions, but only these functions are compatible with the basic intuition of a functional calculus that $f(T)$ should be defined by letting $f$ act on the spectral values of $T$. \u0000Using these tools, we develop a theory of quaternionic spectral operators. In particular, we show the existence of a canonical decomposition of such operator and discuss its behavior under the S-functional calculus. \u0000Finally, we show a relation with complex operator theory: if we embed the complex numbers into the quaternions, then complex and quaternionic operator theory are consistent. The symmetry of intrinsic slice functions guarantees that this compatibility is true for any imbedding of the complex numbers.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"26 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82632674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host–Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host–Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer k ≥ 0 kgeq 0 the joint distribution’s marginals on affine subcubes of dimension k k are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.
{"title":"Nilspace Factors for General Uniformity Seminorms, Cubic Exchangeability and Limits","authors":"P. Candela, B. Szegedy","doi":"10.1090/memo/1425","DOIUrl":"https://doi.org/10.1090/memo/1425","url":null,"abstract":"We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host–Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host–Kra type seminorms for measure-preserving actions of countable nilpotent groups. This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability. These are sequences indexed by the infinite discrete cube, such that for every integer \u0000\u0000 \u0000 \u0000 k\u0000 ≥\u0000 0\u0000 \u0000 kgeq 0\u0000 \u0000\u0000 the joint distribution’s marginals on affine subcubes of dimension \u0000\u0000 \u0000 k\u0000 k\u0000 \u0000\u0000 are all equal. In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43811369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Type II Blow up Manifolds for the Energy Supercritical Semilinear Wave Equation","authors":"Charles Collot","doi":"10.1090/MEMO/1205","DOIUrl":"https://doi.org/10.1090/MEMO/1205","url":null,"abstract":"","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"252 1","pages":"0-0"},"PeriodicalIF":1.9,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43014446","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Coefficient Systems on the Bruhat-Tits Building and Pro-𝑝 Iwahori-Hecke Modules","authors":"Jan Kohlhaase","doi":"10.1090/memo/1374","DOIUrl":"https://doi.org/10.1090/memo/1374","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I\">\u0000 <mml:semantics>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">I</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a pro-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> Iwahori subgroup of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\">\u0000 <mml:semantics>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">G</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper R\">\u0000 <mml:semantics>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">R</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a commutative quasi-Frobenius ring. If <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H equals upper R left-bracket upper I minus upper G slash upper I right-bracket\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>H</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mi>R</mml:mi>\u0000 <mml:mo stretchy=\"false\">[</mml:mo>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">∖<!-- ∖ --></mml:mi>\u0000 <mml:mi>G</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mo stretchy=\"false\">]</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">H=R[Ibackslash G/I]</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> denotes the pro-<inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\">\u0000 <mml:semantics>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">p</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> Iwahori-Hecke algebra of <inline-formula content-type=\"mat","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47901043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser’s motivic vanishing cycles.
{"title":"Motivic Euler Products and Motivic Height Zeta Functions","authors":"Margaret Bilu","doi":"10.1090/memo/1396","DOIUrl":"https://doi.org/10.1090/memo/1396","url":null,"abstract":"A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan’s motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser’s motivic vanishing cycles.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47620196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. Ciocan-Fontanine, David Favero, J'er'emy Gu'er'e, Bumsig Kim, M. Shoemaker
We define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases these invariants recover both the Gromov–Witten type invariants defined by Chang–Li and Fan–Jarvis–Ruan using cosection localization as well as the FJRW type invariants constructed by Polishchuk–Vaintrob. The invariants are defined by constructing a “fundamental factorization” supported on the moduli space of Landau–Ginzburg maps to a convex hybrid model. This gives the kernel of a Fourier–Mukai transform; the associated map on Hochschild homology defines our theory.
{"title":"Fundamental Factorization of a GLSM Part I: Construction","authors":"I. Ciocan-Fontanine, David Favero, J'er'emy Gu'er'e, Bumsig Kim, M. Shoemaker","doi":"10.1090/memo/1435","DOIUrl":"https://doi.org/10.1090/memo/1435","url":null,"abstract":"We define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases these invariants recover both the Gromov–Witten type invariants defined by Chang–Li and Fan–Jarvis–Ruan using cosection localization as well as the FJRW type invariants constructed by Polishchuk–Vaintrob. The invariants are defined by constructing a “fundamental factorization” supported on the moduli space of Landau–Ginzburg maps to a convex hybrid model. This gives the kernel of a Fourier–Mukai transform; the associated map on Hochschild homology defines our theory.","PeriodicalId":49828,"journal":{"name":"Memoirs of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2018-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41743042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Introduction On se propose de generaliser au cas tordu les resultats d’Arthur contenus dans les articles [A1] et [A7]. Soient F un corps local, G un groupe reductif connexe defini sur F et G un espace tordu sur G, au sens de Labesse (cf. 2.1). Nous imposons une condition a G (2.1(2)) qui revient a dire qu’il existe un groupe algebrique non connexe G defini sur F , de composante neutre G, tel que G soit une composante connexe de G. Mais la structure de groupe sur G ne joue aucun role, seules importent les actions a droite et a gauche de G sur G. Notons ZG le centre de G et ZG(F ) θ le sous-groupe des z ∈ ZG(F ) tels que zγ = γz pour tout γ ∈ G. On fixe un caractere unitaire ω de G(F ) dont la restriction a ZG(F ) θ est triviale. On s’interesse aux ”distributions” ω-equivariantes sur G(F ). Ce sont des formes lineaires l : C∞ c (G(F ))→ C telles que, pour tout f ∈ C∞ c (G(F )) et tout g ∈ G(F ), on ait l’egalite l(f) = ω(g)−1l(f), ou f est la fonction f(γ) = f(g−1γg). Il y a deux types basiques de telles distributions. D’abord les integrales orbitales. On fixe γ ∈ G(F ), disons fortement regulier. On note ZG(γ) son commutant dans G et on munit le quotient ZG(γ, F )G(F ) d’une mesure invariante a droite. Pour f ∈ C∞ c (G(F )), l’integrale orbitale de f au point γ est
本文将arthur在[A1]和[A7]文章中给出的结果推广到扭曲情况。设F是局部域，G是定义在F上的相关还原群，G是定义在G上的弯曲空间(参见2.1)。我们强加先决条件a (2) G(2.1)谁说,有了一群algebrique相关非中性定义上的分量,F G, G G或是一个相关的内容,但这样的结构,对G组不发挥作用,只有进口的右边和左边的股票就µG .注意到该中心G和GµG (F)θ分组z∈µG (F)等z z =γγγ每单位ω∈G .固定有脾气的G (F),其限制了µG (F),θ是微不足道的。我们感兴趣的是G(F)上的ω-等变“分布”。它们是线性形式l: C∞C (G(F))→C，因此，对于所有F∈C∞C (G(F))和所有G∈G(F)，我们有相等的l(F) = ω(G)−1l(F)，其中F是函数F (γ) = F (G−1γ G)。这种分布有两种基本类型。首先是轨道积分。固定γ∈G(F)，比方说强正则。我们注意到ZG(γ)在G中的转换，并将商ZG(γ， F)G(F)设为直线不变测度。对于f∈C∞C (G(f))， f在γ处的轨道积分为
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