Journal of the American Mathematical Society最新文献_第6页

Plancherel theory for real spherical spaces: Construction of the Bernstein morphisms 实球面空间的Plancherel理论:Bernstein态射的构造

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-07-19 DOI: 10.1090/jams/971

P. Delorme, F. Knop, Bernhard Krotz, H. Schlichtkrull

This paper lays the foundation for Plancherel theory on real spherical spaces <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z equals upper G slash upper H"> <mml:semantics> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Z=G/H</mml:annotation> </mml:semantics></mml:math></inline-formula>, namely it provides the decomposition of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper Z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(Z)</mml:annotation> </mml:semantics></mml:math></inline-formula> into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics></mml:math></inline-formula> at infinity. In particular, we obtain a generalization of the Maass-Selberg relations. As a corollary we obtain a partial geometric characterization of the discrete spectrum: <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L squared left-parenthesis upper Z right-parenthesis Subscript normal d normal i normal s normal c Baseline not-equals normal empty-set"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>Z</mml:mi> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>≠</mml:mo> <mml:mi mathvariant="normal">∅</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">L^2(Z)_{mathrm {disc}}neq emptyset</mml:annotation> </mml:semantics></mml:math></inline-formula> if <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/Math

本文为实球面空间Z=G/H Z=G/H的Plancherel理论奠定了基础，即通过Bernstein态射提供了l2 (Z) L^2(Z)分解成不同的表示序列。这些级数由球根的子集参数化，这些子集决定了zz在无穷远处的精细几何形状。特别地，我们得到了Maass-Selberg关系的推广。作为推论，我们得到离散谱的部分几何特征:l2 (Z) d sc≠∅L^2(Z)_{ mathfrk {h}^perp如果h⊥ mathfrk {h}^perp内部包含椭圆元素。如果zz是一个实约化群，或者更一般地说，是一个对称空间，我们的结果检索了Harish-Chandra的Plancherel公式(对于群)以及Delorme和van den Ban-Schlichtkrull的Plancherel公式(对于对称空间)，直到诱导基准的离散级数的显式确定。

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引用次数: 15

On the measure of maximal entropy for finite horizon Sinai Billiard maps 有限视界西奈台球图的最大熵测度

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-07-06 DOI: 10.1090/jams/939

V. Baladi, Mark F. Demers

The Sinai billiard map <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics></mml:math></inline-formula> on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">h_*</mml:annotation> </mml:semantics></mml:math></inline-formula> for the topological entropy of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics></mml:math></inline-formula>. We prove that <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>h</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">h_*</mml:annotation> </mml:semantics></mml:math></inline-formula> is not smaller than the value given by the variational principle, and that it is equal to the definitions of Bowen using spanning or separating sets. Under a mild condition of sparse recurrence to the singularities, we get more: First, using a transfer operator acting on a space of anisotropic distributions, we construct an invariant probability measure <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript asterisk"> <mml:semantics> <mml:msub> <mml:mi>μ</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> <mml:annotation encoding="application/x-tex">mu _*</mml:annotation> </mml:semantics></mml:math></inline-formula> of maximal entropy for <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics></mml:math></inline-formula> (i.e., <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript mu Sub Subscript asterisk Baseline left-parenthesis upper T right-parenthesis equals h Subscript asterisk"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>h</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>μ</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> </mml:

双环面上的西奈台球图T T，即周期性洛伦兹气体，是一个不连续图。假设有限视界，我们提出了T T拓扑熵的一个定义h * h_*。我们证明了h * h_*不小于由变分原理给出的值，并且它等于用生成集或分离集定义的Bowen。在奇异点稀疏递归的温和条件下，我们得到:首先，利用作用于各向异性分布空间上的传递算子，构造了T T(即h μ∗(T)=h∗h_{mu _*}(T)=h_*)最大熵的不变概率测度μ∗mu _*)，证明了μ∗mu _*具有完全支持，并且是伯努利的。并证明了μ∗mu _*是最大熵的唯一测度，除了所有非掠带周期轨道的乘子等于h∗h_*外，它与光滑不变测度不同。其次，h * h_*等于限制T T到非紧连续域的Bowen-Pesin-Pitskel拓扑熵。最后，应用Lima和Matheus的结果，得到映射T T对所有n∈n n in mathbb {n}具有至少C e nh * C e^{nh_*}周期为n n的周期点。

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引用次数: 23

On the Ramanujan conjecture for automorphic forms over function fields I. Geometry 函数域上自同构形式的Ramanujan猜想1 .几何

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-05-30 DOI: 10.1090/jams/968

W. Sawin, Nicolas Templier

Let G G be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of G G , subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of Bun G operatorname {Bun}_G . It is independent of the work of Lafforgue on the global Langlands correspondence.

设G G是函数域上的一个分裂半单群。我们证明了G G的自同构表示在非分枝处的调和性，在一个地方服从一个局部假设，强于超可混性，并假设存在具有良好性质的循环基变。我们的方法依赖于Bun G运算符名称的几何结构{Bun}_G。它独立于Lafforgue在全球Langlands通信方面的工作。

引用次数: 9

Tropical curves, graph complexes, and top weight cohomology of ℳ_{ℊ} _{ℊ}的热带曲线、图复形和顶权上同

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-05-25 DOI: 10.1090/jams/965

M. Chan, Søren Galatius, S. Payne

We study the topology of a space <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta Subscript g"> <mml:semantics> <mml:msub> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>g</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">Delta _{g}</mml:annotation> </mml:semantics></mml:math></inline-formula> parametrizing stable tropical curves of genus <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics></mml:math></inline-formula> with volume <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics></mml:math></inline-formula>, showing that its reduced rational homology is canonically identified with both the top weight cohomology of <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M Subscript g"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">mathcal {M}_g</mml:annotation> </mml:semantics></mml:math></inline-formula> and also with the genus <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics></mml:math></inline-formula> part of the homology of Kontsevich’s graph complex. Using a theorem of Willwacher relating this graph complex to the Grothendieck–Teichmüller Lie algebra, we deduce that <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 4 g minus 6 Baseline left-parenthesis script upper M Subscript g Baseline semicolon double-struck upper Q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>4</mml:mn> <mml:mi>g</mml:mi> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:mo>;</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD">

研究了空间Δ g Delta ＿g{的拓扑结构，证明了其简化有理同调与M g }mathcal M＿g{的顶权上同调和Kontsevich图复调中g g的部分同调是正则化的。利用Willwacher关于此图复形与grothendieck - teichm ller李代数的一个定理，我们推导出H 4 g−6 (M g;Q) H^4g}-6({}mathcal M＿g{;}mathbb Q){对于}g=3 g=3, g=5 g=5, g≥7 g geq 7是非零的，事实上它的维数在g中至少呈指数增长。这推翻了Church、Farb和Putman最近的一个猜想，也推翻了Kontsevich一个更古老、更普遍的猜想。我们还独立证明了Willwacher的另一个定理，即复图的同调在负次域中消失。

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引用次数: 64

Algebraicity of the metric tangent cones and equivariant K-stability 度量切锥的代数性与等变k稳定性

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-05-09 DOI: 10.1090/JAMS/974

Chi Li, Xiaowei Wang, Chenyang Xu

We prove two new results on the K K -polystability of Q mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K K -polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

基于纯代数几何变元，我们证明了Qmathbb｛Q｝-Fano变种的K-多稳定性的两个新结果。第一种观点认为，任何K-半稳定log Fano锥都有一个特殊的退化为唯一确定的K-多稳定log Fano-锥。作为推论，我们将其与微分几何结果相结合，完成了唐纳森-孙猜想的证明，该猜想认为，在Kähler-Einstein-Fano流形的Gromov-Hausdorff极限上出现的任何点的度量切锥仅取决于奇点的代数结构。第二个结果表明，对于任何具有环面作用的log Fano变种，K-多稳态等价于等变K-多稳定性，也就是说，为了检验K-多稳定，检验在环面作用下是等变的特殊测试配置就足够了。

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引用次数: 81

Large genus asymptotics for volumes of strata of abelian differentials 阿贝尔微分地层体积的大属渐近性

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-04-15 DOI: 10.1090/jams/947

A. Aggarwal

In this paper we consider the large genus asymptotics for Masur-Veech volumes of arbitrary strata of Abelian differentials. Through a combinatorial analysis of an algorithm proposed in 2002 by Eskin-Okounkov to exactly evaluate these quantities, we show that the volume <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="nu 1 left-parenthesis script upper H 1 left-parenthesis m right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ν</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> <mml:annotation encoding="application/x-tex">nu _1 big ( mathcal {H}_1 (m) big )</mml:annotation> </mml:semantics></mml:math></inline-formula> of a stratum indexed by a partition <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals left-parenthesis m 1 comma m 2 comma ellipsis comma m Subscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">m = (m_1, m_2, ldots , m_n)</mml:annotation> </mml:semantics></mml:math></inline-formula> is <inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 4 plus o left-parenthesis 1 right-parenthesis right-parenthesis product Underscript i equals 1 Overscript n Endscripts left-parenthesis m Subscript i Baseline plus 1 right-parenthesis Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo

本文研究任意Abelian微分地层的Masur-Veech体积的大格渐近性。通过对Eskin-Okounkov在2002年提出的一种算法的组合分析，来精确地评估这些量，我们证明了由分区m = (m1, m2，…)索引的地层的体积ν 1 (h1 (m)) nu ＿1 big (mathcal H＿1{ (m) }big)，M n) M = (m＿1, m＿2, ldots，M＿n) = (4 + o(1))∏I = 1 n (m I + 1) -1 big (4 + o(1) big) prod ＿i = 1{^n (m＿i + 1)^}-{1，当2g−2 =∑I = 1 n m I 2g - 2 }= sum ＿i = 1{^n m I趋于∞}infty。这证实了Eskin-Zorich的一个预测，并推广了Chen-Möller-Zagier和Sauvaget最近的一些结果，他们分别在m = 1 {2g−2 m = 1^2g -} 2和m = (2g−2)m = (2g - 2)的特殊情况下建立了这些极限陈述。我们还包括Anton Zorich的附录，该附录使用我们的主要结果来推断计算某些类型鞍连接的Siegel-Veech常数的大属渐近性。

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引用次数: 18

Positive entropy actions of countable groups factor onto Bernoulli shifts 可数群的正熵作用影响伯努利位移

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-04-14 DOI: 10.1090/JAMS/931

Brandon Seward

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy), then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory.

我们证明了如果可数无限群的自由遍历作用具有正的Rokhlin熵(或者更一般地说，正的sofic熵)，那么它就会因子化到所有较小或相等熵的伯努利位移上。这将经典熵理论中著名的西奈因子定理推广到所有可数无限群。

引用次数: 16

Catalan functions and 𝑘-Schur positivity 加泰罗尼亚函数和𝑘-Schur阳性

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-04-10 DOI: 10.1090/JAMS/921

J. Blasiak, J. Morse, Anna Y. Pun, D. Summers

We prove that graded k k -Schur functions are G G -equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded k k -Schur functions and resolve the Schur positivity and k k -branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.

证明了分次k-舒尔函数是旗变上向量丛的G-等变Euler特征，解决了陈海曼的一个猜想。我们揭示了分次k k-Schur函数的一个新的奇迹移位不变性，并通过使用强标记表提供直接组合公式，以尽可能强的项解决了Schur正性和k k-分支猜想。

引用次数: 12

Categorical joins 分类连接

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-03-31 DOI: 10.1090/jams/963

A. Kuznetsov, Alexander Perry

We introduce the notion of a categorical join, which can be thought of as a categorification of the classical join of two projective varieties. This notion is in the spirit of homological projective duality, which categorifies classical projective duality. Our main theorem says that the homological projective dual category of the categorical join is naturally equivalent to the categorical join of the homological projective dual categories. This categorifies the classical version of this assertion and has many applications, including a nonlinear version of the main theorem of homological projective duality.

我们引入了范畴联接的概念，它可以被认为是两个射影变体的经典联接的范畴化。这个概念是在同调射影对偶的精神中，它对经典射影对偶进行了分类。我们的主要定理是说，范畴联结的同调射影对偶范畴自然地等价于同调射影对偶范畴的范畴联结。这分类了经典版本的这个断言，并有许多应用，包括一个非线性版本的主要定理的同调投影对偶。

引用次数: 21

Tame topology of arithmetic quotients and algebraicity of Hodge loci 算术商的Tame拓扑与Hodge轨迹的代数性

IF 3.9 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society

Pub Date : 2018-03-26 DOI: 10.1090/jams/952

B. Klingler

In this paper we prove the following results: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We prove that the period map associated to any pure polarized variation of integral Hodge structures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathbb {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a smooth complex quasi-projective variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> As a corollary of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http:/

本文证明了以下结果:1)1)证明了齐次空间的任意算术商都存在一个自然实数半代数结构，其Hecke对应是半代数的。一个特别重要的例子是Hodge变量，它参数化了纯极化积分Hodge结构。2) 2)证明了光滑复拟射射角簇S上任意整数Hodge结构V mathbb {V}的纯极化变化所对应的周期映射对于由上述半代数结构诱导的相关Hodge簇上的0 -极小结构是可定义的。3) 3)作为2)2)和Peterzil-Starchenko的o-minimal Chow定理的推论，我们恢复了(S, V) (S， mathbb {V})的Hodge轨迹是S的代数子变种S的可数并，这一结果最初是由cattani - delige - kaplan给出的。我们的方法简化了卡塔尼-德莱尼-卡普兰的证明，因为它没有充分利用卡塔尼-卡普兰-施密德的困难多变量S l2 SL_2轨道定理。

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引用次数: 52