Eigenvalues and dynamical degrees of self-maps on abelian varieties

IF 0.9 1区 数学 Q2 MATHEMATICS
Fei Hu
{"title":"Eigenvalues and dynamical degrees of self-maps on abelian varieties","authors":"Fei Hu","doi":"10.1090/jag/806","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> be a smooth projective variety over an algebraically closed field, and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper X right-arrow upper X\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>f</mml:mi>\n <mml:mo>:<!-- : --></mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:mi>X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">f\\colon X\\to X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> a surjective self-morphism of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. The <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i\">\n <mml:semantics>\n <mml:mi>i</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">i</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-th cohomological dynamical degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi Subscript i Baseline left-parenthesis f right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>χ<!-- χ --></mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\chi _i(f)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is defined as the spectral radius of the pullback <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript asterisk\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">f^{*}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on the étale cohomology group <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper H Subscript ModifyingAbove normal e With acute normal t Superscript i Baseline left-parenthesis upper X comma bold upper Q Subscript script l Baseline right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msubsup>\n <mml:mi>H</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mover>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">e</mml:mi>\n </mml:mrow>\n <mml:mo>´<!-- ´ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">t</mml:mi>\n </mml:mrow>\n </mml:mrow>\n <mml:mi>i</mml:mi>\n </mml:msubsup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">Q</mml:mi>\n </mml:mrow>\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">H^i_{\\acute {\\mathrm {e}}\\mathrm {t}}(X, \\mathbf {Q}_\\ell )</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\n <mml:semantics>\n <mml:mi>k</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-th numerical dynamical degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"lamda Subscript k Baseline left-parenthesis f right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\lambda _k(f)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as the spectral radius of the pullback <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f Superscript asterisk\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>f</mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo>∗<!-- ∗ --></mml:mo>\n </mml:mrow>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">f^{*}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on the vector space <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"sans-serif upper N Superscript k Baseline left-parenthesis upper X right-parenthesis Subscript bold upper R\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"sans-serif\">N</mml:mi>\n </mml:mrow>\n <mml:mi>k</mml:mi>\n </mml:msup>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"bold\">R</mml:mi>\n </mml:mrow>\n </mml:mrow>\n </mml:msub>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\mathsf {N}^k(X)_{\\mathbf {R}}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of real algebraic cycles of codimension <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\n <mml:semantics>\n <mml:mi>k</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\n <mml:semantics>\n <mml:mi>X</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> modulo numerical equivalence. Truong conjectured that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"chi Subscript 2 k Baseline left-parenthesis f right-parenthesis equals lamda Subscript k Baseline left-parenthesis f right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:msub>\n <mml:mi>χ<!-- χ --></mml:mi>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mn>2</mml:mn>\n <mml:mi>k</mml:mi>\n </mml:mrow>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n <mml:mo>=</mml:mo>\n <mml:msub>\n <mml:mi>λ<!-- λ --></mml:mi>\n <mml:mi>k</mml:mi>\n </mml:msub>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>f</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\chi _{2k}(f) = \\lambda _k(f)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for all <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0 less-than-or-equal-to k less-than-or-equal-to dimension upper X\">\n <mml:semantics>\n <mml:mrow>\n <mml:mn>0</mml:mn>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>k</mml:mi>\n <mml:mo>≤<!-- ≤ --></mml:mo>\n <mml:mi>dim</mml:mi>\n <mml:mo>⁡<!-- ⁡ --></mml:mo>\n <mml:mi>X</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">0 \\le k \\le \\dim X</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> as a generalization of Weil’s Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jag/806","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4

Abstract

Let X X be a smooth projective variety over an algebraically closed field, and f : X X f\colon X\to X a surjective self-morphism of X X . The i i -th cohomological dynamical degree χ i ( f ) \chi _i(f) is defined as the spectral radius of the pullback f f^{*} on the étale cohomology group H e ´ t i ( X , Q ) H^i_{\acute {\mathrm {e}}\mathrm {t}}(X, \mathbf {Q}_\ell ) and the k k -th numerical dynamical degree λ k ( f ) \lambda _k(f) as the spectral radius of the pullback f f^{*} on the vector space N k ( X ) R \mathsf {N}^k(X)_{\mathbf {R}} of real algebraic cycles of codimension k k on X X modulo numerical equivalence. Truong conjectured that χ 2 k ( f ) = λ k ( f ) \chi _{2k}(f) = \lambda _k(f) for all 0 k dim X 0 \le k \le \dim X as a generalization of Weil’s Riemann hypothesis. We prove this conjecture in the case of abelian varieties. In the course of the proof we also obtain a new parity result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

阿贝尔变种自映射的特征值和动力度
设X X是代数闭域上的光滑射影变,且f: X→X f \colon X \to X是X X的满射自态射。i i -上同调动力学度χ i(f) \chi _i(f)定义为在上同调群H上的回拉f {* f^*}的谱半径。Q (l) H^i_ {\acute{\mathrm e{}}\mathrm t{(X,}}\mathbf Q_ {}\ell)和k k -数值动力度λ k(f) \lambda _k(f)作为回拉f {* f^*}在向量空间N k(X) R \mathsf N{^k(X)_ }{\mathbf R{上的谱半径余维k k在X X模数值等价上的代数循环。作为Weil黎曼假设的推广,Truong推测χ }}2k(f) = λ k(f) \chi _2k(f) = {}\lambda _k(f)对于所有0≤k≤dim (X) 0 \le k \le\dim X。我们在阿贝尔变的情况下证明了这个猜想。在证明过程中,我们还得到了关于素数特征的阿贝尔变体的自映射的特征值的一个新的奇偶性结果,这是一个独立的研究方向。
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来源期刊
CiteScore
2.70
自引率
5.60%
发文量
23
审稿时长
>12 weeks
期刊介绍: The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology. This journal, published quarterly with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.
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