We prove Bloch’s formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in codimension one. Several applications, including Bloch’s formula for 0-cycles with modulus, are derived.
{"title":"𝐾-theory and 0-cycles on schemes","authors":"Rahul Gupta, A. Krishna","doi":"10.1090/jag/744","DOIUrl":"https://doi.org/10.1090/jag/744","url":null,"abstract":"We prove Bloch’s formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in codimension one. Several applications, including Bloch’s formula for 0-cycles with modulus, are derived.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/744","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48564794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Martin G. Gulbrandsen, L. H. Halle, K. Hulek, Ziyu Zhang
Given a strict simple degeneration f:X→Cf colon Xto C the first three authors previously constructed a degeneration IX/Cn→CI^n_{X/C} to C of the relative degree nn Hilbert scheme of 00-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of ff is at most 22. In this case we show that IX
给定一个严格的简单退化f:X→ 前三位作者先前构建了一个退化的I X/C n→ C I^n_｛X/C｝到0维子项的相对次数n的Hilbert格式的C。在本文中，我们研究了这种退化的几何结构，特别是当f的纤维尺寸至多为2 2时。在这种情况下，我们证明了I X/C n→ ｛X/C｝ to C是一个dlt模型。如果f:X，这甚至是一个很好的最小dlt模型→ C f 冒号X 到C具有此属性。我们计算了中心纤维（IX/Cn）0（I^n_{X/C}）_0的对偶复形，并将其与一般纤维的基本骨架联系起来。对于K3表面的II型退化，我们证明了堆叠I X/C n→ C｛mathcal I｝^n_｛X/C｝to C具有无退化的相对对数2-形式。最后，我们讨论了我们的堕落与永井建筑的关系。
{"title":"The geometry of degenerations of Hilbert schemes of points","authors":"Martin G. Gulbrandsen, L. H. Halle, K. Hulek, Ziyu Zhang","doi":"10.1090/jag/765","DOIUrl":"https://doi.org/10.1090/jag/765","url":null,"abstract":"<p>Given a strict simple degeneration <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper X right-arrow upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">f colon Xto C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> the first three authors previously constructed a degeneration <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I Subscript upper X slash upper C Superscript n Baseline right-arrow upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mi>C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 </mml:msubsup>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>C</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">I^n_{X/C} to C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of the relative degree <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n\">\u0000 <mml:semantics>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">n</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> Hilbert scheme of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"0\">\u0000 <mml:semantics>\u0000 <mml:mn>0</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is at most <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\u0000 <mml:semantics>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. In this case we show that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper I Subscript upper X slash upper C Superscript n Baseline right-arrow upper C\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:msubsup>\u0000 <mml:mi>I</mml:mi>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>X</mml:mi>\u0000 ","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46900861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato’s logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory and the Milnor formula for the canonical lifting.
{"title":"Characteristic cycle of a rank one sheaf and ramification theory","authors":"Yuri Yatagawa","doi":"10.1090/jag/758","DOIUrl":"https://doi.org/10.1090/jag/758","url":null,"abstract":"We compute the characteristic cycle of a rank one sheaf on a smooth surface over a perfect field of positive characteristic. We construct a canonical lifting on the cotangent bundle of Kato’s logarithmic characteristic cycle using ramification theory and prove the equality of the characteristic cycle and the canonical lifting. As corollaries, we obtain a computation of the singular support in terms of ramification theory and the Milnor formula for the canonical lifting.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43724040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a class of surfaces corresponding to the A D E ADE root lattices and construct compactifications of their moduli spaces as quotients of projective varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We exhibit modular families over these moduli spaces, which extend to families of stable pairs over the compactifications. One simple application is a geometric compactification of the moduli of rational elliptic surfaces that is a finite quotient of a projective toric variety.
我们定义了一类对应于a DE ADE根格的曲面，并构造了它们的模空间的紧化，作为Coxeter扇形的射影变商，推广了曲线的Losev-Manin空间。我们在这些模空间上展示了模族，并将其推广到紧化上的稳定对族。一个简单的应用是有理椭圆曲面模的几何紧化，它是一个射影环变的有限商。
{"title":"ADE surfaces and their moduli","authors":"V. Alexeev, A. Thompson","doi":"10.1090/jag/762","DOIUrl":"https://doi.org/10.1090/jag/762","url":null,"abstract":"We define a class of surfaces corresponding to the \u0000\u0000 \u0000 \u0000 A\u0000 D\u0000 E\u0000 \u0000 ADE\u0000 \u0000\u0000 root lattices and construct compactifications of their moduli spaces as quotients of projective varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We exhibit modular families over these moduli spaces, which extend to families of stable pairs over the compactifications. One simple application is a geometric compactification of the moduli of rational elliptic surfaces that is a finite quotient of a projective toric variety.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41275473","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $C_{2}$ be a smooth and projective curve over the ring of dual numbers of a field $k.$ Given non-zero rational functions $f,g,$ and $h$ on $C_{2},$ we define an invariant $rho(fwedge g wedge h) in k.$ This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture. Also using $rho,$ we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus.
设$C_{2}$是域$k对偶数环上的光滑投影曲线。给定C_{2}上的非零有理函数$f,g，$和$h$，我们在k上定义了一个不变量$rho(fwedge g wedge h) 。这是对真实解析周二对数的类比，以及对非线性循环的可加二对数的推广。利用这个构造，我们陈述并证明了强互易猜想的一个无穷小版本。同样使用$rho，$，我们定义了权为2的代数环上的一个无穷小调节器，它推广了Park在有模环情况下的构造。
{"title":"Infinitesimal Chow Dilogarithm","authors":"Sı̇nan Ünver","doi":"10.1090/JAG/746","DOIUrl":"https://doi.org/10.1090/JAG/746","url":null,"abstract":"Let $C_{2}$ be a smooth and projective curve over the ring of dual numbers of a field $k.$ Given non-zero rational functions $f,g,$ and $h$ on $C_{2},$ we define an invariant $rho(fwedge g wedge h) in k.$ This is an analog of the real analytic Chow dilogarithm and the extension to non-linear cycles of the additive dilogarithm. Using this construction we state and prove an infinitesimal version of the strong reciprocity conjecture. Also using $rho,$ we define an infinitesimal regulator on algebraic cycles of weight two which generalizes Park's construction in the case of cycles with modulus.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550832","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let kk be a non-Archimedean complete valued field and let XX be a kk-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension k′k’ of kk, every coherent sheaf on X×kk′X times _{k} k’ is acyclic; (2) XX is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, XX
设k k是一个非阿基米德完备值域，设X X是Berkovich意义上的k k-解析空间。在本文中，我们证明了三个性质之间的等价性：（1）对于k k的每个全值扩展k′k′，X×k′Xtimes_｛k｝k′上的每个相干簇都是非循环的；（2） X X是复几何意义上的Stein（全纯分离，全纯凸），并且结构sheaf的更高上同调群消失（例如，如果X X是紧致的，则后一个假设是关键的）；（3） X X承认刘在其反例中考虑的紧致分析域对仿射的上同调标准的适当穷尽。当X X没有边界时，表征更简单：在（2）结构sheaf的上同调群的消失不再需要，因此我们恢复了复杂几何中Stein空间的通常概念；在（3）刘认为的域可以被仿射域取代，这使我们回到了基尔对Stein空间的定义。
{"title":"Notions of Stein spaces in non-Archimedean geometry","authors":"Marco Maculan, Jérôme Poineau","doi":"10.1090/jag/764","DOIUrl":"https://doi.org/10.1090/jag/764","url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a non-Archimedean complete valued field and let <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> be a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k prime\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">k’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\">\u0000 <mml:semantics>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">k</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, every coherent sheaf on <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X times Subscript k Baseline k prime\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:msub>\u0000 <mml:mo>×<!-- × --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>k</mml:mi>\u0000 </mml:mrow>\u0000 </mml:msub>\u0000 <mml:msup>\u0000 <mml:mi>k</mml:mi>\u0000 <mml:mo>′</mml:mo>\u0000 </mml:msup>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">X times _{k} k’</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is acyclic; (2) <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper X\">\u0000 <mml:semantics>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">X</mml:","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46400035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic p > 0 p>0 . Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.
{"title":"The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen-Macaulay","authors":"B. Totaro","doi":"10.1090/JAG/724","DOIUrl":"https://doi.org/10.1090/JAG/724","url":null,"abstract":"We show that the Kodaira vanishing theorem can fail on smooth Fano varieties of any characteristic \u0000\u0000 \u0000 \u0000 p\u0000 >\u0000 0\u0000 \u0000 p>0\u0000 \u0000\u0000. Taking cones over some of these varieties, we give the first examples of terminal singularities which are not Cohen-Macaulay. By a different method, we construct a terminal singularity of dimension 3 (the lowest possible) in characteristic 2 which is not Cohen-Macaulay.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/724","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42351274","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work continues the study of residually wild morphisms f:Y→Xfcolon Yto X of Berkovich curves initiated in [Adv. Math. 303 (2016), pp. 800-858]. The different function δfdelta _f introduced in that work is the primary discrete invariant of such covers. When ff is not residually tame, it provides a non-trivial enhancement of the classical invariant of ff consisting of morphisms of reductions f~:Y~→X~widetilde fcolon widetilde Yto widetilde X and metric skeletons
本工作继续研究Berkovich曲线的剩余野生态态f: Y→X f colon Y to X[数学学报，303 (2016)，pp. 800-858]。在那项工作中引入的不同函数δ f delta ＿f是这些覆盖的主要离散不变量。当f不是剩余驯服时，它提供了由约化f的态射组成的f的经典不变量的非平凡增强:Y→X widetilde f colonwidetilde Y towidetilde X和公制骨架Γ f: Γ Y→Γ X Gamma ＿f colonGamma ＿Y toGamma ＿X。在本文中，我们将δ f delta ＿f解释为对偶束ω f omega ＿f的典型迹段τ f tau ＿f的范数，并引入一个更精细的约化不变量τ f widetildetau ＿f，也就是ω f log omega ＿ {widetilde f}^{operatorname log{的一部分。我们的主要结果将amni - baker - brugall - rabinoff的一个提升定理从剩余驯服态射推广到最小剩余野性态射。对于这样的态射，我们描述了所有的限制:基准(f， Γ f， δ | Γ Y， τ f) (}}widetilde f, Gamma ＿f, delta |＿ {Gamma ＿Y，}widetildetau ＿f)满足并证明，反过来，任何满足这些限制的四重元都可以提升为Berkovich曲线的态射。
{"title":"Lifting problem for minimally wild covers of Berkovich curves","authors":"Uri Brezner, M. Temkin","doi":"10.1090/jag/728","DOIUrl":"https://doi.org/10.1090/jag/728","url":null,"abstract":"<p>This work continues the study of residually wild morphisms <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f colon upper Y right-arrow upper X\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mi>X</mml:mi>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">fcolon Yto X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of Berkovich curves initiated in [Adv. Math. 303 (2016), pp. 800-858]. The different function <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"delta Subscript f\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mi>δ<!-- δ --></mml:mi>\u0000 <mml:mi>f</mml:mi>\u0000 </mml:msub>\u0000 <mml:annotation encoding=\"application/x-tex\">delta _f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> introduced in that work is the primary discrete invariant of such covers. When <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> is not residually tame, it provides a non-trivial enhancement of the classical invariant of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f\">\u0000 <mml:semantics>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">f</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> consisting of morphisms of reductions <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"f overTilde colon upper Y overTilde right-arrow upper X overTilde\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>f</mml:mi>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mo>:<!-- : --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mover>\u0000 <mml:mi>X</mml:mi>\u0000 <mml:mo>~<!-- ~ --></mml:mo>\u0000 </mml:mover>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">widetilde fcolon widetilde Yto widetilde X</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> and metric skeletons <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Gamma Subscript f Baseline colon normal upper Gamma Subscript upper Y Baseline right-arrow normal upper Gamma Subscript u","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/728","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47665941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.
彼得·奥沙利文(Peter O 'Sullivan)的一个显著成果断言，从阿贝变的有理Chow环到它的有理Chow环模数值等价的代数上的外胚允许一个(正则)截面。在Beauville分裂原理的激励下，我们给出了一个猜想的截面性质，该性质预测了对于光滑射影全纯辛变种存在这样一个代数截面，其象包含了该变种的所有chen类。本文研究了具有阿贝尔型Chow动机的(不一定是辛的)变量的这一性质。我们引入了对称区分阿贝尔动机的概念，并利用它为光滑射影变种允许这样的截面提供了充分条件。然后，我们给出了一系列我们的理论适用的变种的例子。例如，我们证明了具有有限秩的Chow群的变种、阿贝尔变种、超椭圆曲线、费马三次超曲面、阿贝尔曲面、Kummer曲面或Picard数至少为19的K3曲面上点的Hilbert格式和广义Kummer变种的任意积的存在性。后一种情况为推测的截面性质提供了证据，并举例说明了全纯辛变量的动机应该表现为阿贝尔变量的动机，作为代数对象。
{"title":"Distinguished cycles on varieties with motive of abelian type and the Section Property","authors":"L. Fu, Charles Vial","doi":"10.1090/jag/729","DOIUrl":"https://doi.org/10.1090/jag/729","url":null,"abstract":"A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/729","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48729398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The strong version of the Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic 0 0 , (T) implies (S). In characteristic p p an analogous result is true under stronger assumptions.
{"title":"A remark on the Tate Conjecture","authors":"B. Moonen","doi":"10.1090/JAG/720","DOIUrl":"https://doi.org/10.1090/JAG/720","url":null,"abstract":"The strong version of the Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic \u0000\u0000 \u0000 0\u0000 0\u0000 \u0000\u0000, (T) implies (S). In characteristic \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000 an analogous result is true under stronger assumptions.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/720","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44716510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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