The purpose of this paper is to prove a basic pp-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure CC of a pp-adic field: pp-adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over BdR+{mathbf B}^+_{operatorname {dR} }). The key computation is the passage from absolute crystalline cohomology to Hyodo–Kato cohomology and the construction of the related Hyodo–Kato isomorphism. We also “geometrize” our comparison theorem by turning pp-adic pro-étale and syntomic cohomologies into sheaves on the category Pe
本文的目的是证明在一个 p p -adic 场的代数闭包 C C 上的光滑刚性解析变种和匕首变种的一个基本 p p -adic 比较定理：在一个稳定范围内，p p -adic pro-étale cohomology 可以表示为 de Rham cohomology (over B dR + {mathbf B}^+_{operatorname {dR}) 的滤波 Frobenius 特征空间。} ).关键的计算是从绝对晶体同调到兵多-加藤同调的过程，以及相关的兵多-加藤同构的构造。我们还 "几何化 "了我们的比较定理，把 p p -adic pro-étale 和 syntomic cohomologies 转化为在 C C 上的完形空间类别 P e r f C {mathrm {Perf}}_C 上的剪子，并把周期态变转化为这些剪子之间的映射（这种几何化将在我们对 C s t C_{mathrm {st}} 的研究中起到关键作用）。 -猜想以及几何 p p -adic pro-étale cohomology 的对偶性表述中至关重要）。
{"title":"On the cohomology of 𝑝-adic analytic spaces, I: The basic comparison theorem","authors":"Pierre Colmez, Wiesława Nizioł","doi":"10.1090/jag/835","DOIUrl":"https://doi.org/10.1090/jag/835","url":null,"abstract":"<p>The purpose of this paper is to prove a basic <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>-adic comparison theorem for smooth rigid analytic and dagger varieties over the algebraic closure <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper C\">\u0000 <mml:semantics>\u0000 <mml:mi>C</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">C</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> of a <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>-adic field: <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>-adic pro-étale cohomology, in a stable range, can be expressed as a filtered Frobenius eigenspace of de Rham cohomology (over <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"bold upper B Subscript d upper R Superscript plus\">\u0000 <mml:semantics>\u0000 <mml:msubsup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"bold\">B</mml:mi>\u0000 </mml:mrow>\u0000 </mml:mrow>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi>dR</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mo>+</mml:mo>\u0000 </mml:msubsup>\u0000 <mml:annotation encoding=\"application/x-tex\">{mathbf B}^+_{operatorname {dR} }</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>). The key computation is the passage from absolute crystalline cohomology to Hyodo–Kato cohomology and the construction of the related Hyodo–Kato isomorphism. We also “geometrize” our comparison theorem by turning <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>-adic pro-étale and syntomic cohomologies into sheaves on the category <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper P normal e normal r normal f Subscript upper C\">\u0000 <mml:semantics>\u0000 <mml:msub>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"normal\">P</mml:mi>\u0000 <mml:mi mathvariant=\"normal\">e</mml:mi>\u0000 <mml:mi mathv","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141800760","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}
For a rank 1 local system on the complement of a reduced divisor on a complex manifold X X , its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we study necessary or sufficient conditions for a quasi-isomorphism from its twisted logarithmic subcomplex, called the logarithmic comparison theorem (LCT), by using a stronger version in terms of the associated complex of D X D_X -modules. In case the connection is a pullback by a defining function f f of the divisor and the residue is α alpha , we prove among others that if LCT holds, the annihilator of f α − 1 f^{alpha -1} in D X D_X is generated by first order differential operators and α − 1 − j alpha -1-j is not a root of the Bernstein-Sato polynomial for any positive integer j j . The converse holds assuming either of the two conditions in case the associated complex of D X D_X -modules is acyclic except for the top degree. In the case where the local system is constant, the divisor is defined by a homogeneous polynomial, and the associated projective hypersurface has only weighted homogeneous isolated singularities, we show that LCT is equivalent to that − 1 -1 is the unique integral root of the Bernstein-Sato polynomial. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues, which is an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity. Here the zero-extension case is also treated.
对于复流形 X X 上还原除数的补集上的 1 级局部系统，其同调是通过扭曲对数 de Rham 复数计算的。假设除数处处都是正向加权同调，我们研究了从其扭曲对数子复数（称为对数比较定理（LCT））出发的准同调的必要或充分条件，并使用了与 D X D_X 模块相关复数的更强版本。如果连接是除数的定义函数 f f 的回拉，并且残差是 α alpha ，我们证明了如果 LCT 成立，那么在 D X D_X 中 f α - 1 f^{alpha -1} 的湮没器由一阶微分算子生成，并且 α - 1 - j alpha -1-j 对于任何正整数 j j 都不是伯恩斯坦-萨托多项式的根。反过来，假设这两个条件中的任何一个，D X D_X 模块的相关复数除了顶层之外都是非循环的，那么反过来也成立。在局部系统恒定、除数由同次多项式定义、相关投影超曲面只有加权同次孤立奇点的情况下，我们证明 LCT 等价于 - 1 -1 是伯恩斯坦-萨托多项式的唯一积分根。我们还给出了超平面排列情况下 LCT 的简单证明，该证明是与卡斯特努沃-蒙福德正则性相关的高同调消失的直接推论。这里还处理了零扩展情况。
{"title":"Twisted logarithmic complexes of positively weighted homogeneous divisors","authors":"Daniel Bath, M. Saito","doi":"10.1090/jag/833","DOIUrl":"https://doi.org/10.1090/jag/833","url":null,"abstract":"For a rank 1 local system on the complement of a reduced divisor on a complex manifold \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000, its cohomology is calculated by the twisted meromorphic de Rham complex. Assuming the divisor is everywhere positively weighted homogeneous, we study necessary or sufficient conditions for a quasi-isomorphism from its twisted logarithmic subcomplex, called the logarithmic comparison theorem (LCT), by using a stronger version in terms of the associated complex of \u0000\u0000 \u0000 \u0000 D\u0000 X\u0000 \u0000 D_X\u0000 \u0000\u0000-modules. In case the connection is a pullback by a defining function \u0000\u0000 \u0000 f\u0000 f\u0000 \u0000\u0000 of the divisor and the residue is \u0000\u0000 \u0000 α\u0000 alpha\u0000 \u0000\u0000, we prove among others that if LCT holds, the annihilator of \u0000\u0000 \u0000 \u0000 f\u0000 \u0000 α\u0000 −\u0000 1\u0000 \u0000 \u0000 f^{alpha -1}\u0000 \u0000\u0000 in \u0000\u0000 \u0000 \u0000 D\u0000 X\u0000 \u0000 D_X\u0000 \u0000\u0000 is generated by first order differential operators and \u0000\u0000 \u0000 \u0000 α\u0000 −\u0000 1\u0000 −\u0000 j\u0000 \u0000 alpha -1-j\u0000 \u0000\u0000 is not a root of the Bernstein-Sato polynomial for any positive integer \u0000\u0000 \u0000 j\u0000 j\u0000 \u0000\u0000. The converse holds assuming either of the two conditions in case the associated complex of \u0000\u0000 \u0000 \u0000 D\u0000 X\u0000 \u0000 D_X\u0000 \u0000\u0000-modules is acyclic except for the top degree. In the case where the local system is constant, the divisor is defined by a homogeneous polynomial, and the associated projective hypersurface has only weighted homogeneous isolated singularities, we show that LCT is equivalent to that \u0000\u0000 \u0000 \u0000 −\u0000 1\u0000 \u0000 -1\u0000 \u0000\u0000 is the unique integral root of the Bernstein-Sato polynomial. We also give a simple proof of LCT in the hyperplane arrangement case under appropriate assumptions on residues, which is an immediate corollary of higher cohomology vanishing associated with Castelnuovo-Mumford regularity. Here the zero-extension case is also treated.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141652266","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 introduce and study the notion of atomic sheaves and complexes on higher-dimensional hyper-Kähler manifolds and show that they share many of the intriguing properties of simple sheaves on K3 surfaces. For example, we prove formality of the dg algebra of derived endomorphisms for stable atomic bundles. We further demonstrate the characteristics of atomic objects by studying atomic Lagrangian submanifolds. In the appendix, we prove nonexistence results for spherical objects on hyper-Kähler manifolds.
{"title":"Atomic objects on hyper-Kähler manifolds","authors":"Thorsten Beckmann","doi":"10.1090/jag/830","DOIUrl":"https://doi.org/10.1090/jag/830","url":null,"abstract":"We introduce and study the notion of atomic sheaves and complexes on higher-dimensional hyper-Kähler manifolds and show that they share many of the intriguing properties of simple sheaves on K3 surfaces. For example, we prove formality of the dg algebra of derived endomorphisms for stable atomic bundles. We further demonstrate the characteristics of atomic objects by studying atomic Lagrangian submanifolds. In the appendix, we prove nonexistence results for spherical objects on hyper-Kähler manifolds.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140683496","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 develop a framework to construct moduli spaces of Q {mathbb {Q}} -Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of Q {mathbb {Q}} -stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than 1 2 frac {1}{2} . Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.
{"title":"Moduli of ℚ-Gorenstein pairs and applications","authors":"Stefano Filipazzi, Giovanni Inchiostro","doi":"10.1090/jag/823","DOIUrl":"https://doi.org/10.1090/jag/823","url":null,"abstract":"We develop a framework to construct moduli spaces of \u0000\u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 \u0000 {mathbb {Q}}\u0000 \u0000\u0000-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of \u0000\u0000 \u0000 \u0000 \u0000 Q\u0000 \u0000 \u0000 {mathbb {Q}}\u0000 \u0000\u0000-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than \u0000\u0000 \u0000 \u0000 1\u0000 2\u0000 \u0000 frac {1}{2}\u0000 \u0000\u0000. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139452253","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 prove a splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert.
{"title":"Splitting of Gromov–Witten invariants with toric gluing strata","authors":"Yixian Wu","doi":"10.1090/jag/826","DOIUrl":"https://doi.org/10.1090/jag/826","url":null,"abstract":"We prove a splitting formula that reconstructs the logarithmic Gromov–Witten invariants of simple normal crossing varieties from the punctured Gromov–Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov–Witten theory developed by Abramovich, Chen, Gross, and Siebert.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993320","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}
Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a kk-rational isolated singularity is kk-Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the kk-Du Bois and kk-rational singularities in terms of standard invariants of singularities. In particular, we show that kk-Du Bois singularities are (k−1)(k-1)-rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.
高有理数和高杜波依斯奇点最近被引入作为有理数和杜波依斯奇点标准定义的自然推广。在这篇笔记中，我们讨论了孤立奇点的这些性质，特别是在局部完全交集(lci)情况下。首先，我们在没有任何lci假设的情况下，证明了kk -有理孤立奇点是kk -Du Bois的事实。对于孤立的lci奇点，我们用奇点的标准不变量给出了k k -杜波依斯奇点和k k -有理奇点的完整刻画。特别地，我们证明了k k -Du Bois奇点对于孤立的lci奇点是(k−1)(k-1) -有理的。在证明过程中，我们建立了孤立lci奇点不变量之间的一些新关系，并证明了其中许多不变量是消失的。该方法还能快速证明孤立lci情况下附加定理的反演。最后，我们讨论了一些特定于超曲面情况的结果。
{"title":"The higher Du Bois and higher rational properties for isolated singularities","authors":"Robert Friedman, Radu Laza","doi":"10.1090/jag/824","DOIUrl":"https://doi.org/10.1090/jag/824","url":null,"abstract":"Higher rational and higher Du Bois singularities have recently been introduced as natural generalizations of the standard definitions of rational and Du Bois singularities. In this note, we discuss these properties for isolated singularities, especially in the locally complete intersection (lci) case. First, we reprove the fact that a <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational isolated singularity is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Du Bois without any lci assumption. For isolated lci singularities, we give a complete characterization of the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Du Bois and <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational singularities in terms of standard invariants of singularities. In particular, we show that <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Du Bois singularities are <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis k minus 1 right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(k-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-rational for isolated lci singularities. In the course of the proof, we establish some new relations between invariants of isolated lci singularities and show that many of these vanish. The methods also lead to a quick proof of an inversion of adjunction theorem in the isolated lci case. Finally, we discuss some results specific to the hypersurface case.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192781","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}
In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic R{mathbb {R}}-Cartier divisor D¯overline {D} are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized R{mathbb {R}}-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of D¯overline {D} coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.
{"title":"Arithmetic Okounkov bodies and positivity of adelic Cartier divisors","authors":"François Ballaÿ","doi":"10.1090/jag/821","DOIUrl":"https://doi.org/10.1090/jag/821","url":null,"abstract":"In algebraic geometry, theorems of Küronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Cartier divisor <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D overbar\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\"application/x-tex\">overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{mathbb {R}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D overbar\"> <mml:semantics> <mml:mover> <mml:mi>D</mml:mi> <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo> </mml:mover> <mml:annotation encoding=\"application/x-tex\">overline {D}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the infimum of the Boucksom–Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135993452","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 introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen 2 2 -form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the 2017 results of Mikhalkin to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.
{"title":"Refined count of oriented real rational curves","authors":"Thomas Blomme","doi":"10.1090/jag/801","DOIUrl":"https://doi.org/10.1090/jag/801","url":null,"abstract":"We introduce a quantum index for oriented real curves inside toric varieties. This quantum index is related to the computation of the area of the amoeba of the curve for some chosen \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000-form. We then make a refined signed count of oriented real rational curves solution to some enumerative problem. This generalizes the 2017 results of Mikhalkin to higher dimension. Finally, we use the tropical approach to relate these new refined invariants to previously known tropical refined invariants.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43704449","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 A1mathbb {A}^1-connectedness of a large class of varieties over a field kk can be characterized as the condition that their generic point can be connected to a kk-rational point using (not necessarily naive) A1mathbb {A}^1-homotopies. We also show that symmetric powers of A1mathbb {A}^1-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic 00) are A1
{"title":"Geometric criteria for 𝔸¹-connectedness and applications to norm varieties","authors":"Chetan T. Balwe, A. Hogadi, Anand Sawant","doi":"10.1090/jag/790","DOIUrl":"https://doi.org/10.1090/jag/790","url":null,"abstract":"<p>We show that <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-connectedness of a large class of varieties over a field <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> can be characterized as the condition that their generic point can be connected to 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>-rational point using (not necessarily naive) <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-homotopies. We also show that symmetric powers of <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"application/x-tex\">mathbb {A}^1</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-connected smooth projective varieties (over an arbitrary field) as well as smooth proper models of them (over an algebraically closed field of characteristic <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>) are <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper A Superscript 1\">\u0000 <mml:semantics>\u0000 <mml:msup>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mi mathvariant=\"double-struck\">A</mml:mi>\u0000 </mml:mrow>\u0000 <mml:mn>1</mml:mn>\u0000 </mml:msup>\u0000 <mml:annotation encoding=\"applicatio","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42482940","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 give a criterion to test geometric properties such as Whitney equisingularity and Thom’s a f a_f condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
{"title":"Nondegenerate locally tame complete intersection varieties and geometry of nonisolated hypersurface singularities","authors":"C. Eyral, M. Oka","doi":"10.1090/jag/784","DOIUrl":"https://doi.org/10.1090/jag/784","url":null,"abstract":"We give a criterion to test geometric properties such as Whitney equisingularity and Thom’s \u0000\u0000 \u0000 \u0000 a\u0000 f\u0000 \u0000 a_f\u0000 \u0000\u0000 condition for new families of (possibly nonisolated) hypersurface singularities that “behave well” with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41639855","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：