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":"23 ","pages":""},"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":"1 1","pages":""},"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":" ","pages":""},"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":" ","pages":""},"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}
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":" ","pages":""},"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":" ","pages":""},"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}
We construct the moduli stack of torsors over the formal punctured disk in characteristic p > 0 p>0 for a finite group isomorphic to the semidirect product of a p p -group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.
构造了特征为p>0 p b> 0的形式刺破盘上的模堆，得到了与p p -群与单调循环群半直积同构的有限群。证明了该叠是具有有限泛内射跃迁映射的分离delign - mumford叠的极限。
{"title":"Moduli of formal torsors","authors":"F. Tonini, Takehiko Yasuda","doi":"10.1090/jag/771","DOIUrl":"https://doi.org/10.1090/jag/771","url":null,"abstract":"We construct the moduli stack of torsors over the formal punctured disk in characteristic \u0000\u0000 \u0000 \u0000 p\u0000 >\u0000 0\u0000 \u0000 p>0\u0000 \u0000\u0000 for a finite group isomorphic to the semidirect product of a \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-group and a tame cyclic group. We prove that the stack is a limit of separated Deligne-Mumford stacks with finite and universally injective transition maps.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43555407","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}
Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases that these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary.
给定一个有限群作用的变量，比较了它的等变范畴测度，即相应商栈的范畴测度与扩展商的范畴测度。利用轨道的弱分解，我们证明了在很多情况下这两个度量是一致的。这特别暗示了Galkin和Shinder关于种类的范畴和动机的ζ函数的猜想。我们提供的例子表明，在一般情况下，这两个措施是不相等的。我们还举了一个与Polishchuk和Van den Bergh的一个猜想有关的例子，证明了这个猜想中的某个条件确实是必要的。
{"title":"Categorical measures for finite group actions","authors":"Daniel Bergh, S. Gorchinskiy, M. Larsen, V. Lunts","doi":"10.1090/JAG/768","DOIUrl":"https://doi.org/10.1090/JAG/768","url":null,"abstract":"Given a variety with a finite group action, we compare its equivariant categorical measure, that is, the categorical measure of the corresponding quotient stack, and the categorical measure of the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases that these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal. We also give an example related to a conjecture of Polishchuk and Van den Bergh, showing that a certain condition in this conjecture is indeed necessary.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48691631","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 present a new method to solve certain ∂ ¯ bar partial -equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a ∂ ∂ ¯ partial bar partial -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at E 1 E_1 -level, as well as a certain injectivity theorem on compact Kähler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic ( n , q ) (n,q) -form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.
我们提出了一种求解某∂¯的新方法 bar partial -利用谐波积分理论求解Kähler流形上电流的对数微分形式方程。结果可以看作是∂∂¯ partial bar partial 对数形式的引理。作为应用，我们推广了Deligne关于对数形式的闭性的结果，给出了Deligne关于对数Hodge的简并定理在e1e＿1能级上对de Rham谱序列的几何证明和更简单的证明，以及紧形Kähler流形上的一个注入定理。此外，对于Kähler流形上的一类复杂结构的对数变形，我们构造了在中心纤维上任意对数(n,q) (n,q) -形式的可拓，从而通过将迭代方法推广到对数形式，推导出对数Calabi-Yau结构的局部稳定性。最后用微分几何方法证明了对数Calabi-Yau对和Calabi-Yau流形上的一对变形的无障碍性。
{"title":"Geometry of logarithmic forms and deformations of complex structures","authors":"Kefeng Liu, S. Rao, Xueyuan Wan","doi":"10.1090/JAG/723","DOIUrl":"https://doi.org/10.1090/JAG/723","url":null,"abstract":"We present a new method to solve certain \u0000\u0000 \u0000 \u0000 \u0000 ∂\u0000 ¯\u0000 \u0000 \u0000 bar partial\u0000 \u0000\u0000-equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a \u0000\u0000 \u0000 \u0000 ∂\u0000 \u0000 \u0000 ∂\u0000 ¯\u0000 \u0000 \u0000 \u0000 partial bar partial\u0000 \u0000\u0000-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at \u0000\u0000 \u0000 \u0000 E\u0000 1\u0000 \u0000 E_1\u0000 \u0000\u0000-level, as well as a certain injectivity theorem on compact Kähler manifolds.\u0000\u0000Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic \u0000\u0000 \u0000 \u0000 (\u0000 n\u0000 ,\u0000 q\u0000 )\u0000 \u0000 (n,q)\u0000 \u0000\u0000-form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2017-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/723","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550963","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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