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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.
{"title":"Parallel transport for vector bundles on 𝑝-adic varieties","authors":"C. Deninger, A. Werner","doi":"10.1090/jag/747","DOIUrl":"https://doi.org/10.1090/jag/747","url":null,"abstract":"We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/747","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49655222","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 X X be a simply connected projective manifold with nef anticanonical bundle. We prove that X X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC-fibration of any projective manifold with nef anticanonical bundle.
{"title":"A decomposition theorem for projective manifolds with nef anticanonical bundle","authors":"Junyan Cao, A. Horing","doi":"10.1090/JAG/715","DOIUrl":"https://doi.org/10.1090/JAG/715","url":null,"abstract":"Let \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 be a simply connected projective manifold with nef anticanonical bundle. We prove that \u0000\u0000 \u0000 X\u0000 X\u0000 \u0000\u0000 is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC-fibration of any projective manifold with nef anticanonical bundle.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/715","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44568805","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 this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms. Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2). The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).
{"title":"Boundedness questions for Calabi–Yau threefolds","authors":"P. Wilson","doi":"10.1090/JAG/781","DOIUrl":"https://doi.org/10.1090/JAG/781","url":null,"abstract":"In this paper, we study boundedness questions for (simply connected) smooth Calabi–Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms.\u0000\u0000Crucial to this question is the study of rigid non-movable surfaces on the threefold, which are the irreducible surfaces that deform with any small deformation of the complex structure of the threefold but for which no multiple moves on the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2).\u0000\u0000The paper starts by proving general results on these rigid non-movable surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi–Yau threefolds (Theorem 0.3). This appears to be the first non-trivial case where a general boundedness result for Calabi–Yau threefolds has been proved (without the assumption of a special structure).","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44857653","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}
{"title":"PD Higgs crystals and Higgs cohomology in characteristic","authors":"Hideto Oyama","doi":"10.1090/JAG/699","DOIUrl":"https://doi.org/10.1090/JAG/699","url":null,"abstract":"","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/699","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47044659","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 cone of primitive Heegner divisors is finitely generated for many orthogonal Shimura varieties, including the moduli space of polarized K 3 K3 -surfaces. The proof relies on the growth of coefficients of modular forms.
{"title":"Cones of Heegner divisors","authors":"J. Bruinier, M. Moller","doi":"10.1090/JAG/734","DOIUrl":"https://doi.org/10.1090/JAG/734","url":null,"abstract":"We show that the cone of primitive Heegner divisors is finitely generated for many orthogonal Shimura varieties, including the moduli space of polarized \u0000\u0000 \u0000 \u0000 K\u0000 3\u0000 \u0000 K3\u0000 \u0000\u0000-surfaces. The proof relies on the growth of coefficients of modular forms.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/734","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48079677","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 this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The Main Theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that p p -adic Hopf varieties will never have a projective reduction. The proof of our Main Theorem uses the theory of moduli of semistable coherent sheaves.
{"title":"On rigid varieties with projective reduction","authors":"Shizhang Li","doi":"10.1090/jag/740","DOIUrl":"https://doi.org/10.1090/jag/740","url":null,"abstract":"In this paper, we study smooth proper rigid varieties which admit formal models whose special fibers are projective. The Main Theorem asserts that the identity components of the associated rigid Picard varieties will automatically be proper. Consequently, we prove that \u0000\u0000 \u0000 p\u0000 p\u0000 \u0000\u0000-adic Hopf varieties will never have a projective reduction. The proof of our Main Theorem uses the theory of moduli of semistable coherent sheaves.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/jag/740","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49053961","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 associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type ( d , … , d ) (d,dots , d) is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.
{"title":"Stability of associated forms","authors":"M. Fedorchuk, A. Isaev","doi":"10.1090/JAG/719","DOIUrl":"https://doi.org/10.1090/JAG/719","url":null,"abstract":"We show that the associated form, or, equivalently, a Macaulay inverse system, of an Artinian complete intersection of type \u0000\u0000 \u0000 \u0000 (\u0000 d\u0000 ,\u0000 …\u0000 ,\u0000 d\u0000 )\u0000 \u0000 (d,dots , d)\u0000 \u0000\u0000 is polystable. As an application, we obtain an invariant-theoretic variant of the Mather-Yau theorem for homogeneous hypersurface singularities.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/719","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49202850","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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