We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author. For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.
{"title":"Donaldson–Thomas invariants of abelian threefolds and Bridgeland stability conditions","authors":"G. Oberdieck, D. Piyaratne, Yukinobu Toda","doi":"10.1090/JAG/788","DOIUrl":"https://doi.org/10.1090/JAG/788","url":null,"abstract":"We study the reduced Donaldson–Thomas theory of abelian threefolds using Bridgeland stability conditions. The main result is the invariance of the reduced Donaldson–Thomas invariants under all derived autoequivalences, up to explicitly given wall-crossing terms. We also present a numerical criterion for the absence of walls in terms of a discriminant function. For principally polarized abelian threefolds of Picard rank one, the wall-crossing contributions are discussed in detail. The discussion yields evidence for a conjectural formula for curve counting invariants by Bryan, Pandharipande, Yin, and the first author.\u0000\u0000For the proof we strengthen several known results on Bridgeland stability conditions of abelian threefolds. We show that certain previously constructed stability conditions satisfy the full support property. In particular, the stability manifold is non-empty. We also prove the existence of a Gieseker chamber and determine all wall-crossing contributions. A definition of reduced generalized Donaldson–Thomas invariants for arbitrary Calabi–Yau threefolds with abelian actions is given.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46311841","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 XX be a smooth rigid analytic space. We prove that the category of co-admissible wideparen {mathcal {D}_X}-modules supported on a closed smooth subvariety YY of XX is naturally equivalent to the category of co-admissible wideparen {mathcal {D}_Y}-modules and use this result to construct a large family of pairwise non-isomorphic simple co-admissible wideparen {mathcal {D}_X}-modules.
{"title":"wideparen{𝒟}-modules on rigid analytic spaces II: Kashiwara’s equivalence","authors":"K. Ardakov, S. Wadsley","doi":"10.1090/JAG/709","DOIUrl":"https://doi.org/10.1090/JAG/709","url":null,"abstract":"<p>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 smooth rigid analytic space. We prove that the category of co-admissible <inline-formula content-type=\"math/tex\">\u0000<tex-math>\u0000wideparen {mathcal {D}_X}</tex-math></inline-formula>-modules supported on a closed smooth subvariety <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper Y\">\u0000 <mml:semantics>\u0000 <mml:mi>Y</mml:mi>\u0000 <mml:annotation encoding=\"application/x-tex\">Y</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=\"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 naturally equivalent to the category of co-admissible <inline-formula content-type=\"math/tex\">\u0000<tex-math>\u0000wideparen {mathcal {D}_Y}</tex-math></inline-formula>-modules and use this result to construct a large family of pairwise non-isomorphic simple co-admissible <inline-formula content-type=\"math/tex\">\u0000<tex-math>\u0000wideparen {mathcal {D}_X}</tex-math></inline-formula>-modules.</p>","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAG/709","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45772115","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 extend the derived algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraic K 0 K^0 analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.
{"title":"Bivariant derived algebraic cobordism","authors":"Toni Annala","doi":"10.1090/jag/754","DOIUrl":"https://doi.org/10.1090/jag/754","url":null,"abstract":"We extend the derived algebraic bordism of Lowrey and Schürg to a bivariant theory in the sense of Fulton and MacPherson and establish some of its basic properties. As a special case, we obtain a completely new theory of cobordism rings of singular quasi-projective schemes. The extended cobordism is shown to specialize to algebraic \u0000\u0000 \u0000 \u0000 K\u0000 0\u0000 \u0000 K^0\u0000 \u0000\u0000 analogously to the Conner-Floyd theorem in topology. We also give a candidate for the correct definition of Chow rings of singular schemes.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49028393","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 study the Picard groups of connected linear algebraic groups and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.
{"title":"Translation-invariant line bundles on linear algebraic groups","authors":"Zev Rosengarten","doi":"10.1090/jag/753","DOIUrl":"https://doi.org/10.1090/jag/753","url":null,"abstract":"We study the Picard groups of connected linear algebraic groups and especially the subgroup of translation-invariant line bundles. We prove that this subgroup is finite over every global function field. We also utilize our study of these groups in order to construct various examples of pathological behavior for the cohomology of commutative linear algebraic groups over local and global function fields.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41736229","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 give a recipe for producing infinitely many nondivisible codimension 2 2 cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of “field of definition” for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension 2 2 cycles, that we call “primitive cycles”, the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori’s isogeny method (modified by Totaro) to produce infinitely many nondivisible cycles. As an application, we prove the Chow group modulo a prime of a product of three or more very general elliptic curves is infinite, generalizing work of Schoen.
{"title":"Nondivisible cycles on products of very general Abelian varieties","authors":"H. A. Diaz","doi":"10.1090/jag/775","DOIUrl":"https://doi.org/10.1090/jag/775","url":null,"abstract":"In this paper, we give a recipe for producing infinitely many nondivisible codimension \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000 cycles on a product of two or more very general Abelian varieties. In the process, we introduce the notion of “field of definition” for cycles in the Chow group modulo (a power of) a prime. We show that for a quite general class of codimension \u0000\u0000 \u0000 2\u0000 2\u0000 \u0000\u0000 cycles, that we call “primitive cycles”, the field of definition is a ramified extension of the function field of a modular variety. This ramification allows us to use Nori’s isogeny method (modified by Totaro) to produce infinitely many nondivisible cycles. As an application, we prove the Chow group modulo a prime of a product of three or more very general elliptic curves is infinite, generalizing work of Schoen.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49040785","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 that every quasi-log canonical pair has only Du Bois singularities. Note that our arguments are free from the minimal model program.
我们证明了每一个拟对数正则对都只有杜波依斯奇点。请注意，我们的参数不受最小模型程序的约束。
{"title":"Quasi-log canonical pairs are Du Bois","authors":"O. Fujino, Haidong Liu","doi":"10.1090/jag/756","DOIUrl":"https://doi.org/10.1090/jag/756","url":null,"abstract":"We prove that every quasi-log canonical pair has only Du Bois singularities. Note that our arguments are free from the minimal model program.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46959908","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 characterize K. Kato’s log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.
我们用对数余切复数的（co）同调的消失来刻画K.Kato的对数正则性。
{"title":"Homological characterization of regularity in logarithmic algebraic geometry","authors":"J. Conde-Lago, J. Majadas","doi":"10.1090/jag/787","DOIUrl":"https://doi.org/10.1090/jag/787","url":null,"abstract":"We characterize K. Kato’s log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42179311","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 first construct varieties of any dimension n>2n>2 fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [Springer Proc. Math. Stat. 71 (2014), pp. 1–40].
Led by their conjecture, we focus on finding the lowest possible slope when n=3n=3. Based on a characteristic p>0p > 0 method, we prove that the sharp lower bound of the slope of fibered 33-folds over curves is 4/34/3, and it occurs only when the general fiber is a (1,2)(1, 2)-surface. Otherwise, the sharp lower bound is 2
{"title":"Fibered varieties over curves with low slope and sharp bounds in dimension three","authors":"Yong Hu, Tongde Zhang","doi":"10.1090/jag/739","DOIUrl":"https://doi.org/10.1090/jag/739","url":null,"abstract":"<p>In this paper, we first construct varieties of any dimension <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n greater-than 2\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n>2</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> fibered over curves with low slopes. These examples violate the conjectural slope inequality of Barja and Stoppino [Springer Proc. Math. Stat. 71 (2014), pp. 1–40].</p>\u0000\u0000<p>Led by their conjecture, we focus on finding the lowest possible slope when <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n equals 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>n</mml:mi>\u0000 <mml:mo>=</mml:mo>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">n=3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>. Based on a characteristic <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p greater-than 0\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mi>p</mml:mi>\u0000 <mml:mo>></mml:mo>\u0000 <mml:mn>0</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">p > 0</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula> method, we prove that the sharp lower bound of the slope of fibered <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\u0000 <mml:semantics>\u0000 <mml:mn>3</mml:mn>\u0000 <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-folds over curves is <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"4 slash 3\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mn>4</mml:mn>\u0000 <mml:mrow class=\"MJX-TeXAtom-ORD\">\u0000 <mml:mo>/</mml:mo>\u0000 </mml:mrow>\u0000 <mml:mn>3</mml:mn>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">4/3</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>, and it occurs only when the general fiber is a <inline-formula content-type=\"math/mathml\">\u0000<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis 1 comma 2 right-parenthesis\">\u0000 <mml:semantics>\u0000 <mml:mrow>\u0000 <mml:mo stretchy=\"false\">(</mml:mo>\u0000 <mml:mn>1</mml:mn>\u0000 <mml:mo>,</mml:mo>\u0000 <mml:mn>2</mml:mn>\u0000 <mml:mo stretchy=\"false\">)</mml:mo>\u0000 </mml:mrow>\u0000 <mml:annotation encoding=\"application/x-tex\">(1, 2)</mml:annotation>\u0000 </mml:semantics>\u0000</mml:math>\u0000</inline-formula>-surface. Otherwise, the sharp lower bound is <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 encodi","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43991984","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 study the problem of existence of Kähler–Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree 22 22 that admit a faithful action of the multiplicative group C ∗ mathbb {C}^ast . We prove that, with the possible exception of two explicitly described cases, all such smooth Fano threefolds are Kähler–Einstein.
{"title":"Kähler–Einstein Fano threefolds of degree 22","authors":"I. Cheltsov, C. Shramov","doi":"10.1090/jag/812","DOIUrl":"https://doi.org/10.1090/jag/812","url":null,"abstract":"We study the problem of existence of Kähler–Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree \u0000\u0000 \u0000 22\u0000 22\u0000 \u0000\u0000 that admit a faithful action of the multiplicative group \u0000\u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 ∗\u0000 \u0000 mathbb {C}^ast\u0000 \u0000\u0000. We prove that, with the possible exception of two explicitly described cases, all such smooth Fano threefolds are Kähler–Einstein.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42756039","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}
On smooth varieties, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. In dimension three, we reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.
{"title":"On equivalent conjectures for minimal log discrepancies on smooth threefolds","authors":"M. Kawakita","doi":"10.1090/jag/757","DOIUrl":"https://doi.org/10.1090/jag/757","url":null,"abstract":"On smooth varieties, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. In dimension three, we reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.","PeriodicalId":54887,"journal":{"name":"Journal of Algebraic Geometry","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2018-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45766995","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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