Pub Date : 2022-06-06DOI: 10.46298/epiga.2023.volume7.9730
J'anos Koll'ar
We show that flat families of stable 3-folds do not lead to proper moduli spaces in any characteristic $p>0$. As a byproduct, we obtain log canonical 4-fold pairs, whose log canonical centers are not weakly normal.
{"title":"Families of stable 3-folds in positive characteristic","authors":"J'anos Koll'ar","doi":"10.46298/epiga.2023.volume7.9730","DOIUrl":"https://doi.org/10.46298/epiga.2023.volume7.9730","url":null,"abstract":"We show that flat families of stable 3-folds do not lead to proper moduli\u0000spaces in any characteristic $p>0$. As a byproduct, we obtain log canonical\u00004-fold pairs, whose log canonical centers are not weakly normal.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-27DOI: 10.46298/epiga.2023.volume7.9657
Christopher Lazda, A. Skorobogatov
We obtain necessary and sufficient conditions for the good reduction of Kummer surfaces attached to abelian surfaces with non-supersingular reduction when the residue field is perfect of characteristic 2. In this case, good reduction with an algebraic space model is equivalent to good reduction with a scheme model, which we explicitly construct.
{"title":"Reduction of Kummer surfaces modulo 2 in the non-supersingular case","authors":"Christopher Lazda, A. Skorobogatov","doi":"10.46298/epiga.2023.volume7.9657","DOIUrl":"https://doi.org/10.46298/epiga.2023.volume7.9657","url":null,"abstract":"We obtain necessary and sufficient conditions for the good reduction of\u0000Kummer surfaces attached to abelian surfaces with non-supersingular reduction\u0000when the residue field is perfect of characteristic 2. In this case, good\u0000reduction with an algebraic space model is equivalent to good reduction with a\u0000scheme model, which we explicitly construct.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47826078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-25DOI: 10.46298/epiga.2023.10321
Olivier Benoist, O. Debarre
We give new examples of algebraic integral cohomology classes on smooth projective complex varieties that are not integral linear combinations of classes of smooth subvarieties. Some of our examples have dimension 6, the lowest possible. The classes that we consider are minimal cohomology classes on Jacobians of very general curves. Our main tool is complex cobordism.
{"title":"Smooth subvarieties of Jacobians","authors":"Olivier Benoist, O. Debarre","doi":"10.46298/epiga.2023.10321","DOIUrl":"https://doi.org/10.46298/epiga.2023.10321","url":null,"abstract":"We give new examples of algebraic integral cohomology classes on smooth\u0000projective complex varieties that are not integral linear combinations of\u0000classes of smooth subvarieties. Some of our examples have dimension 6, the\u0000lowest possible. The classes that we consider are minimal cohomology classes on\u0000Jacobians of very general curves. Our main tool is complex cobordism.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-30DOI: 10.46298/epiga.2022.9758
Salvatore Floccari
We prove that the rational Chow motive of a six dimensional hyper-K"{a}hler variety obtained as symplectic resolution of O'Grady type of a singular moduli space of semistable sheaves on an abelian surface $A$ belongs to the tensor category of motives generated by the motive of $A$. We in fact give a formula for the rational Chow motive of such a variety in terms of that of the surface. As a consequence, the conjectures of Hodge and Tate hold for many hyper-K"{a}hler varieties of OG6-type.
证明了在阿贝曲面上半稳定轴的奇异模空间O'Grady型的辛分解得到的六维超k {a}的有理Chow动机属于由a $的动机所产生的动机的张量范畴。事实上，我们用表面的理性周氏动机给出了一个公式。因此，Hodge和Tate的猜想对og6型的许多hyper- k {a}hler变种都成立。
{"title":"On the motive of O'Grady's six dimensional hyper-K\"{a}hler varieties","authors":"Salvatore Floccari","doi":"10.46298/epiga.2022.9758","DOIUrl":"https://doi.org/10.46298/epiga.2022.9758","url":null,"abstract":"We prove that the rational Chow motive of a six dimensional hyper-K\"{a}hler\u0000variety obtained as symplectic resolution of O'Grady type of a singular moduli\u0000space of semistable sheaves on an abelian surface $A$ belongs to the tensor\u0000category of motives generated by the motive of $A$. We in fact give a formula\u0000for the rational Chow motive of such a variety in terms of that of the surface.\u0000As a consequence, the conjectures of Hodge and Tate hold for many\u0000hyper-K\"{a}hler varieties of OG6-type.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48026650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-27DOI: 10.46298/epiga.2023.9617
Junliang Shen, Qizheng Yin
Perverse-Hodge complexes are objects in the derived category of coherent sheaves obtained from Hodge modules associated with Saito's decomposition theorem. We study perverse-Hodge complexes for Lagrangian fibrations and propose a symmetry between them. This conjectural symmetry categorifies the "Perverse = Hodge" identity of the authors and specializes to Matsushita's theorem on the higher direct images of the structure sheaf. We verify our conjecture in several cases by making connections with variations of Hodge structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.
{"title":"Perverse-Hodge complexes for Lagrangian fibrations","authors":"Junliang Shen, Qizheng Yin","doi":"10.46298/epiga.2023.9617","DOIUrl":"https://doi.org/10.46298/epiga.2023.9617","url":null,"abstract":"Perverse-Hodge complexes are objects in the derived category of coherent\u0000sheaves obtained from Hodge modules associated with Saito's decomposition\u0000theorem. We study perverse-Hodge complexes for Lagrangian fibrations and\u0000propose a symmetry between them. This conjectural symmetry categorifies the\u0000\"Perverse = Hodge\" identity of the authors and specializes to Matsushita's\u0000theorem on the higher direct images of the structure sheaf. We verify our\u0000conjecture in several cases by making connections with variations of Hodge\u0000structures, Hilbert schemes, and Looijenga-Lunts-Verbitsky Lie algebras.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485591","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-16DOI: 10.46298/epiga.2023.9382
Kuang-Yu Wu
We define a new notion of affine subspace concentration conditions for lattice polytopes, and prove that they hold for smooth and reflexive polytopes with barycenter at the origin. Our proof involves considering the slope stability of the canonical extension of the tangent bundle by the trivial line bundle and with the extension class $c_1(mathcal{T}_X)$ on Fano toric varieties.
{"title":"Affine Subspace Concentration Conditions","authors":"Kuang-Yu Wu","doi":"10.46298/epiga.2023.9382","DOIUrl":"https://doi.org/10.46298/epiga.2023.9382","url":null,"abstract":"We define a new notion of affine subspace concentration conditions for\u0000lattice polytopes, and prove that they hold for smooth and reflexive polytopes\u0000with barycenter at the origin. Our proof involves considering the slope\u0000stability of the canonical extension of the tangent bundle by the trivial line\u0000bundle and with the extension class $c_1(mathcal{T}_X)$ on Fano toric\u0000varieties.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-13DOI: 10.46298/epiga.2023.10073
Ana-Maria Brecan, H. Franzen
We study the locus of fixed points of a torus action on a GIT quotient of a complex vector space by a reductive complex algebraic group which acts linearly. We show that, under the assumption that $G$ acts freely on the stable locus, the components of the fixed point locus are again GIT quotients of linear subspaces by Levi subgroups.
{"title":"Torus Actions on Quotients of Affine Spaces","authors":"Ana-Maria Brecan, H. Franzen","doi":"10.46298/epiga.2023.10073","DOIUrl":"https://doi.org/10.46298/epiga.2023.10073","url":null,"abstract":"We study the locus of fixed points of a torus action on a GIT quotient of a\u0000complex vector space by a reductive complex algebraic group which acts\u0000linearly. We show that, under the assumption that $G$ acts freely on the stable\u0000locus, the components of the fixed point locus are again GIT quotients of\u0000linear subspaces by Levi subgroups.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-23DOI: 10.46298/epiga.2022.8910
Z. Reichstein, F. Scavia
Let $A$ be a discrete valuation ring with generic point $eta$ and closed point $s$. We show that in a family of torsors over $operatorname{Spec}(A)$, the essential dimension of the torsor above $s$ is less than or equal to the essential dimension of the torsor above $eta$. We give two applications of this result, one in mixed characteristic, the other in equal characteristic.
{"title":"The behavior of essential dimension under specialization","authors":"Z. Reichstein, F. Scavia","doi":"10.46298/epiga.2022.8910","DOIUrl":"https://doi.org/10.46298/epiga.2022.8910","url":null,"abstract":"Let $A$ be a discrete valuation ring with generic point $eta$ and closed\u0000point $s$. We show that in a family of torsors over $operatorname{Spec}(A)$,\u0000the essential dimension of the torsor above $s$ is less than or equal to the\u0000essential dimension of the torsor above $eta$. We give two applications of\u0000this result, one in mixed characteristic, the other in equal characteristic.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70484321","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-13DOI: 10.46298/epiga.2023.specialvolumeinhonourofclairevoisin.9626
Benjamin Bakker, Thomas W. Grimm, C. Schnell, Jacob Tsimerman
We generalize the finiteness theorem for the locus of Hodge classes with fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge classes to self-dual classes. The proof uses the definability of period mappings in the o-minimal structure $mathbb{R}_{mathrm{an},exp}$.
我们将Cattani, Deligne, and Kaplan的自交数固定的Hodge类轨迹的有限性定理从Hodge类推广到自对偶类。该证明使用了0最小结构$mathbb{R}_{ mathm {an}，exp}$中周期标记的可定义性。
{"title":"Finiteness for self-dual classes in integral variations of Hodge structure","authors":"Benjamin Bakker, Thomas W. Grimm, C. Schnell, Jacob Tsimerman","doi":"10.46298/epiga.2023.specialvolumeinhonourofclairevoisin.9626","DOIUrl":"https://doi.org/10.46298/epiga.2023.specialvolumeinhonourofclairevoisin.9626","url":null,"abstract":"We generalize the finiteness theorem for the locus of Hodge classes with\u0000fixed self-intersection number, due to Cattani, Deligne, and Kaplan, from Hodge\u0000classes to self-dual classes. The proof uses the definability of period\u0000mappings in the o-minimal structure $mathbb{R}_{mathrm{an},exp}$.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70485166","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-22DOI: 10.46298/epiga.2023.8793
Erwan Brugall'e, Florent Schaffhauser
We prove that moduli spaces of semistable vector bundles of coprime rank and degree over a non-singular real projective curve are maximal real algebraic varieties if and only if the base curve itself is maximal. This provides a new family of maximal varieties, with members of arbitrarily large dimension. We prove the result by comparing the Betti numbers of the real locus to the Hodge numbers of the complex locus and showing that moduli spaces of vector bundles over a maximal curve actually satisfy a property which is stronger than maximality and that we call Hodge-expressivity. We also give a brief account on other varieties for which this property was already known.
{"title":"Maximality of moduli spaces of vector bundles on curves","authors":"Erwan Brugall'e, Florent Schaffhauser","doi":"10.46298/epiga.2023.8793","DOIUrl":"https://doi.org/10.46298/epiga.2023.8793","url":null,"abstract":"We prove that moduli spaces of semistable vector bundles of coprime rank and\u0000degree over a non-singular real projective curve are maximal real algebraic\u0000varieties if and only if the base curve itself is maximal. This provides a new\u0000family of maximal varieties, with members of arbitrarily large dimension. We\u0000prove the result by comparing the Betti numbers of the real locus to the Hodge\u0000numbers of the complex locus and showing that moduli spaces of vector bundles\u0000over a maximal curve actually satisfy a property which is stronger than\u0000maximality and that we call Hodge-expressivity. We also give a brief account on\u0000other varieties for which this property was already known.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46978816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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