Pub Date : 2018-04-24DOI: 10.46298/epiga.2019.volume3.4460
I. Biswas, Sorin Dumitrescu, B. McKay
International audience We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structures. Nous montrons que toute variété complexe compacte de dimension algébrique nulle possédant une géométrie de Cartan holomorphe de type algébrique doit avoir un groupe fondamental infini. Il s’agit d’une généralisation du théorème principal de [DM] où le même résultat était montré dans le cas particulier des connexions affines holomorphes et des structures conformes holomorphes.
{"title":"CARTAN GEOMETRIES ON COMPLEX MANIFOLDS OF ALGEBRAIC DIMENSION ZERO","authors":"I. Biswas, Sorin Dumitrescu, B. McKay","doi":"10.46298/epiga.2019.volume3.4460","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.4460","url":null,"abstract":"International audience\u0000 \u0000 We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structures.\u0000 \u0000 \u0000 Nous montrons que toute variété complexe compacte de dimension algébrique nulle possédant une géométrie de Cartan holomorphe de type algébrique doit avoir un groupe fondamental infini. Il s’agit d’une généralisation du théorème principal de [DM] où le même résultat était montré dans le cas particulier des connexions affines holomorphes et des structures conformes holomorphes.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482685","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 : 2018-04-13DOI: 10.46298/epiga.2020.volume4.5940
Dario Beraldo
We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding.
{"title":"The spectral gluing theorem revisited","authors":"Dario Beraldo","doi":"10.46298/epiga.2020.volume4.5940","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5940","url":null,"abstract":"We strengthen the gluing theorem occurring on the spectral side of the\u0000geometric Langlands conjecture. While the latter embeds $IndCoh_N(LS_G)$ into a\u0000category glued out of 'Fourier coefficients' parametrized by standard\u0000parabolics, our refinement explicitly identifies the essential image of such\u0000embedding.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47726850","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 : 2018-04-02DOI: 10.46298/epiga.2020.volume4.5662
L. Borisov, Sai-Kee Yeung
We construct explicit equations of Cartwright-Steger and related surfaces. Comment: 16 pages, LaTeX
构造了Cartwright-Steger曲面及相关曲面的显式方程。评论:16页，LaTeX
{"title":"Explicit equations of the Cartwright-Steger surface","authors":"L. Borisov, Sai-Kee Yeung","doi":"10.46298/epiga.2020.volume4.5662","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5662","url":null,"abstract":"We construct explicit equations of Cartwright-Steger and related surfaces.\u0000\u0000 Comment: 16 pages, LaTeX","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48502364","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 : 2018-02-20DOI: 10.46298/epiga.2019.volume3.4792
M. Romagny, D. Tossici
International audience We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 to text{Lie}(G, I) to E to G to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$. Nous construisons une équivalence entre la catégorie des schémas en groupes affines et lisses sur l'anneau des nombres duaux généralisés k[I], et la catégorie des extensions de la forme 1 → Lie(G, I) → E → G → 1 où G est un schéma en groupes affine, lisse sur k. Ici k est un anneau commutatif arbitraire et k[I] = k ⊕ I avec I 2 = 0. L'équivalence est donnée par la restriction de Weil, et nous construisons un foncteur quasi-inverse explicite que nous appelons extension de Weil. Ces foncteurs sont compatibles avec les structures exactes et avec les structures de champs en O k-modules des deux catégories. Nos constructions s'appuient sur le schéma en algèbres de groupe d'un schéma en groupes affines, que nous introduisons et dont nous donnons les propriétés principales. En application, nous donnons une classification de Dieudonné pour les schémas en groupes commutatifs, lisses, unipotents sur k[I] lorsque k est un corps parfait.
我们提供了广义对偶数$k[I]$环上仿射光滑群方案的范畴与$1到$ text{Lie}(G, I) 到E 到G 到1$的形式的扩展范畴之间的等价性，其中G是k上的仿射光滑群方案。这里k是一个任意交换环，$k[I] = k o + I$且$I^2 = 0$。该等价由Weil限制给出，并给出一个拟逆，我们称之为Weil扩展。它与两个类别上的精确结构和$mathbb{O}_k$-模块堆栈结构兼容。我们的构造依赖于仿射群方案的群代数方案的使用;我们介绍了这个对象，并确定了它的主要性质。作为应用，我们建立了$k[I]$上光滑、可交换、幂偶群方案的dieudonn分类。有两个构式，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换，一个是单质交换。L'的等效性不是基于限制de Weil的，而是基于特征的准逆显式的基于扩展de Weil的。ce的特点是具有较强的兼容性，包括双通道通道结构、双通道通道结构和双通道通道模块。没有结构的适用范围，没有结构的适用范围，没有结构的适用范围，没有结构的介绍，没有结构的适用范围，没有结构的适用范围，没有结构的适用范围，没有结构的适用范围。在申请中，nous donnons one classification de dieudononnous()将其归类为可交换者、无能力者、无能力者和无能力者([I])。
{"title":"Smooth affine group schemes over the dual numbers","authors":"M. Romagny, D. Tossici","doi":"10.46298/epiga.2019.volume3.4792","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.4792","url":null,"abstract":"International audience\u0000 We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 to text{Lie}(G, I) to E to G to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.\u0000 Nous construisons une équivalence entre la catégorie des schémas en groupes affines et lisses sur l'anneau des nombres duaux généralisés k[I], et la catégorie des extensions de la forme 1 → Lie(G, I) → E → G → 1 où G est un schéma en groupes affine, lisse sur k. Ici k est un anneau commutatif arbitraire et k[I] = k ⊕ I avec I 2 = 0. L'équivalence est donnée par la restriction de Weil, et nous construisons un foncteur quasi-inverse explicite que nous appelons extension de Weil. Ces foncteurs sont compatibles avec les structures exactes et avec les structures de champs en O k-modules des deux catégories. Nos constructions s'appuient sur le schéma en algèbres de groupe d'un schéma en groupes affines, que nous introduisons et dont nous donnons les propriétés principales. En application, nous donnons une classification de Dieudonné pour les schémas en groupes commutatifs, lisses, unipotents sur k[I] lorsque k est un corps parfait.","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483239","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 : 2018-02-08DOI: 10.46298/epiga.2019.volume3.4895
Anne Lonjou
To reinforce the analogy between the mapping class group and the Cremona group of rank $2$ over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. A candidate is a graph introduced by D. Wright. However, we demonstrate that it is not Gromov-hyperbolic. This answers a question of A. Minasyan and D. Osin. Then, we construct two graphs associated to a Vorono"i tesselation of the Cremona group introduced in a previous work of the autor. We show that one is quasi-isometric to the Wright graph. We prove that the second one is Gromov-hyperbolic. Comment: 29 pages, en Franc{c}ais
{"title":"Sur l'hyperbolicit'e de graphes associ'es au groupe de Cremona","authors":"Anne Lonjou","doi":"10.46298/epiga.2019.volume3.4895","DOIUrl":"https://doi.org/10.46298/epiga.2019.volume3.4895","url":null,"abstract":"To reinforce the analogy between the mapping class group and the Cremona\u0000group of rank $2$ over an algebraic closed field, we look for a graph\u0000analoguous to the curve graph and such that the Cremona group acts on it\u0000non-trivially. A candidate is a graph introduced by D. Wright. However, we\u0000demonstrate that it is not Gromov-hyperbolic. This answers a question of A.\u0000Minasyan and D. Osin. Then, we construct two graphs associated to a Vorono\"i\u0000tesselation of the Cremona group introduced in a previous work of the autor. We\u0000show that one is quasi-isometric to the Wright graph. We prove that the second\u0000one is Gromov-hyperbolic.\u0000\u0000 Comment: 29 pages, en Franc{c}ais","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483421","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 : 2017-12-20DOI: 10.46298/epiga.2018.volume2.4454
A. Beauville
We attempt to describe the rank 2 vector bundles on a curve C which are specializations of the trivial bundle. We get a complete classifications when C is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit vector bundles are decomposable. We give examples of indecomposable limit bundles for some special curves. Comment: Final version, published in Epiga
{"title":"Limits of the trivial bundle on a curve","authors":"A. Beauville","doi":"10.46298/epiga.2018.volume2.4454","DOIUrl":"https://doi.org/10.46298/epiga.2018.volume2.4454","url":null,"abstract":"We attempt to describe the rank 2 vector bundles on a curve C which are\u0000specializations of the trivial bundle. We get a complete classifications when C\u0000is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit\u0000vector bundles are decomposable. We give examples of indecomposable limit\u0000bundles for some special curves.\u0000\u0000 Comment: Final version, published in Epiga","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483102","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 : 2017-11-27DOI: 10.46298/epiga.2020.volume4.5506
A. Marian, Xiaolei Zhao
For a moduli space of Bridgeland-stable objects on a K3 surface, we show that the Chow class of a point is determined by the Chern class of the corresponding object on the surface. This establishes a conjecture of Junliang Shen, Qizheng Yin, and the second author.
{"title":"On the group of zero-cycles of holomorphic symplectic varieties","authors":"A. Marian, Xiaolei Zhao","doi":"10.46298/epiga.2020.volume4.5506","DOIUrl":"https://doi.org/10.46298/epiga.2020.volume4.5506","url":null,"abstract":"For a moduli space of Bridgeland-stable objects on a K3 surface, we show that\u0000the Chow class of a point is determined by the Chern class of the corresponding\u0000object on the surface. This establishes a conjecture of Junliang Shen, Qizheng\u0000Yin, and the second author.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47642085","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 : 2017-11-22DOI: 10.46298/epiga.2018.volume2.4179
A. Kuznetsov, Yuri Prokhorov
We give an explicit construction of prime Fano threefolds of genus 12 with a $G_m$-action, describe their isomorphism classes and automorphism groups. Comment: 14 pages, LaTeX, updated version, to appear in 'Epijournal de G'eom'etrie Alg'ebrique, Vol. 2 (2018), Article Nr. 3
给出了具有$G_m$-作用的素数Fano三重格12的显式构造，描述了它们的同构类和自同构群。评论:14页，LaTeX，更新版本，将出现在《Epijournal de G' em 'etrie Alg'ebrique》第2卷(2018)，第3期
{"title":"Prime Fano threefolds of genus 12 with a $G_m$-action","authors":"A. Kuznetsov, Yuri Prokhorov","doi":"10.46298/epiga.2018.volume2.4179","DOIUrl":"https://doi.org/10.46298/epiga.2018.volume2.4179","url":null,"abstract":"We give an explicit construction of prime Fano threefolds of genus 12 with a\u0000$G_m$-action, describe their isomorphism classes and automorphism groups.\u0000\u0000 Comment: 14 pages, LaTeX, updated version, to appear in 'Epijournal de\u0000 G'eom'etrie Alg'ebrique, Vol. 2 (2018), Article Nr. 3","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482506","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 : 2017-11-21DOI: 10.46298/epiga.2018.volume2.4126
Philipp Jell, Johannes Rau, Kristin M. Shaw
For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincar'e duality for integral tropical homology. Comment: 27 pages, 6 figures, published version
{"title":"Lefschetz (1,1)-theorem in tropical geometry","authors":"Philipp Jell, Johannes Rau, Kristin M. Shaw","doi":"10.46298/epiga.2018.volume2.4126","DOIUrl":"https://doi.org/10.46298/epiga.2018.volume2.4126","url":null,"abstract":"For a tropical manifold of dimension n we show that the tropical homology\u0000classes of degree (n-1, n-1) which arise as fundamental classes of tropical\u0000cycles are precisely those in the kernel of the eigenwave map. To prove this we\u0000establish a tropical version of the Lefschetz (1, 1)-theorem for rational\u0000polyhedral spaces that relates tropical line bundles to the kernel of the wave\u0000homomorphism on cohomology. Our result for tropical manifolds then follows by\u0000combining this with Poincar'e duality for integral tropical homology.\u0000\u0000 Comment: 27 pages, 6 figures, published version","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70482420","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 : 2017-11-01DOI: 10.46298/epiga.2018.volume2.4209
I. Biswas, Vamsi Pingali
A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact complex manifold admitting a Gauduchon astheno-Kahler metric.
{"title":"A characterization of finite vector bundles on Gauduchon astheno-Kahler\u0000 manifolds","authors":"I. Biswas, Vamsi Pingali","doi":"10.46298/epiga.2018.volume2.4209","DOIUrl":"https://doi.org/10.46298/epiga.2018.volume2.4209","url":null,"abstract":"A vector bundle E on a projective variety X is called finite if it satisfies\u0000a nontrivial polynomial equation with integral coefficients. A theorem of Nori\u0000implies that E is finite if and only if the pullback of E to some finite etale\u0000Galois covering of X is trivial. We prove the same statement when X is a\u0000compact complex manifold admitting a Gauduchon astheno-Kahler metric.\u0000","PeriodicalId":41470,"journal":{"name":"Epijournal de Geometrie Algebrique","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70483021","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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