Pub Date : 2020-05-04DOI: 10.1007/978-3-030-84721-0_18
A. Gonzalez-Prieto, Marina Logares, V. Muñoz
{"title":"Representation variety for the rank one affine group","authors":"A. Gonzalez-Prieto, Marina Logares, V. Muñoz","doi":"10.1007/978-3-030-84721-0_18","DOIUrl":"https://doi.org/10.1007/978-3-030-84721-0_18","url":null,"abstract":"","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"212 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124174915","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}
In this paper we give a complete description of the integral Chow ring of the stack $mathscr{H}_{g,1}$ of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of $mathscr{H}_g$. We also give a geometric interpretation for the generators.
{"title":"The Integral Chow Ring of the Stack of 1-Pointed Hyperelliptic Curves","authors":"Michele Pernice","doi":"10.1093/IMRN/RNAB072","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB072","url":null,"abstract":"In this paper we give a complete description of the integral Chow ring of the stack $mathscr{H}_{g,1}$ of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of $mathscr{H}_g$. We also give a geometric interpretation for the generators.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116471640","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}
F. Bogomolov, N. Kurnosov, A. Kuznetsova, E. Yasinsky
We consider the only one known class of non-Kahler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kahler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan property.
{"title":"Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds","authors":"F. Bogomolov, N. Kurnosov, A. Kuznetsova, E. Yasinsky","doi":"10.1093/IMRN/RNAB043","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB043","url":null,"abstract":"We consider the only one known class of non-Kahler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover of some non-Kahler manifold $W_F$ which we call the base of $Q$. We show that the algebraic reduction of $Q$ and its base is the projective space of dimension $n-1$. Besides, we give a partial classification of submanifolds in $Q$, describe the degeneracy locus of its algebraic reduction, and prove that the automorphism group of $Q$ satisfies the Jordan property.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126012195","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 : 2020-04-30DOI: 10.3929/ETHZ-B-000314055
A. Puttick
Let $mathbb{F}_q$ be a finite field with $q$ elements and let $V$ be a vector space over $mathbb{F}_q$ of dimension $n>0$. Let $Omega_V$ be the Drinfeld period domain over $mathbb{F}_q$. This is an affine scheme of finite type over $mathbb{F}_q$, and its base change to $mathbb{F}_q(t)$ is the moduli space of Drinfeld $mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. In this thesis, we give a new modular interpretation to Pink and Schieder's smooth compactification $B_V$ of $Omega_V$. Let $hat V$ be the set $Vcup{infty}$ for a new symbol $infty$. We define the notion of a $V$-fern over an $mathbb{F}_q$-scheme $S$, which consists of a stable $hat V$-marked curve of genus $0$ over $S$ endowed with a certain action of the finite group $Vrtimes mathbb{F}_q^times$. Our main result is that the scheme $B_V$ represents the functor that associates an $mathbb{F}_q$-scheme $S$ to the set of isomorphism classes of $V$-ferns over $S$. Thus $V$-ferns over $mathbb{F}_q(t)$-schemes can be regarded as generalizations of Drinfeld $mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. To prove this theorem, we construct an explicit universal $V$-fern over $B_V$. We then show that any $V$-fern over a scheme $S$ determines a unique morphism $Sto B_V$, depending only its isomorphism class, and that the $V$-fern is isomorphic to the pullback of the universal $V$-fern along this morphism. We also give several functorial constructions involving $V$-ferns, some of which are used to prove the main result. These constructions correspond to morphisms between various modular compactifications of Drinfeld period domains over $mathbb{F}_q$. We describe these morphisms explicitly.
{"title":"Compactification of the finite Drinfeld period domain as a moduli space of ferns","authors":"A. Puttick","doi":"10.3929/ETHZ-B-000314055","DOIUrl":"https://doi.org/10.3929/ETHZ-B-000314055","url":null,"abstract":"Let $mathbb{F}_q$ be a finite field with $q$ elements and let $V$ be a vector space over $mathbb{F}_q$ of dimension $n>0$. Let $Omega_V$ be the Drinfeld period domain over $mathbb{F}_q$. This is an affine scheme of finite type over $mathbb{F}_q$, and its base change to $mathbb{F}_q(t)$ is the moduli space of Drinfeld $mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. In this thesis, we give a new modular interpretation to Pink and Schieder's smooth compactification $B_V$ of $Omega_V$. Let $hat V$ be the set $Vcup{infty}$ for a new symbol $infty$. We define the notion of a $V$-fern over an $mathbb{F}_q$-scheme $S$, which consists of a stable $hat V$-marked curve of genus $0$ over $S$ endowed with a certain action of the finite group $Vrtimes mathbb{F}_q^times$. Our main result is that the scheme $B_V$ represents the functor that associates an $mathbb{F}_q$-scheme $S$ to the set of isomorphism classes of $V$-ferns over $S$. Thus $V$-ferns over $mathbb{F}_q(t)$-schemes can be regarded as generalizations of Drinfeld $mathbb{F}_q[t]$-modules with level $(t)$ structure and rank $n$. To prove this theorem, we construct an explicit universal $V$-fern over $B_V$. We then show that any $V$-fern over a scheme $S$ determines a unique morphism $Sto B_V$, depending only its isomorphism class, and that the $V$-fern is isomorphic to the pullback of the universal $V$-fern along this morphism. We also give several functorial constructions involving $V$-ferns, some of which are used to prove the main result. These constructions correspond to morphisms between various modular compactifications of Drinfeld period domains over $mathbb{F}_q$. We describe these morphisms explicitly.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123837586","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}
We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product $Stimes M$ of a profinite set $S$ with a locally noetherian formal scheme $M$ and study intersections thereon. Our application is to the Arithmetic Fundamental Lemma of W. Zhang where the result helps to remove a restriction in its recent proof, cf. arXiv:1909.02697.
{"title":"Local Constancy of Intersection Numbers","authors":"A. Mihatsch","doi":"10.2140/ant.2022.16.505","DOIUrl":"https://doi.org/10.2140/ant.2022.16.505","url":null,"abstract":"We prove that, in certain situations, intersection numbers on formal schemes that come in profinite families vary locally constantly in the parameter. To this end, we define the product $Stimes M$ of a profinite set $S$ with a locally noetherian formal scheme $M$ and study intersections thereon. Our application is to the Arithmetic Fundamental Lemma of W. Zhang where the result helps to remove a restriction in its recent proof, cf. arXiv:1909.02697.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128649705","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}
We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following the same steps as in the general case, treated next.
{"title":"The Bogomolov-Beauville-Yau decomposition for klt projective varieties with trivial first Chern class - without tears","authors":"F. Campana","doi":"10.24033/BSMF.2823","DOIUrl":"https://doi.org/10.24033/BSMF.2823","url":null,"abstract":"We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following the same steps as in the general case, treated next.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116660771","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}
We prove homological mirror symmetry for Milnor fibers of simple singularities, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix factorizations and Calabi--Yau completions. As an application, we give an explicit computation of the symplectic cohomology group of the Milnor fiber of a simple singularity in all dimensions.
{"title":"Homological mirror symmetry for Milnor fibers of simple singularities","authors":"Yankı Lekili, K. Ueda","doi":"10.14231/AG-2021-017","DOIUrl":"https://doi.org/10.14231/AG-2021-017","url":null,"abstract":"We prove homological mirror symmetry for Milnor fibers of simple singularities, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix factorizations and Calabi--Yau completions. As an application, we give an explicit computation of the symplectic cohomology group of the Milnor fiber of a simple singularity in all dimensions.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121942191","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}
In this paper, we prove a Kawamata--Viehweg type vanishing theorem for smooth Fano threefolds, canonical del Pezzo surfaces and del Pezzo fibrations in positive characteristic.
{"title":"On Kawamata-Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic","authors":"Tatsuro Kawakami","doi":"10.1090/tran/8369","DOIUrl":"https://doi.org/10.1090/tran/8369","url":null,"abstract":"In this paper, we prove a Kawamata--Viehweg type vanishing theorem for smooth Fano threefolds, canonical del Pezzo surfaces and del Pezzo fibrations in positive characteristic.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124974569","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 : 2020-04-10DOI: 10.1142/s1664360720500149
I. Biswas
Let $X$ be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map $varphi: {rm Sym}^d(X) rightarrow {rm Pic}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $varphi$, of the flat metric on ${rm Pic}^d(X)$. In particular, we show that when $d=1$, the curvature is strictly negative everywhere if $X$ is not hyperelliptic, and when $X$ is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of $X$ fixed by the hyperelliptic involution.
{"title":"Abel–Jacobi map and curvature of the pulled back metric","authors":"I. Biswas","doi":"10.1142/s1664360720500149","DOIUrl":"https://doi.org/10.1142/s1664360720500149","url":null,"abstract":"Let $X$ be a compact connected Riemann surface of genus at least two. The Abel-Jacobi map $varphi: {rm Sym}^d(X) rightarrow {rm Pic}^d(X)$ is an embedding if $d$ is less than the gonality of $X$. We investigate the curvature of the pull-back, by $varphi$, of the flat metric on ${rm Pic}^d(X)$. In particular, we show that when $d=1$, the curvature is strictly negative everywhere if $X$ is not hyperelliptic, and when $X$ is hyperelliptic, the curvature is nonpositive with vanishing exactly on the points of $X$ fixed by the hyperelliptic involution.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126223742","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}
Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge--Ampere equations, and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.
{"title":"Differentiability of Relative Volumes Over an Arbitrary Non-Archimedean Field","authors":"S. Boucksom, Walter Gubler, Florent Martin","doi":"10.1093/imrn/rnaa314","DOIUrl":"https://doi.org/10.1093/imrn/rnaa314","url":null,"abstract":"Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge--Ampere equations, and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.","PeriodicalId":278201,"journal":{"name":"arXiv: Algebraic Geometry","volume":"64 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130571890","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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