We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F-regular type. As a consequence, all higher cohomologies of the theta line bundle vanish. During the proof, we develop a method to estimate codimension.
我们证明了光滑曲线上抛物线交/正交束的模空间是全局 F 不规则型的。因此，θ线束的所有高次同调都消失了。在证明过程中，我们开发了一种估计标度的方法。
{"title":"Globally F-regular type of the moduli spaces of parabolic symplectic/orthogonal bundles on curves","authors":"Jianping Wang, Xueqing Wen","doi":"10.1017/fms.2024.57","DOIUrl":"https://doi.org/10.1017/fms.2024.57","url":null,"abstract":"We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F-regular type. As a consequence, all higher cohomologies of the theta line bundle vanish. During the proof, we develop a method to estimate codimension.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"23 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141170524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{mathrm {reg}}$ , and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.
凯勒-爱因斯坦流（又称奇异凯勒-爱因斯坦度量）是在十多年前被提出和构建的。这些气流存在于轻度奇异紧凑的凯勒空间 X 上，它们的两个决定性性质如下：它们是 $X_{mathrm {reg}}$ 上真正的凯勒-爱因斯坦度量，而且它们在 X 的奇点附近承认局部有界势能。在本论文中，我们证明了当 X 承认全局平滑或 X 具有孤立的可平滑奇点时，这些电流在奇点位置附近支配凯勒形式。我们的结果适用于 klt 对，并允许我们证明，如果 X 是任何具有对数末端奇点的三维紧凑凯勒空间，那么任何非正曲率的凯勒-爱因斯坦奇异度量都会支配一个凯勒形式。
{"title":"Strict positivity of Kähler–Einstein currents","authors":"Vincent Guedj, Henri Guenancia, Ahmed Zeriahi","doi":"10.1017/fms.2024.54","DOIUrl":"https://doi.org/10.1017/fms.2024.54","url":null,"abstract":"Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces <jats:italic>X</jats:italic> and their two defining properties are the following: They are genuine Kähler–Einstein metrics on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000549_inline1.png\"/> <jats:tex-math> $X_{mathrm {reg}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, and they admit local bounded potentials near the singularities of <jats:italic>X</jats:italic>. In this note, we show that these currents dominate a Kähler form near the singular locus, when either <jats:italic>X</jats:italic> admits a global smoothing, or when <jats:italic>X</jats:italic> has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if <jats:italic>X</jats:italic> is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141170674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline2.png"/> <jats:tex-math> $gge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline3.png"/> <jats:tex-math> $nge 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline4.png"/> <jats:tex-math> $mathcal {H}_{g,n}subset mathcal {M}_{g,n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the complex moduli stack of <jats:italic>n</jats:italic>-marked smooth hyperelliptic curves of genus <jats:italic>g</jats:italic>. A normal crossings compactification of this space is provided by the theory of pointed admissible <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline5.png"/> <jats:tex-math> $mathbb {Z}/2mathbb {Z}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline6.png"/> <jats:tex-math> $mathcal {H}_{g, n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using this graph complex, we give a sum-over-graphs formula for the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline7.png"/> <jats:tex-math> $S_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-equivariant weight zero compactly supported Euler characteristic of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline8.png"/> <jats:tex-math> $mathcal {H}_{g, n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This formula allows for the computer-aided calculation, for each <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline9.png"/> <jats:tex-math> $gle 7$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, of the generating function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000537_inline10.png"/> <jats:tex-math> $mathsf {h}_g$ </jats:tex-math> </jats:alternativ
对于$g/ge 2$和$n/ge 0$，让$mathcal {H}_{g,n}subset mathcal {M}_{g,n}$ 表示属g的n标记光滑超椭圆曲线的复模数堆栈。尖可容许$mathbb {Z}/2mathbb {Z}$ -覆盖的理论提供了这个空间的法向交叉紧凑性。我们明确地确定了由此产生的对偶复数，并以此定义了一个图复数，它可以计算 $mathcal {H}_{g, n}$ 的权重为零的紧凑支持同调。利用这个图复数，我们给出了 $S_n$ 的权重零紧凑支持的 $mathcal {H}_{g, n}$ 的欧拉特征的过图总和公式。这个公式允许计算机辅助计算每个 $gle 7$ 的生成函数 $mathsf {h}_g$ 对于所有 n 的这些等变欧拉特征。更一般地说，当 G 是无性的时候，我们确定在任何零属曲线的尖可容许 G 笼的模空间中边界的对偶复数为对称的 $Delta $ 复数。我们利用这些复数将我们的 $mathsf {h}_g$ 公式推广到零属曲线的 n 点光滑无常盖的模空间。
{"title":"On the weight zero compactly supported cohomology of","authors":"Madeline Brandt, Melody Chan, Siddarth Kannan","doi":"10.1017/fms.2024.53","DOIUrl":"https://doi.org/10.1017/fms.2024.53","url":null,"abstract":"For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline2.png\"/> <jats:tex-math> $gge 2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline3.png\"/> <jats:tex-math> $nge 0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline4.png\"/> <jats:tex-math> $mathcal {H}_{g,n}subset mathcal {M}_{g,n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> denote the complex moduli stack of <jats:italic>n</jats:italic>-marked smooth hyperelliptic curves of genus <jats:italic>g</jats:italic>. A normal crossings compactification of this space is provided by the theory of pointed admissible <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline5.png\"/> <jats:tex-math> $mathbb {Z}/2mathbb {Z}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline6.png\"/> <jats:tex-math> $mathcal {H}_{g, n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Using this graph complex, we give a sum-over-graphs formula for the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline7.png\"/> <jats:tex-math> $S_n$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-equivariant weight zero compactly supported Euler characteristic of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline8.png\"/> <jats:tex-math> $mathcal {H}_{g, n}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. This formula allows for the computer-aided calculation, for each <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline9.png\"/> <jats:tex-math> $gle 7$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, of the generating function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000537_inline10.png\"/> <jats:tex-math> $mathsf {h}_g$ </jats:tex-math> </jats:alternativ","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"201 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141167644","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 prove some orthogonality relations for representations arising from deep level Deligne–Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig [Lus04], and of Chan and the second author [CI21b]. Applications include the study of smooth representations of p-adic groups in the cohomology of p-adic Deligne–Lusztig spaces and their relation to the local Langlands correspondences. Also, the geometry of deep level Deligne–Lusztig schemes gets accessible, in the spirit of Lusztig’s work [Lus76].
{"title":"Orthogonality relations for deep level Deligne–Lusztig schemes of Coxeter type","authors":"Olivier Dudas, Alexander B. Ivanov","doi":"10.1017/fms.2024.55","DOIUrl":"https://doi.org/10.1017/fms.2024.55","url":null,"abstract":"In this paper, we prove some orthogonality relations for representations arising from deep level Deligne–Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig [Lus04], and of Chan and the second author [CI21b]. Applications include the study of smooth representations of <jats:italic>p</jats:italic>-adic groups in the cohomology of <jats:italic>p</jats:italic>-adic Deligne–Lusztig spaces and their relation to the local Langlands correspondences. Also, the geometry of deep level Deligne–Lusztig schemes gets accessible, in the spirit of Lusztig’s work [Lus76].","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"19 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141147037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Divisorial stability of a polarised variety is a stronger – but conjecturally equivalent – variant of uniform K-stability introduced by Boucksom–Jonsson. Whereas uniform K-stability is defined in terms of test configurations, divisorial stability is defined in terms of convex combinations of divisorial valuations on the variety. We consider the behaviour of divisorial stability under finite group actions and prove that equivariant divisorial stability of a polarised variety is equivalent to log divisorial stability of its quotient. We use this and an interpolation technique to give a general construction of equivariantly divisorially stable polarised varieties.
极化变元的除法稳定性是布克索姆-琼森（Boucksom-Jonsson）提出的均匀 K 稳定性的一个更强--但在猜想上等价--的变种。统一 K 稳定性是根据检验配置定义的，而分裂稳定性则是根据极化变体上分裂值的凸组合定义的。我们考虑了有限群作用下的分裂稳定性行为，并证明了极化变体的等变分裂稳定性等价于其商数的对数分裂稳定性。我们利用这一点和插值技术给出了等变分稳定极化变种的一般构造。
{"title":"On divisorial stability of finite covers","authors":"Ruadhaí Dervan, Theodoros Stylianos Papazachariou","doi":"10.1017/fms.2024.47","DOIUrl":"https://doi.org/10.1017/fms.2024.47","url":null,"abstract":"Divisorial stability of a polarised variety is a stronger – but conjecturally equivalent – variant of uniform K-stability introduced by Boucksom–Jonsson. Whereas uniform K-stability is defined in terms of test configurations, divisorial stability is defined in terms of convex combinations of divisorial valuations on the variety. We consider the behaviour of divisorial stability under finite group actions and prove that equivariant divisorial stability of a polarised variety is equivalent to log divisorial stability of its quotient. We use this and an interpolation technique to give a general construction of equivariantly divisorially stable polarised varieties.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"78 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider cyclic unramified coverings of degree d of irreducible complex smooth genus $2$ curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2, 3, 5, 7$ are quite well understood. Nevertheless, very little is known for higher values of d. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $dge 11$ prime such that $frac {d-1}2$ is also prime. We use results of arithmetic nature on $GL_2$ -type abelian varieties combined with theta-duality techniques.
我们考虑了不可还原的复杂光滑 2$ 属曲线的 d 阶循环无ramified 覆盖及其相应的 Prym 变项。它们提供了具有 d 阶自形性的极化无性变种的自然范例。相关普赖姆图的丰富几何学内容已在多篇论文中进行了研究，并且对 $d=2、3、5、7$ 的情况有了很好的理解。然而，对于更高的 d 值，我们所知甚少。在本文中，我们将研究覆盖是否可以从其 Prym 变体中重建，也就是说，通用 Prym Torelli 定理对于这些覆盖是否成立。我们证明，对于所谓的索菲-热尔曼素数，即对于 $dge 11$ 素数，且 $frac {d-1}2$ 也是素数，这一点是成立的。我们使用了关于 $GL_2$ 类型无性变体的算术性质结果，并结合了 Theta 对偶技术。
{"title":"Cyclic coverings of genus curves of Sophie Germain type","authors":"J.C. Naranjo, A. Ortega, I. Spelta","doi":"10.1017/fms.2024.42","DOIUrl":"https://doi.org/10.1017/fms.2024.42","url":null,"abstract":"We consider cyclic unramified coverings of degree <jats:italic>d</jats:italic> of irreducible complex smooth genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000422_inline2.png\"/> <jats:tex-math> $2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order <jats:italic>d</jats:italic>. The rich geometry of the associated Prym map has been studied in several papers, and the cases <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000422_inline3.png\"/> <jats:tex-math> $d=2, 3, 5, 7$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are quite well understood. Nevertheless, very little is known for higher values of <jats:italic>d</jats:italic>. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000422_inline4.png\"/> <jats:tex-math> $dge 11$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> prime such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000422_inline5.png\"/> <jats:tex-math> $frac {d-1}2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is also prime. We use results of arithmetic nature on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000422_inline6.png\"/> <jats:tex-math> $GL_2$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-type abelian varieties combined with theta-duality techniques.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"52 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141146863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a smooth genus three curve <jats:italic>C</jats:italic>, the moduli space of rank two stable vector bundles on <jats:italic>C</jats:italic> with trivial determinant embeds in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline1.png"/> <jats:tex-math> ${mathbb {P}}^8$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as a hypersurface whose singular locus is the Kummer threefold of <jats:italic>C</jats:italic>; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline2.png"/> <jats:tex-math> $operatorname {mathrm {SU}}_C(2,L)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the moduli space of rank two stable vector bundles on <jats:italic>C</jats:italic> with fixed determinant of odd degree <jats:italic>L</jats:italic>, as a subvariety of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline3.png"/> <jats:tex-math> $G(2,8)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In fact, each point <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline4.png"/> <jats:tex-math> $pin C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> defines a natural embedding of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline5.png"/> <jats:tex-math> $operatorname {mathrm {SU}}_C(2,{mathcal {O}}(p))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline6.png"/> <jats:tex-math> $G(2,8)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050509424000525_inline7.png"/> <jats:tex-math> $operatorname {mathrm {SU}}_C(2,{mathcal {O}}(p))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and thus deserves to be coined the Coble quadric of the pointed curve <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999
{"title":"The Coble quadric","authors":"Vladimiro Benedetti, Michele Bolognesi, Daniele Faenzi, Laurent Manivel","doi":"10.1017/fms.2024.52","DOIUrl":"https://doi.org/10.1017/fms.2024.52","url":null,"abstract":"Given a smooth genus three curve <jats:italic>C</jats:italic>, the moduli space of rank two stable vector bundles on <jats:italic>C</jats:italic> with trivial determinant embeds in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline1.png\"/> <jats:tex-math> ${mathbb {P}}^8$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> as a hypersurface whose singular locus is the Kummer threefold of <jats:italic>C</jats:italic>; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline2.png\"/> <jats:tex-math> $operatorname {mathrm {SU}}_C(2,L)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, the moduli space of rank two stable vector bundles on <jats:italic>C</jats:italic> with fixed determinant of odd degree <jats:italic>L</jats:italic>, as a subvariety of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline3.png\"/> <jats:tex-math> $G(2,8)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. In fact, each point <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline4.png\"/> <jats:tex-math> $pin C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> defines a natural embedding of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline5.png\"/> <jats:tex-math> $operatorname {mathrm {SU}}_C(2,{mathcal {O}}(p))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline6.png\"/> <jats:tex-math> $G(2,8)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000525_inline7.png\"/> <jats:tex-math> $operatorname {mathrm {SU}}_C(2,{mathcal {O}}(p))$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and thus deserves to be coined the Coble quadric of the pointed curve <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"189 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study quasimaps to moduli spaces of sheaves on a $K3$ surface S. We construct a surjective cosection of the obstruction theory of moduli spaces of $epsilon $ -stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on S and the reduced Donaldson–Thomas theory of $Stimes C$ , where C is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on $Stimes C$ , if $g(C)leq 1$ ; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on $Stimes mathbb {P}^1$ .
在这篇文章中，我们研究了$K3$曲面S上剪子的模空间的准映射。我们构建了$epsilon$稳定准映射的模空间的阻塞理论的投射共截。然后，我们建立了还原壁交公式，将 S 上剪切的模空间的还原格罗莫夫-维滕理论与 $Stimes C$ 的还原唐纳森-托马斯理论联系起来，其中 C 是一条节点曲线。作为应用，我们证明了伊古萨尖顶形式猜想的希尔伯特结构部分；如果 $g(C)leq 1$ ，在 $Stimes C$ 上与相对插入的高阶/秩一唐纳森-托马斯对应关系；在 $Stimes mathbb {P}^1$ 上与相对插入的唐纳森-托马斯/潘达里潘德-托马斯对应关系。
{"title":"Quasimaps to moduli spaces of sheaves on a surface","authors":"Denis Nesterov","doi":"10.1017/fms.2024.48","DOIUrl":"https://doi.org/10.1017/fms.2024.48","url":null,"abstract":"In this article, we study quasimaps to moduli spaces of sheaves on a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000483_inline2.png\"/> <jats:tex-math> $K3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> surface <jats:italic>S</jats:italic>. We construct a surjective cosection of the obstruction theory of moduli spaces of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000483_inline3.png\"/> <jats:tex-math> $epsilon $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-stable quasimaps. We then establish reduced wall-crossing formulas which relate the reduced Gromov–Witten theory of moduli spaces of sheaves on <jats:italic>S</jats:italic> and the reduced Donaldson–Thomas theory of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000483_inline4.png\"/> <jats:tex-math> $Stimes C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:italic>C</jats:italic> is a nodal curve. As applications, we prove the Hilbert-schemes part of the Igusa cusp form conjecture; higher-rank/rank-one Donaldson–Thomas correspondence with relative insertions on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000483_inline5.png\"/> <jats:tex-math> $Stimes C$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000483_inline6.png\"/> <jats:tex-math> $g(C)leq 1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>; Donaldson–Thomas/Pandharipande–Thomas correspondence with relative insertions on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000483_inline7.png\"/> <jats:tex-math> $Stimes mathbb {P}^1$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"70 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_infty $ -coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.
{"title":"Adams’ cobar construction as a monoidal -coalgebra model of the based loop space","authors":"Anibal M. Medina-Mardones, Manuel Rivera","doi":"10.1017/fms.2024.50","DOIUrl":"https://doi.org/10.1017/fms.2024.50","url":null,"abstract":"We prove that the classical map comparing Adams’ cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000501_inline2.png\"/> <jats:tex-math> $E_infty $ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams’ map preserves monoidal coalgebra structures.","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"37 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062675","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element. This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees (intervals).
{"title":"Limit pretrees for free group automorphisms: existence","authors":"Jean Pierre Mutanguha","doi":"10.1017/fms.2024.38","DOIUrl":"https://doi.org/10.1017/fms.2024.38","url":null,"abstract":"To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element. This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees (intervals).","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140930086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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