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The $${{,textrm{K},}}$$ -theory of the moduli stacks $${{mathcal {M}}}_2$$ and $$overline{{{mathcal {M}}}}_2$$ 模数堆栈 $${{mathcal {M}}_2$$ 和 $$overline{{{mathcal {M}}}}_2$$ 的理论
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-07-05 DOI: 10.1007/s00229-024-01581-z
Dan Edidin, Zhengning Hu
{"title":"The $${{,textrm{K},}}$$ -theory of the moduli stacks $${{mathcal {M}}}_2$$ and $$overline{{{mathcal {M}}}}_2$$","authors":"Dan Edidin, Zhengning Hu","doi":"10.1007/s00229-024-01581-z","DOIUrl":"https://doi.org/10.1007/s00229-024-01581-z","url":null,"abstract":"<p>We compute the integral Grothendieck rings of the moduli stacks, <span>({{mathcal {M}}}_2)</span>, <span>(overline{{{mathcal {M}}}}_2)</span> of smooth and stable curves of genus two respectively. We compute <span>({{,textrm{K},}}_0({{mathcal {M}}}_2))</span> by using the presentation of <span>({{mathcal {M}}}_2)</span> as a global quotient stack given by Vistoli (Invent Math 131(3):635–644, 1998). To compute the Grothendieck ring <span>({{,textrm{K},}}_0(overline{{{mathcal {M}}}}_2))</span> we decompose <span>(overline{{{mathcal {M}}}}_2)</span> as <span>(Delta _1)</span> and its complement <span>(overline{{{mathcal {M}}}}_2 setminus Delta _1)</span> and use their presentations as quotient stacks given by Larson (Algebr Geom 8 (3):286–318, 2021) to compute the Grothendieck rings. We show that they are torsion-free and this, together with the Riemann–Roch isomorphism allows us to ultimately give a presentation for the integral Grothendieck ring <span>({{,textrm{K},}}_0(overline{{{mathcal {M}}}}_2))</span>.\u0000</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141547789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On weaker notions for Kähler-Ricci solitons 关于 Kähler-Ricci 孤子的较弱概念
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-07-02 DOI: 10.1007/s00229-024-01577-9
Nefton Pali
{"title":"On weaker notions for Kähler-Ricci solitons","authors":"Nefton Pali","doi":"10.1007/s00229-024-01577-9","DOIUrl":"https://doi.org/10.1007/s00229-024-01577-9","url":null,"abstract":"<p>We show that shrinking Kähler-Ricci solitons over a compact Kähler manifold are gradient shrinking Kähler-Ricci solitons. The proof relies on a remarkable identity on the kernels of a real and a complex elliptic operator proved in our solution of the variational stability problem for gradient shrinking Kähler-Ricci solitons in Pali (Complex Manifolds 3(1):41–144, 2016).</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Non-thin rank jumps for double elliptic K3 surfaces 双椭圆 K3 曲面的非薄级跃迁
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-28 DOI: 10.1007/s00229-024-01554-2
Hector Pasten, Cecília Salgado
{"title":"Non-thin rank jumps for double elliptic K3 surfaces","authors":"Hector Pasten, Cecília Salgado","doi":"10.1007/s00229-024-01554-2","DOIUrl":"https://doi.org/10.1007/s00229-024-01554-2","url":null,"abstract":"<p>For an elliptic surface <span>(pi :Xrightarrow mathbb {P}^1)</span> defined over a number field <i>K</i>, a theorem of Silverman shows that for all but finitely many fibres above <i>K</i>-rational points, the resulting elliptic curve over <i>K</i> has Mordell-Weil rank at least as large as the rank of the group of sections of <span>(pi )</span>. When <i>X</i> is a <i>K</i>3 surface with two distinct elliptic fibrations, we show that the set of <i>K</i>-rational points of <span>(mathbb {P}^1)</span> for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components 使用规则曲面的光滑投影曲线希尔伯特方案的成分 II：非还原成分的存在
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-28 DOI: 10.1007/s00229-024-01580-0
Youngook Choi, Hristo Iliev, Seonja Kim
{"title":"Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components","authors":"Youngook Choi, Hristo Iliev, Seonja Kim","doi":"10.1007/s00229-024-01580-0","DOIUrl":"https://doi.org/10.1007/s00229-024-01580-0","url":null,"abstract":"<p>Let <span>(mathcal {I}_{d,g,r})</span> be the union of irreducible components of the Hilbert scheme whose general points represent smooth, irreducible, non-degenerate curves of degree <i>d</i> and genus <i>g</i> in <span>(mathbb {P}^r)</span>. Using a family of curves found on ruled surfaces over smooth curves of genus <span>(gamma )</span>, we show that for <span>(gamma ge 7)</span> and <span>(g ge 6 gamma + 5)</span>, the scheme <span>(mathcal {I}_{2g-4gamma + 1, g, g - 3gamma + 1})</span> acquires a non-reduced component <span>(mathcal {D}^{prime })</span> such that <span>({text {dim}}T_{[X^{prime }]} mathcal {D}^{prime } = {text {dim}}mathcal {D}^{prime } + 1)</span> for a general point <span>([X^{prime }] in mathcal {D}^{prime })</span>.\u0000</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the 1- adjoint canonical divisor of a foliation 关于叶形的 1- 邻接正典除数
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-24 DOI: 10.1007/s00229-024-01579-7
Jun Lu, Xiao Hang Wu
{"title":"On the 1- adjoint canonical divisor of a foliation","authors":"Jun Lu, Xiao Hang Wu","doi":"10.1007/s00229-024-01579-7","DOIUrl":"https://doi.org/10.1007/s00229-024-01579-7","url":null,"abstract":"<p>In this paper, we describe the structure of the negative part of a Zariski decomposition of <span>(K_X+K_{{{mathcal {F}}}})</span> for a relatively minimal foliation <span>((X,{{mathcal {F}}}))</span> whenever <span>(K_X+K_{{{mathcal {F}}}})</span> is pseudoeffective.\u0000</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local Langlands correspondences in $$ell $$ -adic coefficients $$ell $$ -adic系数中的本地朗兰兹对应关系
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-21 DOI: 10.1007/s00229-024-01582-y
Naoki Imai
{"title":"Local Langlands correspondences in $$ell $$ -adic coefficients","authors":"Naoki Imai","doi":"10.1007/s00229-024-01582-y","DOIUrl":"https://doi.org/10.1007/s00229-024-01582-y","url":null,"abstract":"<p>Let <span>(ell )</span> be a prime number different from the residue characteristic of a non-archimedean local field <i>F</i>. We give formulations of <span>(ell )</span>-adic local Langlands correspondences for connected reductive algebraic groups over <i>F</i>, which we conjecture to be independent of a choice of an isomorphism between the <span>(ell )</span>-adic coefficient field and the complex number field.</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Arithmetic fundamental lemma for the spherical Hecke algebra 球面赫克代数的算术基本定理
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-20 DOI: 10.1007/s00229-024-01572-0
Chao Li, Michael Rapoport, Wei Zhang
{"title":"Arithmetic fundamental lemma for the spherical Hecke algebra","authors":"Chao Li, Michael Rapoport, Wei Zhang","doi":"10.1007/s00229-024-01572-0","DOIUrl":"https://doi.org/10.1007/s00229-024-01572-0","url":null,"abstract":"<p>We define Hecke correspondences and Hecke operators on unitary RZ spaces and study their basic geometric properties, including a commutativity conjecture on Hecke operators. Then we formulate the arithmetic fundamental lemma conjecture for the spherical Hecke algebra. We also formulate a conjecture on the abundance of spherical Hecke functions with identically vanishing first derivative of orbital integrals. We prove these conjectures for the case <span>(textrm{U} (1)times textrm{U} (2))</span>.\u0000</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141510171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The relatively perfect Greenberg transform and cycle class maps 相对完美的格林伯格变换和循环类图
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-17 DOI: 10.1007/s00229-024-01576-w
Alessandra Bertapelle, Takashi Suzuki
{"title":"The relatively perfect Greenberg transform and cycle class maps","authors":"Alessandra Bertapelle, Takashi Suzuki","doi":"10.1007/s00229-024-01576-w","DOIUrl":"https://doi.org/10.1007/s00229-024-01576-w","url":null,"abstract":"<p>Given a scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field, the Greenberg transform produces a new scheme over the residue field thicker than the special fiber. In this paper, we will generalize this transform to the case of imperfect residue field. We will then construct a certain kind of cycle class map defined on this generalized Greenberg transform applied to the Néron model of a semi-abelian variety, which takes values in the relatively perfect nearby cycle functor defined by Kato and the second author.</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Pureté de l’approximation forte sur le corps des fonctions d’une courbe algébrique complexe 复代数曲线函数场强近似的纯度
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-17 DOI: 10.1007/s00229-024-01560-4
Elyes Boughattas
{"title":"Pureté de l’approximation forte sur le corps des fonctions d’une courbe algébrique complexe","authors":"Elyes Boughattas","doi":"10.1007/s00229-024-01560-4","DOIUrl":"https://doi.org/10.1007/s00229-024-01560-4","url":null,"abstract":"<p>Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension 2 closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension 2 closed subset in an affine smooth complete intersection of low degree.\u0000</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141530093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Root stacks and periodic decompositions 根堆栈和周期分解
IF 0.6 4区 数学
Manuscripta Mathematica Pub Date : 2024-06-15 DOI: 10.1007/s00229-024-01574-y
A. Bodzenta, W. Donovan
{"title":"Root stacks and periodic decompositions","authors":"A. Bodzenta, W. Donovan","doi":"10.1007/s00229-024-01574-y","DOIUrl":"https://doi.org/10.1007/s00229-024-01574-y","url":null,"abstract":"<p>For an effective Cartier divisor <i>D</i> on a scheme <i>X</i> we may form an <span>({n}^{text {th}})</span> root stack. Its derived category is known to have a semiorthogonal decomposition with components given by <i>D</i> and <i>X</i>. We show that this decomposition is <span>(2n)</span>-periodic. For <span>(n=2)</span> this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of <i>D</i>. For <span>(n &gt; 2)</span> we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (<i>N</i>-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.</p>","PeriodicalId":49887,"journal":{"name":"Manuscripta Mathematica","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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