Pub Date : 2022-01-01DOI: 10.4310/mrl.2022.v29.n2.a8
Kai-Wen Lan
We explain that any closed immersion between Shimura varieties defined by morphisms of Shimura data extends to some closed immersion between their projective smooth toroidal compactifications, up to refining the choices of cone decompositions. We also explain that the same holds for many closed immersions between integral models of Shimura varieties and their toroidal compactifications available in the literature.
{"title":"Closed immersions of toroidal compactifications of Shimura varieties","authors":"Kai-Wen Lan","doi":"10.4310/mrl.2022.v29.n2.a8","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n2.a8","url":null,"abstract":"We explain that any closed immersion between Shimura varieties defined by morphisms of Shimura data extends to some closed immersion between their projective smooth toroidal compactifications, up to refining the choices of cone decompositions. We also explain that the same holds for many closed immersions between integral models of Shimura varieties and their toroidal compactifications available in the literature.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70517025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-05DOI: 10.4310/mrl.2022.v29.n6.a11
C. Rivera, B. Viray
Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3nmid[L:k]$. Therefore, the conjecture of Colliot-Th'el`ene and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement is a special case of a conjecture of Cassels and Swinnerton-Dyer.
{"title":"Persistence of the Brauer–Manin obstruction on cubic surfaces","authors":"C. Rivera, B. Viray","doi":"10.4310/mrl.2022.v29.n6.a11","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n6.a11","url":null,"abstract":"Let $X$ be a cubic surface over a global field $k$. We prove that a Brauer-Manin obstruction to the existence of $k$-points on $X$ will persist over every extension $L/k$ with degree relatively prime to $3$. In other words, a cubic surface has nonempty Brauer set over $k$ if and only if it has nonempty Brauer set over some extension $L/k$ with $3nmid[L:k]$. Therefore, the conjecture of Colliot-Th'el`ene and Sansuc on the sufficiency of the Brauer-Manin obstruction for cubic surfaces implies that $X$ has a $k$-rational point if and only if $X$ has a $0$-cycle of degree $1$. This latter statement is a special case of a conjecture of Cassels and Swinnerton-Dyer.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48999770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-28DOI: 10.4310/mrl.2022.v29.n5.a3
Beomjun Choi, Robert Haslhofer
We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.
{"title":"A note on blowup limits in 3d Ricci flow","authors":"Beomjun Choi, Robert Haslhofer","doi":"10.4310/mrl.2022.v29.n5.a3","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n5.a3","url":null,"abstract":"We prove that Perelman's ancient ovals occur as blowup limit in 3d Ricci flow through singularities if and only if there is an accumulation of spherical singularities.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42511546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-08-18DOI: 10.4310/mrl.2021.v28.n5.a13
Sai-Kee Yeung
. We study the problem of non-existence of totally geodesic complex ball quotients in the open Torelli locus in a moduli space of principally polarized Abelian varieties using analytic techniques.
{"title":"Open Torelli locus and complex ball quotients","authors":"Sai-Kee Yeung","doi":"10.4310/mrl.2021.v28.n5.a13","DOIUrl":"https://doi.org/10.4310/mrl.2021.v28.n5.a13","url":null,"abstract":". We study the problem of non-existence of totally geodesic complex ball quotients in the open Torelli locus in a moduli space of principally polarized Abelian varieties using analytic techniques.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47168530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-16DOI: 10.4310/mrl.2023.v30.n1.a7
D. Kazhdan, A. Polishchuk
Let $k$ be a perfect field and let $Xsubset {mathbb P}^N$ be a hypersurface of degree $d$ defined over $k$ and containing a linear subspace $L$ defined over an algebraic closure $overline{k}$ with $mathrm{codim}_{{mathbb P}^N}L=r$. We show that $X$ contains a linear subspace $L_0$ defined over $k$ with $mathrm{codim}_{{mathbb P}^N}Lle dr$. We conjecture that the intersection of all linear subspaces (over $overline{k}$) of minimal codimension $r$ contained in $X$, has codimension bounded above only in terms of $r$ and $d$. We prove this when either $dle 3$ or $rle 2$.
{"title":"Linear subspaces of minimal codimension in hypersurfaces","authors":"D. Kazhdan, A. Polishchuk","doi":"10.4310/mrl.2023.v30.n1.a7","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a7","url":null,"abstract":"Let $k$ be a perfect field and let $Xsubset {mathbb P}^N$ be a hypersurface of degree $d$ defined over $k$ and containing a linear subspace $L$ defined over an algebraic closure $overline{k}$ with $mathrm{codim}_{{mathbb P}^N}L=r$. We show that $X$ contains a linear subspace $L_0$ defined over $k$ with $mathrm{codim}_{{mathbb P}^N}Lle dr$. We conjecture that the intersection of all linear subspaces (over $overline{k}$) of minimal codimension $r$ contained in $X$, has codimension bounded above only in terms of $r$ and $d$. We prove this when either $dle 3$ or $rle 2$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47721817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-05DOI: 10.4310/MRL.2010.v17.n3.a6
Thalia D. Jeffres, J. Rowlett
We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the ``link'' of the singular set. Within this class of ``conic metrics,'' we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature $-1$; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.
{"title":"Conformal deformations of conic metrics to constant scalar curvature","authors":"Thalia D. Jeffres, J. Rowlett","doi":"10.4310/MRL.2010.v17.n3.a6","DOIUrl":"https://doi.org/10.4310/MRL.2010.v17.n3.a6","url":null,"abstract":"We consider conformal deformations within a class of incomplete Riemannian metrics which generalize conic orbifold singularities by allowing both warping and any compact manifold (not just quotients of the sphere) to be the ``link'' of the singular set. Within this class of ``conic metrics,'' we determine obstructions to the existence of conformal deformations to constant scalar curvature of any sign (positive, negative, or zero). For conic metrics with negative scalar curvature, we determine sufficient conditions for the existence of a conformal deformation to a conic metric with constant scalar curvature $-1$; moreover, we show that this metric is unique within its conformal class of conic metrics. Our work is in dimensions three and higher.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"17 1","pages":"449-465"},"PeriodicalIF":1.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49035802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-08DOI: 10.4310/mrl.2023.v30.n1.a11
Y. Zarhin
Let $K$ be a field of characteristic different from $2$, $bar{K}$ its algebraic closure. Let $n ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $bar{K}$ then both jacobians are abelian varieties of CM type with multiplication by the field of $n$th roots of $1$. We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if $char(K)=0$, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order $n$, then $J(C_f)$ and $J(C_h)$ are not isogenous over $bar{K}$.
{"title":"Non-isogenous elliptic curves and hyperelliptic jacobians","authors":"Y. Zarhin","doi":"10.4310/mrl.2023.v30.n1.a11","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n1.a11","url":null,"abstract":"Let $K$ be a field of characteristic different from $2$, $bar{K}$ its algebraic closure. Let $n ge 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f: y^2=f(x)$ and $C_h: y^2=h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$. Suppose that one of the polynomials is irreducible and the other reducible. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $bar{K}$ then both jacobians are abelian varieties of CM type with multiplication by the field of $n$th roots of $1$. We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if $char(K)=0$, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order $n$, then $J(C_f)$ and $J(C_h)$ are not isogenous over $bar{K}$.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43489372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-22DOI: 10.4310/mrl.2022.v29.n1.a2
I. Belegradek, Beibei Liu
Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL embedded closed surface. One can arrange the embedding to have at most one non-locally-flat point, and near the point the topology of the embedding is encoded in the singularity knot $K$. If $K$ is slice, then $W$ has a smooth spine, i.e., deformation retracts onto a smoothly embedded surface. Using the obstructions from the Heegaard Floer homology and the high-dimensional surgery theory, we show that $W$ has no smooth spines if $K$ is a knot with nonzero Arf invariant, a nontrivial L-space knot, the connected sum of nontrivial L-space knots, or an alternating knot of signature $<-4$. We also discuss examples where the interior of $W$ is negatively curved.
{"title":"Four manifolds with no smooth spines","authors":"I. Belegradek, Beibei Liu","doi":"10.4310/mrl.2022.v29.n1.a2","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n1.a2","url":null,"abstract":"Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL embedded closed surface. One can arrange the embedding to have at most one non-locally-flat point, and near the point the topology of the embedding is encoded in the singularity knot $K$. If $K$ is slice, then $W$ has a smooth spine, i.e., deformation retracts onto a smoothly embedded surface. Using the obstructions from the Heegaard Floer homology and the high-dimensional surgery theory, we show that $W$ has no smooth spines if $K$ is a knot with nonzero Arf invariant, a nontrivial L-space knot, the connected sum of nontrivial L-space knots, or an alternating knot of signature $<-4$. We also discuss examples where the interior of $W$ is negatively curved.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48470998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-16DOI: 10.4310/mrl.2022.v29.n6.a12
Mads Bach Villadsen
We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on Simpson's results on the relation between rank one Higgs bundles and local systems of one-dimensional complex vectors spaces, and the structure of the cohomology jump loci in their moduli spaces.
{"title":"Kodaira dimension and zeros of holomorphic one-forms, revisited","authors":"Mads Bach Villadsen","doi":"10.4310/mrl.2022.v29.n6.a12","DOIUrl":"https://doi.org/10.4310/mrl.2022.v29.n6.a12","url":null,"abstract":"We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on Simpson's results on the relation between rank one Higgs bundles and local systems of one-dimensional complex vectors spaces, and the structure of the cohomology jump loci in their moduli spaces.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42100811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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