Pub Date : 2024-03-05DOI: 10.4007/annals.2024.199.2.1
Ana Caraiani, Peter Scholze
We prove that the generic part of the $mathrm{mod}, ell$ cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge–Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.
{"title":"On the generic part of the cohomology of non-compact unitary Shimura varieties | Annals of Mathematics","authors":"Ana Caraiani, Peter Scholze","doi":"10.4007/annals.2024.199.2.1","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.1","url":null,"abstract":"<p>We prove that the generic part of the $mathrm{mod}, ell$ cohomology of Shimura varieties associated to quasi-split unitary groups of even dimension is concentrated above the middle degree, extending our previous work to a non-compact case. The result applies even to Eisenstein cohomology classes coming from the locally symmetric space of the general linear group, and has been used in joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Taylor and Thorne to get good control on these classes and deduce potential automorphy theorems without any self-duality hypothesis. Our main geometric result is a computation of the fibers of the Hodge–Tate period map on compactified Shimura varieties, in terms of similarly compactified Igusa varieties.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"19 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-05DOI: 10.4007/annals.2024.199.2.5
Gal Binyamini, Dmitry Novikov, Benny Zak
We prove an effective form of Wilkie’s conjecture in the structure generated by restricted sub-Pfaffian functions: the number of rational points of height $H$ lying in the transcendental part of such a set grows no faster than some power of $log H$. Our bounds depend only on the Pfaffian complexity of the sets involved. As a corollary we deduce Wilkie’s original conjecture for $mathbb{R}_{rm exp}$ in full generality.
{"title":"Wilkie’s conjecture for Pfaffian structures | Annals of Mathematics","authors":"Gal Binyamini, Dmitry Novikov, Benny Zak","doi":"10.4007/annals.2024.199.2.5","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.5","url":null,"abstract":"<p>We prove an effective form of Wilkie’s conjecture in the structure generated by restricted sub-Pfaffian functions: the number of rational points of height $H$ lying in the transcendental part of such a set grows no faster than some power of $log H$. Our bounds depend only on the Pfaffian complexity of the sets involved. As a corollary we deduce Wilkie’s original conjecture for $mathbb{R}_{rm exp}$ in full generality.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-05DOI: 10.4007/annals.2024.199.2.6
Aaron Landesman, Daniel Litt
Let $Sigma _{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group $mathrm {Mod}_{g,n}$ of $Sigma _{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$rho : pi _1(Sigma _{g,n})to mathrm {GL}_r(mathbb {C})$$ is a representation whose conjugacy class has finite orbit under $mathrm {Mod}_{g,n}$, and $rlt sqrt {g+1}$, then $rho $ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson’s integrality conjecture for cohomologically rigid local systems.
{"title":"Canonical representations of surface groups | Annals of Mathematics","authors":"Aaron Landesman, Daniel Litt","doi":"10.4007/annals.2024.199.2.6","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.6","url":null,"abstract":"<p>Let $Sigma _{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group $mathrm {Mod}_{g,n}$ of $Sigma _{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$rho : pi _1(Sigma _{g,n})to mathrm {GL}_r(mathbb {C})$$ is a representation whose conjugacy class has finite orbit under $mathrm {Mod}_{g,n}$, and $rlt sqrt {g+1}$, then $rho $ has finite image. This answers questions of Junho Peter Whang and Mark Kisin. We give applications of our methods to the Putman-Wieland conjecture, the Fontaine-Mazur conjecture, and a question of Esnault-Kerz. The proofs rely on non-abelian Hodge theory, our earlier work on semistability of isomonodromic deformations, and recent work of Esnault-Groechenig and Klevdal-Patrikis on Simpson’s integrality conjecture for cohomologically rigid local systems.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"8 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-05DOI: 10.4007/annals.2024.199.2.8
Sam Mattheus, Jacques Verstraete
For integers $s,t geq 2$, the Ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. In this paper we prove [ r(4,t) = OmegaBigl(frac{t^3}{log^4 ! t}Bigr) quad quad mbox{ as }t rightarrow infty,] which determines $r(4,t)$ up to a factor of order $log^2 ! t$, and solves a conjecture of Erdős.
{"title":"The asymptotics of $r(4,t)$ | Annals of Mathematics","authors":"Sam Mattheus, Jacques Verstraete","doi":"10.4007/annals.2024.199.2.8","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.8","url":null,"abstract":"<p>For integers $s,t geq 2$, the Ramsey number $r(s,t)$ denotes the minimum $n$ such that every $n$-vertex graph contains a clique of order $s$ or an independent set of order $t$. In this paper we prove [ r(4,t) = OmegaBigl(frac{t^3}{log^4 ! t}Bigr) quad quad mbox{ as }t rightarrow infty,] which determines $r(4,t)$ up to a factor of order $log^2 ! t$, and solves a conjecture of Erdős.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"67 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-05DOI: 10.4007/annals.2024.199.2.7
Yuta Kusakabe
Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $mathbb{C}^{n}$ $(n>1)$ is Oka. Furthermore, we obtain new examples of non-elliptic Oka manifolds which negatively answer Gromov’s question. The relative version of the main theorem is also proved. As an application, we show that the complement $mathbb{C}^{n}setminus mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)neq (2,1),(2,2),(3,3)$.
{"title":"Oka properties of complements of holomorphically convex sets | Annals of Mathematics","authors":"Yuta Kusakabe","doi":"10.4007/annals.2024.199.2.7","DOIUrl":"https://doi.org/10.4007/annals.2024.199.2.7","url":null,"abstract":"<p>Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $mathbb{C}^{n}$ $(n>1)$ is Oka. Furthermore, we obtain new examples of non-elliptic Oka manifolds which negatively answer Gromov’s question. The relative version of the main theorem is also proved. As an application, we show that the complement $mathbb{C}^{n}setminus mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)neq (2,1),(2,2),(3,3)$.</p>","PeriodicalId":8134,"journal":{"name":"Annals of Mathematics","volume":"33 1","pages":""},"PeriodicalIF":4.9,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-29DOI: 10.4007/annals.2024.199.1.3
Daniel Cristofaro-Gardiner, Vincent Humilière, Sobhan Seyfaddini
In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for hbox $n ge 3$, asked if the same statement holds in dimension two. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles what is known as the “simplicity conjecture” in the affirmative. In fact, we prove the a priori stronger statement that this group is not perfect.