We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in the Hitchin component $mathcal{H}_{3}(S)$ of surface group representations into $PSL(3,mathbb{R})$. The proof consists of the following elements: we compute first derivatives of the pressure metric using the thermodynamic formalism. We invoke a gauge-theoretic formula to compute first and second variations of reparametrization functions by studying flat connections from Hitchin's equations and their parallel transports. We then extend these expressions of integrals over closed geodesics to integrals over the two-dimensional surface. Symmetries of the Liouville measure then provide cancellations, which show that the first derivatives of the pressure metric tensors vanish at the Fuchsian locus.
{"title":"Geodesic coordinates for the pressure metric at the Fuchsian locus","authors":"X. Dai","doi":"10.2140/gt.2023.27.1391","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1391","url":null,"abstract":"We prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in the Hitchin component $mathcal{H}_{3}(S)$ of surface group representations into $PSL(3,mathbb{R})$. \u0000The proof consists of the following elements: we compute first derivatives of the pressure metric using the thermodynamic formalism. We invoke a gauge-theoretic formula to compute first and second variations of reparametrization functions by studying flat connections from Hitchin's equations and their parallel transports. We then extend these expressions of integrals over closed geodesics to integrals over the two-dimensional surface. Symmetries of the Liouville measure then provide cancellations, which show that the first derivatives of the pressure metric tensors vanish at the Fuchsian locus.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117351886","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}
A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove that Einstein manifolds sufficiently close, in the Gromov-Hausdorff sense, to an orbifold are actually close to a gluing of model spaces in suitable weighted Holder spaces. The proof consists in controlling the metric in the neck regions thanks to the construction of optimal coordinates. This refined convergence is the cornerstone of our subsequent work on the degeneration of Einstein metrics or, equivalently, on the desingularization of Einstein orbifolds in which we show that all Einstein metrics Gromov-Hausdorff close to an Einstein orbifold are the result of a gluing-perturbation procedure. This procedure turns out to be generally obstructed, and this provides the first obstructions to a Gromov-Hausdorff desingularization of Einstein orbifolds.
{"title":"Noncollapsed degeneration of Einstein\u00004–manifolds, II","authors":"Tristan Ozuch","doi":"10.2140/gt.2022.26.1529","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1529","url":null,"abstract":"A theorem of Anderson and Bando-Kasue-Nakajima from 1989 states that to compactify the set of normalized Einstein metrics with a lower bound on the volume and an upper bound on the diameter in the Gromov-Hausdorff sense, one has to add singular spaces called Einstein orbifolds, and the singularities form as blow-downs of Ricci-flat ALE spaces. This raises some natural issues, in particular: can all Einstein orbifolds be Gromov-Hausdorff limits of smooth Einstein manifolds? Can we describe more precisely the smooth Einstein metrics close to a given singular one? In this first paper, we prove that Einstein manifolds sufficiently close, in the Gromov-Hausdorff sense, to an orbifold are actually close to a gluing of model spaces in suitable weighted Holder spaces. The proof consists in controlling the metric in the neck regions thanks to the construction of optimal coordinates. This refined convergence is the cornerstone of our subsequent work on the degeneration of Einstein metrics or, equivalently, on the desingularization of Einstein orbifolds in which we show that all Einstein metrics Gromov-Hausdorff close to an Einstein orbifold are the result of a gluing-perturbation procedure. This procedure turns out to be generally obstructed, and this provides the first obstructions to a Gromov-Hausdorff desingularization of Einstein orbifolds.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128385673","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 article, we consider the $G$-character variety of a compact Riemann surface of genus $g > 0$, when $G$ is $mathrm{SL}(n,mathbb{C})$ or $mathrm{GL}(n,mathbb{C})$. We show that these varieties are symplectic singularities and classify when they admit symplectic resolutions: they do when $g = 1$ or $n = 1$ or $(g,n)=(2,2)$.
{"title":"Symplectic resolutions of character varieties","authors":"G. Bellamy, T. Schedler","doi":"10.2140/gt.2023.27.51","DOIUrl":"https://doi.org/10.2140/gt.2023.27.51","url":null,"abstract":"In this article, we consider the $G$-character variety of a compact Riemann surface of genus $g > 0$, when $G$ is $mathrm{SL}(n,mathbb{C})$ or $mathrm{GL}(n,mathbb{C})$. We show that these varieties are symplectic singularities and classify when they admit symplectic resolutions: they do when $g = 1$ or $n = 1$ or $(g,n)=(2,2)$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121619027","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}
of Einstein metrics the of orbifolds in which we show that metrics Gromov Hausdorffclose to an Einstein orbifold are the result of a gluing perturbation procedure. This out to be generally obstructed, and this provides the first obstructions to a Gromov Hausdorff desingularization of
{"title":"Noncollapsed degeneration of Einstein\u00004–manifolds, I","authors":"Tristan Ozuch","doi":"10.2140/gt.2022.26.1483","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1483","url":null,"abstract":"of Einstein metrics the of orbifolds in which we show that metrics Gromov Hausdorffclose to an Einstein orbifold are the result of a gluing perturbation procedure. This out to be generally obstructed, and this provides the first obstructions to a Gromov Hausdorff desingularization of","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124271308","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}
Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({rover c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Misha Gromov that this result would imply two famous Gromov's inequalities: $Fill Rad(M^n)leq c(n)vol(M^n)^{1over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)leq 6c(n)vol(M^n)^{1over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible curve in $M^n$. Here we prove that these results hold with $c(n)=n$. All previously known upper bounds for $c(n)$ were exponential in $n$. The proof uses ideas of Guth from [Gu 10] and of Panos Papasoglu from his recent work [P].
{"title":"Linear bounds for constants in Gromov’s\u0000systolic inequality and related results","authors":"A. Nabutovsky","doi":"10.2140/gt.2022.26.3123","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3123","url":null,"abstract":"Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({rover c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Misha Gromov that this result would imply two famous Gromov's inequalities: $Fill Rad(M^n)leq c(n)vol(M^n)^{1over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)leq 6c(n)vol(M^n)^{1over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible curve in $M^n$. \u0000Here we prove that these results hold with $c(n)=n$. All previously known upper bounds for $c(n)$ were exponential in $n$. The proof uses ideas of Guth from [Gu 10] and of Panos Papasoglu from his recent work [P].","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124697587","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 joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formula for the behaviour of these immersed curves under cabling.
{"title":"Cabling in terms of immersed curves","authors":"J. Hanselman, Liam Watson","doi":"10.2140/gt.2023.27.925","DOIUrl":"https://doi.org/10.2140/gt.2023.27.925","url":null,"abstract":"In joint work with J. Rasmussen, we gave an interpretation of Heegaard Floer homology for manifolds with torus boundary in terms of immersed curves in a punctured torus. In particular, knot Floer homology is captured by this invariant. Appealing to earlier work of the authors on bordered Floer homology, we give a formula for the behaviour of these immersed curves under cabling.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116992363","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}
This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee a J-bilipschitz map on epsilon-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation. Our quantitative results have two immediate applications. First, we relate the Margulis number of N to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed 3-manifold M whose Margulis number is less than 0.29. Combined with Shalen's upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of 3-manifolds whose Margulis numbers are below 0.29. Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold N, we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on N. This reduces the cosmetic surgery conjecture on N to an explicit finite search.
{"title":"Effective bilipschitz bounds on drilling and filling","authors":"D. Futer, J. Purcell, S. Schleimer","doi":"10.2140/gt.2022.26.1077","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1077","url":null,"abstract":"This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee a J-bilipschitz map on epsilon-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infinitesimal change in metric during the cone deformation. \u0000Our quantitative results have two immediate applications. First, we relate the Margulis number of N to the Margulis numbers of its Dehn fillings. In particular, we give a lower bound on the systole of any closed 3-manifold M whose Margulis number is less than 0.29. Combined with Shalen's upper bound on the volume of such a manifold, this gives a procedure to compute the finite list of 3-manifolds whose Margulis numbers are below 0.29. \u0000Our second application is to the cosmetic surgery conjecture. Given the systole of a one-cusped hyperbolic manifold N, we produce an explicit upper bound on the length of a slope involved in a cosmetic surgery on N. This reduces the cosmetic surgery conjecture on N to an explicit finite search.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117128474","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 use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.
{"title":"Invariants of 4–manifolds from Khovanov–Rozansky\u0000link homology","authors":"S. Morrison, K. Walker, Paul Wedrich","doi":"10.2140/gt.2022.26.3367","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3367","url":null,"abstract":"We use Khovanov-Rozansky gl(N) link homology to define invariants of oriented smooth 4-manifolds, as skein modules constructed from certain 4-categories with well-behaved duals. The technical heart of this construction is a proof of the sweep-around property, which makes these link homologies well defined in the 3-sphere.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121050972","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 show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Szekelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.
{"title":"Optimal destabilization of K–unstable Fano\u0000varieties via stability thresholds","authors":"Harold Blum, Yuchen Liu, Chuyu Zhou","doi":"10.2140/gt.2022.26.2507","DOIUrl":"https://doi.org/10.2140/gt.2022.26.2507","url":null,"abstract":"We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fano variety. We also show that the stability threshold can be approximated by divisorial valuations induced by special test configurations. As an application of the above results and the analytic work of Datar, Szekelyhidi, and Ross, we deduce that greatest Ricci lower bounds of Fano manifolds of fixed dimension form a finite set of rational numbers. As a key step in the proofs, we adapt the process of Li and Xu producing special test configurations to twisted K-stability in the sense of Dervan.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128692316","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}
Fix a scheme $S$ of characteristic $p$. Let $mathscr{M}$ be an $S$-algebraic stack and let $mbox{Fdiv}(mathscr{M})$ be the stack of $mbox{F}$-divided objects, that is sequences of objects $x_iinmathscr{M}$ with isomorphisms $sigma_i:x_ito mbox{F}^*x_{i+1}$. Let $mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $mathscr{X}to Pi_1(mathscr{X}/S)$ the 'etale fundamental pro-groupoid, constructed in the present text. We prove that if $mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $mathscr{X}to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks [mathscr{H}!om(Pi_1(mathscr{X}/S),mathscr{M}) simeq mathscr{H}!om(mathscr{X},mbox{Fdiv}(mathscr{M})).] In particular, the system of relative Frobenius morphisms $mathscr{X}to mathscr{X}^{p/S}to mathscr{X}^{p^2/S}todots$ allows to recover the space of connected components $pi_0(mathscr{X}/S)$ and the relative 'etale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.
{"title":"Unramified F–divided objects and the étale\u0000fundamental pro-groupoid in positive characteristic","authors":"Yuliang Huang, G. Orecchia, M. Romagny","doi":"10.2140/gt.2022.26.3221","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3221","url":null,"abstract":"Fix a scheme $S$ of characteristic $p$. Let $mathscr{M}$ be an $S$-algebraic stack and let $mbox{Fdiv}(mathscr{M})$ be the stack of $mbox{F}$-divided objects, that is sequences of objects $x_iinmathscr{M}$ with isomorphisms $sigma_i:x_ito mbox{F}^*x_{i+1}$. Let $mathscr{X}$ be a flat, finitely presented $S$-algebraic stack and $mathscr{X}to Pi_1(mathscr{X}/S)$ the 'etale fundamental pro-groupoid, constructed in the present text. We prove that if $mathscr{M}$ is a quasi-separated Deligne-Mumford stack and $mathscr{X}to S$ has geometrically reduced fibres, there is a bifunctorial isomorphism of stacks [mathscr{H}!om(Pi_1(mathscr{X}/S),mathscr{M}) simeq mathscr{H}!om(mathscr{X},mbox{Fdiv}(mathscr{M})).] In particular, the system of relative Frobenius morphisms $mathscr{X}to mathscr{X}^{p/S}to mathscr{X}^{p^2/S}todots$ allows to recover the space of connected components $pi_0(mathscr{X}/S)$ and the relative 'etale fundamental gerbe. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic $p$.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114934976","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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