André Belotto da Silva, Lorenzo Fantini, A. Pichon
Given a complex analytic germ (X, 0) in (C n , 0), the standard Hermitian metric of C n induces a natural arc-length metric on (X, 0), called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity (X, 0) by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of (X, 0). We deduce in particular that the global data consisting of the topology of (X, 0), together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of (X, 0), completely determine all the inner rates on (X, 0), and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.
{"title":"Inner geometry of complex surfaces: a valuative approach","authors":"André Belotto da Silva, Lorenzo Fantini, A. Pichon","doi":"10.2140/gt.2022.26.163","DOIUrl":"https://doi.org/10.2140/gt.2022.26.163","url":null,"abstract":"Given a complex analytic germ (X, 0) in (C n , 0), the standard Hermitian metric of C n induces a natural arc-length metric on (X, 0), called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity (X, 0) by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of (X, 0). We deduce in particular that the global data consisting of the topology of (X, 0), together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of (X, 0), completely determine all the inner rates on (X, 0), and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131119131","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}
Ilya Kapovich, Joseph Maher, Catherine Pfaff, Samuel J. Taylor
We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This answers a question of M. Bestvina.
{"title":"Random trees in the boundary of outer space","authors":"Ilya Kapovich, Joseph Maher, Catherine Pfaff, Samuel J. Taylor","doi":"10.2140/gt.2022.26.127","DOIUrl":"https://doi.org/10.2140/gt.2022.26.127","url":null,"abstract":"We prove that for the harmonic measure associated to a random walk on Out$(F_r)$ satisfying some mild conditions, a typical tree in the boundary of Outer space is trivalent and nongeometric. This answers a question of M. Bestvina.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115093408","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 construct various modular compactifications of the space of elliptic K3 surfaces using tools from the minimal model program, and explicitly describe the surfaces parametrized by their boundaries. The coarse spaces of our constructed compactifications admit morphisms to the Satake-Baily-Borel compactification. Finally, we show that one of our spaces is smooth with coarse space the GIT quotient of pairs of Weierstrass K3 surfaces with a chosen fiber.
{"title":"Compact moduli of elliptic K3 surfaces","authors":"Kenneth Ascher, Dori Bejleri","doi":"10.2140/gt.2023.27.1891","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1891","url":null,"abstract":"We construct various modular compactifications of the space of elliptic K3 surfaces using tools from the minimal model program, and explicitly describe the surfaces parametrized by their boundaries. The coarse spaces of our constructed compactifications admit morphisms to the Satake-Baily-Borel compactification. Finally, we show that one of our spaces is smooth with coarse space the GIT quotient of pairs of Weierstrass K3 surfaces with a chosen fiber.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123029198","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}
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.
{"title":"A quantum categorification of the Alexander polynomial","authors":"Louis-Hadrien Robert, E. Wagner","doi":"10.2140/gt.2022.26.1985","DOIUrl":"https://doi.org/10.2140/gt.2022.26.1985","url":null,"abstract":"Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133594012","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}
Given $din mathbb{N}$, $gin mathbb{N} cup{0}$, and an integral vector $kappa=(k_1,dots,k_n)$ such that $k_i>-d$ and $k_1+dots+k_n=d(2g-2)$, let $Omega^dmathcal{M}_{g,n}(kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $kappa$. We show that $Omega^dmathcal{M}_{g,n}(kappa)$ carries a volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $mathbb{P}Omega^dmathcal{M}_{g,n}(kappa)=Omega^dmathcal{M}_{g,n}/mathbb{C}^*$ with respect to the measure induced by this volume form is finite.
{"title":"Volume forms on moduli spaces of\u0000d–differentials","authors":"Duc-Manh Nguyen","doi":"10.2140/gt.2022.26.3173","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3173","url":null,"abstract":"Given $din mathbb{N}$, $gin mathbb{N} cup{0}$, and an integral vector $kappa=(k_1,dots,k_n)$ such that $k_i>-d$ and $k_1+dots+k_n=d(2g-2)$, let $Omega^dmathcal{M}_{g,n}(kappa)$ denote the moduli space of meromorphic $d$-differentials on Riemann surfaces of genus $g$ whose zeros and poles have orders prescribed by $kappa$. We show that $Omega^dmathcal{M}_{g,n}(kappa)$ carries a volume form that is parallel with respect to its affine complex manifold structure, and that the total volume of $mathbb{P}Omega^dmathcal{M}_{g,n}(kappa)=Omega^dmathcal{M}_{g,n}/mathbb{C}^*$ with respect to the measure induced by this volume form is finite.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122495881","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 construct non-K"{a}hler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves. We also give an example of a family of K3 surfaces with involutions which do not lift biregularly over the total space.
{"title":"Examples of non-Kähler Calabi–Yau 3–folds\u0000with arbitrarily large b2","authors":"Kenji Hashimoto, Taro Sano","doi":"10.2140/gt.2023.27.131","DOIUrl":"https://doi.org/10.2140/gt.2023.27.131","url":null,"abstract":"We construct non-K\"{a}hler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves. \u0000We also give an example of a family of K3 surfaces with involutions which do not lift biregularly over the total space.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125561577","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}
Given a group $G$ splitting as a free product $G=G_1astdotsast G_kast F_N$, we establish classification results for subgroups of the group $Out(G,mathcal{F})$ of all automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $Hsubseteq Out(G,mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $mathrm{FF}$ and the $mathcal{Z}$-factor graph $mathcal{Z}mathrm{F}$, as spaces of equivalence classes of arational trees (respectively relatively free arational trees). We also identify the loxodromic isometries of $mathrm{FF}$ with the fully irreducible elements of $Out(G,mathcal{F})$, and loxodromic isometries of $mathcal{Z}mathrm{F}$ with the fully irreducible atoroidal outer automorphisms.
{"title":"Boundaries of relative factor graphs and subgroup classification for automorphisms of free products","authors":"Vincent Guirardel, Camille Horbez","doi":"10.2140/gt.2022.26.71","DOIUrl":"https://doi.org/10.2140/gt.2022.26.71","url":null,"abstract":"Given a group $G$ splitting as a free product $G=G_1astdotsast G_kast F_N$, we establish classification results for subgroups of the group $Out(G,mathcal{F})$ of all automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $Hsubseteq Out(G,mathcal{F})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $mathrm{FF}$ and the $mathcal{Z}$-factor graph $mathcal{Z}mathrm{F}$, as spaces of equivalence classes of arational trees (respectively relatively free arational trees). We also identify the loxodromic isometries of $mathrm{FF}$ with the fully irreducible elements of $Out(G,mathcal{F})$, and loxodromic isometries of $mathcal{Z}mathrm{F}$ with the fully irreducible atoroidal outer automorphisms.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"27 44","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120835927","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 homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.
{"title":"Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds","authors":"Christoph Bohm, Ramiro A. Lafuente","doi":"10.2140/gt.2022.26.899","DOIUrl":"https://doi.org/10.2140/gt.2022.26.899","url":null,"abstract":"We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133049773","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}
Every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infrasolv fibers which is called its infrasolv tower. The tower arises from the solvable radicals of isometry group actions on the universal covers. Its length and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. We say that the manifold has large symmetry if it admits an infrasolv tower whose base is a locally homogeneous space. We construct examples of aspherical manifolds with large symmetry, which do not support any locally homogeneous Riemannian metrics.
{"title":"Isometry groups with radical, and aspherical Riemannian manifolds with large symmetry, I","authors":"O. Baues, Y. Kamishima","doi":"10.2140/gt.2023.27.1","DOIUrl":"https://doi.org/10.2140/gt.2023.27.1","url":null,"abstract":"Every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infrasolv fibers which is called its infrasolv tower. The tower arises from the solvable radicals of isometry group actions on the universal covers. Its length and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. We say that the manifold has large symmetry if it admits an infrasolv tower whose base is a locally homogeneous space. We construct examples of aspherical manifolds with large symmetry, which do not support any locally homogeneous Riemannian metrics.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127932102","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}
Author(s): Gerig, Chris | Advisor(s): Hutchings, Michael | Abstract: For a closed oriented smooth 4-manifold X with $b^2_+(X)g0$, the Seiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This thesis describes well-defined counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2-forms, and it is shown that they recover the Seiberg-Witten invariants (modulo 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds.The main results are the following. Given a suitable near-symplectic form w and tubular neighborhood N of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form "near-symplectic" Gromov invariants as a map on the set of spin-c structures of X. They are furthermore equal to the Seiberg-Witten invariants with mod 2 coefficients, where w determines the "chamber" for defining the latter invariants when $b^2_+(X)=1$.In the final chapter, as a non sequitur, a new proof of the Fredholm index formula for punctured pseudoholomorphic curves is sketched. This generalizes Taubes' proof of the Riemann-Roch theorem for compact Riemann surfaces.
{"title":"Seiberg–Witten and Gromov invariants for\u0000self-dual harmonic 2–forms","authors":"Chris Gerig","doi":"10.2140/gt.2022.26.3307","DOIUrl":"https://doi.org/10.2140/gt.2022.26.3307","url":null,"abstract":"Author(s): Gerig, Chris | Advisor(s): Hutchings, Michael | Abstract: For a closed oriented smooth 4-manifold X with $b^2_+(X)g0$, the Seiberg-Witten invariants are well-defined. Taubes' \"SW=Gr\" theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This thesis describes well-defined counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2-forms, and it is shown that they recover the Seiberg-Witten invariants (modulo 2). This is an extension of Taubes' \"SW=Gr\" theorem to non-symplectic 4-manifolds.The main results are the following. Given a suitable near-symplectic form w and tubular neighborhood N of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form \"near-symplectic\" Gromov invariants as a map on the set of spin-c structures of X. They are furthermore equal to the Seiberg-Witten invariants with mod 2 coefficients, where w determines the \"chamber\" for defining the latter invariants when $b^2_+(X)=1$.In the final chapter, as a non sequitur, a new proof of the Fredholm index formula for punctured pseudoholomorphic curves is sketched. This generalizes Taubes' proof of the Riemann-Roch theorem for compact Riemann surfaces.","PeriodicalId":254292,"journal":{"name":"Geometry & Topology","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134408170","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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