Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Fr{o}yshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.
{"title":"On the Frøyshov invariant and monopole Lefschetz number","authors":"Jianfeng Lin, Daniel Ruberman, N. Saveliev","doi":"10.4310/jdg/1683307008","DOIUrl":"https://doi.org/10.4310/jdg/1683307008","url":null,"abstract":"Given an involution on a rational homology 3-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten's conjecture relating the Donaldson and Seiberg--Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Fr{o}yshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46174205","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}
Jaroslaw Buczy'nski, Jaroslaw A. Wi'sniewski, Andrzej Weber
We prove LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most (dim X+5)/2 with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.
{"title":"Algebraic torus actions on contact manifolds","authors":"Jaroslaw Buczy'nski, Jaroslaw A. Wi'sniewski, Andrzej Weber","doi":"10.4310/jdg/1659987892","DOIUrl":"https://doi.org/10.4310/jdg/1659987892","url":null,"abstract":"We prove LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most (dim X+5)/2 with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46260595","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}
In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $mathbb{B}_n$ of $mathbb{C}^n$ $(nge 2)$ is a level set of a noncritical holomorphic function on $mathbb{B}_n$ all of whose level sets are complete. This shows that $mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $qin{1,ldots,n-1}$.
{"title":"Complete complex hypersurfaces in the ball come in foliations","authors":"A. Alarcón","doi":"10.4310/jdg/1656005494","DOIUrl":"https://doi.org/10.4310/jdg/1656005494","url":null,"abstract":"In this paper we prove that every smooth complete closed complex hypersurface in the open unit ball $mathbb{B}_n$ of $mathbb{C}^n$ $(nge 2)$ is a level set of a noncritical holomorphic function on $mathbb{B}_n$ all of whose level sets are complete. This shows that $mathbb{B}_n$ admits a nonsingular holomorphic foliation by smooth complete closed complex hypersurfaces and, what is the main point, that every hypersurface in $mathbb{B}_n$ of this type can be embedded into such a foliation. We establish a more general result in which neither completeness nor smoothness of the given hypersurface is required. \u0000Furthermore, we obtain a similar result for complex submanifolds of arbitrary positive codimension and prove the existence of a nonsingular holomorphic submersion foliation of $mathbb{B}_n$ by smooth complete closed complex submanifolds of any pure codimension $qin{1,ldots,n-1}$.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47629902","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}
We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity C^n/G for a finite subgroup G in SL(n,C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we construct a novel filtration on the symplectic chain complex for any non-exact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology.
{"title":"The McKay correspondence for isolated singularities via Floer theory","authors":"Mark McLean, Alexander F. Ritter","doi":"10.4310/jdg/1685121321","DOIUrl":"https://doi.org/10.4310/jdg/1685121321","url":null,"abstract":"We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity C^n/G for a finite subgroup G in SL(n,C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we construct a novel filtration on the symplectic chain complex for any non-exact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46218303","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}
In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the K"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,epsilon}$ regularity.
{"title":"Smoothly bounded domains covering finite volume manifolds","authors":"Andrew M. Zimmer","doi":"10.4310/jdg/1631124346","DOIUrl":"https://doi.org/10.4310/jdg/1631124346","url":null,"abstract":"In this paper we prove: if a bounded domain with $C^2$ boundary covers a manifold which has finite volume with respect to either the Bergman volume, the K\"ahler-Einstein volume, or the Kobayashi-Eisenman volume, then the domain is biholomorphic to the unit ball. This answers an old question of Yau. Further, when the domain is convex we can assume that the boundary only has $C^{1,epsilon}$ regularity.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45000668","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}
Zhijie Chen, Ting-Jung Kuo, Changshou Lin, Chin-Lung Wang
We study the problem: How many singular points of a solution λ(t) to the Painleve VI equation with parameter ( 8 , −1 8 , 1 8 , 3 8 ) might have in C {0, 1}? Here t0 ∈ C {0, 1} is called a singular point of λ(t) if λ(t0) ∈ {0, 1, t0, ∞}. Based on Hitchin’s formula, we explore the connection of this problem with Green function and the Eisenstein series of weight one. Among other things, we prove: (i) There are only three solutions which have no singular points in C {0, 1}. (ii) For a special type of solutions (called real solutions here), any branch of a solution has at most two singular points (in particular, at most one pole) in C {0, 1}. (iii) Any Riccati solution has singular points in C {0, 1}. (iv) For each N ≥ 5 and N 6= 6, we calculate the number of the real j-values of zeros of the Eisenstein series EN 1 (τ; k1, k2) of weight one, where (k1, k2) runs over [0, N − 1]2 with gcd(k1, k2, N) = 1. The geometry of the critical points of the Green function on a flat torus Eτ , as τ varies in the moduliM1, plays a fundamental role in our analysis of the Painleve IV equation. In particular, the conjectures raised in [22] on the shape of the domain Ω5 ⊂ M1, which consists of tori whose Green function has extra pair of critical points, are completely solved here.
{"title":"Green function, Painlevé VI equation, and Eisenstein series of weight one","authors":"Zhijie Chen, Ting-Jung Kuo, Changshou Lin, Chin-Lung Wang","doi":"10.4310/JDG/1518490817","DOIUrl":"https://doi.org/10.4310/JDG/1518490817","url":null,"abstract":"We study the problem: How many singular points of a solution λ(t) to the Painleve VI equation with parameter ( 8 , −1 8 , 1 8 , 3 8 ) might have in C {0, 1}? Here t0 ∈ C {0, 1} is called a singular point of λ(t) if λ(t0) ∈ {0, 1, t0, ∞}. Based on Hitchin’s formula, we explore the connection of this problem with Green function and the Eisenstein series of weight one. Among other things, we prove: (i) There are only three solutions which have no singular points in C {0, 1}. (ii) For a special type of solutions (called real solutions here), any branch of a solution has at most two singular points (in particular, at most one pole) in C {0, 1}. (iii) Any Riccati solution has singular points in C {0, 1}. (iv) For each N ≥ 5 and N 6= 6, we calculate the number of the real j-values of zeros of the Eisenstein series EN 1 (τ; k1, k2) of weight one, where (k1, k2) runs over [0, N − 1]2 with gcd(k1, k2, N) = 1. The geometry of the critical points of the Green function on a flat torus Eτ , as τ varies in the moduliM1, plays a fundamental role in our analysis of the Painleve IV equation. In particular, the conjectures raised in [22] on the shape of the domain Ω5 ⊂ M1, which consists of tori whose Green function has extra pair of critical points, are completely solved here.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":"108 1","pages":"185-241"},"PeriodicalIF":2.5,"publicationDate":"2018-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4310/JDG/1518490817","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42697151","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}
We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $alpha'$. Actually we obtain at the same time a solution of the open case $alpha'>0$, an improved solution of the known case $alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.
{"title":"Fu–Yau Hessian equations","authors":"D. Phong, Sebastien Picard, Xiangwen Zhang","doi":"10.4310/JDG/1620272943","DOIUrl":"https://doi.org/10.4310/JDG/1620272943","url":null,"abstract":"We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $alpha'$. Actually we obtain at the same time a solution of the open case $alpha'>0$, an improved solution of the known case $alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44830497","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}
Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 ge 5$ and in the hypersurface case when $n=2$, a case that was completely solved by Yau for $n ge 3$ but only partially solved by Du and Yau for $n=2$. As an application, we determine the existence of a link-theoretic invariant of normal isolated singularities that distinguishes smooth points from singular ones.
{"title":"Smooth solutions to the complex plateau problem","authors":"T. Fernex","doi":"10.4310/JDG/1622743141","DOIUrl":"https://doi.org/10.4310/JDG/1622743141","url":null,"abstract":"Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 ge 5$ and in the hypersurface case when $n=2$, a case that was completely solved by Yau for $n ge 3$ but only partially solved by Du and Yau for $n=2$. As an application, we determine the existence of a link-theoretic invariant of normal isolated singularities that distinguishes smooth points from singular ones.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48297981","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}
Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat K"ahler metric with cone singularities along the lines asymptotic to a polyhedral K"ahler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expresses the energy of the metric as a `logarithmic' Euler characteristic with points weighted according to the volume density of the metric.
给定投影平面上的加权线排列，在权值满足自然约束条件的情况下，我们证明了一个Ricci-flat K ahler度规的存在性，该度规在每个多点上沿多面体K ahler圆锥渐近的直线上具有锥奇点。此外，我们讨论了一个chen - weil公式，该公式将度规的能量表示为根据度规的体积密度加权的点的“对数”欧拉特征。
{"title":"Calabi–Yau metrics with conical singularities along line arrangements","authors":"Martin de Borbon, Cristiano Spotti","doi":"10.4310/jdg/1680883576","DOIUrl":"https://doi.org/10.4310/jdg/1680883576","url":null,"abstract":"Given a weighted line arrangement in the projective plane, with weights satisfying natural constraint conditions, we show the existence of a Ricci-flat K\"ahler metric with cone singularities along the lines asymptotic to a polyhedral K\"ahler cone at each multiple point. Moreover, we discuss a Chern-Weil formula that expresses the energy of the metric as a `logarithmic' Euler characteristic with points weighted according to the volume density of the metric.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2017-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45290916","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}
Tristan C. Collins, Tomoyuki Hisamoto, Ryosuke Takahashi
We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 pi lambda c_1(X)$ for $lambda=pm 1$. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with negative Ricci curvature. In the Fano case, assuming $X$ admits a Kahler-Einstein metric, we prove the weak convergence of the flow to a Kahler-Einstein metric. In general, we expect that the limit of the flow is related with the optimally destabilizing test configuration for the $L^2$-normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.
{"title":"The inverse Monge–Ampère flow and applications to Kähler–Einstein metrics","authors":"Tristan C. Collins, Tomoyuki Hisamoto, Ryosuke Takahashi","doi":"10.4310/jdg/1641413788","DOIUrl":"https://doi.org/10.4310/jdg/1641413788","url":null,"abstract":"We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in $2 pi lambda c_1(X)$ for $lambda=pm 1$. We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with negative Ricci curvature. In the Fano case, assuming $X$ admits a Kahler-Einstein metric, we prove the weak convergence of the flow to a Kahler-Einstein metric. In general, we expect that the limit of the flow is related with the optimally destabilizing test configuration for the $L^2$-normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.","PeriodicalId":15642,"journal":{"name":"Journal of Differential Geometry","volume":" ","pages":""},"PeriodicalIF":2.5,"publicationDate":"2017-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48555351","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}
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