Pub Date : 2019-05-04DOI: 10.4310/ajm.2021.v25.n6.a3
Li Guo, Zongzhu Lin
We give a broad study of representation and module theory of Rota-Baxter algebras. Regular-singular decompositions of Rota-Baxter algebras and Rota-Baxter modules are obtained under the condition of quasi-idempotency. Representations of an Rota-Baxter algebra are shown to be equivalent to the representations of the ring of Rota-Baxter operators whose categorical properties are obtained and explicit constructions are provided. Representations from coalgebras are investigated and their algebraic Birkhoff factorization is given. Representations of Rota-Baxter algebras in the tensor category context is also formulated.
{"title":"Representations and modules of Rota–Baxter algebras","authors":"Li Guo, Zongzhu Lin","doi":"10.4310/ajm.2021.v25.n6.a3","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n6.a3","url":null,"abstract":"We give a broad study of representation and module theory of Rota-Baxter algebras. Regular-singular decompositions of Rota-Baxter algebras and Rota-Baxter modules are obtained under the condition of quasi-idempotency. Representations of an Rota-Baxter algebra are shown to be equivalent to the representations of the ring of Rota-Baxter operators whose categorical properties are obtained and explicit constructions are provided. Representations from coalgebras are investigated and their algebraic Birkhoff factorization is given. Representations of Rota-Baxter algebras in the tensor category context is also formulated.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48984992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-18DOI: 10.4310/ajm.2020.v24.n2.a4
A. Dkabrowski, Tomasz Jkedrzejak, L. Szymaszkiewicz
Let $K=Bbb Q(sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-beta)}$ denote its quadratic twist, with $beta=sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. We use Magma to calculate the values $L(E/H,1)$ for all such $q$'s up to some reasonable ranges (different for $qequiv 7 , text{mod} , 8$ and $qequiv 3 , text{mod} , 8$). All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of $sza(E/H)$ in these cases. Our calculations extend those given by J. Choi and J. Coates [{it Iwasawa theory of quadratic twists of $X_0(49)$}, Acta Mathematica Sinica(English Series) {bf 34} (2017), 19-28] for the case $q=7$.
{"title":"Critical $L$-values for some quadratic twists of gross curves","authors":"A. Dkabrowski, Tomasz Jkedrzejak, L. Szymaszkiewicz","doi":"10.4310/ajm.2020.v24.n2.a4","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n2.a4","url":null,"abstract":"Let $K=Bbb Q(sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-beta)}$ denote its quadratic twist, with $beta=sqrt{-q}$. The curve $E$ is defined over the Hilbert class field $H$ of $K$. We use Magma to calculate the values $L(E/H,1)$ for all such $q$'s up to some reasonable ranges (different for $qequiv 7 , text{mod} , 8$ and $qequiv 3 , text{mod} , 8$). All these values are non-zero, and using the Birch and Swinnerton-Dyer conjecture, we can calculate hypothetical orders of $sza(E/H)$ in these cases. Our calculations extend those given by J. Choi and J. Coates [{it Iwasawa theory of quadratic twists of $X_0(49)$}, Acta Mathematica Sinica(English Series) {bf 34} (2017), 19-28] for the case $q=7$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49028999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-18DOI: 10.4310/ajm.2021.v25.n2.a4
Ming Xu, Yu.G. Nikonorov
In this paper, we consider a connected Riemannian manifold $M$ where a connected Lie group $G$ acts effectively and isometrically. Assume $Xinmathfrak{g}=mathrm{Lie}(G)$ defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition $X=X_r+X_s$ according to a Levi decomposition $mathfrak{g}=mathfrak{r}(mathfrak{g})+mathfrak{s}$, where $mathfrak{r}(mathfrak{g})$ is the radical, and $mathfrak{s}=mathfrak{s}_coplusmathfrak{s}_{nc}$ is a Levi subalgebra. The decomposition $X=X_r+X_s$ coincides with the abstract Jordan decomposition of $X$, and is unique in the sense that it does not depend on the choice of $mathfrak{s}$. By these properties, we prove that the eigenvalues of $mathrm{ad}(X):mathfrak{g}rightarrowmathfrak{g}$ are all imaginary. Furthermore, when $M=G/H$ is a Riemannian homogeneous space, we can completely determine all bounded Killing vector fields induced by vectors in $mathfrak{g}$. We prove that the space of all these bounded Killing vector fields, or equivalently the space of all bounded vectors in $mathfrak{g}$ for $G/H$, is a compact Lie subalgebra, such that its semi-simple part is the ideal $mathfrak{c}_{mathfrak{s}_c}(mathfrak{r}(mathfrak{g}))$ of $mathfrak{g}$, and its Abelian part is the sum of $mathfrak{c}_{mathfrak{c}(mathfrak{r}(mathfrak{g}))} (mathfrak{s}_{nc})$ and all two-dimensional irreducible $mathrm{ad}(mathfrak{r}(mathfrak{g}))$-representations in $mathfrak{c}_{mathfrak{c}(mathfrak{n})}(mathfrak{s}_{nc})$ corresponding to nonzero imaginary weights, i.e. $mathbb{R}$-linear functionals $lambda:mathfrak{r}(mathfrak{g})rightarrow mathfrak{r}(mathfrak{g})/mathfrak{n}(mathfrak{g}) rightarrowmathbb{R}sqrt{-1}$, where $mathfrak{n}(mathfrak{g})$ is the nilradical.
{"title":"Algebraic properties of bounded killing vector fields","authors":"Ming Xu, Yu.G. Nikonorov","doi":"10.4310/ajm.2021.v25.n2.a4","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a4","url":null,"abstract":"In this paper, we consider a connected Riemannian manifold $M$ where a connected Lie group $G$ acts effectively and isometrically. Assume $Xinmathfrak{g}=mathrm{Lie}(G)$ defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition $X=X_r+X_s$ according to a Levi decomposition $mathfrak{g}=mathfrak{r}(mathfrak{g})+mathfrak{s}$, where $mathfrak{r}(mathfrak{g})$ is the radical, and $mathfrak{s}=mathfrak{s}_coplusmathfrak{s}_{nc}$ is a Levi subalgebra. The decomposition $X=X_r+X_s$ coincides with the abstract Jordan decomposition of $X$, and is unique in the sense that it does not depend on the choice of $mathfrak{s}$. By these properties, we prove that the eigenvalues of $mathrm{ad}(X):mathfrak{g}rightarrowmathfrak{g}$ are all imaginary. Furthermore, when $M=G/H$ is a Riemannian homogeneous space, we can completely determine all bounded Killing vector fields induced by vectors in $mathfrak{g}$. We prove that the space of all these bounded Killing vector fields, or equivalently the space of all bounded vectors in $mathfrak{g}$ for $G/H$, is a compact Lie subalgebra, such that its semi-simple part is the ideal $mathfrak{c}_{mathfrak{s}_c}(mathfrak{r}(mathfrak{g}))$ of $mathfrak{g}$, and its Abelian part is the sum of $mathfrak{c}_{mathfrak{c}(mathfrak{r}(mathfrak{g}))} (mathfrak{s}_{nc})$ and all two-dimensional irreducible $mathrm{ad}(mathfrak{r}(mathfrak{g}))$-representations in $mathfrak{c}_{mathfrak{c}(mathfrak{n})}(mathfrak{s}_{nc})$ corresponding to nonzero imaginary weights, i.e. $mathbb{R}$-linear functionals $lambda:mathfrak{r}(mathfrak{g})rightarrow mathfrak{r}(mathfrak{g})/mathfrak{n}(mathfrak{g}) rightarrowmathbb{R}sqrt{-1}$, where $mathfrak{n}(mathfrak{g})$ is the nilradical.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-08DOI: 10.4310/ajm.2020.v24.n2.a3
Andr'es Pedroza
We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of (CP^2,omega) of the proper transform of the Clifford torus.
{"title":"Lagrangian Floer homology on symplectic blow ups","authors":"Andr'es Pedroza","doi":"10.4310/ajm.2020.v24.n2.a3","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n2.a3","url":null,"abstract":"We show how to compute the Lagrangian Floer homology in the one-point blow up of the proper transform of Lagrangians submanifolds, solely in terms of information of the base manifold. As an example we present an alternative computation of the Lagrangian quantum homology in the one-point blow up of (CP^2,omega) of the proper transform of the Clifford torus.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43132969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-04-04DOI: 10.4310/ajm.2022.v26.n6.a1
Xiaoling Han, Hikaru Yamamoto
In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kahler manifold. The goal of this paper is to give an $varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the $varepsilon$-regularity theorem.
{"title":"An $varepsilon$-regularity theorem for line bundle mean curvature flow","authors":"Xiaoling Han, Hikaru Yamamoto","doi":"10.4310/ajm.2022.v26.n6.a1","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n6.a1","url":null,"abstract":"In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The line bundle mean curvature flow is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on a given Kahler manifold. The goal of this paper is to give an $varepsilon$-regularity theorem for the line bundle mean curvature flow. To establish the theorem, we provide a scale invariant monotone quantity. As a critical point of this quantity, we define self-shrinker solution of the line bundle mean curvature flow. The Liouville type theorem for self-shrinkers is also given. It plays an important role in the proof of the $varepsilon$-regularity theorem.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47635900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-26DOI: 10.4310/ajm.2021.v25.n4.a2
M. Greenfield, L. Ji
Using the identification of the symmetric space $mathrm{SL}(n,mathbb{R})/mathrm{SO}(n)$ with the Teichm"uller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichm"uller theory and symmetric spaces. We define and study analogs of the Thurston, Teichm"uller, and Weil-Petersson metrics. We show the Teichm"uller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil-Petersson metric is the Riemannian metric of $mathrm{SL}(n,mathbb{R})/mathrm{SO}(n)$ as a symmetric space. We also construct a Thurston-type compactification using measured foliations on $n$-tori, and show that the horofunction compactification with respect to the Thurston metric is isomorphic to it, as well as to a minimal Satake compactification.
{"title":"Metrics and compactifications of Teichmüller spaces of flat tori","authors":"M. Greenfield, L. Ji","doi":"10.4310/ajm.2021.v25.n4.a2","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a2","url":null,"abstract":"Using the identification of the symmetric space $mathrm{SL}(n,mathbb{R})/mathrm{SO}(n)$ with the Teichm\"uller space of flat $n$-tori of unit volume, we explore several metrics and compactifications of these spaces, drawing inspiration both from Teichm\"uller theory and symmetric spaces. We define and study analogs of the Thurston, Teichm\"uller, and Weil-Petersson metrics. We show the Teichm\"uller metric is a symmetrization of the Thurston metric, which is a polyhedral Finsler metric, and the Weil-Petersson metric is the Riemannian metric of $mathrm{SL}(n,mathbb{R})/mathrm{SO}(n)$ as a symmetric space. We also construct a Thurston-type compactification using measured foliations on $n$-tori, and show that the horofunction compactification with respect to the Thurston metric is isomorphic to it, as well as to a minimal Satake compactification.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47253785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-03-19DOI: 10.4310/ajm.2021.v25.n6.a2
N. Dutertre, V. Grandjean
Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{sin mathbb{R}}$ be a globally definable one parameter family of $C^2$-hypersurfaces of $mathbb{R}^n$. Upon defining the notion of generalized critical value for such a family we show that the functions $s to |K(s)|$ and $sto K(s)$, respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of $T_s$, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.
{"title":"Gauss–Kronecker curvature and equisingularity at infinity of definable families","authors":"N. Dutertre, V. Grandjean","doi":"10.4310/ajm.2021.v25.n6.a2","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n6.a2","url":null,"abstract":"Assume given a polynomially bounded o-minimal structure expanding the real numbers. Let $(T_s)_{sin mathbb{R}}$ be a globally definable one parameter family of $C^2$-hypersurfaces of $mathbb{R}^n$. Upon defining the notion of generalized critical value for such a family we show that the functions $s to |K(s)|$ and $sto K(s)$, respectively the total absolute Gauss-Kronecker and total Gauss-Kronecker curvature of $T_s$, are continuous in any neighbourhood of any value which is not generalized critical. In particular this provides a necessary criterion of equisingularity for the family of the levels of a real polynomial.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42435792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-02-11DOI: 10.4310/ajm.2021.v25.n2.a8
L. D. Cerbo, L. Lombardi
Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian etale covers of $X$ are arbitrarily large. As an application, given any integer $kgeq 1$, there exists an abelian etale cover $pcolon X' to X$ such that the adjoint system $big|K_{X'} + p^*L big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.
设$X$为极大艾博年维数的光滑投影复变，设$L to X$为一个大线束。证明了$L$对$X$的合适有限阿贝列盖的回拉的运动Seshadri常数是任意大的。作为一个应用，给定任意整数$kgeq 1$，存在一个阿贝尔覆盖$pcolon X' to X$，使得伴随系统$big|K_{X'} + p^*L big|$将$k$ -射流与$p^*L$的增广基轨迹和$p$下$X$的艾博地图的回拉的例外轨迹分开。
{"title":"Moving Seshadri constants, and coverings of varieties of maximal Albanese dimension","authors":"L. D. Cerbo, L. Lombardi","doi":"10.4310/ajm.2021.v25.n2.a8","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a8","url":null,"abstract":"Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian etale covers of $X$ are arbitrarily large. As an application, given any integer $kgeq 1$, there exists an abelian etale cover $pcolon X' to X$ such that the adjoint system $big|K_{X'} + p^*L big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44111120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-23DOI: 10.4310/ajm.2020.v24.n6.a3
Nathan Grieve
We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.
{"title":"Divisorial instability and Vojta’s main conjecture for $mathbb{Q}$-Fano varieties","authors":"Nathan Grieve","doi":"10.4310/ajm.2020.v24.n6.a3","DOIUrl":"https://doi.org/10.4310/ajm.2020.v24.n6.a3","url":null,"abstract":"We study Diophantine arithmetic properties of birational divisors in conjunction with concepts that surround $mathrm{K}$-stability for Fano varieties. There is also an interpretation in terms of the barycentres of Newton-Okounkov bodies. Our main results show how the notion of divisorial instability, in the sense of K. Fujita, implies instances of Vojta's Main Conjecture for Fano varieties. A main tool in the proof of these results is an arithmetic form of Cartan's Second Main Theorem that has been obtained by M. Ru and P. Vojta.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42089497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2019-01-08DOI: 10.4310/ajm.2021.v25.n1.a7
Youngjin Bae, Seonhwa Kim, Y. Oh
This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M setminus K$. We equip $M setminus K$ with a hyperbolic metric $h$ and its cotangent bundle $T^*(M setminus K)$ with the induced kinetic energy Hamiltonian $H_h = frac{1}{2} |p|_h^2$ and Sasakian almost complex structure $J_h$, and associate a wrapped Fukaya category to $T^*(Msetminus K)$ whose wrapping is given by $H_h$. We then consider the conormal $nu^*T$ of a horo-torus $T$ as its object. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus $T$, and so that the structure maps satisfy $widetilde{mathfrak m}^k = 0$ unless $k neq 2$ and an $A_infty$-algebra associated to $nu^*T$ is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology $HW(nu^*T; H_h)$ with respect to $H_h$ is well-defined and isomorphic to the Knot Floer cohomology $HW(partial_infty(M setminus K))$ that was introduced in [BKO] for arbitrary knot $K subset M$. We also define a reduced cohomology, denoted by $widetilde{HW}^d(partial_infty(M setminus K))$, by modding out constant chords and prove that if $widetilde{HW}^d(partial_infty(M setminus K))neq 0$ for some $d geq 1$, then $K$ cannot be hyperbolic. On the other hand, we prove that all torus knots have $widetilde{HW}^1(partial_infty(M setminus K)) neq 0$.
{"title":"Formality of Floer complex of the ideal boundary of hyperbolic knot complement","authors":"Youngjin Bae, Seonhwa Kim, Y. Oh","doi":"10.4310/ajm.2021.v25.n1.a7","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n1.a7","url":null,"abstract":"This is a sequel to the authors' article [BKO](arXiv:1901.02239). We consider a hyperbolic knot $K$ in a closed 3-manifold $M$ and the cotangent bundle of its complement $M setminus K$. We equip $M setminus K$ with a hyperbolic metric $h$ and its cotangent bundle $T^*(M setminus K)$ with the induced kinetic energy Hamiltonian $H_h = frac{1}{2} |p|_h^2$ and Sasakian almost complex structure $J_h$, and associate a wrapped Fukaya category to $T^*(Msetminus K)$ whose wrapping is given by $H_h$. We then consider the conormal $nu^*T$ of a horo-torus $T$ as its object. We prove that all non-constant Hamiltonian chords are transversal and of Morse index 0 relative to the horo-torus $T$, and so that the structure maps satisfy $widetilde{mathfrak m}^k = 0$ unless $k neq 2$ and an $A_infty$-algebra associated to $nu^*T$ is reduced to a noncommutative algebra concentrated to degree 0. We prove that the wrapped Floer cohomology $HW(nu^*T; H_h)$ with respect to $H_h$ is well-defined and isomorphic to the Knot Floer cohomology $HW(partial_infty(M setminus K))$ that was introduced in [BKO] for arbitrary knot $K subset M$. We also define a reduced cohomology, denoted by $widetilde{HW}^d(partial_infty(M setminus K))$, by modding out constant chords and prove that if $widetilde{HW}^d(partial_infty(M setminus K))neq 0$ for some $d geq 1$, then $K$ cannot be hyperbolic. On the other hand, we prove that all torus knots have $widetilde{HW}^1(partial_infty(M setminus K)) neq 0$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41689793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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