In this article, we study the cyclicity problem of elliptic curves $E/mathbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for elliptic curves E, which also presents a generalization of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg and Wu.
{"title":"Cyclicity and Exponent of Elliptic Curves Modulo p in Arithmetic Progressions","authors":"Peng-Jie Wong","doi":"10.1093/qmath/haae029","DOIUrl":"https://doi.org/10.1093/qmath/haae029","url":null,"abstract":"In this article, we study the cyclicity problem of elliptic curves $E/mathbb{Q}$ modulo primes in a given arithmetic progression. We extend the recent work of Akbal and Güloğlu by proving an unconditional asymptotic for such a cyclicity problem over arithmetic progressions for elliptic curves E, which also presents a generalization of the previous works of Akbary, Cojocaru, M.R. Murty, V.K. Murty and Serre. In addition, we refine the conditional estimates of Akbal and Güloğlu, which gives log-power savings (for small moduli) and consequently improves the work of Cojocaru and M.R. Murty. Moreover, we study the average exponent of E modulo primes in a given arithmetic progression and obtain several conditional and unconditional estimates, extending the previous works of Freiberg, Kim, Kurlberg and Wu.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"162 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141149193","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}
In this paper, we show some nonexistence results of radial solutions for the following Minkowski curvature problems in an exterior domain: $$ begin{cases} -text{div} big(phi(nabla v(x))big)=k(x)f(v(x)), quadquad xinOmega, v=0 text{on} partialOmega, qquadlimlimits_{xrightarrowinfty}v(x)=0 end{cases} $$ for R sufficiently large, where $phi(s)=frac{s}{sqrt{1-s^{2}}}$ for $sin{mathbb R}$ with $s^2lt1,$ $Omega={xin{{mathbb R}^{N}}: |x| gt R}$, $Ngeq3$ is an integer, $|cdot|$ denotes the Euclidean norm on $mathbb{R}^{N}$, R is a positive parameter, $f:mathbb{R}rightarrowmathbb{R}$ is an odd and locally Lipschitz continuous function and $k in C^{1}(mathbb{R}^{+}, mathbb{R}^{+})$ with $mathbb{R}^{+}=(0, +infty)$. We also apply the fixed-point index theory to establish the existence of positive radial solutions of the above problems for R sufficiently small.
在本文中,我们展示了以下闵科夫斯基曲率问题在外部域中径向解的一些不存在结果: $$ (开始{案例} big(phi(nabla v(x))/big)=k(x)f(v(x)), quadquad xinOmega,v=0text{on} $$ for R sufficiently large、where $phi(s)=frac{s}{sqrt{1-s^{2}}$ for $sin{mathbb R}$ with $s^2lt1,$ $Omega={xin{{mathbb R}^{N}}:|x| gt R}$, $Ngeq3$ 是整数, $|cdot|$ 表示 $mathbb{R}^{N}$ 上的欧氏规范, R 是一个正参数, $f:mathbb{R}rightarrowmathbb{R}$是奇数局部利普齐兹连续函数,$k在C^{1}(mathbb{R}^{+},mathbb{R}^{+})$中,$mathbb{R}^{+}=(0, +infty)$。我们还应用定点索引理论建立了上述问题在 R 足够小时的正径向解的存在性。
{"title":"Existence and Nonexistence of Solutions of Minkowski-Curvature Problems in Exterior Domains","authors":"Tianlan Chen, Haiyi Wu","doi":"10.1093/qmath/haae023","DOIUrl":"https://doi.org/10.1093/qmath/haae023","url":null,"abstract":"In this paper, we show some nonexistence results of radial solutions for the following Minkowski curvature problems in an exterior domain: $$ begin{cases} -text{div} big(phi(nabla v(x))big)=k(x)f(v(x)), quadquad xinOmega, v=0 text{on} partialOmega, qquadlimlimits_{xrightarrowinfty}v(x)=0 end{cases} $$ for R sufficiently large, where $phi(s)=frac{s}{sqrt{1-s^{2}}}$ for $sin{mathbb R}$ with $s^2lt1,$ $Omega={xin{{mathbb R}^{N}}: |x| gt R}$, $Ngeq3$ is an integer, $|cdot|$ denotes the Euclidean norm on $mathbb{R}^{N}$, R is a positive parameter, $f:mathbb{R}rightarrowmathbb{R}$ is an odd and locally Lipschitz continuous function and $k in C^{1}(mathbb{R}^{+}, mathbb{R}^{+})$ with $mathbb{R}^{+}=(0, +infty)$. We also apply the fixed-point index theory to establish the existence of positive radial solutions of the above problems for R sufficiently small.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"181 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936068","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}
Geraldo Botelho, José Lucas P Luiz, Vinícius C C Miranda
New characterizations of the disjoint Dunford–Pettis property of order p (disjoint DPPp) are proved and applied to show that a Banach lattice of cotype p has the disjoint DPPp whenever its dual has this property. The disjoint Dunford–Pettis$^*$ property of order p (disjoint $DP^*P_p$) is thoroughly investigated. Close connections with the positive Schur property of order p, with the disjoint DPPp, with the p-weak $DP^*$ property and with the positive $DP^*$ property of order p are established. In a final section, we study the polynomial versions of the disjoint DPPp and of the disjoint $DP^*P_p$.
本文证明并应用了阶 p 的邓福德-佩蒂斯不相交属性(DPPp 不相交)的新特征,以说明只要对偶具有该属性,阶 p 的巴拿赫网格就具有 DPPp 不相交属性。阶 p 的不相交邓福德-佩蒂斯$^*$ 性质(不相交 $DP^*P_p$)得到了深入研究。我们建立了阶 p 的正舒尔性质、不相交 DPPp、p 弱 $DP^*$ 性质以及阶 p 的正 $DP^*$ 性质之间的密切联系。在最后一节中,我们研究了多项式版本的不相交 DPPp 和不相交 $DP^*P_p$。
{"title":"Disjoint Dunford–Pettis-Type Properties in Banach Lattices","authors":"Geraldo Botelho, José Lucas P Luiz, Vinícius C C Miranda","doi":"10.1093/qmath/haae024","DOIUrl":"https://doi.org/10.1093/qmath/haae024","url":null,"abstract":"New characterizations of the disjoint Dunford–Pettis property of order p (disjoint DPPp) are proved and applied to show that a Banach lattice of cotype p has the disjoint DPPp whenever its dual has this property. The disjoint Dunford–Pettis$^*$ property of order p (disjoint $DP^*P_p$) is thoroughly investigated. Close connections with the positive Schur property of order p, with the disjoint DPPp, with the p-weak $DP^*$ property and with the positive $DP^*$ property of order p are established. In a final section, we study the polynomial versions of the disjoint DPPp and of the disjoint $DP^*P_p$.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936011","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}
Bo Li, Jinxia Li, Qingze Lin, Tianjun Shen, Chao Zhang
Let $({mathcal{M}},d,mu)$ be the metric measure space with a Dirichlet form $mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${mathcal{M}}$ to its Poisson integral Ptu on ${mathcal{M}}timesmathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].
{"title":"On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES","authors":"Bo Li, Jinxia Li, Qingze Lin, Tianjun Shen, Chao Zhang","doi":"10.1093/qmath/haae019","DOIUrl":"https://doi.org/10.1093/qmath/haae019","url":null,"abstract":"Let $({mathcal{M}},d,mu)$ be the metric measure space with a Dirichlet form $mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${mathcal{M}}$ to its Poisson integral Ptu on ${mathcal{M}}timesmathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"19 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140838462","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}
We prove that the loop space of the moment-angle complex associated with the k-skeleton of a flag complex belongs to the class $mathcal{P}$ of spaces homotopy equivalent to a finite-type product of spheres and loops on simply connected spheres. To do this, a general result showing $mathcal{P}$ is closed under retracts is proved.
我们证明了与旗状复数的 k 骨架相关的矩角复数的环空间属于 $mathcal{P}$ 这类同调等价于简单相连球面上球面与环的有限类型乘积的空间。为此,我们证明了一个显示 $mathcal{P}$ 在收回条件下是封闭的一般结果。
{"title":"Loop Space Decompositions of Moment-Angle Complexes Associated to Flag Complexes","authors":"Lewis Stanton","doi":"10.1093/qmath/haae020","DOIUrl":"https://doi.org/10.1093/qmath/haae020","url":null,"abstract":"We prove that the loop space of the moment-angle complex associated with the k-skeleton of a flag complex belongs to the class $mathcal{P}$ of spaces homotopy equivalent to a finite-type product of spheres and loops on simply connected spheres. To do this, a general result showing $mathcal{P}$ is closed under retracts is proved.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140838563","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}
In this paper, we consider generalized Möbius functions associated with two types of L-functions: Rankin–Selberg L-functions of symmetric powers of distinct holomorphic cusp forms and L-functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.
在本文中,我们考虑了与两类 L 函数相关的广义莫比乌斯函数:不同全形尖点形式的对称幂的 Rankin-Selberg L 函数,以及从 Maass 尖点形式导出的 L 函数。我们证明这些广义莫比乌斯函数与有界序列弱正交。作为直接推论,这两类莫比乌斯函数的广义萨尔纳克猜想成立。
{"title":"On Möbius Functions from Automorphic Forms and a Generalized Sarnak’s Conjecture","authors":"Zhining Wei, Shifan Zhao","doi":"10.1093/qmath/haae018","DOIUrl":"https://doi.org/10.1093/qmath/haae018","url":null,"abstract":"In this paper, we consider generalized Möbius functions associated with two types of L-functions: Rankin–Selberg L-functions of symmetric powers of distinct holomorphic cusp forms and L-functions derived from Maass cusp forms. We show that these generalized Möbius functions are weakly orthogonal to bounded sequences. As a direct corollary, a generalized Sarnak’s conjecture holds for these two types of Möbius functions.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"52 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140629520","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}
We prove algebraicity results for critical L-values attached to the group ${rm GSp}_4 times {rm GL}_2$, and for Gan–Gross–Prasad periods which are conjecturally related to central L-values for ${rm GSp}_4 times {rm GL}_2 times {rm GL}_2$. Our result for ${rm GSp}_4 times {rm GL}_2$ overlaps substantially with recent results of Morimoto, but our methods are very different; these results will be used in a sequel paper to construct a new p-adic L-function for ${rm GSp}_4 times {rm GL}_2$. The results for Gan–Gross–Prasad periods appear to be new. A key aspect is the computation of certain Archimedean zeta integrals, whose p-adic counterparts are also studied in this note.
我们证明了附着于${rm GSp}_4 times {rm GL}_2$组的临界L值的代数性结果,以及与${rm GSp}_4 times {rm GL}_2 times {rm GL}_2$的中心L值猜想相关的甘-格罗斯-普拉萨德周期的代数性结果。我们关于 ${rm GSp}_4 times {rm GL}_2$ 的结果与森本(Morimoto)的最新结果有很大重叠,但我们的方法却截然不同;这些结果将在续篇论文中用于构建 ${rm GSp}_4 times {rm GL}_2$ 的新 p-adic L 函数。关于甘-格罗斯-普拉萨德周期的结果似乎是新的。其中一个关键方面是某些阿基米德zeta积分的计算,本注释也研究了其p-adic对应物。
{"title":"Algebraicity of L-values for GSP4 X GL2 and G","authors":"David Loeffler, Óscar Rivero","doi":"10.1093/qmath/haae016","DOIUrl":"https://doi.org/10.1093/qmath/haae016","url":null,"abstract":"We prove algebraicity results for critical L-values attached to the group ${rm GSp}_4 times {rm GL}_2$, and for Gan–Gross–Prasad periods which are conjecturally related to central L-values for ${rm GSp}_4 times {rm GL}_2 times {rm GL}_2$. Our result for ${rm GSp}_4 times {rm GL}_2$ overlaps substantially with recent results of Morimoto, but our methods are very different; these results will be used in a sequel paper to construct a new p-adic L-function for ${rm GSp}_4 times {rm GL}_2$. The results for Gan–Gross–Prasad periods appear to be new. A key aspect is the computation of certain Archimedean zeta integrals, whose p-adic counterparts are also studied in this note.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"13 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561808","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}
We give a short proof of the fact that each homogeneous linear differential operator $mathscr{A}$ of constant rank admits a homogeneous potential operator $mathscr{B}$, meaning that $$kermathscr{A}(xi)=mathrm{im,}mathscr{B}(xi) quadtext{for }xiinmathbb{R}^nbackslash{0}.$$ We make some refinements of the original result and some related remarks.
{"title":"A simple construction of potential operators for compensated compactness","authors":"Bogdan Raiță","doi":"10.1093/qmath/haae008","DOIUrl":"https://doi.org/10.1093/qmath/haae008","url":null,"abstract":"We give a short proof of the fact that each homogeneous linear differential operator $mathscr{A}$ of constant rank admits a homogeneous potential operator $mathscr{B}$, meaning that $$kermathscr{A}(xi)=mathrm{im,}mathscr{B}(xi) quadtext{for }xiinmathbb{R}^nbackslash{0}.$$ We make some refinements of the original result and some related remarks.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"46 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140562206","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}
For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.
{"title":"Higher-degree Artin conjecture","authors":"Olli Järviniemi","doi":"10.1093/qmath/haae012","DOIUrl":"https://doi.org/10.1093/qmath/haae012","url":null,"abstract":"For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"48 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561989","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}
Let $(M,I,J,K,Omega)$ be a compact HKT manifold, and let us denote with $partial$ the conjugate Dolbeault operator with respect to I, $partial_J:=J^{-1}overlinepartial J$, $partial^Lambda:=[partial,Lambda]$, where Λ is the adjoint of $L:=Omegawedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{bullet,0},partial,partial_J)$ and $(A^{bullet,0},partial,partial^Lambda)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $mathrm{SL}(n,mathbb{H})$-manifold, the differential graded algebra $(A^{bullet,0},partial)$ is formal and this will lead to an obstruction for the existence of an HKT $mathrm{SL}(n,mathbb{H})$ structure $(I,J,K,Omega)$ on a compact complex manifold (M, I). Finally, balanced HKT structures on solvmanifolds are studied.
{"title":"HKT Manifolds: Hodge Theory, Formality and Balanced Metrics","authors":"Giovanni Gentili, Nicoletta Tardini","doi":"10.1093/qmath/haae013","DOIUrl":"https://doi.org/10.1093/qmath/haae013","url":null,"abstract":"Let $(M,I,J,K,Omega)$ be a compact HKT manifold, and let us denote with $partial$ the conjugate Dolbeault operator with respect to I, $partial_J:=J^{-1}overlinepartial J$, $partial^Lambda:=[partial,Lambda]$, where Λ is the adjoint of $L:=Omegawedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{bullet,0},partial,partial_J)$ and $(A^{bullet,0},partial,partial^Lambda)$ showing a similar behavior to Kähler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $mathrm{SL}(n,mathbb{H})$-manifold, the differential graded algebra $(A^{bullet,0},partial)$ is formal and this will lead to an obstruction for the existence of an HKT $mathrm{SL}(n,mathbb{H})$ structure $(I,J,K,Omega)$ on a compact complex manifold (M, I). Finally, balanced HKT structures on solvmanifolds are studied.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140561941","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}