Pub Date : 2021-01-01DOI: 10.4310/ajm.2021.v25.n2.a5
B. Cheng, M. Melgaard
We study the self-adjoint Dirac operators D = D0 + V (x), where D0 is the free three-dimensional Dirac operator and V (x) is a smooth compactly supported Hermitian matrix potential. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. By analyzing the resolvent behaviour at the spectrum edges ±m, we establish a generalized Birman-Krein formula, taking into account possible resonances at ±m. As an application of the new Birman-Krein formula we establish the Poisson wave trace formula in its full generality. The Poisson wave trace formula links the resonances with the trace of the difference of the wave groups. The Poisson wave trace formula, in conjunction with asymptotics of the scattering phase, allows us to prove that, under certain natural assumptions on V , the perturbed Dirac operator has infinitely many resonances; a result similar in nature to Melrose’s classic 1995 result for Schr¨odinger operators.
研究了自伴随狄拉克算子D = D0 + V (x)，其中D0是自由三维狄拉克算子，V (x)是光滑紧支持厄米矩阵势。我们将D的共振定义为其截止解的亚纯延拓的极点。通过分析光谱边缘±m处的解析行为，我们建立了一个广义的Birman-Krein公式，考虑了±m处可能的共振。作为新Birman-Krein公式的一个应用，我们建立了具有完全普遍性的泊松波迹公式。泊松波迹公式将共振与波群的差迹联系起来。泊松波迹公式，结合散射相位的渐近性，允许我们证明，在V上的某些自然假设下，扰动狄拉克算子具有无限多个共振;本质上类似于梅尔罗斯1995年关于薛定谔算子的经典结果。
{"title":"Poisson wave trace formula for Dirac resonances at spectrum edges and applications","authors":"B. Cheng, M. Melgaard","doi":"10.4310/ajm.2021.v25.n2.a5","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a5","url":null,"abstract":"We study the self-adjoint Dirac operators D = D0 + V (x), where D0 is the free three-dimensional Dirac operator and V (x) is a smooth compactly supported Hermitian matrix potential. We define resonances of D as poles of the meromorphic continuation of its cut-off resolvent. By analyzing the resolvent behaviour at the spectrum edges ±m, we establish a generalized Birman-Krein formula, taking into account possible resonances at ±m. As an application of the new Birman-Krein formula we establish the Poisson wave trace formula in its full generality. The Poisson wave trace formula links the resonances with the trace of the difference of the wave groups. The Poisson wave trace formula, in conjunction with asymptotics of the scattering phase, allows us to prove that, under certain natural assumptions on V , the perturbed Dirac operator has infinitely many resonances; a result similar in nature to Melrose’s classic 1995 result for Schr¨odinger operators.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392222","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 : 2021-01-01DOI: 10.4310/ajm.2021.v25.n1.a6
F. Ivorra, J. Sebag
{"title":"Quasi-unipotent motives and motivic nearby sheaves","authors":"F. Ivorra, J. Sebag","doi":"10.4310/ajm.2021.v25.n1.a6","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n1.a6","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391903","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 : 2021-01-01DOI: 10.4310/ajm.2021.v25.n2.a2
Q. Cheng, G. Wei, Yuting Zeng
{"title":"Area of minimal hypersurfaces in the unit sphere","authors":"Q. Cheng, G. Wei, Yuting Zeng","doi":"10.4310/ajm.2021.v25.n2.a2","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a2","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70391926","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 : 2021-01-01DOI: 10.4310/ajm.2021.v25.n2.a6
Dong Pu, Hong-wei Xu
{"title":"Global pinching theorems for minimal submanifolds in a complex projective space","authors":"Dong Pu, Hong-wei Xu","doi":"10.4310/ajm.2021.v25.n2.a6","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n2.a6","url":null,"abstract":"","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70392240","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 : 2020-11-30DOI: 10.4310/ajm.2022.v26.n5.a3
S. Shivaprasad
Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$. We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of $overline{mathcal{M}_g}$ in such a way that these measures form a continuous family of measures on the universal curve over $overline{mathcal{M}_g}$.
{"title":"Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces","authors":"S. Shivaprasad","doi":"10.4310/ajm.2022.v26.n5.a3","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n5.a3","url":null,"abstract":"Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$. We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of $overline{mathcal{M}_g}$ in such a way that these measures form a continuous family of measures on the universal curve over $overline{mathcal{M}_g}$.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48147817","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 : 2020-11-06DOI: 10.4310/ajm.2022.v26.n4.a2
S. Rao, I. Tsai
This paper answers a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera affirmatively, and proves this more generally for a flat family of varieties with only canonical singularities and uncountable ones therein being of general type and also two Chow-type lemmata on the structure of a family of projective complex analytic spaces.
{"title":"Invariance of plurigenera and Chow-type lemma","authors":"S. Rao, I. Tsai","doi":"10.4310/ajm.2022.v26.n4.a2","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n4.a2","url":null,"abstract":"This paper answers a question of Demailly whether a smooth family of nonsingular projective varieties admits the deformation invariance of plurigenera affirmatively, and proves this more generally for a flat family of varieties with only canonical singularities and uncountable ones therein being of general type and also two Chow-type lemmata on the structure of a family of projective complex analytic spaces.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45150576","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 : 2020-10-10DOI: 10.4310/ajm.2022.v26.n5.a6
Robert E. Gompf
We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented 3-manifold M has a TPC embedding in a compact, complex surface (without boundary) realizing any homotopy class of almost-complex structures (the analogue of the homotopy class of the contact plane field in the smooth case). We prove our tool theorems with invariants that classify almost-complex structures on any 4-manifold homotopy equivalent to M. These invariants are amenable to computation and respected by homeomorphisms (not necessarily smooth). We study the two equivalence classes of smoothings on the product of a 3-manifold with a line, and on collared ends. Both classes of smoothings are realized by holomorphic embeddings exhibiting any preassigned homotopy class of almost-complex structures. One class arises from TPC embedded 3-manifolds, while the other likely does not.
{"title":"Topological convexity in complex surfaces","authors":"Robert E. Gompf","doi":"10.4310/ajm.2022.v26.n5.a6","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n5.a6","url":null,"abstract":"We study a notion of strict pseudoconvexity in the context of topologically (often unsmoothably) embedded 3-manifolds in complex surfaces. Topologically pseudoconvex (TPC) 3-manifolds behave similarly to their smooth analogues, cutting out open domains of holomorphy (Stein surfaces), but they are much more common. We provide tools for constructing TPC embeddings, and show that every closed, oriented 3-manifold M has a TPC embedding in a compact, complex surface (without boundary) realizing any homotopy class of almost-complex structures (the analogue of the homotopy class of the contact plane field in the smooth case). We prove our tool theorems with invariants that classify almost-complex structures on any 4-manifold homotopy equivalent to M. These invariants are amenable to computation and respected by homeomorphisms (not necessarily smooth). We study the two equivalence classes of smoothings on the product of a 3-manifold with a line, and on collared ends. Both classes of smoothings are realized by holomorphic embeddings exhibiting any preassigned homotopy class of almost-complex structures. One class arises from TPC embedded 3-manifolds, while the other likely does not.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48062904","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 : 2020-09-01DOI: 10.4310/ajm.2022.v26.n6.a4
Adam Jacob, Norman Sheu
We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.
{"title":"The deformed Hermitian–Yang–Mills equation on the blowup of $mathbb{P}^n$","authors":"Adam Jacob, Norman Sheu","doi":"10.4310/ajm.2022.v26.n6.a4","DOIUrl":"https://doi.org/10.4310/ajm.2022.v26.n6.a4","url":null,"abstract":"We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41717607","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 : 2020-08-21DOI: 10.4310/ajm.2021.v25.n4.a5
S. Ghosh, Somnath Jha, Sudhanshu Shekhar
A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $mathbb{Z}_p[[Gamma ]]$ with $Gamma cong mathbb{Z}_p, exists$ a continuous character $theta: Gamma rightarrow mathbb{Z}_p^times$ such that, the $ Gamma^{n}$-Euler characteristic of the twist $M(theta)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$-adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $mathcal{T}[[G]]$ modules, where $G$ is a compact $p$-adic Lie group and $mathcal{T}$ is a finite extension of $mathbb{Z}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic application by considering the twisted Euler Characteristic of the big Selmer (respectively fine Selmer) group of a $Lambda$-adic form over a $p$-adic Lie extension.
{"title":"Twisting lemma for $lambda$-adic modules","authors":"S. Ghosh, Somnath Jha, Sudhanshu Shekhar","doi":"10.4310/ajm.2021.v25.n4.a5","DOIUrl":"https://doi.org/10.4310/ajm.2021.v25.n4.a5","url":null,"abstract":"A classical twisting lemma says that given a finitely generated torsion module $M$ over the Iwasawa algebra $mathbb{Z}_p[[Gamma ]]$ with $Gamma cong mathbb{Z}_p, exists$ a continuous character $theta: Gamma rightarrow mathbb{Z}_p^times$ such that, the $ Gamma^{n}$-Euler characteristic of the twist $M(theta)$ is finite for every $n$. This twisting lemma has been generalized for the Iwasawa algebra of a general compact $p$-adic Lie group $G$. In this article, we consider a further generalization of the twisting lemma to $mathcal{T}[[G]]$ modules, where $G$ is a compact $p$-adic Lie group and $mathcal{T}$ is a finite extension of $mathbb{Z}_p[[X]]$. Such modules naturally occur in Hida theory. We also indicate arithmetic application by considering the twisted Euler Characteristic of the big Selmer (respectively fine Selmer) group of a $Lambda$-adic form over a $p$-adic Lie extension.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48666783","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 : 2020-06-05DOI: 10.4310/ajm.2023.v27.n1.a3
Xianjing Dong
We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete K"ahler manifold into a complex projective manifold. When certain curvature conditions are imposed, the Nevanlinna-type defect relation based on heat diffusion is derived.
{"title":"Nevanlinna-type theory based on heat diffusion","authors":"Xianjing Dong","doi":"10.4310/ajm.2023.v27.n1.a3","DOIUrl":"https://doi.org/10.4310/ajm.2023.v27.n1.a3","url":null,"abstract":"We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete K\"ahler manifold into a complex projective manifold. When certain curvature conditions are imposed, the Nevanlinna-type defect relation based on heat diffusion is derived.","PeriodicalId":55452,"journal":{"name":"Asian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42537439","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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