Pub Date : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a3
Chris Peters
The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite. Examples include the Godeaux surfaces, the Kunev surface and a specific Horikawa surface. There are also some results concerning (negative) definite primitive lattices, especially for canonically polarized surfaces of general type.
{"title":"A note on the primitive cohomology lattice of a projective surface","authors":"Chris Peters","doi":"10.4310/pamq.2023.v19.n6.a3","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a3","url":null,"abstract":"The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite. Examples include the Godeaux surfaces, the Kunev surface and a specific Horikawa surface. There are also some results concerning (negative) definite primitive lattices, especially for canonically polarized surfaces of general type.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656318","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a10
Xiuxiong Chen, Jingrui Cheng
In this paper, we consider a version of parabolic complex Monge–Ampère equations, and use a PDE approach similar to Phong et al. to establish $L^infty$ and Hölder estimates. We also generalize the $L^infty$ estimates to parabolic Hessian equations.
{"title":"The $L^infty$ estimates for parabolic complex Monge–Ampère and Hessian equations","authors":"Xiuxiong Chen, Jingrui Cheng","doi":"10.4310/pamq.2023.v19.n6.a10","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a10","url":null,"abstract":"In this paper, we consider a version of parabolic complex Monge–Ampère equations, and use a PDE approach similar to Phong <i>et al.</i> to establish $L^infty$ and Hölder estimates. We also generalize the $L^infty$ estimates to parabolic Hessian equations.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656645","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a16
Mark Green, Phillip Griffiths
The global behavior of period mappings defined on generally non-complete algebraic varieties $B$ as well as their local behavior around points in the boundary $Z = overline{B}setminus B$ of smooth completions of $B$ have been extensively investigated. In this paper we shall study the global behavior of period mappings in neighborhoods of the entire boundary $Z$ when $dim B = 2$. One method will be to decompose the dual graph of the boundary into basic building blocks of cycles and trees and analyze these separately. A main tool will be a global version of the classical nilpotent orbit theorem of Schmid.
{"title":"Global behavior at infinity of period mappings defined on algebraic surface","authors":"Mark Green, Phillip Griffiths","doi":"10.4310/pamq.2023.v19.n6.a16","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a16","url":null,"abstract":"The global behavior of period mappings defined on generally non-complete algebraic varieties $B$ as well as their local behavior around points in the boundary $Z = overline{B}setminus B$ of smooth completions of $B$ have been extensively investigated. In this paper we shall study the <i>global</i> behavior of period mappings in neighborhoods of the entire boundary $Z$ when $dim B = 2$. One method will be to decompose the dual graph of the boundary into basic building blocks of cycles and trees and analyze these separately. A main tool will be a global version of the classical nilpotent orbit theorem of Schmid.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656127","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a4
Mikhail Khovanov, Robert Laugwitz
A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.
{"title":"Planar diagrammatics of self-adjoint functors and recognizable tree series","authors":"Mikhail Khovanov, Robert Laugwitz","doi":"10.4310/pamq.2023.v19.n5.a4","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a4","url":null,"abstract":"A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, then one assigns an element of that field to a diagram of nested circles. We focus on the self-adjoint functor case of this construction and study the reverse problem of recovering such a functor and a category given values associated to diagrams of nested circles.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647318","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a8
Hans Wenzl
$defEnd{operatorname{End}}$$defRep{operatorname{Rep}}$$defsl{mathfrak{sl}}$Let $V = mathbb{C}^N$ with $N$ odd.We construct a $q$-deformation of $End_{Sp(N-1)}(V^{otimes n})$ which contains $End_{U_q sl_N} (V^{otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest the existence of module categories of $Rep(U_q sl_N)$ which may not come from already known coideal subalgebras of $ U_q sl_N$. We moreover indicate how this can be used to construct module categories of the associated fusion tensor categories as well as subfactors, along the lines of previous work for inclusions $Sp(N) subset SL(N)$.
{"title":"On module categories related to $Sp(N-1) subset Sl(N)$","authors":"Hans Wenzl","doi":"10.4310/pamq.2023.v19.n5.a8","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a8","url":null,"abstract":"$defEnd{operatorname{End}}$$defRep{operatorname{Rep}}$$defsl{mathfrak{sl}}$Let $V = mathbb{C}^N$ with $N$ odd.We construct a $q$-deformation of $End_{Sp(N-1)}(V^{otimes n})$ which contains $End_{U_q sl_N} (V^{otimes n})$. It is a quotient of an abstract two-variable algebra which is defined by adding one more generator to the generators of the Hecke algebras $H_n$. These results suggest the existence of module categories of $Rep(U_q sl_N)$ which may not come from already known coideal subalgebras of $ U_q sl_N$. We moreover indicate how this can be used to construct module categories of the associated fusion tensor categories as well as subfactors, along the lines of previous work for inclusions $Sp(N) subset SL(N)$.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647720","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n5.a9
Feng Xu
This paper contains my previously unpublished work on three problems proposed by Vaughan Jones.
本文包含我之前未发表的关于沃恩-琼斯提出的三个问题的研究成果。
{"title":"On three homework problems from Vaughan Jones","authors":"Feng Xu","doi":"10.4310/pamq.2023.v19.n5.a9","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n5.a9","url":null,"abstract":"This paper contains my previously unpublished work on three problems proposed by Vaughan Jones.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139647313","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 : 2024-01-30DOI: 10.4310/pamq.2023.v19.n6.a4
Barbara Drinovec Drnovšek, Franc Forstnerič
The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb{R}^n$, $n geq 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi’s pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb{R}^n$, this minimal metric coincides with the classical Beltrami–Cayley–Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine $2$-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.
{"title":"Hyperbolic domains in real Euclidean spaces","authors":"Barbara Drinovec Drnovšek, Franc Forstnerič","doi":"10.4310/pamq.2023.v19.n6.a4","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n6.a4","url":null,"abstract":"The second named author and David Kalaj introduced a pseudometric on any domain in the real Euclidean space $mathbb{R}^n$, $n geq 3$, defined in terms of conformal harmonic discs, by analogy with Kobayashi’s pseudometric on complex manifolds, which is defined in terms of holomorphic discs. They showed that on the unit ball of $mathbb{R}^n$, this minimal metric coincides with the classical Beltrami–Cayley–Klein metric. In the present paper we investigate properties of the minimal pseudometric and give sufficient conditions for a domain to be (complete) hyperbolic, meaning that the minimal pseudometric is a (complete) metric. We show in particular that a convex domain is complete hyperbolic if and only if it does not contain any affine $2$-planes. One of our main results is that a domain with a negative minimal plurisubharmonic exhaustion function is hyperbolic, and a bounded strongly minimally convex domain is complete hyperbolic. We also prove a localization theorem for the minimal pseudometric.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139656087","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 : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a13
Yi Lin, Yiannis Loizides, Reyer Sjamaar, Yanli Song
We extend the Marsden–Weinstein reduction theorem and the Darboux–Moser–Weinstein theorem to symplectic Lie algebroids. We also obtain a coisotropic embedding theorem for symplectic Lie algebroids.
{"title":"Symplectic reduction and a Darboux–Moser–Weinstein theorem for Lie algebroids","authors":"Yi Lin, Yiannis Loizides, Reyer Sjamaar, Yanli Song","doi":"10.4310/pamq.2023.v19.n4.a13","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a13","url":null,"abstract":"We extend the Marsden–Weinstein reduction theorem and the Darboux–Moser–Weinstein theorem to symplectic Lie algebroids. We also obtain a coisotropic embedding theorem for symplectic Lie algebroids.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495287","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 : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a11
Peer Christian Kunstmann, Eric Todd Quinto, Andreas Rieder
Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset.
{"title":"Seismic imaging with generalized Radon transforms: stability of the Bolker condition","authors":"Peer Christian Kunstmann, Eric Todd Quinto, Andreas Rieder","doi":"10.4310/pamq.2023.v19.n4.a11","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a11","url":null,"abstract":"Generalized Radon transforms are Fourier integral operators which are used, for instance, as imaging models in geophysical exploration. They appear naturally when linearizing about a known background compression wave speed. In this work we first consider a linearly increasing background velocity in two spatial dimensions. We verify the Bolker condition for the zero-offset scanning geometry and provide meaningful arguments for it to hold even if the common offset is positive. Based on this result we suggest an imaging operator for which we calculate the top order symbol in the zero-offset case to study how it maps singularities. Second, to support the usage of background models obtained from linear regression we present a stability result for the Bolker condition under perturbations of the background velocity and of the offset.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138495289","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 : 2023-11-20DOI: 10.4310/pamq.2023.v19.n4.a10
David Kazhdan, Alexander Polishchuk
We prove an efficient version of the Wagner’s theorem on almost invariant subspaces (see $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1608090}{[5]}$) and deduce some consequences in the context of Galois extensions.
{"title":"Almost invariant subspaces and operators","authors":"David Kazhdan, Alexander Polishchuk","doi":"10.4310/pamq.2023.v19.n4.a10","DOIUrl":"https://doi.org/10.4310/pamq.2023.v19.n4.a10","url":null,"abstract":"We prove an efficient version of the Wagner’s theorem on almost invariant subspaces (see $href{https://mathscinet.ams.org/mathscinet/relay-station?mr=1608090}{[5]}$) and deduce some consequences in the context of Galois extensions.","PeriodicalId":54526,"journal":{"name":"Pure and Applied Mathematics Quarterly","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517075","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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