{"title":"Uniqueness of irreducible desingularization of singularities associated to negative vector bundles","authors":"Fusheng Deng, Yinji Li, Qunhuan Liu, Xiangyu Zhou","doi":"arxiv-2409.09402","DOIUrl":"https://doi.org/arxiv-2409.09402","url":null,"abstract":"We prove that the irreducible desingularization of a singularity given by the\u0000Grauert blow down of a negative holomorphic vector bundle over a compact\u0000complex manifold is unique up to isomorphism, and as an application, we show\u0000that two negative line bundles over compact complex manifolds are isomorphic if\u0000and only if their Grauert blow downs have isomorphic germs near the\u0000singularities. We also show that there is a unique way to modify a submanifold\u0000of a complex manifold to a hypersurface, namely, the blow up of the ambient\u0000manifold along the submanifold.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Very weak solutions of quadratic Hessian equations","authors":"Sławomir Dinew, Szymon Myga","doi":"arxiv-2409.08852","DOIUrl":"https://doi.org/arxiv-2409.08852","url":null,"abstract":"We extend the methods of Lewicka - Pakzad, Sz'ekelyhidi - Cao and Li - Qiu\u0000to study the notion of very weak solutions to the complex $sigma_2$ equation\u0000in domains in $mathbb C^n, ngeq 2$. As a by-product we sharpen the\u0000regularity threshold of the counterexamples obtained by Li and Qiu in the real\u0000case.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142265800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Black Holes, Complex Curves, and Graph Theory: Revising a Conjecture by Kasner","authors":"Yen Chin Ong","doi":"arxiv-2409.08236","DOIUrl":"https://doi.org/arxiv-2409.08236","url":null,"abstract":"The ratios $sqrt{8/9}=2sqrt{2}/3approx 0.9428$ and $sqrt{3}/2 approx\u00000.866$ appear in various contexts of black hole physics, as values of the\u0000charge-to-mass ratio $Q/M$ or the rotation parameter $a/M$ for\u0000Reissner-Nordstr\"om and Kerr black holes, respectively. In this work, in the\u0000Reissner-Nordstr\"om case, I relate these ratios with the quantization of the\u0000horizon area, or equivalently of the entropy. Furthermore, these ratios are\u0000related to a century-old work of Kasner, in which he conjectured that certain\u0000sequences arising from complex analysis may have a quantum interpretation.\u0000These numbers also appear in the case of Kerr black holes, but the explanation\u0000is not as straightforward. The Kasner ratio may also be relevant for\u0000understanding the random matrix and random graph approaches to black hole\u0000physics, such as fast scrambling of quantum information, via a bound related to\u0000Ramanujan graph. Intriguingly, some other pure mathematical problems in complex\u0000analysis, notably complex interpolation in the unit disk, appear to share some\u0000mathematical expressions with the black hole problem and thus also involve the\u0000Kasner ratio.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frequently hypercyclic meromorphic curves with slow growth","authors":"Bin Guo, Song-Yan Xie, Zhangchi Chen","doi":"arxiv-2409.08048","DOIUrl":"https://doi.org/arxiv-2409.08048","url":null,"abstract":"We construct entire curves in projective spaces, which are frequently\u0000hypercyclic simultaneously for countably many given translations, with optimal\u0000slow growth rates.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Blow-up analysis and degree theory for the Webster curvature prescription problem in three dimensions","authors":"Claudio Afeltra","doi":"arxiv-2409.07334","DOIUrl":"https://doi.org/arxiv-2409.07334","url":null,"abstract":"Given a strictly pseudoconvex CR manifold $M$ of dimension three and positive\u0000CR Yamabe class, and a positive smooth function $K:Mtomathbf{R}$ verifying\u0000some mild and generic hypotheses, we prove the compactness of the set of\u0000solutions of the Webster curvature prescription problem associated to $K$, and\u0000we compute the Leray-Schauder degree in terms of the critical points of $K$. As\u0000a corollary, we get an existence result which generalizes the ones existent in\u0000the literature.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201811","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Steven R. Bell, Loredana Lanzani, Nathan A. Wagner
{"title":"A new way to express boundary values in terms of holomorphic functions on planar Lipschitz domains","authors":"Steven R. Bell, Loredana Lanzani, Nathan A. Wagner","doi":"arxiv-2409.06611","DOIUrl":"https://doi.org/arxiv-2409.06611","url":null,"abstract":"We decompose $p$ - integrable functions on the boundary of a simply connected\u0000Lipschitz domain $Omega subset mathbb C$ into the sum of the boundary values\u0000of two, uniquely determined holomorphic functions, where one is holomorphic in\u0000$Omega$ while the other is holomorphic in $mathbb Csetminus\u0000overline{Omega}$ and vanishes at infinity. This decomposition has been\u0000described previously for smooth functions on the boundary of a smooth domain.\u0000Uniqueness of the decomposition is elementary in the smooth case, but extending\u0000it to the $L^p$ setting relies upon a regularity result for the holomorphic\u0000Hardy space $h^p(bOmega)$ which appears to be new even for smooth $Omega$. An\u0000immediate consequence of our result will be a new characterization of the\u0000kernel of the Cauchy transform acting on $L^p(bOmega)$. These results give a\u0000new perspective on the classical Dirichlet problem for harmonic functions and\u0000the Poisson formula even in the case of the disc. Further applications are\u0000presented along with directions for future work.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on the Wiman-Valiron inequality","authors":"Karl-G. Grosse-Erdmann","doi":"arxiv-2409.06499","DOIUrl":"https://doi.org/arxiv-2409.06499","url":null,"abstract":"The Wiman-Valiron inequality relates the maximum modulus of an analytic\u0000function to its Taylor coefficients via the maximum term. After a short\u0000overview of the known results, we obtain a general version of this inequality\u0000that seems to have been overlooked in the literature so far. We end the paper\u0000with an open problem.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform $L^2$-estimates for flat nontrivial line bundles on compact complex manifolds","authors":"Yoshinori Hashimoto, Takayuki Koike, Shin-ichi Matsumura","doi":"arxiv-2409.05300","DOIUrl":"https://doi.org/arxiv-2409.05300","url":null,"abstract":"In this paper, we extend the uniform $L^2$-estimate of\u0000$bar{partial}$-equations for flat nontrivial line bundles, proved for compact\u0000K\"ahler manifolds in the previous work, to compact complex manifolds. In the\u0000proof, by tracing the Dolbeault isomorphism in detail, we derive the desired\u0000$L^2$-estimate directly from Ueda's lemma.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries","authors":"Simone Gallivanone, Roberto Paoletti","doi":"arxiv-2409.04753","DOIUrl":"https://doi.org/arxiv-2409.04753","url":null,"abstract":"Let $(M,kappa)$ be a closed and connected real-analytic Riemannian manifold,\u0000acted upon by a compact Lie group of isometries $G$. We consider the following\u0000two kinds of equivariant asymptotics along a fixed Grauer tube boundary\u0000$X^tau$ of $(M,kappa)$. 1): Given the induced unitary representation of $G$ on the eigenspaces of the\u0000Laplacian of $(M,kappa)$, these split over the irreducible representations of\u0000$G$. On the other hand, the eigenfunctions of the Laplacian of $(M,kappa)$\u0000admit a simultaneous complexification to some Grauert tube. We study the\u0000asymptotic concentration along $X^tau$ of the complexified eigenfunctions\u0000pertaining to a fixed isotypical component. 2): There are furthermore an induced action of $G$ as a group of CR and\u0000contact automorphisms on $X^tau$, and a corresponding unitary representation\u0000on the Hardy space $H(X^tau)$. The action of $G$ on $X^tau$ commutes with the\u0000homogeneous lq geogesic flowrq, and the representation on the Hardy space\u0000commutes with the elliptic self-adjoint Toeplitz operator induced by the\u0000generator of the goedesic flow. Hence each eigenspace of the latter also splits\u0000over the irreducible representations of $G$. We study the asymptotic\u0000concentration of the eigenfunctions in a given isotypical component. We also give some applications of these asymptotics.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Bieberbach conjecture, Bohr radius, Bloch constant and Alexander's theorem in infinite dimensions","authors":"Hidetaka Hamada, Gabriela Kohr, Mirela Kohr","doi":"arxiv-2409.04028","DOIUrl":"https://doi.org/arxiv-2409.04028","url":null,"abstract":"In this paper, we investigate holomorphic mappings $F$ on the unit ball\u0000$mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a\u0000holomorphic function on $mathbb{B}$. First, we investigate criteria for\u0000univalence, starlikeness and quasi-convexity of type $B$ on $mathbb{B}$. Next,\u0000we investigate a generalized Bieberbach conjecture, a covering theorem and a\u0000distortion theorem, the Fekete-Szeg\"{o} inequality, lower bound for the Bloch\u0000constant, and Alexander's type theorem for such mappings.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}