{"results":[{"id":"arxiv_2508.10151","title":"Sharp bounds for the valence of certain logharmonic polynomials","authors":[{"name":"Kirill Lazebnik"},{"name":"Erik Lundberg"}],"abstract":"Consider a logharmonic polynomial; that is, a product of the form $p(z)\\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner.","source":"arXiv","year":2025,"language":"en","subjects":["math.CV","math.DS"],"url":"https://arxiv.org/abs/2508.10151","pdf_url":"https://arxiv.org/pdf/2508.10151","is_open_access":true,"published_at":"2025-08-13T19:29:57Z","score":69},{"id":"ss_b8710bc1a38e0ba28956a71b250d0f30c78ba199","title":"Sharp bounds for the valence of certain logharmonic polynomials","authors":[{"name":"Kirill Lazebnik"},{"name":"Erik Lundberg"}],"abstract":"Consider a logharmonic polynomial; that is, a product of the form $p(z)\\overline{q(z)}$, where $p$, $q$ are holomorphic polynomials. Assume $q$ is linear and denote by $n$ the degree of $p$. It was recently shown in arXiv:2302.04339 [math.CV] that the valence of such a logharmonic polynomial is at most $3n-1$; in this paper we show that their $3n-1$ upper bound is sharp. Together with the work of arXiv:2302.04339 [math.CV], this resolves a conjecture of Bshouty and Hengartner.","source":"Semantic Scholar","year":2025,"language":"en","subjects":["Mathematics"],"doi":"10.1090/proc/17674","url":"https://www.semanticscholar.org/paper/b8710bc1a38e0ba28956a71b250d0f30c78ba199","is_open_access":true,"published_at":"","score":69},{"id":"crossref_10.1090/bull/1849","title":"Ray–Singer torsion, topological strings, and black holes","authors":[{"name":"Cumrun Vafa"}],"abstract":"Genus one amplitude for topological strings on Calabi–Yau 3-folds can be computed using mirror symmetry: The partition function at genus one gets mapped to a holomorphic version of Ray–Singer torsion on the mirror Calabi–Yau. On the other hand it can be shown by a physical argument that this gives a curvature squared correction term to the gravitational action. This in paticular leads to an effective quantum gravity cutoff known as the species scale, which varies over moduli space of Calabi–Yau manifolds. This resolves some of the puzzles associated to the entropy of small black holes when there are a large number of light species of particles. Thus Ray–Singer torsion, via its connection to topological strings at genus one, provides a measure of light degrees of freedom of four dimensional N = 2 \\mathcal {N}=2 supergravity theories.","source":"CrossRef","year":2024,"language":"en","subjects":null,"doi":"10.1090/bull/1849","url":"https://doi.org/10.1090/bull/1849","is_open_access":true,"citations":1,"published_at":"","score":68.03},{"id":"arxiv_2101.09872","title":"Chow's Theorem Revisited","authors":[{"name":"Carlos Martínez Aguilar"},{"name":"Alberto Verjovsky"}],"abstract":"We present a proof of Chow's theorem using two results of Errett Bishop retated to volumes and limits of analytic varieties. We think this approach suggested a long time ago in the beautiful book by Gabriel Stolzenberg, is very attractive and easier for students and newcomers to understand, also the theory presented here is linked to areas of mathematics that are not usually associated with Chow's theorem. Furthermore, Bishop's results imply both Chow's and Remmert-Stein's theorems directly, meaning that this approach is more economic and just as profound as Remmert-Stein's proof. At the end of the paper there is a comparison table that explains how Bishop's theorems generalize to several complex variables classical results of one complex variable.","source":"arXiv","year":2021,"language":"en","subjects":["math.AG","math.CV"],"url":"https://arxiv.org/abs/2101.09872","pdf_url":"https://arxiv.org/pdf/2101.09872","is_open_access":true,"published_at":"2021-01-25T03:13:03Z","score":65},{"id":"ss_b2c335327f043a07ea667bb1827684e7bfd54d0f","title":"Fatou and Julia like sets II","authors":[{"name":"K. S. Charak"},{"name":"Ashutosh Kumar Singh"},{"name":"Manish Kumar"}],"abstract":"This paper is a continuation of authors work: Fatou and Julia like sets, Ukranian Math. J., to appear/arXiv:2006.08308[math.CV](see [5]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic functions on an arbitrary domain, and establish some distinctive properties of these sets. The connectedness of the Julia like set is also proved.","source":"Semantic Scholar","year":2020,"language":"en","subjects":["Mathematics"],"doi":"10.2298/fil2108721c","url":"https://www.semanticscholar.org/paper/b2c335327f043a07ea667bb1827684e7bfd54d0f","pdf_url":"http://www.doiserbia.nb.rs/ft.aspx?id=0354-51802108721C","is_open_access":true,"citations":1,"published_at":"","score":64.03},{"id":"arxiv_2006.09095","title":"Fatou and Julia like sets II","authors":[{"name":"Kuldeep Singh Charak"},{"name":"Anil Singh"},{"name":"Manish Kumar"}],"abstract":"This paper is a continuation of authors work: Fatou and Julia like sets,Ukranian J. Math., to appear/arXiv:2006.08308[math.CV](see [4]). Here, we introduce escaping like set and generalized escaping like set for a family of holomorphic functions on an arbitrary domain, and establish some distinctive properties of these sets. The connectedness of the Julia like set is also proved.","source":"arXiv","year":2020,"language":"en","subjects":["math.CV"],"url":"https://arxiv.org/abs/2006.09095","pdf_url":"https://arxiv.org/pdf/2006.09095","is_open_access":true,"published_at":"2020-06-16T11:59:36Z","score":64},{"id":"ss_fbefe4473806fa7ae9279672606ca5ad659fb18c","title":"Roots of Random Polynomials","authors":null,"abstract":"","source":"Semantic Scholar","year":2020,"language":"en","subjects":null,"url":"https://www.semanticscholar.org/paper/fbefe4473806fa7ae9279672606ca5ad659fb18c","is_open_access":true,"published_at":"","score":64},{"id":"ss_74ef7f1d564409c65ca4c251484e4c0a5376025e","title":"Cauchy and Bergman projection, sharp gradient estimates and certain operator norm equalities","authors":[{"name":"Petar Melentijević"}],"abstract":"ABSTRACT We get sharp pointwise estimates for the gradient of Pf, where P is Bergman projection in terms of -norm of function f defined in . Using limiting argument we transfer this result to Cauchy projection on and hence, the optimal gradient estimates of solution of -problem, thus extending results from Kalaj, Vujadinović [Norm of the Bergman projection onto the Bloch space. J Oper Theory. 2015;73(1):113–126], Kalaj, Marković [Optimal estimates for the gradient of harmonic functions in the unit disk. Complex Anal Oper Theory. 2013;7:1167–1183], Melentijević [Norm of the Bergman projection onto the Bloch space with -invariant gradient norm. arXiv 1711.08719[math.CV]]. As corollaries we get the sharp gradient estimate of a function in Hardy and Bergman spaces and exact norms of Cauchy projection acting into the Bloch space equipped with several (quasi)-norms.","source":"Semantic Scholar","year":2019,"language":"en","subjects":["Mathematics"],"doi":"10.1080/17476933.2019.1574771","url":"https://www.semanticscholar.org/paper/74ef7f1d564409c65ca4c251484e4c0a5376025e","is_open_access":true,"citations":1,"published_at":"","score":63.03},{"id":"ss_eff35d852fdbd76b4946dffd82d15b6ec411abdc","title":"On Hölder continuity of solutions of the Beltrami equations on the boundary","authors":[{"name":"V. Ryazanov"},{"name":"R. Salimov"}],"abstract":"In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for the Beltrami equations. In a series of recent papers, under the study of the boundary value problems of Dirichlet, Hilbert, Neumann, Poincare and Riemann with arbitrary measurable boundary data for the Beltrami equations as well as for the generalizations of the Laplace equation in anisotropic and inhomogeneous media, it was applied the logarithmic capacity, see e.g. Gutlyanskii V., Ryazanov V., Yefimushkin A. On the boundary value problems for quasiconformal functions in the plane // Ukr. Mat. Visn. - 2015. - 12, no. 3. - P. 363-389; transl. in J. Math. Sci. (N.Y.) - 2016. - 214, no. 2. - P. 200-219; Gutlyanskii V., Ryazanov V., Yefimushkin A. On a new approach to the study of plane boundary-value problems // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. - 2017. - No. 4. - P. 12-18; Yefimushkin A. On Neumann and Poincare Problems in A-harmonic Analysis // Advances in Analysis. - 2016. - 1, no. 2. - P. 114-120; Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations in quasidisks // Ukr. Mat. Visn. - 2015. - 12, no. 2. - P. 190–209; transl. in J. Math. Sci. (N.Y.) - 2015. - 211, no. 5. - P. 646–659; Yefimushkin A., Ryazanov V. On the Riemann–Hilbert Problem for the Beltrami Equations // Contemp. Math. - 2016. - 667. - P. 299-316; Gutlyanskii V., Ryazanov V., Yakubov E., Yefimushkin A. On Hilbert problem for Beltrami equation in quasihyperbolic domains // ArXiv.org: 1807.09578v3 [math.CV] 1 Nov 2018, 28 pp. As well known, the logarithmic capacity of a set coincides with the so-called transfinite diameter of the set. This geometric characteristic implies that sets of logarithmic capacity zero and, as a consequence, measurable functions with respect to logarithmic capacity are invariant under mappings that are continuous by Hölder. That circumstance is a motivation of our research. Let \\(D\\) be a domain in the complex plane \\(\\mathbb C\\) and let \\(\\mu: D\\to\\mathbb C\\) be a measurable function with \\( |\\mu(z)| \\lt 1 \\) a.e. The equation of the form \\(f_{\\bar{z}}\\ =\\ \\mu(z) f_z \\) where \\( f_{\\bar z}={\\bar\\partial}f=(f_x+if_y)/2 \\), \\(f_{z}=\\partial f=(f_x-if_y)/2\\), \\(z=x+iy\\), \\( f_x \\) and \\( f_y \\) are partial derivatives of the function \\(f\\) in \\(x\\) and \\(y\\), respectively, is said to be a Beltrami equation. The function \\(\\mu\\) is called its complex coefficient, and \\( K_{\\mu}(z)=\\frac{1+|\\mu(z)|}{1-|\\mu(z)|}\\) is called its dilatation quotient. The Beltrami equation is said to be degenerate if \\({\\rm ess}\\,{\\rm sup}\\,K_{\\mu}(z)=\\infty\\). The existence of homeomorphic solutions in the Sobolev class \\(W^{1,1}_{\\rm loc}\\) has been recently established for many degenerate Beltrami equations under the corresponding conditions on the dilatation quotient \\(K_{\\mu}\\), see e.g. the monograph Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York, 2012 and the further references therein. The main theorem of the paper, Theorem 1, states that a homeomorphic solution \\( f:\\mathbb D\\to\\mathbb D \\) in the Sobolev class \\( W^{1,1}_{\\rm loc} \\) of the Beltrami equation in the unit disk \\(\\mathbb D\\) has a homeomorphic extension to the boundary that is Hölder continuous if \\(K_{\\mu}\\in L^1(\\Bbb D)\\) and, for some \\(\\varepsilon_0\\in(0,1)\\) and \\(C\\in[1,\\infty)\\), $$ \\sup\\limits_{\\varepsilon\\in(0,\\varepsilon_0)} \\int_{\\mathbb D\\cap D(\\zeta,\\varepsilon)}K_{\\mu}(z) dm(z) \\lt C \\qquad \\forall \\zeta \\in \\partial \\mathbb{D} $$ where \\(D(\\zeta,\\varepsilon)=\\left\\{z\\in{\\Bbb C}: |z-\\zeta| \\lt \\varepsilon\\right\\}\\).","source":"Semantic Scholar","year":2018,"language":"en","subjects":["Physics"],"doi":"10.37069/1683-4720-2018-32-11","url":"https://www.semanticscholar.org/paper/eff35d852fdbd76b4946dffd82d15b6ec411abdc","pdf_url":"http://dspace.nbuv.gov.ua/xmlui/bitstream/123456789/169129/1/11-Ryazanov.pdf","is_open_access":true,"published_at":"","score":62},{"id":"ss_1be3d2ca75cb590dfd7d22866d1dad3932033128","title":"Schmidt's subspace theorem for moving hypersurface targets","authors":[{"name":"N. T. Son"},{"name":"T. Tan"},{"name":"N. Thin"}],"abstract":"Abstract It was discovered that there is a formal analogy between Nevanlinna theory and Diophantine approximation. Via Vojta's dictionary, the Second Main Theorem in Nevanlinna theory corresponds to Schmidt's Subspace Theorem in Diophantine approximation. Recently, Cherry, Dethloff, and Tan ( arXiv:1503.08801v2 [math.CV] ) obtained a Second Main Theorem for moving hypersurfaces intersecting projective varieties. In this paper, we shall give the counterpart of their Second Main Theorem in Diophantine approximation.","source":"Semantic Scholar","year":2017,"language":"en","subjects":["Mathematics"],"doi":"10.1016/j.jnt.2017.10.008","url":"https://www.semanticscholar.org/paper/1be3d2ca75cb590dfd7d22866d1dad3932033128","pdf_url":"https://doi.org/10.1016/j.jnt.2017.10.008","is_open_access":true,"citations":9,"published_at":"","score":61.27},{"id":"ss_df68b8e7e6defe51df05ecb949149af250f7fba1","title":"The conformal dilatation and Beltrami forms over quadratic field extensions","authors":[{"name":"N. V. Ivanov"}],"abstract":"The paper is devoted to an algebraic analogue of a geometric approach to the classical notion of complex dilatation suggested in the paper arXiv:1701.06259 [math.CV] by the author. At the same time it provides an invariant version of this geometric approach. From the algebraic point of view it is only natural to work with a general field extension K/k of degree 2 instead of the fields of real and complex number (under the assumption that the characteristic is not equal to 2). Given a k-linear map between two K-vector spaces of dimension 1 over K, there are two natural measures of deviation of this map from being K-linear: its conformal dilatation, defined in terms of quadratic forms over k, and its Beltrami form, directly generalizing the classical complex dilatation. It turns out that these two measures are related in the same way as in the classical case. Working with a general field extension does not lead to any new difficulties compared to the classical case, but only clarifies the algebraic aspects of the theory.","source":"Semantic Scholar","year":2017,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/df68b8e7e6defe51df05ecb949149af250f7fba1","is_open_access":true,"citations":1,"published_at":"","score":61.03},{"id":"doaj_10.46298/epiga.2017.volume1.1521","title":"Finiteness of the space of n-cycles for a reduced (n − 2)-concave complex space","authors":[{"name":"Daniel Barlet"}],"abstract":"EPIGA, Volume 1 (2017), Nr. 5","source":"DOAJ","year":2017,"language":"","subjects":["Mathematics"],"doi":"10.46298/epiga.2017.volume1.1521","url":"https://epiga.episciences.org/1521/pdf","pdf_url":"https://epiga.episciences.org/1521/pdf","is_open_access":true,"published_at":"","score":61},{"id":"arxiv_1403.2858","title":"Computational analytic continuation","authors":[{"name":"Stefan Kranich"}],"abstract":"Holomorphic functions are amazing because their values in an ever so small disk in the complex plane completely determine the function values at arbitrary points in their maximum possible domain. The process of extending such a function beyond its initial domain is called analytic continuation. We attempt to make this theoretic result tractable by computers. In the present article, we first prove that any algorithm for analytic continuation can generally not depend on finitely many function values only, without closer inspection of the function itself. We then derive a computable local bound on the step size between sampling points which yields an algorithm for analytic continuation of complex plane algebraic curves. Finally, we provide a numerical example demonstrating its practical use.","source":"arXiv","year":2014,"language":"en","subjects":["math.CV","math.AG"],"url":"https://arxiv.org/abs/1403.2858","pdf_url":"https://arxiv.org/pdf/1403.2858","is_open_access":true,"published_at":"2014-03-12T09:20:49Z","score":58},{"id":"arxiv_1302.1261","title":"Second main theorem and unicity of meromorphic mappings for hypersurfaces of projective varieties in subgeneral position","authors":[{"name":"Si Duc Quang"}],"abstract":"The purpose of this article is twofold. The first is to prove a second main theorem for meromorphic mappings of $\\C^m$ into a complex projective variety intersecting hypersurfaces in subgeneral position with truncated counting functions. The second is to show a uniqueness theorem for these mappings which share few hypersurfaces without counting multiplicity.","source":"arXiv","year":2013,"language":"en","subjects":["math.CV"],"url":"https://arxiv.org/abs/1302.1261","pdf_url":"https://arxiv.org/pdf/1302.1261","is_open_access":true,"published_at":"2013-02-06T04:24:10Z","score":57},{"id":"ss_9665f01cd0d24e9d0a777bf686a18efd89e67728","title":"Geometry Behind Chordal Loewner Chains","authors":[{"name":"Manuel D. Contreras"},{"name":"S. Díaz-Madrigal"},{"name":"P. Gumenyuk"}],"abstract":"Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach’s conjecture and well-celebrated Schramm’s stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv 0807.1594; Bracci F et al. in Math Ann 344:947–962, 2009; Contreras MD et al. in Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al. in Complex Analysis. PWN, Warsaw, pp 7–32, 1979; Bauer RO in J Math Anal Appl 302: 484–501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209–1217, 1992).","source":"Semantic Scholar","year":2010,"language":"en","subjects":["Mathematics"],"doi":"10.1007/S11785-010-0057-6","url":"https://www.semanticscholar.org/paper/9665f01cd0d24e9d0a777bf686a18efd89e67728","pdf_url":"http://arxiv.org/pdf/1002.0826.pdf","is_open_access":true,"citations":10,"published_at":"","score":54.3},{"id":"ss_c75bcc9b36c9464d61bd04634073c02ffe11a5c4","title":"Holomorphic curves in complex spaces","authors":[{"name":"Barbara Drinovec-Drnovšek"},{"name":"F. Forstnerič"}],"abstract":"We study the existence of topologically closed complex curves normalized by bordered Riemann surfaces in complex spaces. Our main result is that such curves abound in any noncompact complex space admitting an exhaustion function whose Levi form has at least two positive eigenvalues at every point outside a compact set, and this condition is essential. The proof involves a lifting method for the boundary of the curve and a newly developed technique of gluing holomorphic sprays over Cartan pairs in Stein manifolds whose value lie in a complex space, with control up to the boundary of the domains. (The latter technique is also exploited in the subsequent papers math.CV/0607185 and math.CV/0609706.) We also prove that any compact complex curve with C^2 boundary in a complex space admits a basis of open Stein neighborhoods. In particular, an embedded disc of class C^2 with holomorphic interior in a complex manifold admits a basis of open polydisc neighborhoods.","source":"Semantic Scholar","year":2006,"language":"en","subjects":["Mathematics"],"doi":"10.1215/S0012-7094-07-13921-8","url":"https://www.semanticscholar.org/paper/c75bcc9b36c9464d61bd04634073c02ffe11a5c4","pdf_url":"https://arxiv.org/pdf/math/0604118v1.pdf","is_open_access":true,"citations":77,"published_at":"","score":52.31},{"id":"ss_13f711c46ab84101ef2d3024cbb3b82f841e7ca9","title":"The Oka principle for sections of stratified fiber bundles","authors":[{"name":"F. Forstnerič"}],"abstract":"A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces.","source":"Semantic Scholar","year":2007,"language":"en","subjects":["Mathematics"],"doi":"10.4310/PAMQ.2010.V6.N3.A11","url":"https://www.semanticscholar.org/paper/13f711c46ab84101ef2d3024cbb3b82f841e7ca9","pdf_url":"https://www.intlpress.com/site/pub/files/_fulltext/journals/pamq/2010/0006/0003/PAMQ-2010-0006-0003-a011.pdf","is_open_access":true,"citations":39,"published_at":"","score":52.17},{"id":"ss_696f7ab66b7a50fda9c3413168a7453eac658267","title":"Moduli of Continuity of Quasiregular Mappings","authors":[{"name":"V. Manojlović"}],"abstract":"This thesis consists of Chapters 1 and 2. The main results are contained in the two preprints and two published papers, listed below. Chapter 1 deals with conformal invariants in the euclidean space Rn; n \u003e= 2; and their interrelation. In particular, conformally invariant metrics and balls of the respective metric spaces are studied. Another theme in Chapter 1 is the study of quasiconformal maps with identity boundary values in two diferent cases, the unit ball and the whole space minus two points. These results are based on the two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion of two point normalized quasiconformal mappings, arXiv:0808.1219[math.CV], 13 pp., V. Manojlovic and M. Vuorinen: On quasiconformal maps with identity boundary values, arXiv:0807.4418[math.CV], 16 pp. Chapter 2 deals with harmonic quasiregular maps. Topics studied are: Preservation of modulus of continuity, in particular Lipschitz continuity, from the boundary to the interior of domain in case of harmonic quasiregular maps and quasiisometry property of harmonic quasiconformal maps. Chapter 2 is based mainly on the two published papers: M. Arsenovic, V. Kojic and M. Mateljevic: On Lipschitz continuity of harmonic quasiregular maps on the unit ball in Rn., Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 315-318. V. Kojic and M. Pavlovic: Subharmonicity of jfjp for quasiregular harmonic functions, with applications, J. Math. Anal. Appl. 342 (2008) 742-746","source":"Semantic Scholar","year":2008,"language":"en","subjects":["Mathematics"],"url":"https://www.semanticscholar.org/paper/696f7ab66b7a50fda9c3413168a7453eac658267","is_open_access":true,"citations":5,"published_at":"","score":52.15},{"id":"ss_38753d8588aebdda89d4f8b9af9c7b8a0bab66ee","title":"Oka's principle for holomorphic submersions with sprays","authors":[{"name":"F. Forstnerič"},{"name":"Jasna Prezelj"}],"abstract":"We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromov's result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201.","source":"Semantic Scholar","year":2001,"language":"en","subjects":["Mathematics"],"doi":"10.1007/s002080100249","url":"https://www.semanticscholar.org/paper/38753d8588aebdda89d4f8b9af9c7b8a0bab66ee","pdf_url":"http://arxiv.org/pdf/math/0101040","is_open_access":true,"citations":65,"published_at":"","score":51.95}],"total":1103858,"page":1,"page_size":20,"sources":["CrossRef","DOAJ","arXiv","Semantic Scholar"],"query":"math.CV"}