Holomorphic Flexibility Properties of Complements and Mapping Spaces
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Generalized Polar Varieties and an Efficient Real Elimination Procedure1
KYBERNETIKA — VOLUME 4 0 (2004), NUMBER 5, PAGES 519-550 GENERALIZED POLAR VARIETIES AND AN EFFICIENT REAL ELIMINATION PROCEDURE1 BERND BANK, MARC GIUSTI, JOOS HEINTZ AND LUIS M. PARDO Dedicated to our friend František Nožička. Let V be a closed algebraic subvariety of the n-dimensional projective space over the com plex or real numbers and suppose that V is non-empty and equidimensional. In this paper we generalize the classic notion of polar variety of V associated with a given linear subva riety of the ambient space of V. As particular instances of this new notion of generalized polar variety we reobtain the classic ones and two new types of polar varieties, called dual and (in case that V is affine) conic. We show that for a generic choice of their parame ters the generalized polar varieties of V are either empty or equidimensional and, if V is smooth, that their ideals of definition are Cohen-Macaulay. In the case that the variety V is affine and smooth and has a complete intersection ideal of definition, we are able, for a generic parameter choice, to describe locally the generalized polar varieties of V by explicit equations. Finally, we use this description in order to design a new, highly efficient elimination procedure for the following algorithmic task: In case, that the variety V is Q-definable and affine, having a complete intersection ideal of definition, and that the real trace of V is non-empty and smooth, find for each connected component of the real trace of V a representative point. -
Cones in Complex Affine Space Are Topologically Singular
CONES IN COMPLEX AFFINE SPACE ARE TOPOLOGICALLY SINGULAR DAVID PRILL I. Introduction. A point of a complex analytic variety will be called a topological regular point (TRP), resp. an analytical regular point (ARP), if it has an open neighborhood which is homeomorphic, resp. biholomorphic, to an open set in a finite-dimensional complex vector space. This paper shows one type of TRP is an ARP. A subvariety of C", the w-dimensional complex vector space, 0<w< °°, is called a cone if it is a union of one-dimensional linear subspaces of C™. Let 0 be the zero vector in C". We shall prove the following: Theorem. If 0 is a TRP of a cone, then the cone is a linear subspace. Thus, for cones: If 0 is a TRP, then it is an ARP. The idea of the proof is as follows: For XCCn, let X* = X- {o}. The map p: (C)*—>CPn_1 onto (w—1)-dimensional complex projec- tive space is the projection map of a principal C*-bundle. If V is a cone, V* is a subbundle of (C™)*. Propositions 1 and 2 derive topo- logical properties of the subbundle V* from the assumption that 0 is a TRP of V. These properties guarantee p(V*) is a projective variety of order one. It is classical that if p(V*) is irreducible and of order one, then V is a linear space. The lemma preceding the proof of the theorem enables us to avoid any assumptions of irreducibility. In contrast to our result, 0 is a TRP and not an ARP for the n- dimensional variety {(zi, • • • ,xn+1) G C"+1| x\ = xl\. -
Arxiv:1003.6025V1 [Math.HO] 31 Mar 2010 Karl Stein (1913-2000)
Karl Stein (1913-2000) Alan Huckleberry Karl Stein was born on the first of January 1913 in Hamm in Westfalen, grew up there, received his Abitur in 1932 and immediately thereafter be- gan his studies in M¨unster. Just four years later, under the guidance of Heinrich Behnke, he passed his Staatsexam, received his promotion and became Behnke’s assistant. Throughout his life, complex analysis, primarily in higher dimensions (“mehrere Ver¨anderliche”), was the leitmotif of Stein’s mathematics. As a fresh Ph.D. in M¨unster in 1936, under the leadership of the master Behnke, he had already been exposed to the fascinating developments in this area. The brilliant young Peter Thullen was proving fundamental theorems, Henri Cartan had visited M¨unster, and Behnke and Thullen had just writ- ten the book on the subject. It must have been clear to Stein that this was the way to go. Indeed it was! The amazing phenomenon of analytic continuation in higher dimensions had already been exemplified more than 20 years be- fore in the works of Hartogs and E. E. Levi. Thullen’s recent work had gone much further. In the opposite direction, Cartan and Thullen had arXiv:1003.6025v1 [math.HO] 31 Mar 2010 proved their characterization of domains in Cn which admit a holomor- phic function which can not be continued any further. Behnke himself was also an active participant in mathematics research, always bringing new ideas to M¨unster. This was indeed an exciting time for the young researcher, Karl Stein. Even though the pest of the Third Reich was already invading academia, Behnke kept things going for as long as possible. -
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗ We must always have old memories and young hopes Abstract This paper analyzes the development of the theory of Riemann surfaces and complex spaces, with emphasis on the work of Rie- mann, Klein and Poincar´e in the nineteenth century and on the work of Behnke-Stein and Cartan-Serre in the middle of this cen- tury. R´esum´e Cet article analyse le d´eveloppement de la th´eorie des surfaces de Riemann et des espaces analytiques complexes, en ´etudiant notamment les travaux de Riemann, Klein et Poincar´eauXIXe si`ecle et ceux de Behnke-Stein et Cartan-Serre au milieu de ce si`ecle. Table of Contents 1. Riemann surfaces from 1851 to 1912 1.1. Georg Friedrich Bernhard Riemann and the covering principle 1.1∗. Riemann’s doctorate 1.2. Christian Felix Klein and the atlas principle 1.3. Karl Theodor Wilhelm Weierstrass and analytic configurations AMS 1991 Mathematics Subject Classification: 01A55, 01A60, 30-03, 32-03 ∗Westf¨alische Wilhelms–Universit¨at, Mathematisches Institut, D–48149 Munster,¨ Deutschland This expos´e is an enlarged version of my lecture given in Nice. Gratias ago to J.-P. Serre for critical comments. A detailed exposition of sections 1 and 2 will appear elsewhere. SOCIET´ EMATH´ EMATIQUE´ DE FRANCE 1998 204 R. REMMERT 1.4. The feud between G¨ottingen and Berlin 1.5. Jules Henri Poincar´e and automorphic functions 1.6. The competition between Klein and Poincar´e 1.7. Georg Ferdinand Ludwig Philipp Cantor and countability of the topology 1.8. -
The Cousin Problems and the Emergence of the Sheaf Concept Author(S): Renaud Chorlay Source: Archive for History of Exact Sciences, Vol
From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept Author(s): Renaud Chorlay Source: Archive for History of Exact Sciences, Vol. 64, No. 1 (January 2010), pp. 1-73 Published by: Springer Stable URL: http://www.jstor.org/stable/41342411 Accessed: 23-05-2017 00:50 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://about.jstor.org/terms Springer is collaborating with JSTOR to digitize, preserve and extend access to Archive for History of Exact Sciences This content downloaded from 132.248.9.8 on Tue, 23 May 2017 00:50:12 UTC All use subject to http://about.jstor.org/terms Arch. Hist. Exact Sci. (2010) 64:1-73 DOI 10. 1007/s00407-009-0052-3 From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept Renaud Chorlay Received: 19 November 2008 / Published online: 30 July 2009 © Springer- Verlag 2009 Abstract Historical work on the emergence of sheaf theory has mainly concen- trated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. -
4-Manifolds with Indefinite Intersection Form
4-MANIFOLDS WITH INDEFINITE INTERSECTION FORM S.K. Donaldson All Souls College Oxford, England In writing up this lecture I shall not concentrate so much on des- cribing problems of 4-manifold topology; instead I shall explain how a simple topological construction has applications in two different direc- tions. First I will recall that, just as bundles over a single space have homotopy invariants, so do families of bundles, and that these define cor- responding invariants in families of connections. Next I will sketch the way in which such a topological invariant, when endowed with a geometric realisation, becomes important for studying holomorphic bundles over al- gebraic varieties. Last I will indicate how this same homotopy invariant of families of connections, combined with arguments involving moduli spa- ces of self-dual connections over a Riemannian 4-manifold, gives restric- tions on the possible homotopy types of smooth 4-manifolds and I will speculate on possible future progress in this area. Topology of bundles. This is standard material that may be found in [2] for example. Con- sider a fixed manifold X and a family of bundles over X parametrised by some auxiliary space T , so we have a bundle P over the product X × T with structure group G (compact and connected, say). Take first the case when T is a point so we have a single bundle over X , deter- mined up to equivalence by a homotopy class of maps from X to BG . This may be non-trivial, detected for example by characteristic classes in the cohomology of X . -
Contributions to the Theory of Density Properties for Stein Manifolds
Contributions to the theory of density properties for Stein manifolds Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Alexandre Ramos-Peon | downloaded: 23.9.2021 von Mexiko Leiter der Arbeit: Prof. Dr. F. Kutzschebauch Mathematisches Institut der Universität Bern https://doi.org/10.7892/boris.99221 source: Contributions to the theory of density properties for Stein manifolds Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Alexandre Ramos-Peon von Mexiko Leiter der Arbeit: Prof. Dr. F. Kutzschebauch Mathematisches Institut der Universität Bern Von der Philosophisch-naturwissenschaftlichen Fakultät angenommen. Bern, 27. Mai 2016. Der Dekan: Prof. Dr. G. Colangelo Contents Introduction 1 Main results . .3 Acknowledgments . .7 1 Preliminary notions & context 8 1.1 Flows of holomorphic vector fields . .8 1.2 Automorphisms of complex affine space . 11 1.2.1 Shears on Cn and the Andersén-Lempert theorem . 11 1.2.2 Isotopic and parametric Andersén-Lempert theorems . 15 1.2.3 The push-out method: compositions of automorphisms . 17 1.3 Stein manifolds . 19 1.3.1 Sheaf cohomology and applications . 21 1.3.2 Remarks on holomorphic volume forms . 24 1.4 Density properties . 26 1.4.1 Some consequences of the density property . 28 1.4.2 Known examples . 30 1.4.3 Algebraic criteria . 33 1.5 Oka theory . 36 1.5.1 Historical Oka-Grauert principle . 36 1.5.2 Gromov’s ellipticity and Oka manifolds . 37 2 An Oka principle for a parametric infinite transitivity property 41 2.1 Summary of results . 41 2.2 A parametric Andersén-Lempert theorem . -
Arxiv:1907.08545V3 [Math.CO] 4 Mar 2021
POSITIVELY HYPERBOLIC VARIETIES, TROPICALIZATION, AND POSITROIDS FELIPE RINCÓN, CYNTHIA VINZANT, AND JOSEPHINE YU Abstract. A variety of codimension c in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension c. Positively hyperbolic hypersurfaces are defined by stable polynomials. We give a new characterization of positively hyperbolic varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. We prove that positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type A hyperplane arrangement defined by xi = xj, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of some results of Choe–Oxley–Sokal–Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we also give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans. 1. Introduction There are many beautiful appearances of real rooted-ness in combinatorics, see e.g. [Brä15, Vis12]. Hyperbolicity and stability are generalizations of real rooted-ness for multivariate polynomials. A polynomial f 2 C[x1; : : : ; xn] is called stable if f(z) 6= 0 for every point n n z 2 C with Im(z) 2 R+. Here Im(z) = (Im(z1);:::; Im(zn)) denotes the imaginary part of z, and R+ denotes the set of positive real numbers. -
Riemann Surfaces Lecture Notes
Introduction to Riemann Surfaces Lecture Notes Armin Rainer 0 0 1 dim H (X; L−D) − dim H (X; LD) = 1 − g − deg D June 17, 2018 PREFACE i Preface These are lecture notes for the course Riemann surfaces held in Vienna in Spring 2018 (three semester hours). The presentation is primarily based on the book [4] which is followed quite closely. Also [13] had some influence. Apart from some familiarity with basic complex analysis, general topology, and basic algebra no other prerequisites are demanded. All necessary tools will be developed when needed. Riemann surfaces were originally conceived in complex analysis in order to deal with multivalued functions. The analytic continuation of a given holomorphic function element along different paths leads in general to different branches of that function. Riemann replaced the domain of the function by a multiple-sheeted covering of the complex plane to get a single valued function on the covering space. Abstract Riemann surfaces are by definition connected complex one- dimensional manifolds. They are the natural domains of definitions of holomorphic functions in one variable. In chapter 1 we introduce Riemann surfaces and discuss basic properties. We develop the fundamentals of the theory of topological covering spaces including the fundamental group, the universal covering, and deck transformations. It will turn out that non-constant holomorphic maps between Riemann surfaces are covering maps, possibly with branch points. In chapter 2 we get acquainted with the language of sheaves. It proves very useful in the construction of Riemann surfaces which arise from the analytic con- tinuation of germs of holomorphic functions. -
Seminar on Fundamentals of Algebraic Geometry I
SEMINAR ON FUNDAMENTALS OF ALGEBRAIC GEOMETRY I HONGˆ VANˆ LEˆ ∗ AND PETR SOMBERG ∗∗ Abstract. Algebraic geometry is one of the central subjects of mathematics. Mathematical physicists, homotopy theorists, com- plex analysts, symplectic geometers, representation theorists speak the language of algebraic geometry. In this seminar we shall discuss some basic topics of algebraic geometry and their relation with current problems in mathematics. Recommended textbook: J. Harris, Algebraic Geometry: A First Course, I. R. Shafarevich: Basic Algebraic Geometry 1,2, Dol- gachev: Introduction to Algebraic Geometry (lecture notes) Contents 1. Overview and proposed topics 2 1.1. Overview of our plan 2 1.2. A brief history 3 1.3. Tentative plan of our seminar 4 2. Algebraic sets and the Hilbert basic theorem 4 2.1. Algebraic sets 5 2.2. The Hilbert basic theorem 5 3. Hilbert’s Nullstellensatz and Zariski topology 6 3.1. Hilbert’s Nullstellensatz 7 3.2. Zariski topology 9 4. Tangent space - a motivation + definition ... 10 4.1. Tangent space and the ring of dual numbers 11 4.2. Tangent space and the Hensel’s Lemma 12 4.3. Hensel’s lemma for algebraic varieties of dimension 0 13 5. The Hilbert basis theorem and Groebner bases 14 5.1. Groebner basis and Buchberger’s algorithm 14 5.2. Some applications of Groebner bases 16 ∗ Institute of Mathematics of ASCR, Zitna 25, 11567 Praha 1, email: [email protected] ∗∗ Mathematical Institute, Charles University, Sokolovska 83, 180 00 Praha 8, Czech Republic, email: [email protected]ff.cuni.cz. 1 2 HONGˆ VANˆ LEˆ ∗ AND PETR SOMBERG ∗∗ 6. -
Sheaf Cohomology and Its Applications in Analytic Geometry
Sheaf Cohomology and its Applications in Analytic Geometry by Jean Michel Menjanahary ([email protected]) 17th May 2020 A THESIS SUBMITTED TO CENTRAL EUROPEAN UNIVERSITY IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF A MASTER OF SCIENCE IN MATHEMATICS AND ITS APPLICATIONS CEU eTD Collection Declaration As a work carried out at Central European University in partial fulfilment of the requirements for a Master of Science in Mathematics, I hereby declare that the work contained in this thesis is my original work. The work done by others has been acknowledged and referenced accordingly. Student: Jean Michel Menjanahary Supervisor: Prof. András Némethi CEU eTD Collection i Acknowledgments I would like to appreciate the CEU founder and philanthropist, George Soros and the CEU community, who trusted and gave me the opportunity to continue my studies at CEU. My appreciation goes particularly to our head of department, Prof Károly Böröczky and our MSc program director, Prof Pál Hegedüs. They helped me a lot to join this program. I am so grateful for all that they have done for me. Many thanks also to our department coordinators Elvira Kadvány and Melinda Balázs for their great helps during my stay in Budapest. I would also like to express my thanks to my supervisor Prof András Némethi. He agreed to supervise me on this wonderful topic. He worked a lot in guiding and teaching me throughout the academic year. He also helped me a lot to write this work in a better way. I am really thankful to him. Finally, it has been a wonderful and fun time with my Malagasy friends Manana, Mahenina, Mahefa, Ny Aina and Masera Vicky in Budapest for two years. -
On the Geometry of Polar Varieties1
On the geometry of polar varieties1 B. Bank 2, M. Giusti 3, J. Heintz 4, M. Safey El Din 5, E. Schost 6 April 28, 2009 Abstract The aim of this paper is a comprehensive presentation of the geometrical tools which are necessary to prove the correctness of several up to date algorithms with intrinsic complexity bounds for the problem of real root finding in (mainly) smooth real algebraic varieties given by reduced regular sequences of polynomial equations (see [4, 5, 6, 7], [30] and [31]). The results exposed in this paper form also the geometrical main ingredients for the computational treatment of singular hypersurfaces (see [8]). In particular, we show the non–emptiness of suitable fully generic dual polar varieties of (possibly singular) real varieties, show that fully generic polar varieties may become singular at smooth points of the original variety, give a sufficient criterion when this is not the case, give an intrinsic degree estimate for polar varieties and introduce the new concept of a sufficiently generic polar variety. Our statements are illustrated by examples and a computer experiment. Keywords: Real polynomial equation solving; singularities; classic polar varieties; dual polar varieties; fully generic polar varieties; sufficiently generic polar varieties MSC: 14P05, 14B05, 14Q10, 14Q15, 68W30 1Research partially supported by the following Argentinian, Canadian, French and Spanish agencies and grants: UBACYT X-098, UBACYT X-113, PICT–2006–02067, the Canada Re- search Chair Program, NSERC, BLAN NT05-4-45732 (projet GECKO), MTM 2007-62799. 2Humboldt-Universit¨at zu Berlin, Institut f¨ur Mathematik, D–10099 Berlin, Germany. [email protected] 3CNRS, Ecole´ Polytechnique, Laboratoire LIX, F–91228 Palaiseau Cedex, France.