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Topic: Complex manifold

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Hodge theory

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	PlanetMath: complex analytic manifold
		Examples of complex analytic manifolds are for example the Stein manifolds or the Riemann surfaces.
		Complex analytic manifolds can also be considered as a special case of CR manifolds where the CR dimension is maximal.
		This is version 2 of complex analytic manifold, born on 2005-02-22, modified 2005-03-05.
	planetmath.org /encyclopedia/ComplexAnalyticManifold.html (221 words)

	Lie group
		In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps.
		Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures.
		The set of all vector fields on an analytic manifold is a Lie algebra over F.
	www.sciencedaily.com /encyclopedia/lie_group (1429 words)

	[No title]
		Given a vector space of functions of a parameter or functions on a manifold, an operator may have a kernel or matrix whose rows and columns are indexed by the parameter or by points on the manifold.
		Orbifolds are manifolds with singularities such as reflection surfaces, where they resemble manifolds with boundary, and cone lines, where they are modelled (in the direction perpendicular to the cone line) by a cone with an angle of 360/n degrees for some n.
		PL flow A "piecewise linear" motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	UC Davis Math: Glossary (Site not responding. Last check: 2007-09-17)
		A manifold with the property that each tangent space has the structure of a complex vector space, but the complex structures are not necessarily compatible with true complex coordinates as they are for a complex manifold.
		A motion on a space or a manifold, akin to a flow given by a vector field, in which every particle in a given simplex of some triangulation moves with constant velocity and in the same direction, so that the particle trajectories are polygons.
		A surface with a conformal structure; a complex manifold with one complex dimension.
	www.math.ucdavis.edu /profiles/glossary.html (9932 words)

	PlanetMath: Stein manifold
		This is a generalization of the concept of the domain of holomorphy to manifolds.
		Further Stein manifolds are the generalizations of Riemann surfaces in higher dimensions.
		This is version 2 of Stein manifold, born on 2005-02-22, modified 2005-03-07.
	planetmath.org /encyclopedia/SteinManifold.html (107 words)

	Présentation (Site not responding. Last check: 2007-09-17)
		Complex Analysis and Analytic Geometry belong closely together and are one of the few fields in the center of pure mathematics with many applications to other areas of pure mathematics (algebraic geometry, differential geometry, dynamical systems, P.D.E., topology, number theory, etc.) and applied Mathematics (theoritical physics, geophysics, mathematical economy, tomography).
		Complex Monge-Ampère equation and canonical bundle of a complex manifold.
		Differential geometry of vector bundles on complex manifolds: complex and holomorphic structures, invariants and singularities, gauge theories and moduli, twistor spaces.
	www.math.jussieu.fr /projets/ac/Reseau/presentation.htm (2291 words)

	Introduction to Complex Manifolds (Site not responding. Last check: 2007-09-17)
		Complex manifolds play a great role in many areas of modern mathematics.
		Complex manifold are of importance also in the transcendental approach to algebraic geometry in higher dimension; a pioneer here was Hodge.
		A basic question is then to find conditions for a complex manifold to admit imbedding in a complex projective space of sufficiently high dimension -- an answer to this question is provided by the Kodaira imbedding theorem.
	www.maths.lth.se /matematiklu/personal/jaak/Complex-Manifolds.html (218 words)

	M&W Custom Hydraulic Manifolds
		Depending on the quantity, size and complexity of the manifold, multiple parts may be machined on multiple sides of the fixturing that accurately secures the manifold relative to the known reference system within the machining center.
		Manifolds are placed in the wash racks to facilitate liquid flow through the manifold for maximum cleaning action.
		The assembled manifolds are then packaged as required and correctly labeled externally with special MandW computer tags showing job number, p/n, qty, ship date requested, name of individual doing the packaging and count, the number of packages in the batch, and finally the customer name.
	www.mandwmfg.com /faq.htm (7914 words)

	[No title]
		It may be phr* ased by saying that on a Stein manifold* (complex submanifold of Euclidean space), an* *alytic problems of a cohomological nature have only topological obstructions.
		C(S; X) is a weak equivalence for all Stein manifolds S, where the spaces of holomorphi* *c and continuous maps from S to X carry the compact-open topology.
		Note, finally, that a nondiscrete complex manifold X is never flabby, let al* *one coarsely or finely fibrant.
	hopf.math.purdue.edu /Larusson/excision.txt (2966 words)

	Lie Balls and Relativistic Quantum Fields (Site not responding. Last check: 2007-09-17)
		D is in particular a complex manifold with (integrable) complex structure j0.
		It is also a non compact Hermitian homogeneous manifold for the action of the conformal group S0(4, 2) of Space-Time and is of rank two as a homogeneous space.
		What we suggest here is a kind of different game: analytic continuation from the real line to the complex plane, with its flat euchdean geometry, is not the same as going from the real line (or from the circle) to the Poincare upper half-plane (or to the disk), with its curved Lobatchevskian geometry.
	quantumfuture.net /quantum_future/lieballs.html (800 words)

	Symplectic geometry seminar (Site not responding. Last check: 2007-09-17)
		Abstract: An almost complex manifold is a manifold M with a complex structure J on the fibers of the tangent bundle TM.
		We explore the geometry of almost complex manifolds by means of model theory.
		In model theory, a "structure" is an infinite set D together with a collection of subsets of D^n closed under intersections, complements, projections and their inverses, and containing the diagonals.
	www.math.toronto.edu /symplec/fall02/sem120202.html (153 words)

	Calabi-Yau manifold (Site not responding. Last check: 2007-09-17)
		In mathematics, a Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class.
		The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau in 1977 and became Yau's theorem.
		In one complex dimension, the only examples are family of tori.
	www.worldhistory.com /wiki/C/Calabi-Yau-manifold.htm (379 words)

	Abstracts of the Research Papers (Site not responding. Last check: 2007-09-17)
		It is also shown that a Fueter structure on a smooth manifold defines a foliation with a canonical complex structure on leaves.
		On an almost complex manifold with an arbitary metric a (1,2)-tensor A is defined.
		A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.
	math.iisc.ernet.in /~dattab/wwwabs.html (1960 words)

	Real and Complex Manifolds (Site not responding. Last check: 2007-09-17)
		A real manifold (or smooth manifold) is a topological space which is locally homeomorphic to real n-dimensional space and whose transition functions are differentiable.
		A complex manifold is locally like an open set in complex n-dimensional space with holomorphic transition functions.
		For example, the set of all complex structures on a given real manifold varies continuously and can itself be given the structure of a complex manifold.
	www.math.harvard.edu /~deepee/tutorial03/tutorial2.html (336 words)

	Papers by M. Verbitsky
		We obtain that a complex manifold of hyperkaehler type is mirror dual to itself.
		In this paper the instanton equation is being generalized to hyperkaehler manifolds of arbitrary dimension.
		If the complex structure on this manifold is "generic enough" in its deformation space (we describe the sufficient conditions explicitly) these subvarieties turn out to be holomorphically symplectic as well.
	imperium.lenin.ru /~verbit/Math.HTML (1123 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		It is simultaneously a complex manifold, an algebraic
		A surface with a conformal structure; a complex manifold with one complex
		A subset of a manifold that is itself a manifold.
	www.mcs.vuw.ac.nz /~hong/defs.html (4965 words)

	Complex geometry (Site not responding. Last check: 2007-09-17)
		A manifold with this property is called an almost complex manifold.
		When it satisfies a certain integrability condition it is possible that the manifold is isomorphic to a complex manifold, i.e.
		A Kähler manifold which is in addition Ricci-flat, that is the Ricci tensor vanishes, is called a Calabi-Yau manifold.
	www.phys.uu.nl /~hofman/scriptie/duality/node53.html (658 words)

	RESEARCH in DIFFERENTIAL GEOMETRY
		Kähler manifolds (a class of complex manifolds which includes algebraic manifolds) possess a differential 2-form, w, which is non-degenerate (i.e., wÙw is never 0), and closed (i.e., dw = 0).
		This was very hopeful because all (real 4-dimensional) locally complex 2-manifolds, including the algebraic surfaces, were somewhat classified by K. Kodaira in the 1960’s using the Kodaira dimension, k.
		), is a differentiable invariant of the manifold.
	facpub.stjohns.edu /~watsonw/diffgeom.htm (1548 words)

	Amazon.com: Books: Principles of Algebraic Geometry (Site not responding. Last check: 2007-09-17)
		Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds.
		This book would probably be one the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers.
		The complex analytic point of view however is the best way of learning the material from a practical point of view, and mastery of this book will pave the way for indulging oneself in its many applications.
	www.amazon.com /exec/obidos/tg/detail/-/0471050598?v=glance (2463 words)

	week195
		When O is a 6-dimensional spin manifold with a nonzero covariantly constant spinor field, its spinor bundle becomes an octonion bundle, while its tangent bundle plus a trivial line bundle becomes an imaginary octonion bundle.
		When P is a 5-dimensional spin manifold with a nonzero covariantly constant spinor field, its spinor bundle becomes an octonion bundle, while its tangent bundle plus two trivial line bundles becomes an imaginary octonion bundle.
		It sounds like he's saying that a Kaehler manifold is a *complex* manifold for which J is preserved by parallel transport.
	math.ucr.edu /home/baez/week195.html (5401 words)

	MPI MIS Leipzig - Preprint Nr. 63/1999
		A special symplectic manifold is then defined as a special complex manifold together with a
		Locally, any special complex manifold is realised as the image of a local holomorphic 1-form
		Finally, we discuss the natural geometric structures on the cotangent bundle of a special symplectic manifold, which are generalisations of the known hyper-Kähler structure on the cotangent bundle of a special Kähler manifold.
	www.mis.mpg.de /preprints/1999/prepr6399-abstr.html (193 words)

	[No title]
		The versions of elliptic cohom constructed by Hopkins and Miller, which are now being called rings of topological modular forms and one of which is the proper target for the Witten genus, follow this pattern.
		According to a friend who studies elliptic operators on manifolds, the KO-theory orientation class is, in an appropriate setting, represented by the Dirac operator.
		More precisely, we've got a spectrum MU for complex cobordism theory, and we've got a spectrum MSO for oriented cobordism, but localizing away from 2 MU becomes isomorphic to MSO wedged with the double suspension of MSO.
	www.math.niu.edu /~rusin/known-math/00_incoming/ellip_coho (735 words)

	CATHOLIC ENCYCLOPEDIA: Nature
		It will be convenient to reduce to two classes the various meanings of the term nature according as it applies to the natures of individual beings or to nature in general.
		In an individual being, especially if its constitutive elements and its activities are manifold and complex, the term nature is sometimes applied to the collection of distinctive features, original or acquired, by which such an individual is characterized and distinguished from others.
		For their development out of a primitive germ they require the co-operation of many external factors, yet they have within themselves the principle of activity by which external substances are elaborated and assimilated.
	www.newadvent.org /cathen/10715a.htm (1788 words)

	Existence of close pseudoholomorphic disks for almost complex manifolds and an application to Kobayashi-Royden ... (Site not responding. Last check: 2007-09-17)
		Abstract: It is proved in the paper 1 that near every pseudoholomorphic disk on an almost complex manifold a disk of almost the same size in any close direction passes.
		As an application the Kobayashi-Royden pseudonorm for almost complex manifolds is de ned and studied.
		3 The canonical almost complex structure on the manifold of 1-..
	citeseer.ist.psu.edu /153454.html (504 words)

	Aesthetic theories of David Hume and Immanuel Kant
		Their complex proposals for bringing the various arts under a comprehensive doctrine are an important source of concepts, issues and arguments that underlie debates in our own century.
		We divide the manifold of sensations into various objects by grouping the sensations under empirical concepts: the sound comes from the airplane, but when I look toward the source of the sound, the white patch that I see is cloud, not an airplane.
		As a consequence of this analysis, Kant concludes that the "faculty of taste" is neither a separate faculty nor a passive receptivity to objects.
	www.mnstate.edu /gracyk/courses/phil%20of%20art/hume_and_kant.htm (5866 words)

	[No title] (Site not responding. Last check: 2007-09-17)
		(I.e., it's a complex affine algebraic curve, not a complex projective algebraic curve.) >[Mod.
		Gunning & Rossi, VII.C.13) says that if X is a Stein manifold of complex dimension n, then almost every (2n+2)-tuple of complex-analytic functions on X gives an embedding of X as a closed complex submanifold of C^{2n+2}; by exercising care, one can embed X in C^{2n+1}.
		Conversely, any closed complex submanifold of C^N is a Stein manifold.
	www.math.niu.edu /~rusin/known-math/95/stein.mflds (143 words)

	Real Algebraic and Analytic Geometry (Site not responding. Last check: 2007-09-17)
		It is a category of subsets of real analytic manifolds which extends the category of subanalytic sets.
		The questions are of the following nature: We start with a subset A of a complex analytic manifold M and assume that A is an object of an analytic-geometric category (by viewing M as a real analytic manifold of double dimension).
		In the second part of the paper we consider the notion of a complex S-manifold, which generalizes that of a compact complex manifold.
	www.uni-regensburg.de /Fakultaeten/nat_Fak_I/RAAG/preprints/0139.html (188 words)

	Special Complex Manifolds (ResearchIndex) (Site not responding. Last check: 2007-09-17)
		Abstract: We introduce the notion of a special complex manifold: a complex manifold (M;J) with a flat torsionfree connection r satisfying the condition d r J = 0.
		A special symplectic manifold is then defined as a special complex manifold together with a r-parallel symplectic form !.
		1 On hyper-Kahler manifolds associated to Lagrangian Kahler su..
	citeseer.ist.psu.edu /405754.html (249 words)

	DC MetaData for: Special Complex Manifolds (Site not responding. Last check: 2007-09-17)
		Abstract: We define the notion of a special complex manifold as a complex manifold
		form $\omega$ is given, then we call the manifold special symplectic.
		Special K\"ahler manifolds are realised by complex Lagrangian submanifolds
	www.esi.ac.at /Preprint-shadows/esi779.html (173 words)

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