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2014-11-21T10:08:42Z

Phase space picture

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Collective dephasing

← Older revision		Revision as of 07:10, 22 January 2014
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	I'm a ~~37 years old~~, ~~married~~ and ~~work at the college~~ (~~Optometry~~).<br>~~In my spare time I try~~ to ~~learn Chinese. I have been~~ ~~there and look forward~~ to ~~returning sometime near future. I like~~ to ~~read~~, ~~preferably on my ipad~~. ~~I really love to watch Family Guy~~ and ~~NCIS as well as documentaries about anything astronomical~~. ~~I like~~ [~~http~~://~~Data~~.~~gov~~.uk/~~data~~/~~search?q~~=~~College+football College football~~].<br><br>~~Also visit my web site; [http~~://~~Www~~.~~Galerie~~-~~Indigo~~.ch/~~galerie~~/~~index~~.~~php?album~~=~~europart~~-~~ren~~-~~francillon~~&~~image~~=~~009~~.~~jpg how to get Free fifa 15 coins~~]		In [[mathematics]], '''complex geometry''' is the study of [[complex manifold]]s and functions of many [[complex variable]]s. Application of transcendental methods to [[algebraic geometry]] falls in this category, together with more geometric chapters of [[complex analysis]].

			Throughout this article, "[[Analytic function\|analytic]]" is often dropped for simplicity; for instance, subvarieties or hypersurfaces refer to analytic ones. Following the convention in Wikipedia, varieties are assumed to be irreducible.

			== Definitions ==
			An ''[[Analytic variety\|analytic subset]]'' of a complex-analytic manifold ''M'' is locally the zero-locus of some family of holomorphic functions on ''M''. It is called an analytic subvariety if it is irreducible in the Zariski topology.

			== Line bundles and divisors ==
			Throughout this section, ''X'' denotes a complex manifold.

			Let <math>\operatorname{Pic}(X)</math> be the set of all isomorphism classes of line bundles on ''X''. It is called the [[Picard group]] of ''X'' and is naturally isomorphic to <math>H^1(X, \mathcal{O}^*)</math>. Taking the short exact sequence of
			:<math>0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0</math>
			where the second map is <math>f \mapsto \exp (2\pi i f)</math>
			yields a homomorphism of groups:
			:<math>\operatorname{Pic}(X) \to H^2(X, \mathbb{Z}).</math>
			The image of a line bundle <math>\mathcal{L}</math> under this map is denoted by <math>c_1(\mathcal{L})</math> and is called the first [[Chern class]] of <math>\mathcal{L}</math>.

			A [[divisor (algebraic geometry)\|divisor]] ''D'' on ''X'' is a [[formal sum]] of hypersurfaces (subvariety of codimension one):
			:<math>D = \sum a_i V_i, \quad a_i \in \mathbb{Z}</math>
			that is locally a finite sum.<ref>This last condition is automatic for a noetherian scheme or a compact complex manifold.</ref> The set of all divisors on ''X'' is denoted by <math>\operatorname{Div}(X)</math>. It can be canonically identified with <math>H^0(X, \mathcal{M}^/\mathcal{O}^)</math>. Taking the long exact sequence of the quotient <math>\mathcal{M}^/\mathcal{O}^</math>, one obtains a homomorphism:
			:<math>\operatorname{Div}(X) \to \operatorname{Pic}(X).</math>

			A line bundle is said to be [[positive line bundle\|positive]] if its first Chern class is represented by a closed positive real <math>(1,1)</math>-form. Equivalently, a line bundle is positive if it admits a hermitian structure such that the induced connection has [[Griffiths-positive]] curvature. A complex manifold admitting a positive line bundle is kähler.

			The [[Kodaira embedding theorem]] states that a line bundle on a compact kähler manifold is positive if and only if it is [[ample line bundle\|ample]].

			==Complex vector bundles==
			Let ''X'' be a differentiable manifold. The basic invariant of a complex vector bundle <math>\pi: E \to X</math> is the [[Chern class]] of the bundle. By definition, it is a sequence <math>c_1, c_2, \dots</math> such that <math>c_i(E)</math> is an element of <math>H^{2i}(X, \mathbb{Z})</math> and that satisfies the following axioms:<ref>{{harvnb\|Kobayashi–Nomizu\|1996\|Ch XII}}</ref>
			# <math>c_i(f^(E)) = f^(c_i(E))</math> for any differentiable map <math>f: Z \to X</math>.
			# <math>c(E \oplus F) = c(E) \cup c(F)</math> where ''F'' is another bundle and <math>c = 1 + c_1 + c_2 + \dots.</math>
			# <math>c_i(E) = 0</math> for <math>i > \operatorname{rk}E</math>.
			# <math>-c_1(E_1)</math> generates <math>H^2(\mathbb{C}\mathbf{P}^1, \mathbb{Z})</math> where <math>E_1</math> is the [[canonical line bundle]] over <math>\mathbb{C}\mathbf{P}^1</math>.

			If ''L'' is a line bundle, then the [[Chern character]] of ''L'' is given by
			:<math>\operatorname{ch}(L) = e^{c_1(L)}</math>.
			More generally, if ''E'' is a vector bundle of rank ''r'', then we have the formal factorization:
			<math>\sum c_i(E)t^i = \prod_1^r (1+ \eta_i t)</math> and then we set
			:<math>\operatorname{ch}(E) = \sum e^{\eta_i}</math>.

			== Methods from harmonic analysis ==
			Some deep results in complex geometry are obtained with the aid of harmonic analysis.

			== Vanishing theorem ==
			There are several versions of vanishing theorems in complex geometry for both compact and non-compact complex manifolds. They are however all based on the [[Bochner method]].

			==See also==
			* [[Bivector (complex)]]
			* [[Deformation Theory#Deformations of complex manifolds]]
			* [[Complex analytic space]]
			* [[GAGA]]
			* [[Several complex variables]]
			* [[Complex projective space]]
			* [[List of complex and algebraic surfaces]]
			* [[Enriques–Kodaira classification]]
			* [[Kähler manifold]]
			* [[Stein manifold]]
			* [[Pseudoconvexity]]
			* [[Kobayashi metric]]
			* [[Projective variety]]
			* [[Cousin problems]]
			* [[Cartan's theorems A and B]]
			* [[Hartogs' extension theorem]]
			* [[Calabi–Yau manifold]]
			* [[Reflection symmetry\|Mirror symmetry]]
			* [[Hermitian symmetric space]]
			* [[Complex Lie group]]
			* [[Hopf manifold]]
			* [[Hodge decomposition]]
			* [[Kobayashi-Hitchin correspondence]]
			* [[Holomorphic Higgs pairs]]

			==References==
			{{Reflist}}

			*{{cite book \|title=Complex Geometry: An Introduction\|first=Daniel\|last=Huybrechts
			\|publisher=Springer\|year=2005\|isbn=3-540-21290-6}}
			* {{Citation \| last1=Griffiths \| first1=Phillip \| author1-link=Phillip Griffiths \| last2=Harris \| first2=Joseph \| author2-link=Joe Harris (mathematician) \| title=Principles of algebraic geometry \| publisher=[[John Wiley & Sons]] \| location=New York \| series=Wiley Classics Library \| isbn=978-0-471-05059-9 \| mr=1288523 \| year=1994}}
			*{{Citation
			\| last = Hörmander
			\| first = Lars
			\| author-link = Lars Hörmander
			\| title = An Introduction to Complex Analysis in Several Variables
			\| place = Amsterdam–London–New York–Tokyo
			\| publisher = [[Elsevier\|North-Holland]]
			\| origyear = 1966
			\| year = 1990
			\| series = North–Holland Mathematical Library
			\| volume = 7
			\| edition = 3rd (Revised)
			\| url =
			\| doi =
			\| mr = 1045639
			\| zbl = 0685.32001
			\| isbn = 0-444-88446-7
			}}
			*{{Kobayashi-Nomizu}}

			[[Category:Complex manifolds]]
			[[Category:Several complex variables]]

79.9.95.248: /* Collective dephasing */

2012-08-07T16:18:19Z

Collective dephasing

New page

I'm a 37 years old, married and work at the college (Optometry).<br>In my spare time I try to learn Chinese. I have been there and look forward to returning sometime near future. I like to read, preferably on my ipad. I really love to watch Family Guy and NCIS as well as documentaries about anything astronomical. I like [http://Data.gov.uk/data/search?q=College+football College football].<br><br>Also visit my web site; [http://Www.Galerie-Indigo.ch/galerie/index.php?album=europart-ren-francillon&image=009.jpg how to get Free fifa 15 coins]

Quantum decoherence - Revision history

178.38.76.15: /* Phase space picture */

89.110.30.224: /* Collective dephasing */

79.9.95.248: /* Collective dephasing */