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| 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]].
| | Let's look an actual registry scan plus certain of what we will see whenever we do 1 on a computer. This test was completed on a computer which was not working as it must, running at slow speed and having several issues with freezing up.<br><br>Carry out window's program restore. It is important to do this considering it removes wrong changes that have happened inside the program. Some of the errors result from inability of the system to create restore point frequently.<br><br>The error is basically a result of issue with Windows Installer package. The Windows Installer is a tool employed to install, uninstall plus repair the many programs on your computer. Let you discuss a few items which helped a great deal of folks whom facing the similar issue.<br><br>First, constantly clean a PC and keep it without dust plus dirt. Dirt clogs up all fans and may result the PC to overheat. We have to clean up disk room inside purchase to create a computer run quicker. Delete temporary plus unnecessary files plus unused programs. Empty the recycle bin and remove programs you may be not utilizing.<br><br>Many [http://bestregistrycleanerfix.com/system-mechanic iolo system mechanic] s enable we to download their product for free, thus you can scan the computer oneself. That technique you are able to see how several mistakes it finds, where it finds them, plus how it can fix them. A perfect registry cleaner usually remove your registry problems, plus optimize and accelerate the PC, with little effort on your piece.<br><br>2)Fix a Windows registry to accelerate PC- The registry is a complex section of your computer that holds different kinds of data within the elements we do on a computer every day. Coincidentally, over time the registry usually become cluttered with info and/or may receive some kind of virus. This really is especially important plus you MUST get this problem fixed right away, otherwise you run the risk of the computer being forever damage and/or your sensitive info (passwords, etc.) can be stolen.<br><br>Google Chrome is my lifeline plus for this day fortunately. My all settings and research connected bookmarks were saved in Chrome and stupidly I didn't synchronize them with all the Gmail to shop them online. I can not afford to install modern version plus sacrifice all my function settings. There was no method to retrieve the aged settings. The only way left for me was to miraculously fix it browser inside a way that all the data and settings stored in it are recovered.<br><br>A system and registry cleaner could be downloaded from the internet. It's simple to use and the procedure does not take lengthy. All it does is scan and then when it finds mistakes, it might fix plus clean those errors. An error free registry can safeguard the computer from errors plus give we a slow PC fix. |
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| 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.
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| == Definitions ==
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| 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.
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| == Line bundles and divisors ==
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| Throughout this section, ''X'' denotes a complex manifold.
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| 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
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| :<math>0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0</math>
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| where the second map is <math>f \mapsto \exp (2\pi i f)</math>
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| yields a homomorphism of groups:
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| :<math>\operatorname{Pic}(X) \to H^2(X, \mathbb{Z}).</math>
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| 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>.
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| A [[divisor (algebraic geometry)|divisor]] ''D'' on ''X'' is a [[formal sum]] of hypersurfaces (subvariety of codimension one):
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| :<math>D = \sum a_i V_i, \quad a_i \in \mathbb{Z}</math>
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| 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:
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| :<math>\operatorname{Div}(X) \to \operatorname{Pic}(X).</math>
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| 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.
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| 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]].
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| ==Complex vector bundles==
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| 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>
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| # <math>c_i(f^*(E)) = f^*(c_i(E))</math> for any differentiable map <math>f: Z \to X</math>.
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| # <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>
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| # <math>c_i(E) = 0</math> for <math>i > \operatorname{rk}E</math>.
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| # <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>.
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| If ''L'' is a line bundle, then the [[Chern character]] of ''L'' is given by
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| :<math>\operatorname{ch}(L) = e^{c_1(L)}</math>.
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| More generally, if ''E'' is a vector bundle of rank ''r'', then we have the formal factorization:
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| <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>.
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| == Methods from harmonic analysis ==
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| Some deep results in complex geometry are obtained with the aid of harmonic analysis.
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| == Vanishing theorem ==
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| 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]].
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| ==See also==
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| * [[Bivector (complex)]]
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| * [[Deformation Theory#Deformations of complex manifolds]]
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| * [[Complex analytic space]]
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| * [[GAGA]]
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| * [[Several complex variables]]
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| * [[Complex projective space]]
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| * [[List of complex and algebraic surfaces]]
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| * [[Enriques–Kodaira classification]]
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| * [[Kähler manifold]]
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| * [[Stein manifold]]
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| * [[Pseudoconvexity]]
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| * [[Kobayashi metric]]
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| * [[Projective variety]]
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| * [[Cousin problems]]
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| * [[Cartan's theorems A and B]]
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| * [[Hartogs' extension theorem]]
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| * [[Calabi–Yau manifold]]
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| * [[Reflection symmetry|Mirror symmetry]]
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| * [[Hermitian symmetric space]]
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| * [[Complex Lie group]]
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| * [[Hopf manifold]]
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| * [[Hodge decomposition]]
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| * [[Kobayashi-Hitchin correspondence]]
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| * [[Holomorphic Higgs pairs]]
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| ==References==
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| {{Reflist}}
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| *{{cite book |title=Complex Geometry: An Introduction|first=Daniel|last=Huybrechts
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| |publisher=Springer|year=2005|isbn=3-540-21290-6}}
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| * {{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}}
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| *{{Citation
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| | last = Hörmander
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| | first = Lars
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| | author-link = Lars Hörmander
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| | title = An Introduction to Complex Analysis in Several Variables
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| | place = Amsterdam–London–New York–Tokyo
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| | publisher = [[Elsevier|North-Holland]]
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| | origyear = 1966
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| | year = 1990
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| | series = North–Holland Mathematical Library
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| | volume = 7
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| | edition = 3rd (Revised)
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| | url =
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| | doi =
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| | mr = 1045639
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| | zbl = 0685.32001
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| | isbn = 0-444-88446-7
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| }}
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| *{{Kobayashi-Nomizu}}
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| [[Category:Complex manifolds]]
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| [[Category:Several complex variables]]
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Let's look an actual registry scan plus certain of what we will see whenever we do 1 on a computer. This test was completed on a computer which was not working as it must, running at slow speed and having several issues with freezing up.
Carry out window's program restore. It is important to do this considering it removes wrong changes that have happened inside the program. Some of the errors result from inability of the system to create restore point frequently.
The error is basically a result of issue with Windows Installer package. The Windows Installer is a tool employed to install, uninstall plus repair the many programs on your computer. Let you discuss a few items which helped a great deal of folks whom facing the similar issue.
First, constantly clean a PC and keep it without dust plus dirt. Dirt clogs up all fans and may result the PC to overheat. We have to clean up disk room inside purchase to create a computer run quicker. Delete temporary plus unnecessary files plus unused programs. Empty the recycle bin and remove programs you may be not utilizing.
Many iolo system mechanic s enable we to download their product for free, thus you can scan the computer oneself. That technique you are able to see how several mistakes it finds, where it finds them, plus how it can fix them. A perfect registry cleaner usually remove your registry problems, plus optimize and accelerate the PC, with little effort on your piece.
2)Fix a Windows registry to accelerate PC- The registry is a complex section of your computer that holds different kinds of data within the elements we do on a computer every day. Coincidentally, over time the registry usually become cluttered with info and/or may receive some kind of virus. This really is especially important plus you MUST get this problem fixed right away, otherwise you run the risk of the computer being forever damage and/or your sensitive info (passwords, etc.) can be stolen.
Google Chrome is my lifeline plus for this day fortunately. My all settings and research connected bookmarks were saved in Chrome and stupidly I didn't synchronize them with all the Gmail to shop them online. I can not afford to install modern version plus sacrifice all my function settings. There was no method to retrieve the aged settings. The only way left for me was to miraculously fix it browser inside a way that all the data and settings stored in it are recovered.
A system and registry cleaner could be downloaded from the internet. It's simple to use and the procedure does not take lengthy. All it does is scan and then when it finds mistakes, it might fix plus clean those errors. An error free registry can safeguard the computer from errors plus give we a slow PC fix.