en>Hyacinth: -->Category:Knot theory

2013-05-04T02:51:25Z

← Older revision		Revision as of 04:51, 4 May 2013
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	~~Nestor~~ is the ~~title my mothers and fathers gave me but I don~~'~~t like when individuals use my complete name~~. ~~Delaware~~ is the ~~only location I~~'~~ve been residing in~~. ~~The preferred pastime for him~~ and ~~his children~~ is to ~~play badminton but he~~ is ~~struggling~~ to ~~discover time for it~~. ~~Bookkeeping~~ is ~~what he does.~~<br><br>~~Have~~ a ~~look at my blog~~ ... ~~extended auto warranty (~~[http://~~titon~~.~~net~~/~~UserProfile~~/~~tabid~~/~~43/userId/160343/Default.aspx click to read~~])		In mathematics, more particularly in [[complex geometry]],
			[[algebraic geometry]] and [[complex analysis]], a '''positive current'''
			is a [[positive form\|positive]] (''n-p'',''n-p'')-form over an ''n''-dimensional [[complex manifold]],
			taking values in distributions.

			For a formal definition, consider a manifold ''M''.
			[[Current (mathematics)\|Current]]s on ''M'' are (by definition)
			differential forms with coefficients in distributions. ; integrating
			over ''M'', we may consider currents as "currents of integration",
			that is, functionals

			:<math>\eta \mapsto \int_M \eta\wedge \rho</math>

			on smooth forms with compact support. This way, currents
			are considered as elements in the dual space to the space
			<math>\Lambda_c^*(M)</math> of forms with compact support.

			Now, let ''M'' be a complex manifold.
			The [[De Rham cohomology#Hodge_decomposition\|Hodge decomposition]] <math>\Lambda^i(M)=\bigoplus_{p+q=i}\Lambda^{p,q}(M)</math>
			is defined on currents, in a natural way, the ''(p,q)''-currents being
			functionals on <math>\Lambda_c^{p, q}(M)</math>.

			A '''positive current''' is defined as a real [[Current (mathematics)\|current]]
			of Hodge type ''(p,p)'', taking non-negative values on all [[positive form\|positive]]
			''(p,p)''-forms.



			== Characterization of [[Kähler manifold]]s ==

			Using the [[Hahn–Banach theorem]], Harvey and Lawson proved the following criterion of existence of [[Kähler manifold\|Kähler metrics]].<ref>R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198.</ref>


			'''Theorem:''' Let ''M'' be a compact complex manifold. Then ''M'' does not admit a [[Kähler manifold\|Kähler structure]] if and only if ''M'' admits a non-zero positive (1,1)-current <math>\Theta</math> which is a (1,1)-part of an exact 2-current.


			Note that the [[de Rham cohomology\|de Rham differential]] maps 3-currents to 2-currents, hence <math>\Theta</math> is a differential of a 3-current; if <math>\Theta</math> is a current of integration of a [[Riemann surface\|complex curve]], this means that this curve is a (1,1)-part of a boundary.

			When ''M'' admits a surjective map <math>\pi:\; M \mapsto X</math> to a [[Kähler manifold]] with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry.


			'''Corollary:''' In this situation, ''M'' is non-[[Kähler manifold\|Kähler]] if and only if the [[homology class]] of a generic fiber of <math>\pi</math> is a (1,1)-part of a boundary.

			== Notes ==
			<references />

			==References==
			*Phillip Griffiths and Joseph Harris (1978), ''Principles of Algebraic Geometry'', Wiley. ISBN 0-471-32792-1
			*J.-P. Demailly, ''[http://arxiv.org/abs/alg-geom/9410022 $L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)]''


			{{differential-geometry-stub}}

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

en>Brad7777: removed Category:Geometric topology using HotCat knot theory is a subcat of geometric topology

2012-02-06T00:14:25Z

removed Category:Geometric topology using HotCat knot theory is a subcat of geometric topology

New page

Nestor is the title my mothers and fathers gave me but I don't like when individuals use my complete name. Delaware is the only location I've been residing in. The preferred pastime for him and his children is to play badminton but he is struggling to discover time for it. Bookkeeping is what he does.<br><br>Have a look at my blog ... extended auto warranty ([http://titon.net/UserProfile/tabid/43/userId/160343/Default.aspx click to read])

Knot thickness - Revision history

en>Hyacinth: -->Category:Knot theory

en>Brad7777: removed Category:Geometric topology using HotCat knot theory is a subcat of geometric topology