en>Yobot: WP:CHECKWIKI errors fixed + general fixes using AWB (8961)

2013-03-07T21:15:37Z

WP:CHECKWIKI errors fixed + general fixes using AWB (8961)

← Older revision		Revision as of 23:15, 7 March 2013
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	~~Hi there~~. ~~Let me start~~ by ~~introducing~~ the ~~writer~~, ~~her name~~ is ~~Sophia~~. ~~North Carolina~~ is the ~~place he enjoys most but now he~~ is ~~contemplating other options~~. ~~One~~ of the ~~extremely email psychic readings ([http~~://~~www~~.~~edmposts.com~~/~~build-~~a-~~beautiful~~-~~organic~~-~~garden~~-~~using-these-ideas~~/ ~~www~~.~~edmposts~~.~~com~~]) ~~very best issues~~ in the ~~world for him~~ is ~~[http~~://~~Cspl.postech.ac.kr~~/~~zboard~~/~~Membersonly~~/~~144571 online psychic chat] doing ballet and he'll~~ be ~~starting something else along~~ with ~~it. I am currently~~ a ~~travel agent~~.<br><br>~~My page~~ - ~~online psychics~~ ([~~http:~~//~~www~~.~~prograd~~.~~uff~~.br/~~novo~~/~~facts~~-~~about~~-~~growing~~-~~greater~~-~~organic~~-~~garden~~ www.~~prograd~~.~~uff~~.br])		In [[mathematics]], specifically in [[topology]] and [[geometry]], a '''pseudoholomorphic curve''' (or '''''J''-holomorphic curve''') is a smooth map from a [[Riemann surface]] into an [[almost complex manifold]] that satisfies the [[Cauchy–Riemann equations\|Cauchy–Riemann equation]]. Introduced in 1985 by [[Mikhail Gromov (mathematician)\|Mikhail Gromov]], pseudoholomorphic curves have since revolutionized the study of [[symplectic manifold]]s. In particular, they lead to the [[Gromov–Witten invariant]]s and [[Floer homology]], and play a prominent role in [[string theory]].

			==Definition==
			Let <math>X</math> be an almost complex manifold with almost complex structure <math>J</math>. Let <math>C</math> be a smooth [[Riemann surface]] (also called a [[algebraic curve\|complex curve]]) with complex structure <math>j</math>. A '''pseudoholomorphic curve''' in <math>X</math> is a map <math>f : C \to X</math> that satisfies the Cauchy–Riemann equation
			:<math>\bar \partial_{j, J} f := \frac{1}{2}(df + J \circ df \circ j) = 0.</math>
			Since <math>J^2 = -1</math>, this condition is equivalent to
			:<math>J \circ df = df \circ j,</math>
			which simply means that the differential <math>df</math> is complex-linear, that is, <math>J</math> maps each tangent space
			:<math>T_xf(C)\subseteq T_xX</math>
			to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term <math>\nu</math> and to study maps satisfying the perturbed Cauchy–Riemann equation
			:<math>\bar \partial_{j, J} f = \nu.</math>
			A pseudoholomorphic curve satisfying this equation can be called, more specifically, a '''<math>(j, J, \nu)</math>-holomorphic curve'''. The perturbation <math>\nu</math> is sometimes assumed to be generated by a [[Hamiltonian vector field\|Hamiltonian]] (particularly in Floer theory), but in general it need not be.

			A pseudoholomorphic curve is, by its definition, always parametrized. In applications one is often truly interested in unparametrized curves, meaning embedded (or immersed) two-submanifolds of <math>X</math>, so one mods out by reparametrizations of the domain that preserve the relevant structure. In the case of Gromov-Witten invariants, for example, we consider only [[closed manifold\|closed]] domains <math>C</math> of fixed genus <math>g</math> and we introduce <math>n</math> '''marked points''' (or '''punctures''') on <math>C</math>. As soon as the punctured [[Euler characteristic]] <math>2 - 2 g - n</math> is negative, there are only finitely many holomorphic reparametrizations of <math>C</math> that preserve the marked points. The domain curve <math>C</math> is an element of the [[Deligne–Mumford moduli space of curves]].

			==Analogy with the classical Cauchy–Riemann equations==
			The classical case occurs when <math>X</math> and <math>C</math> are both simply the [[complex number]] plane. In real coordinates
			:<math>j = J = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix},</math>
			and
			:<math>df = \begin{bmatrix} du/dx & du/dy \\ dv/dx & dv/dy \end{bmatrix},</math>
			where <math>f(x, y) = (u(x, y), v(x, y))</math>. After multiplying these matrices in two different orders, one sees immediately that the equation
			:<math>J \circ df = df \circ j</math>
			written above is equivalent to the classical Cauchy–Riemann equations
			:<math>\begin{cases} du/dx = dv/dy \\ dv/dx = -du/dy. \end{cases}</math>

			==Applications in symplectic topology==
			Although they can be defined for any almost complex manifold, pseudoholomorphic curves are especially interesting when <math>J</math> interacts with a [[symplectic form]] <math>\omega</math>. An almost complex structure <math>J</math> is said to be '''<math>\omega</math>-tame''' if and only if
			:<math>\omega(v, J v) > 0</math>
			for all nonzero tangent vectors <math>v</math>. Tameness implies that the formula
			:<math>(v, w) = \frac{1}{2}\left(\omega(v, Jw) + \omega(w, Jv)\right)</math>
			defines a [[Riemannian metric]] on <math>X</math>. Gromov showed that, for a given <math>\omega</math>, the space of <math>\omega</math>-tame <math>J</math> is nonempty and [[contractible]]. He used this theory to prove a [[non-squeezing theorem]] concerning symplectic embeddings of spheres into cylinders.

			Gromov showed that certain [[moduli space]]s of pseudoholomorphic curves (satisfying additional specified conditions) are [[compact space\|compact]], and described the way in which pseudoholomorphic curves can degenerate when only finite energy is assumed. (The finite energy condition holds most notably for curves with a fixed homology class in a symplectic manifold where J is <math>\omega</math>-tame or <math>\omega</math>-compatible). This [[Gromov's compactness theorem (topology)\|Gromov compactness theorem]], now greatly generalized using [[stable map]]s, makes possible the definition of Gromov-Witten invariants, which count pseudoholomorphic curves in symplectic manifolds.

			Compact moduli spaces of pseudoholomorphic curves are also used to construct [[Floer homology]], which [[Andreas Floer]] (and later authors, in greater generality) used to prove the famous conjecture of [[Vladimir Arnol'd]] concerning the number of fixed points of [[Hamiltonian flow]]s.

			==Applications in physics==
			In type II string theory, one considers surfaces traced out by strings as they travel along paths in a [[Calabi-Yau]] 3-fold. Following the [[path integral formulation]] of [[quantum mechanics]], one wishes to compute certain integrals over the space of all such surfaces. Because such a space is infinite-dimensional, these path integrals are not mathematically well-defined in general. However, under the [[A-twist]] one can deduce that the surfaces are parametrized by pseudoholomorphic curves, and so the path integrals reduce to integrals over moduli spaces of pseudoholomorphic curves (or rather stable maps), which are finite-dimensional. In closed type IIA string theory, for example, these integrals are precisely the [[Gromov-Witten invariant]]s.

			==See also==

			* [[Holomorphic curve]]

			==References==
			* [[Dusa McDuff]] and Dietmar Salamon, ''J-Holomorphic Curves and Symplectic Topology'', American Mathematical Society colloquium publications, 2004. ISBN 0-8218-3485-1.
			* M. Gromov, Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae vol. 82, 1985, pgs. 307-347.
			* {{ cite journal
			\| last = Donaldson
			\| first = Simon K.
			\| authorlink = Simon Donaldson
			\| title = What Is...a Pseudoholomorphic Curve?
			\| journal = [[Notices of the American Mathematical Society]]
			\|date=October 2005
			\| volume = 52
			\| issue = 9
			\| pages = pp.1026–1027
			\| url = http://www.ams.org/notices/200509/what-is.pdf
			\| format = [[PDF]]
			\| accessdate = 2008-01-17 }}

			[[Category:Complex manifolds]]
			[[Category:Symplectic topology]]
			[[Category:Algebraic geometry]]
			[[Category:String theory]]
			[[Category:Curves]]

en>Hans Adler: refine stub

2010-04-26T13:39:14Z

refine stub

New page

Hi there. Let me start by introducing the writer, her name is Sophia. North Carolina is the place he enjoys most but now he is contemplating other options. One of the extremely email psychic readings ([http://www.edmposts.com/build-a-beautiful-organic-garden-using-these-ideas/ www.edmposts.com]) very best issues in the world for him is [http://Cspl.postech.ac.kr/zboard/Membersonly/144571 online psychic chat] doing ballet and he'll be starting something else along with it. I am currently a travel agent.<br><br>My page - online psychics ([http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden www.prograd.uff.br])

Global square - Revision history

en>Yobot: WP:CHECKWIKI errors fixed + general fixes using AWB (8961)

en>Hans Adler: refine stub