Adaptive stepsize - Revision history

en>Magioladitis: clean up using AWB (10685)

2015-01-04T21:49:26Z

clean up using AWB (10685)

← Older revision		Revision as of 23:49, 4 January 2015
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	~~In algebraic geometry, the '''Fourier–Mukai transform''' or '''Mukai–Fourier transform''~~', introduced by {{harvs\|txt\|last=Mukai\|year=1981\|authorlink=Shigeru Mukai}}, is an isomorphism between the [[derived categories]] of [[coherent sheaves]] on an [[abelian variety]] and its dual. It is analogous to the classical [[Fourier transform]] that gives an isomorphism between [[Distribution (mathematics)\|tempered distribution]]s ~~on a finite-dimensional real vector space and its dual.~~		The writer's title is Christy Brookins. To climb is some thing she would by no means give up. Distributing manufacturing is where her main income arrives from. Ohio is where her home is.<br><br>My web page [http://www.skullrocker.com/blogs/post/10991 psychic readings]

	~~If the [[Canonical bundle\|canonical class]] of a variety~~ is ~~positive or negative, then the derived category of coherent sheaves determines the variety~~. The Fourier–Mukai transform gives examples of different varieties (with trivial canonical bundle) that have isomorphic derived categories, as in general an abelian variety of dimension greater than 1 is ~~not isomorphic to its dual.~~

	~~==Definition==~~

	~~Let <math>X</math> be an [[abelian variety]] and <math>\hat X</math> be its [[Dual abelian variety\|dual variety]]. We denote~~ by ~~<math>\mathcal P</math> the [[Poincaré bundle]] on~~

	~~:<math>X \times \hat X,</math>~~

	~~normalized to be trivial on the fibers at zero~~. ~~Let <math>p</math> and <math>\hat p</math> be the canonical projections.~~

	~~The Fourier–Mukai functor is then~~
	~~:<math>R\mathcal S: \mathcal F \in D(X) \mapsto R\hat p_\ast (p^\ast \mathcal F \otimes \mathcal P) \in D(\hat X)</math>~~

	~~The notation here: ''D'' means [[derived category]] of [[coherent sheaves]], and ''R''~~ is ~~the [[higher direct image functor]], at the derived category level~~.

	~~There~~ is ~~a similar functor~~

	~~:<math>R\widehat{\mathcal S} : D(\hat X) \to D(X). \, </math>~~

	~~==Properties==~~

	~~Let ''g'' denote the dimension of ''X''.~~

	~~The Fourier–Mukai transformation~~ is ~~nearly involutive :~~
	~~:<math>R\mathcal S \circ R\widehat{\mathcal S} = (-1)^\ast [-g]</math>~~

	~~It transforms [[Pontrjagin product]] in [[tensor product]] and conversely~~.
	~~:<math>R\mathcal S(\mathcal F \ast \mathcal G) = R\mathcal S(\mathcal F) \otimes R\mathcal S(\mathcal G)~~<~~/math~~>
	:<~~math~~>~~R\mathcal S(\mathcal F \otimes \mathcal G) = R\mathcal S(\mathcal F) \ast R\mathcal S(\mathcal G)~~[~~g]</math>~~

	~~==References==~~

	*{{Citation \| last1=Huybrechts \| first1=D. \| title=Fourier–Mukai transforms in algebraic geometry \| url=http://dx.~~doi.org/10~~.~~1093~~/~~acprof:oso~~/~~9780199296866.001.0001 \| publisher=The Clarendon Press Oxford University Press \| series=Oxford Mathematical Monographs \| isbn=978-0-19-929686-6; 978-0-19-929686-6 \| doi=10.1093~~/~~acprof:oso/9780199296866.001.0001 \| id={{MR\|2244106}} \| year=2006}}~~
	*{{cite journal
	~~\| last=Mukai~~
	~~\| first=Shigeru~~
	~~\| authorlink=Shigeru Mukai~~
	~~\| title=Duality between <math>D(X)</math> and <math>D(\hat X)</math> with its application to Picard sheaves~~
	~~\| journal=Nagoya Mathematical Journal~~
	~~\| volume=81~~
	~~\| year=1981~~
	~~\| pages=153–175~~
	~~\| id=ISSN 0027-7630~~
	~~\| url=http://projecteuclid.org/euclid.nmj/1118786312~~
	}}

	~~{{DEFAULTSORT:Fourier-Mukai transform}}~~
	~~[[Category:Abelian varieties]~~]

en>DoctorKubla: rm deprecated wikify tag

2012-12-29T12:13:49Z

rm deprecated wikify tag

← Older revision		Revision as of 14:13, 29 December 2012
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	~~The name~~ of the ~~author~~ is ~~Luther~~. ~~Interviewing~~ is ~~what she does in her working day occupation but soon her husband~~ and ~~her will begin their own company~~. The ~~preferred pastime for him~~ and ~~his kids~~ is ~~to play badminton but he~~ is ~~having difficulties~~ to ~~find time for it~~. ~~Delaware~~ is ~~our beginning location~~.<br><br>~~Stop by my page~~: ~~extended auto warranty -~~ [http://~~www~~.~~Schwaben~~-~~gaming~~.de/~~index~~.~~php?mod~~=~~users&action~~=~~view&~~id=~~9562 linked web page~~],		In algebraic geometry, the '''Fourier–Mukai transform''' or '''Mukai–Fourier transform''', introduced by {{harvs\|txt\|last=Mukai\|year=1981\|authorlink=Shigeru Mukai}}, is an isomorphism between the [[derived categories]] of [[coherent sheaves]] on an [[abelian variety]] and its dual. It is analogous to the classical [[Fourier transform]] that gives an isomorphism between [[Distribution (mathematics)\|tempered distribution]]s on a finite-dimensional real vector space and its dual.

			If the [[Canonical bundle\|canonical class]] of a variety is positive or negative, then the derived category of coherent sheaves determines the variety. The Fourier–Mukai transform gives examples of different varieties (with trivial canonical bundle) that have isomorphic derived categories, as in general an abelian variety of dimension greater than 1 is not isomorphic to its dual.

			==Definition==

			Let <math>X</math> be an [[abelian variety]] and <math>\hat X</math> be its [[Dual abelian variety\|dual variety]]. We denote by <math>\mathcal P</math> the [[Poincaré bundle]] on

			:<math>X \times \hat X,</math>

			normalized to be trivial on the fibers at zero. Let <math>p</math> and <math>\hat p</math> be the canonical projections.

			The Fourier–Mukai functor is then
			:<math>R\mathcal S: \mathcal F \in D(X) \mapsto R\hat p_\ast (p^\ast \mathcal F \otimes \mathcal P) \in D(\hat X)</math>

			The notation here: ''D'' means [[derived category]] of [[coherent sheaves]], and ''R'' is the [[higher direct image functor]], at the derived category level.

			There is a similar functor

			:<math>R\widehat{\mathcal S} : D(\hat X) \to D(X). \, </math>

			==Properties==

			Let ''g'' denote the dimension of ''X''.

			The Fourier–Mukai transformation is nearly involutive :
			:<math>R\mathcal S \circ R\widehat{\mathcal S} = (-1)^\ast [-g]</math>

			It transforms [[Pontrjagin product]] in [[tensor product]] and conversely.
			:<math>R\mathcal S(\mathcal F \ast \mathcal G) = R\mathcal S(\mathcal F) \otimes R\mathcal S(\mathcal G)</math>
			:<math>R\mathcal S(\mathcal F \otimes \mathcal G) = R\mathcal S(\mathcal F) \ast R\mathcal S(\mathcal G)[g]</math>

			==References==

			*{{Citation \| last1=Huybrechts \| first1=D. \| title=Fourier–Mukai transforms in algebraic geometry \| url=http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001 \| publisher=The Clarendon Press Oxford University Press \| series=Oxford Mathematical Monographs \| isbn=978-0-19-929686-6; 978-0-19-929686-6 \| doi=10.1093/acprof:oso/9780199296866.001.0001 \| id={{MR\|2244106}} \| year=2006}}
			*{{cite journal
			\| last=Mukai
			\| first=Shigeru
			\| authorlink=Shigeru Mukai
			\| title=Duality between <math>D(X)</math> and <math>D(\hat X)</math> with its application to Picard sheaves
			\| journal=Nagoya Mathematical Journal
			\| volume=81
			\| year=1981
			\| pages=153–175
			\| id=ISSN 0027-7630
			\| url=http://projecteuclid.org/euclid.nmj/1118786312
			}}

			{{DEFAULTSORT:Fourier-Mukai transform}}
			[[Category:Abelian varieties]]

en>Timrollpickering: clean up using AWB

2012-08-25T17:41:09Z

clean up using AWB

New page

The name of the author is Luther. Interviewing is what she does in her working day occupation but soon her husband and her will begin their own company. The preferred pastime for him and his kids is to play badminton but he is having difficulties to find time for it. Delaware is our beginning location.<br><br>Stop by my page: extended auto warranty - [http://www.Schwaben-gaming.de/index.php?mod=users&action=view&id=9562 linked web page],