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In mathematics, '''Picard–Lefschetz theory''' studies the topology of a [[complex manifold]] by looking at the [[critical point (mathematics)|critical points]] of a [[holomorphic function]] on the manifold. It was introduced by [[Émile Picard]] for complex surfaces in his book {{harvtxt|Picard|Simart|1897}}, and extended to higher dimensions by {{harvs|txt|last=Lefschetz|authorlink=Solomon Lefschetz|year=1924}}. It is a complex analog of [[Morse theory]] that studies the topology of a real [[manifold]] by looking at the critical points of a real function. {{harvtxt|Deligne|Katz|1973}} extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the [[Weil conjectures]].
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==Picard–Lefschetz formula==
 
The '''Picard–Lefschetz formula''' describes the [[monodromy]] at a critical point.
 
Suppose that ''f'' is a holomorphic map from an ''k+1''-dimensional projective complex manifold to the projective line '''P'''<sup>1</sup>.  Also  suppose that all critical points are non-degenerate and lie in different fibers, and have images ''x''<sub>1</sub>,...,''x''<sub>''n''</sub> in '''P'''<sup>1</sup>. Pick any other point ''x'' in '''P'''<sup>1</sup>. The [[fundamental group]] π<sub>1</sub>('''P'''<sup>1</sup>&nbsp;–&nbsp;{''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>},&nbsp;''x'') is generated by loops ''w''<sub>''i''</sub> going around the points ''x''<sub>''i''</sub>, and to each point ''x''<sub>''i''</sub> there is a [[vanishing cycle]] in the homology ''H''<sub>''k''–1</sub>(''Y''<sub>''x''</sub>) of the fiber at&nbsp;''x''.  
There is a monodromy action of π<sub>1</sub>('''P'''<sup>1</sup>&nbsp;–&nbsp;{''x''<sub>1</sub>,&nbsp;...,&nbsp;''x''<sub>''n''</sub>},&nbsp;''x'') on ''H''<sub>''k''–1</sub>(''Y''<sub>''x''</sub>), described as follows by the Picard–Lefschetz formula. (The action of monodromy on other homology groups is trivial.)  The monodromy action of a generator ''w''<sub>''i''</sub> of the fundamental group on ''<math>\gamma</math>''&nbsp;∈&nbsp;''H''<sub>''k''–1</sub>(''Y''<sub>''x''</sub>) is given by
 
:<math>w_i(\gamma) = \gamma+(-1)^{k(k+1)/2}\langle \gamma,\delta_i\rangle \delta_i</math>
 
where δ<sub>''i''</sub> is the vanishing cycle of ''x''<sub>''i''</sub>. This formula appears implicitly for ''n''&nbsp;=&nbsp;2 (without the explicit coefficients of the vanishing cycles δ<sub>''i''</sub>) in {{harvtxt|Picard|Simart|1897|loc=p.95}}. {{harvtxt|Lefschetz|1924|loc=chapters II, V}} gave the explicit formula in all dimensions.
 
==References==
 
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | last2=Katz | first2=Nicholas | author2-link=Nicholas Katz | title=Groupes de monodromie en géométrie algébrique. II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | doi=10.1007/BFb0060505  | id={{MR|0354657}} | year=1973 | volume=340}}
*{{Citation | last1=Lamotke | first1=Klaus | title=The topology of complex projective varieties after S. Lefschetz | url=http://dx.doi.org/10.1016/0040-9383(81)90013-6 | doi=10.1016/0040-9383(81)90013-6 | id={{MR|592569}} | year=1981 | journal=[[Topology (journal)|Topology. An International Journal of Mathematics]] | issn=0040-9383 | volume=20 | issue=1 | pages=15–51}}
*{{Citation | last1=Lefschetz | first1=S. | title=L'analysis situs et la géométrie algébrique | publisher=Gauthier-Villars | id={{MR|0033557}} | year=1924}}
*{{Citation | last1=Lefschetz | first1=Solomon | title=Applications of algebraic topology. Graphs and networks, the Picard-Lefschetz theory and Feynman              integrals | url=http://books.google.com/books?id=eqCyQgAACAAJ | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Applied Mathematical Sciences | isbn=978-0-387-90137-4 | id={{MR|0494126}} | year=1975 | volume=16}}
*{{Citation | last2=Simart | first2=G. | last1=Picard | first1=É. | title=Théorie des fonctions algébriques de deux variables indépendantes. Tome I| publisher=Paris: Gauthier-Villars et Fils. | language=French | year=1897|url=http://www.archive.org/details/thoriedesfoncti00simagoog}}
*{{Citation | last1=Vassiliev | first1=V. A. | title=Applied Picard–Lefschetz theory | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-2948-6 | id={{MR|1930577}} | year=2002 | volume=97}}
 
{{DEFAULTSORT:Picard-Lefschetz theory}}
[[Category:Algebraic geometry]]

Latest revision as of 10:36, 8 July 2014

Set Goals And Make it rewarding Do desire to look better in the swimsuit? Lose that abdominal fat around your middle? Or build your stamina? Pick out a goal and keep it in their mind. Then post those goals where you can obtain them constantly, like on the bathroom mirror or maybe the fridge. When you've met those goals, treat you to ultimately something like a movie or dinner absent. Give yourself a reward every time you complete your physical.

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The truth is, Power Max 360 Review bodybuilding is really an art and science, and as well as more more experts are attempting to figure out its exact nature daily. Sure there are specific routines and practices you actually can follow that permits you to have the greatest success as possible, an individual should understandthat this process is being tested on a day during the day basis.

It vital to increase water intake during your Muscle Building workout routine. To determine necessary number of ounces per day, multiply your bodyweight by.66. The allocate correct consistency of your energy to the while exercising.

In most activities, the you do something, far better the results you get. In gym routines, this actually isn't fact. You do not build muscle while in order to exercising; simply build them while happen to be resting. Workouts are only previously used to stimulate the growth, then you certainly need to provide body you'd like it requires to actually build muscle mass.

The same goes you r diet too. There are different ways to eat just plus there is different ways to exercise. Those that need for losing weight fast would never the same diet plan as people that need obtain weight.

What positive things haven't really been said about eating tuna? It's super high in protein. It's low in fat. The fat it does contain could be the "good kind". And you can get it coming from a market for Power Max 360 Review relatively bad. Wait.

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