Plasma acceleration

2013-11-26T04:54:05Z

208.120.184.170: Undid revision 581043706 by 156.39.10.22 (talk) Original citation is T. Tajima and J.M. Dawson, Phys. Rev. Lett. 43, 267-270 (1979), Tajima as primary author.

In [[mathematics]], the '''arithmetic genus''' of an [[algebraic variety]] is one of some possible generalizations of the [[genus of an algebraic curve]] or [[Riemann surface]].

The arithmetic genus of a [[complex projective manifold]]
of dimension ''n'' can be defined as a combination of [[Hodge number]]s, namely

:''p''''a'' = ''h''''n'',0 − ''h''''n'' − 1, 0 + ... + (−1)''n'' − 1''h''1, 0.

When ''n'' = 1 we have χ = 1 − ''g'' where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible.

By using ''h''''p'',''q'' = ''h''''q'',''p'' for compact Kähler manifolds this can be
reformulated as the [[Euler characteristic]] in [[coherent cohomology]] for the [[structure sheaf]] <math>\mathcal{O}_M</math>:

: <math> p_a=(-1)^n(\chi(\mathcal{O}_M)-1).\,</math>

This definition therefore can be applied to some other
[[locally ringed space]]s.

==See also==
*[[Genus (mathematics)]]
* [[Geometric genus]]

==References==
* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | edition=2nd | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | zbl=0836.14001 | page=494 }}

==Further reading==
* {{cite book | last=Hirzebruch | first=Friedrich | authorlink=Friedrich Hirzebruch | title=Topological methods in algebraic geometry | others=Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel | edition=Reprint of the 2nd, corr. print. of the 3rd | origyear=1978 | series=Classics in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | isbn=3-540-58663-6 | zbl=0843.14009 }}

[[Category:Topological methods of algebraic geometry]]

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Plasma acceleration