arithmetic genus
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the arithmetic genus of an
algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
is one of a few possible generalizations of the genus of an algebraic curve or
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
.


Projective varieties

Let ''X'' be a projective scheme of dimension ''r'' over a field ''k'', the ''arithmetic genus'' p_a of ''X'' is defined asp_a(X)=(-1)^r (\chi(\mathcal_X)-1).Here \chi(\mathcal_X) is the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of the structure sheaf \mathcal_X.


Complex projective manifolds

The arithmetic genus of a complex projective manifold of dimension ''n'' can be defined as a combination of Hodge numbers, namely :p_a=\sum_^ (-1)^j h^. When ''n=1'', the formula becomes p_a=h^. According to the Hodge theorem, h^=h^. Consequently h^=h^1(X)/2=g, where ''g'' is the usual (topological) meaning of genus of a surface, so the definitions are compatible. When ''X'' is a compact Kähler manifold, applying ''h''''p'',''q'' = ''h''''q'',''p'' recovers the earlier definition for projective varieties.


Kähler manifolds

By using ''h''''p'',''q'' = ''h''''q'',''p'' for compact Kähler manifolds this can be reformulated as the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
in coherent cohomology for the structure sheaf \mathcal_M: : p_a=(-1)^n(\chi(\mathcal_M)-1).\, This definition therefore can be applied to some other locally ringed spaces.


See also

* Genus (mathematics) *
Geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...


References

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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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...
, year=1995 , isbn=3-540-58663-6 , zbl=0843.14009 Topological methods of algebraic geometry