• Source: Arithmetic genus

In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.




Projective varieties


Let X be a projective scheme of dimension r over a field k, the arithmetic genus




p

a




{\displaystyle p_{a}}

of X is defined as




p

a


(
X
)
=
(
−
1

)

r


(
χ
(



O



X


)
−
1
)
.


{\displaystyle p_{a}(X)=(-1)^{r}(\chi ({\mathcal {O}}_{X})-1).}

Here



χ
(



O



X


)


{\displaystyle \chi ({\mathcal {O}}_{X})}

is the Euler characteristic of the structure sheaf






O



X




{\displaystyle {\mathcal {O}}_{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


=

∑

j
=
0


n
−
1


(
−
1

)

j



h

n
−
j
,
0


.


{\displaystyle p_{a}=\sum _{j=0}^{n-1}(-1)^{j}h^{n-j,0}.}


When n=1, the formula becomes




p

a


=

h

1
,
0




{\displaystyle p_{a}=h^{1,0}}

. According to the Hodge theorem,




h

0
,
1


=

h

1
,
0




{\displaystyle h^{0,1}=h^{1,0}}

. Consequently




h

0
,
1


=

h

1


(
X
)

/

2
=
g


{\displaystyle h^{0,1}=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 hp,q = hq,p recovers the earlier definition for projective varieties.




Kähler manifolds


By using hp,q = hq,p for compact Kähler manifolds this can be
reformulated as the Euler characteristic in coherent cohomology for the structure sheaf






O



M




{\displaystyle {\mathcal {O}}_{M}}

:





p

a


=
(
−
1

)

n


(
χ
(



O



M


)
−
1
)
.



{\displaystyle p_{a}=(-1)^{n}(\chi ({\mathcal {O}}_{M})-1).\,}


This definition therefore can be applied to some other
locally ringed spaces.




See also


Genus (mathematics)
Geometric genus




References


P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library (2nd ed.). Wiley Interscience. p. 494. ISBN 0-471-05059-8. Zbl 0836.14001.
Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3




Further reading


Hirzebruch, Friedrich (1995) [1978]. Topological methods in algebraic geometry. Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel (Reprint of the 2nd, corr. print. of the 3rd ed.). Berlin: Springer-Verlag. ISBN 3-540-58663-6. Zbl 0843.14009.

Kata Kunci Pencarian:

• Kurva eliptik
• Geometri aritmetika
• Teori bilangan
• Gerd Faltings
• Bentuk modular
• Arithmetic genus
• Genus (mathematics)
• Genus–degree formula
• Glossary of arithmetic and diophantine geometry
• Arithmetic geometry
• Algebraic surface
• Projective variety
• Riemann–Roch theorem
• Genus of a multiplicative sequence
• Number theory