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A numerical birational invariant of a two-dimensional algebraic variety defined over an algebraically closed field  $  k $. 
There are two different genera — the [[Arithmetic genus|arithmetic genus]] and the [[Geometric genus|geometric genus]]. The geometric genus  $  p _ {g} $
of a complete smooth algebraic surface  $  X $
is equal to

$$ 
p _ {g}  =   \mathop{\rm dim} _ {g}  H  ^ {0} ( X , \Omega _ {X}  ^ {2} ) ,
$$

i.e. to the dimension of the space of regular differential  $  2 $-
forms (cf. [[Differential form|Differential form]]) on  $  X $. 
The arithmetic genus  $  p _ {a} $
of a complete smooth algebraic surface  $  X $
is equal to

$$ 
p _ {a}  =  \chi ( X , {\mathcal O} _ {X} ) - 1  = \ 
 \mathop{\rm dim} _ {k}  H  ^ {2} ( X , {\mathcal O} _ {X} ) -
 \mathop{\rm dim} _ {k}  H  ^ {1} ( X , {\mathcal O} _ {X} ) .
$$

The geometric and arithmetic genera of a complete smooth algebraic surface  $  X $
are related by the formula  $  p _ {g} - p _ {a} = q $, 
where  $  q $
is the [[Irregularity|irregularity]] of  $  X $, 
which is equal to the dimension of the space of regular differential  $  1 $-
forms on  $  X $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.R., et al. Shafarevich, "Algebraic surfaces" ''Proc. Steklov Inst. Math.'' , '''75''' (1967) ''Trudy Mat. Inst. Steklov.'' , '''75''' (1965) {{MR|1392959}} {{MR|1060325}} {{ZBL|0830.00008}} {{ZBL|0733.14015}} {{ZBL|0832.14026}} {{ZBL|0509.14036}} {{ZBL|0492.14024}} {{ZBL|0379.14006}} {{ZBL|0253.14006}} {{ZBL|0154.21001}} </TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. van de Ven, "Compact complex surfaces" , Springer (1984) {{MR|}} {{ZBL|0718.14023}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , Wiley (Interscience) (1978) {{MR|0507725}} {{ZBL|0408.14001}} </TD></TR></table>