In algebraic geometry
, the geometric genus
is a basic birational invariant pg
of algebraic varieties
, defined for non-singular
complex projective varieties (and more generally for complex manifolds
) as the Hodge number hn,0
(equal to h0,n
by Serre duality
). In other words for a variety V
of complex dimension n
it is the number of linearly independent holomorphic n
to be found on V
. This definition, as the dimension of
then carries over to any base field, when Ω is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power.
The definition of geometric genus is carried over classically to singular curves C, by decreeing that
is the geometric genus of the normalization C′. That is, since the mapping
- C′ → C
is birational, the definition is extended by birational invariance.
The geometric genus is the first invariant pg = P1 of a sequence of invariants Pn called the plurigenera.