Definition
The geometric genus can be 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), that is, the dimension of the canonical linear system.
In other words for a variety V of complex dimension n it is the number of linearly independent holomorphic n-forms to be found on V. This definition, as the dimension of
- H0(V,Ωn)
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 canonical line bundle.
The geometric genus is the first invariant pg = P1 of a sequence of invariants Pn called the plurigenera.
Read more about this topic: Geometric Genus
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