Geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.
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 plus one.
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
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.
In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree 2g − 2.
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