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.
The geometric genus is the first invariant pg = P1 of a sequence of invariants Pn called the plurigenera.
Case of curves
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.
The notion of genus features prominently in the statement of the Riemann–Roch theorem (see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula. By the Riemann-Roch theorem, an irreducible plane curve of degree d has geometric genus
where s is the number of singularities when properly counted
If C is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree d, then its normal line bundle is the Serre twisting sheaf (d), so by the adjunction formula, the canonical line bundle of C is given by C = [(d)]|C = (d − 3)|C.
Genus of singular varieties
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.
- Danilov & Shokurov (1998), p. 53