Geometric genus

Template:Short description In algebraic geometry, the geometric genus is a basic birational invariant p_g of algebraic varieties and complex manifolds.

Definition

The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number h^n,0 (equal to h^0,n by Serre duality), that is, the dimension of the canonical linear system plus one.

In other words, for a variety Template:Mvar of complex dimension Template:Mvar it is the number of linearly independent holomorphic Template:Mvar-forms to be found on Template:Mvar. 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 p_g = P₁ of a sequence of invariants P_n 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

$g=\frac{(d-1)(d-2)}{2}-s,$

where s is the number of singularities when properly counted.

If Template:Mvar is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree Template:Mvar, then its normal line bundle is the Serre twisting sheaf Template:Tmath(d), so by the adjunction formula, the canonical line bundle of Template:Mvar is given by

${{𝒦}}_C={[{{𝒦}}_{{\mathbb{\mathbb{P}}}^2}+{𝒪}(d)]}_{|C}={𝒪}(d-3{)}_{|C}$

Genus of singular varieties

The definition of geometric genus is carried over classically to singular curves Template:Mvar, by decreeing that

p_g(C)

is the geometric genus of the normalization C′. That is, since the mapping

C′ → C

is birational, the definition is extended by birational invariance.

See also

• Genus (mathematics)
• Arithmetic genus
• Invariants of surfaces