Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant p_g of algebraic varieties, 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). 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 definition of geometric genus is carried over classically to singular curves C, 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.

The geometric genus is the first invariant p_g = P₁ of a sequence of invariants P_n called the plurigenera.

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