The geometric genus of a smooth projective variety is the dimension
of its space of top-degree holomorphic forms,
For an integral projective curve it is the genus of its algebraic normalization (Huybrechts 2016). For surfaces, the condition is the hypothesis in the Bloch
conjecture.
is the dimension
of its space of top-degree holomorphic forms,
is the hypothesis in the Bloch
conjecture.
