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geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant 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 (equal to by Serre duality), that is, the dimension of the canonical linear system plus one.
In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on .〔Danilov & Shokurov (1998), (p. 53 )〕 This definition, as the dimension of
:
then carries over to any base field, when 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 of a sequence of invariants called the plurigenera.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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