Definition
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.
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 canonical line bundle.
The geometric genus is the first invariant pg = P1 of a sequence of invariants Pn called the plurigenera.
Read more about this topic: Geometric Genus
Famous quotes containing the word definition:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)