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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## See also

## References

- H
^{0}(V,Ω^{n})

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} = P_{1} of a sequence of invariants P_{n} called the plurigenera.

- P. Griffiths; J. Harris (1994).
*Principles of Algebraic Geometry*. Wiley Interscience.

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Last updated on Tuesday September 23, 2008 at 10:46:21 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Tuesday September 23, 2008 at 10:46:21 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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