# Geometric genus

In algebraic geometry, the **geometric genus** is a basic birational invariant *p*_{g} 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 *h*^{n,0} (equal to *h*^{0,n} by Serre duality), that is, the dimension of the canonical linear system plus one.

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.[1] This definition, as the dimension of

*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 canonical line bundle.

The geometric genus is the first invariant *p*_{g} = *P*_{1} of a sequence of invariants *P*_{n} called the plurigenera.

## Case of curves

In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical line bundle has degree 2*g* − 2.

The notion of genus features prominently in the statement of the Riemann–Roch theorem (see also Riemann–Roch theorem for algebraic curves) and of the Riemann–Hurwitz formula. By the Riemann-Roch theorem, an irreducible plane curve of degree *d* has geometric genus

where *s* is the number of singularities when properly counted

If C is an irreducible (and smooth) hypersurface in the projective plane cut out by a polynomial equation of degree d, then its normal line bundle is the Serre twisting sheaf (*d*), so by the adjunction formula, the canonical line bundle of C is given by

## Genus of singular varieties

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.

## Notes

- Danilov & Shokurov (1998), p. 53

## References

- P. Griffiths; J. Harris (1994).
*Principles of Algebraic Geometry*. Wiley Classics Library. Wiley Interscience. p. 494. ISBN 0-471-05059-8. - V. I. Danilov; Vyacheslav V. Shokurov (1998).
*Algebraic curves, algebraic manifolds, and schemes*. Springer. ISBN 978-3-540-63705-9.