Serre Duality - Algebraic Curve

Algebraic Curve

The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve C the coherent groups Hi vanish for i > 1; but H1 does enter implicitly. In fact, the basic relation of the theorem involves l(D) and l(K−D), where D is a divisor and K is a divisor of the canonical class. After Serre we recognise l(K−D) as the dimension of H1(D), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H1(D) and H0(KD*), and we are reading off dimensions (notation: K is the canonical line bundle, D* is the dual line bundle, and juxtaposition is the tensor product of line bundles).

In this formulation the Riemann-Roch theorem can be viewed as a computation of the Euler characteristic of a sheaf

h0(D) − h1(D),

in terms of the genus of the curve, which is

h1(C,O_C),

and the degree of D. It is this expression that can be generalised to higher dimensions.

Serre duality of curves is therefore something very classical; but it has an interesting light to cast. For example, in Riemann surface theory, the deformation theory of complex structures is studied classically by means of quadratic differentials (namely sections of L(K2)). The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via H1(T), where T is the tangent bundle sheaf K*. The duality shows why these approaches coincide.