Serre duality

In algebraic geometry, Serre duality is a duality (mathematics) present on non-singular projective algebraic variety V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaf. It shows that a Cohomology group H i is the dual space of another one, H n − i . If the variety is defined over the complex numbers, this is therefore quite distinct from Poincaré duality, which relates H i to H 2 n − i because as a manifold V has dimension 2 n .

The case of of line bundles).

In this formulation the theorem can be rearranged to read as a calculation of the Euler characteristic of a sheaf

: h 0( D ) − h 1( D ),

in terms of the genus (mathematics) of the curve, which is

: h 1( 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 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 ( K 2)). The deformation theory of Kunihiko Kodaira and D. C. Spencer identifies deformations via H 1( T ), where T is the tangent bundle sheaf K *. The duality shows why these approaches coincide.

The origin of the theory lay in Serre s earlier work on several complex variables. In the generalisation of Alexander Grothendieck, Serre duality becomes a part of coherent duality in a much broader setting. While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot just be a single sheaf in the absence of some hypothesis of non-singularity on V . The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves. The statement of the theorem is recognisably Serre s, however.