Serre duality

The Serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. In this setting, the Serre duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators.

These two different interpretations of Serre duality coincide for non-singular projective complex algebraic varieties, by an application of Dolbeault's theorem relating sheaf cohomology to Dolbeault cohomology.

Suppose in addition that X is proper (for example, projective) over k. Then Serre duality says: for an algebraic vector bundle E on X and an integer i, there is a natural isomorphism

The trace map is the analog for coherent sheaf cohomology of integration in de Rham cohomology.

Serre duality is especially relevant to the Riemann–Roch theorem for curves. For a line bundle L of degree d on a curve X of genus g, the Riemann–Roch theorem says that

The latter statement (expressed in terms of divisors) is in fact the original version of the theorem from the 19th century. This is the main tool used to analyze how a given curve can be embedded into projective space and hence to classify algebraic curves.

Another formulation of Serre duality holds for all coherent sheaves, not just vector bundles. As a first step in generalizing Serre duality, Grothendieck showed that this version works for schemes with mild singularities, Cohen–Macaulay schemes, not just smooth schemes.

When X is a local complete intersection of codimension r in a smooth scheme Y, there is a more elementary description: the normal bundle of X in Y is a vector bundle of rank r, and the dualizing sheaf of X is given by

Using the dualizing complex, Serre duality generalizes to any proper scheme X over k. Namely, there is a natural isomorphism of finite-dimensional k-vector spaces

Serre duality holds more generally for proper algebraic spaces over a field.