In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an n-dimensional variety, the theorem says that a cohomology group is the dual space of another one, . Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.
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.
Serre duality for vector bundles
Algebraic theorem
Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle to be the bundle of n-forms on X, the top exterior power of the cotangent bundle:
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:
of finite-dimensional k-vector spaces. Here denotes the tensor product of vector bundles. It follows that the dimensions of the two cohomology groups are equal:
As in Poincaré duality, the isomorphism in Serre duality comes from the cup product in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on is a perfect pairing:
The trace map is the analog for coherent sheaf cohomology of integration in de Rham cohomology.
Differential-geometric theorem
Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle. Here, the Serre duality theorem is a consequence of Hodge theory. Namely, on a compact complex manifold equipped with a Riemannian metric, there is a Hodge star operator:
where . Additionally, since is complex, there is a splitting of the complex differential forms into forms of type . The Hodge star operator (extended complex-linearly to complex-valued differential forms) interacts with this grading as:
Notice that the holomorphic and anti-holomorphic indices have switched places. There is a conjugation on complex differential forms which interchanges forms of type and , and if one defines the conjugate-linear Hodge star operator by then we have:
Using the conjugate-linear Hodge star, one may define a Hermitian -inner product on complex differential forms, by:
where now is an -form, and in particular a complex-valued -form and can therefore be integrated on with respect to its canonical orientation. Furthermore, suppose is a Hermitian holomorphic vector bundle. Then the Hermitian metric gives a conjugate-linear isomorphism between and its dual vector bundle, say . Defining , one obtains an isomorphism:
where consists of smooth -valued complex differential forms. Using the pairing between and given by and , one can therefore define a Hermitian -inner product on such -valued forms by:
where here means wedge product of differential forms and using the pairing between and given by .
The Hodge theorem for Dolbeault cohomology asserts that if we define:
where is the Dolbeault operator of and is its formal adjoint with respect to the inner product, then:
On the left is Dolbeault cohomology, and on the right is the vector space of harmonic -valued differential forms defined by:
Using this description, the Serre duality theorem can be stated as follows: The isomorphism induces a complex linear isomorphism:
This can be easily proved using the Hodge theory above. Namely, if is a cohomology class in with unique harmonic representative , then:
with equality if and only if . In particular, the complex linear pairing:
between and is non-degenerate, and induces the isomorphism in the Serre duality theorem.
The statement of Serre duality in the algebraic setting may be recovered by taking , and applying Dolbeault's theorem, which states that:
where on the left is Dolbeault cohomology and on the right sheaf cohomology, where denotes the sheaf of holomorphic -forms. In particular, we obtain: