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.
Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf.
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.
Let X be a smooth variety of dimension n over a field k. Define the canonical line bundle
Namely, the composition of the cup product with a natural trace map on
[1] Serre also proved the same duality statement for X a compact complex manifold and E a holomorphic vector bundle.
equipped with a Riemannian metric, there is a Hodge star operator: where
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.
means wedge product of differential forms and using the pairing between
The Hodge theorem for Dolbeault cohomology asserts that if we define: where
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.
The statement of Serre duality in the algebraic setting may be recovered by taking
(Over the complex numbers, it is equivalent to consider compact Riemann surfaces.)
For a line bundle L on a smooth projective curve X over a field k, the only possibly nonzero cohomology groups are
For a line bundle L of degree d on a curve X of genus g, the Riemann–Roch theorem says that: Using Serre duality, this can be restated in more elementary terms: The latter statement (expressed in terms of divisors) is in fact the original version of the theorem from the 19th century.
Example: Every global section of a line bundle of negative degree is zero.
This is the basic calculation needed to show that the moduli space of curves of genus g has dimension
Another formulation of Serre duality holds for all coherent sheaves, not just vector bundles.
Namely, for a Cohen–Macaulay scheme X of pure dimension n over a field k, Grothendieck defined a coherent sheaf
Suppose in addition that X is proper over k. For a coherent sheaf E on X and an integer i, Serre duality says that there is a natural isomorphism: of finite-dimensional k-vector spaces.
In order to use this result, one has to determine the dualizing sheaf explicitly, at least in special cases.
More generally, if X is a Cohen–Macaulay subscheme of codimension r in a smooth scheme Y over k, then the dualizing sheaf can be described as an Ext sheaf:[5] 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:[6] In this case, X is a Cohen–Macaulay scheme with
for integers d, with the property that homogeneous polynomials of degree d can be viewed as sections of O(d).
Then the dualizing sheaf of X is the line bundle: by the adjunction formula.
Of course, the last statement depends on the Bogomolev–Tian–Todorov theorem which states every deformation on a Calabi–Yau is unobstructed.
; that is, it is the dualizing sheaf discussed above, viewed as a complex in (cohomological) degree −n.
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: for any object E in