Serre duality

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 formula_1 is the dual space of another one, formula_2. 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 formula_3 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 formula_6 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 formula_8 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 formula_10 equipped with a Riemannian metric, there is a Hodge star operator
where formula_12. Additionally, since formula_10 is complex, there is a splitting of the complex differential forms into forms of type formula_14. 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 formula_14 and formula_17, and if one defines the conjugate-linear Hodge star operator by formula_18 then we have
Using the conjugate-linear Hodge star, one may define a Hermitian formula_20-inner product on complex differential forms, by
where now formula_22 is an formula_23-form, and in particular a complex-valued formula_24-form, and can therefore be integrated on formula_10 with respect to its canonical orientation. Furthermore, suppose formula_26 is a Hermitian holomorphic vector bundle. Then the Hermitian metric formula_27 gives a conjugate-linear isomorphism formula_28 between formula_29 and its dual vector bundle, say formula_30. Defining formula_31, one obtains an isomorphism
where formula_33 consists of smooth formula_29-valued complex differential forms. Using the pairing between formula_29 and formula_36 given by formula_37 and formula_27, one can therefore define a Hermitian formula_20-inner product on such formula_29-valued forms by
where here formula_42 means wedge product of differential forms and using the pairing between formula_29 and formula_36 given by formula_27.
The Hodge theorem for Dolbeault cohomology asserts that if we define
where formula_47 is the Dolbeault operator of formula_29 and formula_49 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 formula_29-valued differential forms defined by
Using this description, the Serre duality theorem can be stated as follows: The isomorphism formula_53 induces a complex linear isomorphism
This can be easily proved using the Hodge theory above. Namely, if formula_55 is a cohomology class in formula_56 with unique harmonic representative formula_57, then
with equality if and only if formula_59. In particular, the complex linear pairing
between formula_61 and formula_62 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 formula_63, and applying Dolbeault's theorem, which states that
where on the left is Dolbeault cohomology and on the right sheaf cohomology, where formula_65 denotes the sheaf of holomorphic formula_66-forms. In particular, we obtain
where we have used that the sheaf of holomorphic formula_68-forms is just the canonical bundle of formula_10.
Algebraic curves.
A fundamental application of Serre duality is to algebraic curves. (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 formula_70 and formula_71. Serre duality describes the formula_72 group in terms of an formula_73 group (for a different line bundle). That is more concrete, since formula_73 of a line bundle is simply its space of sections.
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
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. This is the main tool used to analyze how a given curve can be embedded into projective space and hence to classify algebraic curves.
Example: Every global section of a line bundle of negative degree is zero. Moreover, the degree of the canonical bundle is formula_77. Therefore, Riemann–Roch implies that for a line bundle "L" of degree formula_78, formula_79 is equal to formula_80. When the genus "g" is at least 2, it follows by Serre duality that formula_81. Here formula_82 is the first-order deformation space of "X". This is the basic calculation needed to show that the moduli space of curves of genus "g" has dimension formula_83.
Serre duality for coherent sheaves.
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.
Namely, for a Cohen–Macaulay scheme "X" of pure dimension "n" over a field "k", Grothendieck defined a coherent sheaf formula_84 on "X" called the dualizing sheaf. (Some authors call this sheaf formula_3.) 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. Here the Ext group is taken in the abelian category of formula_87-modules. This includes the previous statement, since formula_88 is isomorphic to formula_89 when "E" is a vector bundle.
In order to use this result, one has to determine the dualizing sheaf explicitly, at least in special cases. When "X" is smooth over "k", formula_84 is the canonical line bundle formula_3 defined above. 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:
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
In this case, "X" is a Cohen–Macaulay scheme with formula_84 a line bundle, which says that "X" is Gorenstein.
Example: Let "X" be a complete intersection in projective space formula_95 over a field "k", defined by homogeneous polynomials formula_96 of degrees formula_97. (To say that this is a complete intersection means that "X" has dimension formula_98.) There are line bundles "O"("d") on formula_95 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. For example, the dualizing sheaf of a plane curve "X" of degree "d" is formula_101.
Complex moduli of Calabi–Yau threefolds.
In particular, we can compute the number of complex deformations, equal to formula_102 for a quintic threefold in formula_103, a Calabi–Yau variety, using Serre duality. Since the Calabi–Yau property ensures formula_104 Serre duality shows us that formula_105 showing the number of complex moduli is equal to formula_106 in the Hodge diamond. Of course, the last statement depends on the Bogomolev–Tian–Todorov theorem which states every deformation on a Calabi–Yau is unobstructed.
Grothendieck duality.
Grothendieck's theory of coherent duality is a broad generalization of Serre duality, using the language of derived categories. For any scheme "X" of finite type over a field "k", there is an object formula_107 of the bounded derived category of coherent sheaves on "X", formula_108, called the dualizing complex of "X" over "k". Formally, formula_107 is the exceptional inverse image formula_110, where "f" is the given morphism formula_111. When "X" is Cohen–Macaulay of pure dimension "n", formula_107 is formula_113; that is, it is the dualizing sheaf discussed above, viewed as a complex in (cohomological) degree −"n". In particular, when "X" is smooth over "k", formula_107 is the canonical line bundle placed in 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 formula_108.
More generally, for a proper scheme "X" over "k", an object "E" in formula_108, and "F" a perfect complex in formula_118, one has the elegant statement: 
Here the tensor product means the derived tensor product, as is natural in derived categories. (To compare with previous formulations, note that formula_88 can be viewed as formula_121.) When "X" is also smooth over "k", every object in formula_108 is a perfect complex, and so this duality applies to all "E" and "F" in formula_108. The statement above is then summarized by saying that formula_124 is a Serre functor on formula_108 for "X" smooth and proper over "k".
Serre duality holds more generally for proper algebraic spaces over a field.