In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, a branch of
mathematics, Serre duality is a
duality for the
coherent sheaf cohomology of algebraic varieties, proved by
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
. The basic version applies to
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s 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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by cons ...
of another one,
. Serre duality is the analog for coherent sheaf cohomology of
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold ( compa ...
in topology, with the
canonical line bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers, it ...
replacing the
orientation sheaf.
The Serre duality theorem is also true in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and ...
more generally, for compact
complex manifolds
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a ...
that are not necessarily
projective complex algebraic varieties
In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers. Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. ''Algebraic Geometry III: Complex Algeb ...
. In this setting, the Serre duality theorem is an application of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
for
Dolbeault cohomology In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohom ...
, and may be seen as a result in the theory of
elliptic operator
In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which imp ...
s.
These two different interpretations of Serre duality coincide for non-singular projective complex algebraic varieties, by an application of
Dolbeault's theorem In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault coho ...
relating sheaf cohomology to Dolbeault cohomology.
Serre duality for vector bundles
Algebraic theorem
Let ''X'' be a
smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smo ...
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
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
:
:
Suppose in addition that ''X'' is
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map fo ...
(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
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...
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 mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commuta ...
in sheaf cohomology. Namely, the composition of the cup product with a natural trace map on
is a
perfect pairing
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear ...
:
:
The trace map is the analog for coherent sheaf cohomology of integration in
de Rham cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapt ...
.
Differential-geometric theorem
Serre also proved the same duality statement for ''X'' a compact
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a c ...
and ''E'' a
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a c ...
.
Here, the Serre duality theorem is a consequence of
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
. Namely, on a compact complex manifold
equipped with a
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
, there is a
Hodge star operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the a ...
:
where
. Additionally, since
is complex, there is a splitting of the
complex differential form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, t ...
s 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 {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...
-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
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
. Furthermore, suppose
is a Hermitian holomorphic vector bundle. Then the Hermitian metric
gives a conjugate-linear isomorphism
between
and its
dual vector bundle In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces.
Definition
The dual bundle of a vector bundle \pi: E \to X is the vector bundle \pi^*: E^* \to X whose fibers are the dual sp ...
, 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