In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space.
The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.
Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels.
On an arbitrary ringed space, quasi-coherent sheaves do not necessarily form an abelian category.
On the other hand, the quasi-coherent sheaves on any scheme form an abelian category, and they are extremely useful in that context.
In particular, the Oka coherence theorem states that the sheaf of holomorphic functions on a complex analytic space
A related fact is that the dimension of the fibers of a coherent sheaf is upper-semicontinuous.
[12] On a general scheme, one cannot determine whether a coherent sheaf is a vector bundle just from its fibers (as opposed to its stalks).
Thus sections of the canonical bundle are algebro-geometric analogs of volume forms on
[14] This contrasts with the simpler case of affine space, where a closed subscheme is simply the zero set of some collection of regular functions.
Over the complex numbers, this means that projective space has a Kähler metric with positive Ricci curvature.
If we use the fact that given an exact sequence of vector bundles with ranks
Moreover, any isomorphism given on the left corresponds to a locally free sheaf in the middle of the extension on the right.
This vector bundle can then be further studied using cohomological invariants to determine if it is stable or not.
This forms the basis for studying moduli of stable vector bundles in many specific cases, such as on principally polarized abelian varieties[17] and K3 surfaces.
[20] These satisfy the same formal properties as Chern classes in topology.
is the quotient of the free abelian group on the set of isomorphism classes of vector bundles on
is hard to compute in general, algebraic K-theory provides many tools for studying it, including a sequence of related groups
is a regular separated Noetherian scheme, using that every coherent sheaf has a finite resolution by vector bundles in that case.
[21] For example, that gives a definition of the Chern classes of a coherent sheaf on a smooth variety over a field.
on a Noetherian scheme is quasi-isomorphic in the derived category to the complex of vector bundles :
with For example, this formula is useful for finding the Chern classes of the sheaf representing a subscheme of
When vector bundles and locally free sheaves of finite constant rank are used interchangeably, care must be given to distinguish between bundle homomorphisms and sheaf homomorphisms.
The quasi-coherent sheaves on any fixed scheme form an abelian category.
(such as an algebraic variety over a field) is determined up to isomorphism by the abelian category of quasi-coherent sheaves on
[23] The fundamental technical tool in algebraic geometry is the cohomology theory of coherent sheaves.
Although it was introduced only in the 1950s, many earlier techniques of algebraic geometry are clarified by the language of sheaf cohomology applied to coherent sheaves.
Broadly speaking, coherent sheaf cohomology can be viewed as a tool for producing functions with specified properties; sections of line bundles or of more general sheaves can be viewed as generalized functions.
In complex analytic geometry, coherent sheaf cohomology also plays a foundational role.
Among the core results of coherent sheaf cohomology are results on finite-dimensionality of cohomology, results on the vanishing of cohomology in various cases, duality theorems such as Serre duality, relations between topology and algebraic geometry such as Hodge theory, and formulas for Euler characteristics of coherent sheaves such as the Riemann–Roch theorem.