Holomorphic Euler characteristic
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.
Much of algebraic geometry and complex analytic geometry is formulated in terms of coherent sheaves and their cohomology.
Coherent sheaves can be seen as a generalization of vector bundles. There is a notion of a coherent analytic sheaf on a complex analytic space, and an analogous notion of a coherent algebraic sheaf on a scheme. In both cases, the given space X comes with a sheaf of rings OX, the sheaf of holomorphic functions or regular functions, and coherent sheaves are defined as a full subcategory of the category of OX-modules (that is, sheaves of OX-modules).
Vector bundles such as the tangent bundle play a fundamental role in geometry. More generally, for a closed subvariety Y of X with inclusion i: Y → X, a vector bundle E on Y determines a coherent sheaf on X, the direct image sheaf i*E, which is zero outside Y. In this way, many questions about subvarieties of X can be expressed in terms of coherent sheaves on X.
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