pseudosections

Pseudosections are a generalized notion of a section used in various areas of geometry and topology. The exact definition of a pseudosection varies by context, but it generally represents a relaxation of the standard concept of a section of a fiber bundle, sheaf, or similar structure. The common idea is that a pseudosection behaves like a local section but may not glue together to a global section due to obstructions or singularities.

In differential geometry, a pseudosection often refers to a locally defined lift or a map that is defined on an open subset and extends with controlled behavior near its boundary, yet cannot be extended to a global section because of topological obstructions such as monodromy or singularities. For example, a choice of local vector field that cannot be extended smoothly to the whole manifold may be described as a pseudosection.

In algebraic geometry and related areas, pseudosections can refer to sections that are defined only on a dense open subset (rational or generically defined sections) or to multivalued or base-changed notions of a section that become genuine after a suitable modification. In the language of stacks and groupoids, a pseudosection may be viewed as a compatible choice of object in a fiber category that does not form a strict global section of a sheaf.

Applications of pseudosections include describing obstructions to the existence of global sections, formulating moduli problems where genuine sections do not exist, and providing flexible tools in the study of fibered structures. See also section, multivalued section, rational section, stack, and gerbe.