# Extension theorems (in analytic geometry)

Statements on the continuation (extension) of functions, sections of analytic sheaves, analytic sheaves, analytic subsets, holomorphic and meromorphic mappings, from the complement in an analytic space of a set (as a rule, also analytic) to the whole space . Two theorems of B. Riemann form the classical results concerning continuation of functions.
Riemann's first theorem states that every analytic function on , where is a normal complex space and an analytic subspace of codimension , can be continued to an analytic function on . Riemann's second theorem states that every analytic function on that is locally bounded on , where is a nowhere-dense analytic subset in a normal complex space , can be continued to an analytic function on . There are generalizations of these theorems to arbitrary complex spaces , as well as to sections of coherent analytic sheaves (cf. Local cohomology).
Important results concerning extension of analytic subsets are the theorems of Remmert–Stein–Shiffman and Bishop. The Remmert–Stein–Shiffman theorem states that every pure -dimensional complex-analytic subset in , where is a complex-analytic space and a closed subset having zero -dimensional Hausdorff measure, can be extended to a pure -dimensional complex-analytic subset in . Bishop's theorem states that every pure -dimensional complex-analytic subset in , where is a complex-analytic space and is a complex-analytic subset, can be extended to a pure -dimensional complex-analytic subset in if has locally finite volume in some neighbourhood of in .
There are criteria for extendability of analytic mappings, generalizing the classical Picard theorem. E.g., every analytic mapping , where is a complex manifold, is an analytic nowhere-dense set and is a hyperbolic compact complex manifold, can be extended to an analytic mapping . Every analytic mapping that is not everywhere-degenerate, where is a complex manifold, is an analytic subset and is a compact complex manifold with negative first Chern class, can be extended to a meromorphic mapping .