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- Exponential_sheaf_sequence abstract "In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphismbecause for a holomorphic function f, exp(f) is a non-vanishing holomorphic function, and exp(f + g) = exp(f)exp(g). Its kernel is the sheaf 2πiZ of locally constant functions on M taking the values 2πin, with n an integer. The exponential sheaf sequence is thereforeThe exponential mapping here is not always a surjective map on sections; this can be seen for example when M is a punctured disk in the complex plane. The exponential map is surjective on the stalks: Given a germ g of an holomorphic function at a point P such that g(P) ≠ 0, one can take the logarithm of g in a neighborhood of P. The long exact sequence of sheaf cohomology shows that we have an exact sequencefor any open set U of M. Here H0 means simply the sections over U, and the sheaf cohomology H1(2πiZ|U) is the singular cohomology of U. The connecting homomorphism is therefore a generalized winding number and measures the failure of U to be contractible. In other words, there is a potential topological obstruction to taking a global logarithm of a non-vanishing holomorphic function, something that is always locally possible.A further consequence of the sequence is the exactness ofHere H1(OM*) can be identified with the Picard group of holomorphic line bundles on M. The connecting homomorphism sends a line bundle to its first Chern class.".
- Exponential_sheaf_sequence wikiPageID "1881339".
- Exponential_sheaf_sequence wikiPageRevisionID "544011278".
- Exponential_sheaf_sequence hasPhotoCollection Exponential_sheaf_sequence.
- Exponential_sheaf_sequence subject Category:Complex_manifolds.
- Exponential_sheaf_sequence subject Category:Sheaf_theory.
- Exponential_sheaf_sequence type Artifact100021939.
- Exponential_sheaf_sequence type ComplexManifolds.
- Exponential_sheaf_sequence type Conduit103089014.
- Exponential_sheaf_sequence type Manifold103717750.
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- Exponential_sheaf_sequence comment "In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian groups. The exponential function gives a sheaf homomorphismbecause for a holomorphic function f, exp(f) is a non-vanishing holomorphic function, and exp(f + g) = exp(f)exp(g).".
- Exponential_sheaf_sequence label "Exponential sheaf sequence".
- Exponential_sheaf_sequence label "Sequência de feixe exponencial".
- Exponential_sheaf_sequence sameAs Sequência_de_feixe_exponencial.
- Exponential_sheaf_sequence sameAs m.063c6p.
- Exponential_sheaf_sequence sameAs Q5421530.
- Exponential_sheaf_sequence sameAs Q5421530.
- Exponential_sheaf_sequence sameAs Exponential_sheaf_sequence.
- Exponential_sheaf_sequence wasDerivedFrom Exponential_sheaf_sequence?oldid=544011278.
- Exponential_sheaf_sequence isPrimaryTopicOf Exponential_sheaf_sequence.