Exponential sheaf sequence
Encyclopedia
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 function
s on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian group
s. The exponential function
gives a sheaf homomorphism
because for an holomorphic function f, exp(f) is a non-vanishing holomorphic function, and exp(f+g) = exp(f)exp(g). Its kernel
can be identified as the sheaf denoted by 2πiZ, meaning the sheaf on M of locally constant function
s taking values which are 2πin, with n an integer
. The exponential sheaf sequence is therefore
The 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; because given a germ g of an holomorphic function at a point P, such that g(P) ≠ 0, one can take the logarithm
of g close enough to P. The long exact sequence of sheaf cohomology
shows that we have an exact sequence
for any open set U of M. Here H0 means simply the sections over U; while the sheaf cohomology H1 in this case is essentially the singular cohomology of U. Therefore there is a kind of winding number
invariant: if U is not contractible, the exponential map on sections may not be surjective. 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 of
Here H1(OM*) can be identified with the Picard group of holomorphic line bundles on M. The homomorphism to the H2 group is essentially the first Chern class
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the exponential sheaf sequence is a fundamental short exact sequence of sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
used in complex geometry
Complex geometry
In mathematics, complex geometry is the study of complex manifolds and functions of many complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric chapters of complex analysis....
.
Let M be a complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
, and write OM for the sheaf of holomorphic function
Holomorphic function
In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
s on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s. The exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
gives a sheaf homomorphism
because for an holomorphic function f, exp(f) is a non-vanishing holomorphic function, and exp(f+g) = exp(f)exp(g). Its kernel
Kernel (mathematics)
In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:* The kernel of a mapping is the set of elements that map to the zero element , as in kernel of a linear operator and kernel of a matrix...
can be identified as the sheaf denoted by 2πiZ, meaning the sheaf on M of locally constant function
Locally constant function
In mathematics, a function f from a topological space A to a set B is called locally constant, if for every a in A there exists a neighborhood U of a, such that f is constant on U....
s taking values which are 2πin, with n an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
. The exponential sheaf sequence is therefore
The 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; because given a germ g of an holomorphic function at a point P, such that g(P) ≠ 0, one can take the logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...
of g close enough to P. The long exact sequence of sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
shows that we have an exact sequence
for any open set U of M. Here H0 means simply the sections over U; while the sheaf cohomology H1 in this case is essentially the singular cohomology of U. Therefore there is a kind of winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
invariant: if U is not contractible, the exponential map on sections may not be surjective. 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 of
Here H1(OM*) can be identified with the Picard group of holomorphic line bundles on M. The homomorphism to the H2 group is essentially the first Chern class
Chern class
In mathematics, in particular in algebraic topology and differential geometry, the Chern classes are characteristic classes associated to complex vector bundles.Chern classes were introduced by .-Basic idea and motivation:...
.