mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the exponential sheaf sequence is a fundamental

short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...

of sheaves used in

complex geometry In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces su ...

. Let ''M'' be a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

, and write ''O''_''M'' for the sheaf of

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

s on ''M''. Let ''O''_''M''* be the subsheaf consisting of the non-vanishing holomorphic functions. These are both sheaves of

abelian group In mathematics, 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 the order in which they are written. That is, the group operation is commu ...

s. The exponential function gives a sheaf homomorphism :

\exp : \mathcal O_M \to \mathcal O_M^*,

because for a holomorphic function ''f'', exp(''f'') is a non-vanishing holomorphic function, and exp(''f'' + ''g'') = exp(''f'')exp(''g''). Its

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...

is the sheaf 2π''i''Z of

locally constant function In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...

s on ''M'' taking the values 2π''in'', with ''n'' an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

. The exponential sheaf sequence is therefore :

0\to 2\pi i\,\mathbb Z \to \mathcal O_M\to\mathcal O_M^*\to 0.

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: Given a

germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embry ...

''g'' of an holomorphic function at a point ''P'' such that ''g''(''P'') ≠ 0, one can take the

logarithm In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...

of ''g'' in a neighborhood of ''P''. The

long exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...

sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem glob ...

shows that we have an exact sequence :

\cdots\to H^0(\mathcal O_U) \to H^0(\mathcal O_U^*)\to H^1(2\pi i\,\mathbb Z, _U) \to \cdots

for any open set ''U'' of ''M''. Here ''H''⁰ means simply the sections over ''U'', and the sheaf cohomology ''H''¹(2π''i''Z, _''U'') is the

singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

of ''U''. One can think of ''H''¹(2π''i''Z, _''U'') as associating an integer to each loop in ''U''. For each section of ''O''_''M''*, the connecting homomorphism to ''H''¹(2π''i''Z, _''U'') gives the

winding number In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ...

for each loop. So this homomorphism is therefore a generalized

and measures the failure of ''U'' to be

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...

. 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 :

\cdots\to H^1(\mathcal O_M)\to H^1(\mathcal O_M^*)\to H^2(2\pi i\,\mathbb Z)\to \cdots.

Here ''H''¹(''O''_''M''*) can be identified with the

Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ver ...

holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...

s on ''M''. The connecting homomorphism sends a line bundle to its first

Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...

References

* , see especially p. 37 and p. 139 {{DEFAULTSORT:Exponential Sheaf Sequence Complex manifolds Sheaf theory