# Differential calculus (on analytic spaces)

A generalization of the classical calculus of differential forms and differential operators to analytic spaces. For the calculus of differential forms on complex manifolds see Differential form. Let $( X, {\mathcal O} _ {X} )$ be an analytic space over a field $k$, let $\Delta$ be the diagonal in $X \times X$, let $J$ be the sheaf of ideals defining $\Delta$ and generated by all germs of the form $\pi _ {1} ^ {*} f - \pi _ {2} ^ {*} f$, where $f$ is an arbitrary germ from ${\mathcal O} _ {X}$, and let $\pi _ {i} : X \times X \rightarrow X$ be projection on the $i$- th factor.

The analytic sheaf $\pi _ {1} ( J / J ^ {2} ) = \Omega _ {X} ^ {1}$ is known as the sheaf of analytic differential forms of the first order on $X$. If $f$ is the germ of an analytic function on $X$, then the germ $\pi _ {1} ^ {*} f - \pi _ {2} ^ {*} f$ belongs to $J$ and defines the element $df$ of $\Omega _ {X} ^ {1}$ known as the differential of the germ $f$. This defines a sheaf homomorphism of vector spaces $d : {\mathcal O} _ {X} \rightarrow \Omega _ {X} ^ {1}$. If $X = k ^ {n}$, then $\Omega _ {X} ^ {1}$ is the free sheaf generated by $dx _ {1} \dots dx _ {n}$, where $x _ {1} \dots x _ {n}$ are the coordinates in $k ^ {n}$. If $X$ is an analytic subspace in $k ^ {n}$, defined by a sheaf of ideals $J$, then

$$\Omega _ {X} ^ {1} \cong \Omega _ {k ^ {n} } ^ {1} / ( J \Omega _ {k ^ {n} } ^ {1} + dJ) \mid _ {X} .$$

Each analytic mapping $f : X \rightarrow Y$ may be related to a sheaf of relative differentials $\Omega _ {X/Y} ^ {1}$. This is the analytic sheaf $\Omega _ {X/Y} ^ {1}$ inducing $\Omega _ {X _ {s} } ^ {1}$ on each fibre $X _ {s}$( $s \in Y$) of $f$; it is defined from the exact sequence

$$f ^ { * } \Omega _ {Y} ^ {1} \rightarrow \Omega _ {X} ^ {1} \rightarrow \Omega _ {X/Y} ^ {1} \rightarrow 0 .$$

The sheaf $\Theta _ {X} = \mathop{\rm Hom} _ { {\mathcal O} _ {X} } ( \Omega _ {X} ^ {1} , {\mathcal O} _ {X} )$ is called the sheaf of germs of analytic vector fields on $X$. If $X$ is a manifold, $\Omega _ {X} ^ {1}$ and $\Theta _ {X}$ are locally free sheaves, which are naturally isomorphic to the sheaf of analytic sections of the cotangent and the tangent bundle over $X$, respectively.

The analytic sheaves $\Omega _ {X} ^ {p} = \wedge _ { {\mathcal O} _ {X} } ^ {p} \Omega _ {X} ^ {1}$ are called sheaves of analytic exterior differential forms of degree $p$ on $X$( if $k = \mathbf C$, they are also called holomorphic forms). For any $p\geq 0$ one may define a sheaf homomorphism of vector spaces $d ^ {p} : \Omega _ {X} ^ {p} \rightarrow \Omega _ {X} ^ {p+} 1$, which for $p= 0$ coincides with the one introduced above, and which satisfies the condition $d ^ {p+} 1 d ^ {p} = 0$. The complex of sheaves $( \Omega _ {X} ^ {*} , d)$ is called the de Rham complex of the space $X$. If $X$ is a manifold and $k = \mathbf C$ or $\mathbf R$, the de Rham complex is an exact complex of sheaves. If $X$ is a Stein manifold or a real-analytic manifold, the cohomology groups of the complex of sections $\Gamma ( \Omega _ {X} ^ {*} )$, which is also often referred to as the de Rham complex, are isomorphic to $H ^ {p} ( X , k)$.

If $X$ has singular points, the de Rham complex need not be exact. If $k = \mathbf C$, a sufficient condition for the exactness of the de Rham complex at a point $x \in X$ is the presence of a complex-analytic contractible neighbourhood at $x$. The hyperhomology groups of the complex $\Gamma ( \Omega _ {X} ^ {*} )$ contain, for $k= \mathbf C$, the cohomology groups of the space $X$ with coefficients in $\mathbf C$ as direct summands, and are identical with them if $X$ is smooth. The sections of the sheaf $\Theta _ {X}$ are called analytic (and if $k= \mathbf C$, also holomorphic) vector fields on $X$. For any open $U \subset X$ the field $Z \in \Gamma ( X , \Theta _ {X} )$ defines a derivation in the algebra of analytic functions $\Gamma ( U , {\mathcal O} _ {X} )$, acting according to the formula $\phi \rightarrow Z _ \phi = Z ( d \phi )$. If $k= \mathbf C$ or $\mathbf R$, then $Z$ defines a local one-parameter group $\mathop{\rm exp} Z$ of automorphisms of the space $X$. If, in addition, $X$ is compact, the group $\mathop{\rm exp} Z$ is globally definable.

The space $\Gamma ( X, \Theta _ {X} )$ provided with the Lie bracket is a Lie algebra over $k$. If $X$ is a compact complex space, $\Gamma ( X, \Theta _ {X} )$ is the Lie algebra of the group $\mathop{\rm Aut} X$.

Differential operators on an analytic space $( X, {\mathcal O} _ {X} )$ are defined in analogy to the differential operators on a module (cf. Differential operator on a module). If $F, G$ are analytic sheaves on $X$, then a linear differential operator of order $\leq l$, acting from $F$ into $G$, is a sheaf homomorphism of vector spaces $F \rightarrow G$ which extends to an analytic homomorphism $F \otimes \pi _ {1} ( {\mathcal O} _ {X \times X } / I ^ {l+} 1 ) \rightarrow G$. If $X$ is smooth and $F$ and $G$ are locally free, this definition gives the usual concept of a differential operator on a vector bundle , .

The germs of the linear differential operators $F \rightarrow G$ form an analytic sheaf $\mathop{\rm Diff} ( F, G)$ with filtration

$$\mathop{\rm Diff} ^ {0} ( F, G) \subset \dots \subset \mathop{\rm Diff} ^ {l} ( F, G) \subset \dots ,$$

where $\mathop{\rm Diff} ^ {l} ( F, G)$ is the sheaf of germs of operators of order $< l$. In particular, $\mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} )$ is a filtered sheaf of associative algebras over $k$ under composition of mappings. One has

$$\mathop{\rm Diff} ^ {0} ( F, G) \cong \mathop{\rm Hom} _ {\mathcal O} ( F, G) ,$$

$$\mathop{\rm Diff} ^ {1} ( {\mathcal O} , {\mathcal O} ) / \mathop{\rm Diff} ^ {0} ( {\mathcal O} , {\mathcal O} ) \cong \Theta _ {X} .$$

The sheaf $\mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} )$ was studied (for the non-smooth case) only for certain special types of singular points. In particular, it was proved in the case of an irreducible one-dimensional complex space $X$ that the sheaf of algebras $\mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} )$ and the corresponding sheaf of graded algebras have finite systems of generators .

How to Cite This Entry:
Differential calculus (on analytic spaces). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_calculus_(on_analytic_spaces)&oldid=46663
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article