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 , [4].

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 [5].

References

[1]	B. Malgrange, "Analytic spaces" Enseign. Math. Ser. 2 , 14 : 1 (1968) pp. 1–28
[2]	W. Kaup, "Infinitesimal Transformationsgruppen komplexer Räume" Math. Ann. , 160 : 1 (1965) pp. 72–92
[3a]	L. Schwartz, "Variedades analiticas complejas elipticas" , Univ. Nac. Colombia (1956)
[3b]	L. Schwartz, "Ecuaciones differenciales parciales" , Univ. Nac. Colombia (1956)
[4]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[5]	Th. Bloom, "Differential operators on curves" Rice Univ. Stud. , 59 : 2 (1973) pp. 13–19
[6]	R. Berger, R. Kiehl, E. Kunz, H.-J. Nastold, "Differentialrechnung in der analytischen Geometrie" , Springer (1967)
[7]	G. Fischer, "Complex analytic geometry" , Springer (1976)