View source for 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|Differential form]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318601.png" /> be an [[Analytic space|analytic space]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318602.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318603.png" /> be the diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318604.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318605.png" /> be the sheaf of ideals defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318606.png" /> and generated by all germs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318608.png" /> is an arbitrary germ from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318609.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186010.png" /> be projection on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186011.png" />-th factor.

The analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186012.png" /> is known as the sheaf of analytic differential forms of the first order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186014.png" /> is the germ of an analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186015.png" />, then the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186016.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186017.png" /> and defines the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186019.png" /> known as the differential of the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186020.png" />. This defines a sheaf homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186023.png" /> is the free sheaf generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186025.png" /> are the coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186027.png" /> is an analytic subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186028.png" />, defined by a sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186029.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186030.png" /></td> </tr></table>

Each analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186031.png" /> may be related to a sheaf of relative differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186032.png" />. This is the analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186033.png" /> inducing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186034.png" /> on each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186036.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186037.png" />; it is defined from the exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186038.png" /></td> </tr></table>

The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186039.png" /> is called the sheaf of germs of analytic vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186041.png" /> is a manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186043.png" /> are locally free sheaves, which are naturally isomorphic to the sheaf of analytic sections of the cotangent and the tangent bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186044.png" />, respectively.

The analytic sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186045.png" /> are called sheaves of analytic exterior differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186048.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186049.png" />, they are also called holomorphic forms). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186050.png" /> one may define a sheaf homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186051.png" />, which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186052.png" /> coincides with the one introduced above, and which satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186053.png" />. The complex of sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186054.png" /> is called the de Rham complex of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186056.png" /> is a manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186057.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186058.png" />, the de Rham complex is an exact complex of sheaves. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186059.png" /> is a Stein manifold or a real-analytic manifold, the cohomology groups of the complex of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186060.png" />, which is also often referred to as the de Rham complex, are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186061.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186062.png" /> has singular points, the de Rham complex need not be exact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186063.png" />, a sufficient condition for the exactness of the de Rham complex at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186064.png" /> is the presence of a complex-analytic contractible neighbourhood at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186065.png" />. The hyperhomology groups of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186066.png" /> contain, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186067.png" />, the cohomology groups of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186068.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186069.png" /> as direct summands, and are identical with them if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186070.png" /> is smooth. The sections of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186071.png" /> are called analytic (and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186072.png" />, also holomorphic) vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186073.png" />. For any open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186074.png" /> the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186075.png" /> defines a derivation in the algebra of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186076.png" />, acting according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186078.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186079.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186080.png" /> defines a local one-parameter group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186081.png" /> of automorphisms of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186082.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186083.png" /> is compact, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186084.png" /> is globally definable.

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186085.png" /> provided with the Lie bracket is a Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186086.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186087.png" /> is a compact complex space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186088.png" /> is the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186089.png" />.

Differential operators on an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186090.png" /> are defined in analogy to the differential operators on a module (cf. [[Differential operator on a module|Differential operator on a module]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186091.png" /> are analytic sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186092.png" />, then a linear differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186094.png" />, acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186095.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186096.png" />, is a sheaf homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186097.png" /> which extends to an analytic homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186098.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186099.png" /> is smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860101.png" /> are locally free, this definition gives the usual concept of a differential operator on a vector bundle , [[#References|[4]]].

The germs of the linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860102.png" /> form an analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860103.png" /> with filtration

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860104.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860105.png" /> is the sheaf of germs of operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860106.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860107.png" /> is a filtered sheaf of associative algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860108.png" /> under composition of mappings. One has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860109.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860110.png" /></td> </tr></table>

The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860111.png" /> 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|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860112.png" /> that the sheaf of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860113.png" /> and the corresponding sheaf of graded algebras have finite systems of generators [[#References|[5]]].

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