scholarly journals Differential and holomorphic differential operators on noncommutative algebras

2015 ◽  
Vol 22 (3) ◽  
pp. 279-300 ◽  
Author(s):  
E. Beggs
Keyword(s):  
Differential Operators ◽  
Holomorphic Differential ◽  
Noncommutative Algebras

scholarly journals DIFFERENTIAL STRUCTURE OF ABELIAN FUNCTIONS

2008 ◽  
Vol 19 (02) ◽  
pp. 145-171 ◽  
Author(s):  
KOJI CHO ◽  
ATSUSHI NAKAYASHIKI
Keyword(s):  
Abelian Variety ◽  
Differential Operators ◽  
Free Resolution ◽  
Holomorphic Differential ◽  
Differential Structure ◽  
Linear Basis ◽  
Higher Dimensional ◽  
Abelian Functions ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring [Formula: see text] of global holomorphic differential operators on J. We construct a [Formula: see text] free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

scholarly journals Differential operators and flat connections on a Riemann surface

2003 ◽  
Vol 2003 (64) ◽  
pp. 4041-4056
Author(s):  
Indranil Biswas
Keyword(s):  
Riemann Surface ◽  
Vector Bundle ◽  
Vector Bundles ◽  
Differential Operators ◽  
Transversality Condition ◽  
Jet Bundle ◽  
Holomorphic Differential ◽  
Isomorphism Classes ◽  
Holomorphic Vector Bundles ◽  
Natural Filtration

We consider filtered holomorphic vector bundles on a compact Riemann surfaceXequipped with a holomorphic connection satisfying a certain transversality condition with respect to the filtration. IfQis a stable vector bundle of rankrand degree(1−genus(X))nr, then any holomorphic connection on the jet bundleJn(Q)satisfies this transversality condition for the natural filtration ofJn(Q)defined by projections to lower-order jets. The vector bundleJn(Q)admits holomorphic connection. The main result is the construction of a bijective correspondence between the space of all equivalence classes of holomorphic vector bundles onXwith a filtration of lengthntogether with a holomorphic connection satisfying the transversality condition and the space of all isomorphism classes of holomorphic differential operators of ordernwhose symbol is the identity map.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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