Alphabeta MathThe Dolbeault Complex and Integral Solutions
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
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Combinatorics
Complex Analysis
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Differential Geometry
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Functional Analysis
Linear Algebra
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Measure Theory
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Partial Differential Equations
Real Analysis
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2 · Logical flowchart
3 · Items
Definition
Bigraded complex forms and the Dolbeault operators
Theorem
The d, partial and dbar identities
Lemma
Stokes for complex forms on a bounded C1 Euclidean domain
Theorem
The Cauchy–Pompeiu formula with fixed signs
Lemma
Local Cauchy transform with smooth parameters
Definition
The normalized Bochner–Martinelli kernel
Theorem
The Bochner–Martinelli formula for C1 functions
Theorem
The local Dolbeault lemma on nested polydiscs
Theorem
Compactly supported dbar solutions on complex Euclidean space
Corollary
Hartogs extension by a compact-support dbar correction
Definition
Dolbeault cohomology of a domain
Theorem
Positive-degree Dolbeault cohomology vanishes on polydiscs
Sources
- Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapters 4–5
- Jabbari, Notes for Analysis and Geometry of Several Complex Variables, §3.2
- Guillemin and Campbell, MIT 18.117 Lecture Notes, Lectures 1–4
- Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapter 4 §4.4
- Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapter 4 §4.1
- Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapter 5 §5.1
- Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapter 4 §4.2
- Jiří Lebl, Tasty Bits of Several Complex Variables, v4.4, Chapter 4 §4.3, Theorem 4.3.1