2014
DOI: 10.1007/s00208-014-1016-8
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$$L^2$$ L 2 -properties of the $$\overline{\partial }$$ ∂ ¯ and the $$\overline{\partial }$$ ∂ ¯ -Neumann operator on spaces with isolated singularities

Abstract: Let X be a Hermitian complex space of pure dimension with only isolated singularities and π : M → X a resolution of singularities. Let Ω ⊂⊂ X be a domain with no singularities in the boundary, Ω * = Ω \ Sing X and Ω ′ = π −1 (Ω). We relate L 2 -properties of the ∂ and the ∂-Neumann operator on Ω * to properties of the corresponding operators on Ω ′ (where the situation is classically well understood). Outside some middle degrees, there are compact solution operators for the ∂-equation on Ω * exactly if there a… Show more

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Cited by 12 publications
(19 citation statements)
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“…By (26) we have the following two orthogonal decompositions for L 2 Ω m,0 (A h , E| A h , h| A h ) : Consider again the setting of Th. 4.1.…”
Section: Main Theoremsmentioning
confidence: 99%
See 1 more Smart Citation
“…By (26) we have the following two orthogonal decompositions for L 2 Ω m,0 (A h , E| A h , h| A h ) : Consider again the setting of Th. 4.1.…”
Section: Main Theoremsmentioning
confidence: 99%
“…Therefore we can conclude that im(∂ t 1,0,max • ∂ 1,0,min ) is closed in L 2 Ω 1,0 (reg(V ), h). By (26) we have the following two orthogonal decompositions for L 2 Ω 1,0 (reg(V ), h):…”
Section: The Hodge-kodaira Laplacian On Complex Projective Surfacesmentioning
confidence: 99%
“…Let D V j = D ⊗ I N be the corresponding operator. As (D + i) −1 is compact [22], the same is true for D V j . Let D AP S be the operator ∂ V + ∂ * V on X − j Z j , with Atiyah-Patodi-Singer boundary conditions [4].…”
Section: Minimal Closure and Compact Resolventmentioning
confidence: 80%
“…Proof. If V is trivial then the lemma is true [22]. We will use a parametrix construction to prove it for general V .…”
Section: Minimal Closure and Compact Resolventmentioning
confidence: 99%
“…To the best of our knowledge, the only known cases of Theorem 1.1 for general surfaces with canonical singularities are the following: Part (i) for p = 2 was proven in [26,Corollary 1.3]. Part (ii) for p = 2 and (0, 2)-forms was proven in [17,Theorem 4.3], which builds on the vanishing result from [28]. Some weaker versions of part (ii) are known as well.…”
Section: Introductionmentioning
confidence: 99%