$$L^2$$ L 2 -properties of the $$\overline{\partial }$$ ∂ ¯ and the $$\overline{\partial }$$ ∂ ¯ -Neumann operator on spaces with isolated singularities

Øvrelid, Nils; Ruppenthal, Jean

doi:10.1007/s00208-014-1016-8

Cited by 12 publications

(19 citation statements)

References 35 publications

(21 reference 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%

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Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Bei

2018

Advances in Mathematics

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Let (X, h) be a compact and irreducible Hermitian complex space of complex dimension m. In this paper we are interested in the Dolbeault operator acting on the space of L 2 sections of the canonical bundle of reg(X), the regular part of X. More precisely let dm,0 : L 2 Ω m,0 (reg(X), h) → L 2 Ω m,1 (reg(X), h) be an arbitrarily fixed closed extension of ∂m,0 : L 2 Ω m,0 (reg(X), h) → L 2 Ω m,1 (reg(X), h) where the domain of the latter operator is Ω m,0 c (reg(X)). We establish various properties such as closed range of dm,0, compactness of the inclusion D(dm,0) → L 2 Ω m,0 (reg(X), h) where D(dm,0), the domain of dm,0, is endowed with the corresponding graph norm, and discreteness of the spectrum of the associated Hodge-Kodaira Laplacian d * m,0 • dm,0 with an estimate for the growth of its eigenvalues. Several corollaries such as trace class property for the heat operator associated to d * m,0 • dm,0, with an estimate for its trace, are derived. Finally in the last part we provide several applications to the Hodge-Kodaira Laplacian in the setting of both compact irreducible Hermitian complex spaces with isolated singularities and complex projective surfaces. IntroductionConsider a complex projective variety V ⊂ CP n . The regular part of V , reg(V ), comes equipped with a natural Kähler metric g, which is the one induced by the Fubini-Study metric of CP n . In particular, whenever V has a nonempty singular set, we get an incomplete Kähler manifold of finite volume. In the seminal papers [10] and [22], given a singular projective variety V , many questions with a rich interaction of topology and analysis, for instance intersection cohomology, L 2 -cohomology and Hodge theory, have been raised for the incomplete Kähler manifold (reg(V ), g). Some of the most important among them are the Cheeger-Goresky-MacPherson's conjecture and the MacPherson's conjecture. The former, which is still open, says that the maximal L 2 -de Rham cohomology groups of (reg(V ), g) are isomorphic to the middle perversity intersection cohomology groups of V while the latter, proved in [29], asks whether the L 2 -∂-cohomology groups in bidegree (0, q) of (reg(V ), g) are 1 arXiv:1607.00289v3 [math.DG] 17 Feb 2018 * E,m,0 )}, has discrete spectrum.We point out explicitly that in the previous theorem the assumption concerning the parabolicity of (A h , g| A h ) does not depend on the particular Hermitian metric g that we fix on M . Indeed if g is another Hermitian metric on M then, since g and g are quasi-isometric on M , we have that (A h , g| A h ) is parabolic if and only if (A h , g | A h ) is parabolic. Consider again the setting of Theorem 0.1 and let ∂ t E,m,0 be the formal adjoint of ∂ E,m,0 with respect to g. Let ∆ ∂,E,m,0 : Ω m,0 (M, E) → Ω m,0 (M, E), ∆ ∂,E,m,0 = ∂ t E,m,0 • ∂ E,m,0 be the Hodge-Kodaira Laplacian in bidegree (m, 0). Since M is compact and ∆ ∂,E,m,0 is elliptic and formally self-adjoint we have that ∆ ∂,E,m,0 , acting on L 2 Ω m,0 (M, E, g) with domain Ω m,0 (M, E), is essentially self-adjoint. With * m,0 )}, has ...

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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%

“…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%

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Bei

2018

Advances in Mathematics

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“…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%

A Dolbeault–Hilbert complex for a variety with isolated singular points

Lott

2019

Ann. K-Th.

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“…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%

Estimates for the $${\bar{\partial }}$$-Equation on Canonical Surfaces

et al. 2019

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We study the solvability in L p of the∂-equation in a neighborhood of a canonical singularity on a complex surface, a so-called du Val singularity. We get a quite complete picture in case p = 2 for two natural closed extensions∂ s and∂ w of∂. For∂ s we have solvability, whereas for∂ w there is solvability if and only if a certain boundary condition ( * ) is fulfilled at the singularity. Our main tool is certain integral operators for solving∂ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.

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$$L^2$$ L 2 -properties of the $$\overline{\partial }$$ ∂ ¯ and the $$\overline{\partial }$$ ∂ ¯ -Neumann operator on spaces with isolated singularities

Cited by 12 publications

References 35 publications

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

Degenerating Hermitian metrics and spectral geometry of the canonical bundle

A Dolbeault–Hilbert complex for a variety with isolated singular points

Estimates for the $${\bar{\partial }}$$-Equation on Canonical Surfaces

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