$L^2 - \overline \partial$-Cohomology of Complex Projective Varieties

Pardon, William; Stern, Mark

doi:10.2307/2939271

Cited by 31 publications

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“…Our result complements results by Pardon [8], Pardon and Stern [9]. In the particular case that X is a complex projective surface with isolated singularities and the set of regular points of X is given the Hermitian (incomplete) metric induced by an embedding of X to a projective space Pardon and Stern [9] identified the (0, q) L 2 -∂-cohomology groups with Neumann boundary conditions of X \ sing X with certain sheaf cohomology groups of its blow-upX. Namely they proved H (0,q) …”

Section: Introductionsupporting

confidence: 79%

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Local $L^2$ results for $\overline{\partial}$ on a singular surface

Diederich¹,

Fornæss²,

Vassiliadou³

2003

MATH. SCAND.

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The Cauchy-Riemann equations are fundamental in complex analysis. This paper contributes to the understanding of these equations on singular spaces. Various methods have been used to overcome the problem of defining forms near singularities. One can blow up the singularity, restrict forms from smooth ambient spaces or work on the regular points. In this paper we use the latter approach to obtain square integrable solutions on singular surfaces. This can be briefly called the Kohn solution up to the singularity to contrast with results in terms of curvature, weights or different function spaces.

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Section: Introductionsupporting

confidence: 79%

“…Similar results (as in [8], [9]) for complex algebraic surfaces in projective space with isolated singularities were obtained by Haskell [3], and Nagase [7].…”

Section: N (X \ Sing X) ∼ = H Nq (X)supporting

confidence: 72%

Local $L^2$ results for $\overline{\partial}$ on a singular surface

Diederich¹,

Fornæss²,

Vassiliadou³

2003

MATH. SCAND.

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“…This problem was studied in [5] for conic singularities and ∂-closed, square integrable (0, 1) forms and later on in [4], for generic surfaces with an isolated singularity at 0 and for the same type of forms. Global analogues of this question for projective surfaces with isolated singularities were studied by Haskell [9], Nagase [12], Pardon [15], Pardon and Stern [16], and for compact Kähler spaces with isolated singularities by Ohsawa [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…Let us recall a particular family of functions that were constructed in [16]. For k ∈ N * , let ρ k : R → [0, 1] be smooth functions such that…”

Section: Introductionmentioning

confidence: 99%

LOCAL L² RESULTS FOR $\bar\partial$: THE ISOLATED SINGULARITIES CASE

2005

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Abstract. Let X be a pure n-dimensional complex analytic set in C N with an isolated singularity at 0. We obtain some L 2 -existence and regularity results for the ∂-operator on a deleted neighborhood of the singular point 0 in X. IntroductionLet X be a pure n-dimensional complex analytic set in C N with an isolated singularity at 0 and letand identify i ∂∂ z 2 with the euclidean metric in C N . Then X * inherits a Kähler metric from its embedding in C N , which we call the ambient metric. Due to the incompleteness of the metric, there are many possible closed L 2 -extensions of the ∂-operator originally acting on smooth forms on X * . We consider the maximal (distributional) ∂-operator. The pointwise norm of a form f defined in X * with respect to the ambient metric will be denoted by |f |, its L 2 -norm by f and the volume element by dV . Let R be a positive number. We set for 0 < r < R, B r := {z ∈ C N ; z < r}, B r := {z ∈ C N ; z ≤ r}, X r := X ∩ B r , X * r := X * ∩ B r and X r * := X * ∩ B r . We shall choose R small enough, so that bB r intersects X transversally for all 0 < r < R.In this paper we address the question of whether it is possible to solve the equation ∂u = f in X * r with L 2 -estimates for f ∈ L 2 p,q (X r ), ∂f = 0 in X * r . This problem was studied in [5] for conic singularities and ∂-closed, square integrable (0, 1) forms and later on in [4], for generic surfaces with an isolated singularity at 0 and for the same type of forms. Global analogues of this question for projective surfaces with isolated singularities were studied by Haskell [9], Nagase [12], Pardon [15], Pardon and Stern [16], and for compact Kähler spaces with isolated singularities by Ohsawa [13,14].There seems to be a "dichotomy of results" for our problem, depending on whether p + q < n or p + q > n. More precisely we have: Theorem 1.1. Let p + q < n, q > 0. There exists a closed subspace H of finite codimension in

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“…Especially on regular sets in singular complex spaces, it is crucial to distinguish the different closed extensions of the ∂-operator for they lead to different Dolbeault cohomology groups (see e.g. [BS], [P], [PS1] or [PS2]). It was realized that investigating the relation between the various extensions is an essential and very fruitful (maybe even indispensable) step in understanding the ∂-equation on singular complex spaces which has to be pursued (see also [R2]).…”

Section: Introductionmentioning

confidence: 99%

Friedrichs' extension lemma with boundary values and applications in complex analysis

Ruppenthal

2011

Complex Variables and Elliptic Equations

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Abstract. Let Q be a first-order differential operator on a compact, smooth oriented Riemannian manifold with smooth boundary. Then, Friedrichs' extension lemma states that the minimal closed extension Q min (the closure of the graph) and the maximal closed extension Q max (in the sense of distributions) of Q in L pspaces (1 ≤ p < ∞) coincide. In the present paper, we show that the same is true for boundary values with respect to Q min and Q max . This gives a useful characterization of weak boundary values, particularly for Q = ∂ the Cauchy-Riemannn operator. As an application, we derive the Bochner-Martinelli-Koppelman formula for L p -forms with weak ∂-boundary values.

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$L^2 - \overline \partial$-Cohomology of Complex Projective Varieties

Cited by 31 publications

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Local $L^2$ results for $\overline{\partial}$ on a singular surface

Local $L^2$ results for $\overline{\partial}$ on a singular surface

LOCAL L² RESULTS FOR $\bar\partial$: THE ISOLATED SINGULARITIES CASE

Friedrichs' extension lemma with boundary values and applications in complex analysis

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$L^2 - \overline \partial$-Cohomology of Complex Projective Varieties

Cited by 31 publications

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Local $L^2$ results for $\overline{\partial}$ on a singular surface

Local $L^2$ results for $\overline{\partial}$ on a singular surface

LOCAL L2 RESULTS FOR $\bar\partial$: THE ISOLATED SINGULARITIES CASE

Friedrichs' extension lemma with boundary values and applications in complex analysis

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LOCAL L² RESULTS FOR $\bar\partial$: THE ISOLATED SINGULARITIES CASE