| 1. |
- Andersson, Mats, 1957, et al.
(author)
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Estimates for the ∂¯ -Equation on Canonical Surfaces
- 2020
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In: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30:3, s. 2974-3001
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Journal article (peer-reviewed)abstract
- We study the solvability in Lp 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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| 2. |
- Lärkäng, Richard, 1985, et al.
(author)
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Chern forms of singular metrics on vector bundles
- 2018
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In: Advances in Mathematics. - : Elsevier BV. - 1090-2082 .- 0001-8708. ; 326, s. 465-489
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Journal article (peer-reviewed)abstract
- We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-Paun. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.
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| 3. |
- Lärkäng, Richard, 1985, et al.
(author)
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Koppelman formulas on affine cones over smooth projective complete intersections
- 2015
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Other publication (other academic/artistic)abstract
- In the present paper, we study regularity of the Andersson-Samuelsson Koppelman integral operator on affine cones over smooth projective complete intersections. Particularly, we prove L^p- and C^\alpha-estimates, and compactness of the operator, when the degree is sufficiently small. As applications, we obtain homotopy formulas for different \dbar-operators acting on L^p-spaces of forms, including the case p=2 if the varieties have canonical singularities. We also prove that the A-forms introduced by Andersson-Samuelsson are C^\alpha for \alpha<1.
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| 4. |
- Lärkäng, Richard, 1985, et al.
(author)
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Koppelman formulas on the A_1-singularity
- 2016
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In: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 437:1, s. 214-240
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Journal article (peer-reviewed)abstract
- In the present paper, we study the regularity of the Andersson-Samuelsson Koppelman integral operator on the A_1-singularity. Particularly, we prove L^p- and C^0-estimates. As applications, we obtain L^p-homotopy formulas for the dbar-equation on the A_1-singularity, and we prove that the A-forms introduced by Andersson-Samuelsson are continuous on the A_1-singularity.
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| 5. |
- Ruppenthal, Jean, et al.
(author)
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Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L²-cohomology classes
- 2015
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In: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 64:2, s. 533-558
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Journal article (peer-reviewed)abstract
- In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface $V$ in a complex manifold $M$. This adjunction formula is used to study the problem of extending $L^2$-cohomology classes of $\bar{\partial}$-closed forms from the singular hypersurface $V$ to the manifold $M$ in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the $L^2$-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of $V$ in $M$. The smooth case has recently been studied by Berndtsson.
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| 6. |
- Ruppenthal, Jean, et al.
(author)
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Explicit Serre duality on complex spaces
- 2017
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In: Advances in Mathematics. - : Elsevier BV. - 1090-2082 .- 0001-8708. ; 305, s. 1320-1355
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Journal article (peer-reviewed)abstract
- In this paper we use recently developed calculus of residue currentstogether with integral formulas to give a new explicit analytic realization, as wellas a new analytic proof of Serre duality on any reduced pure n-dimensional paracompact complex space X. At the core of the paper is the introduction of concretefine sheaves $A^{n,q}_X$ of certain currents on X of bidegree (n,q), such that the corresponding Dolbeault complex becomes, in a certain sense, a dualizing complex. Inparticular, if X is Cohen-Macaulay (e.g., Gorenstein or a complete intersection)then this Dolbeault complex becomes an explicit fine resolution of the Grothendieck dualizing sheaf.
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| 7. |
- Samuelsson Kalm, Håkan, 1976, et al.
(author)
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Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L^2-cohomology classes
- 2012
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Reports (other academic/artistic)abstract
- In the present paper, we derive an adjunction formula for the Grauert-Riemenschneider canonical sheaf of a singular hypersurface V in a complex manifold M. This adjunction formula is used to study the problem of extending L2-cohomology classes of dbar-closed forms from the singular hypersurface V to the manifold M in the spirit of the Ohsawa-Takegoshi-Manivel extension theorem. We do that by showing that our formulation of the L2-extension problem is invariant under bimeromorphic modifications, so that we can reduce the problem to the smooth case by use of an embedded resolution of V in M. The smooth case has recently been studied by Berndtsson.
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