| 1. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Chern forms of singular metrics on vector bundles
- 2018
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 1090-2082 .- 0001-8708. ; 326, s. 465-489
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Tidskriftsartikel (refereegranskat)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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| 2. |
- Andersson, Mats, 1957, et al.
(författare)
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Estimates for the ∂¯ -Equation on Canonical Surfaces
- 2020
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 30:3, s. 2974-3001
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Tidskriftsartikel (refereegranskat)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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| 3. |
- Andersson, Mats, 1957, et al.
(författare)
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Norm Estimates for the ∂¯ -Equation on a Non-reduced Space
- 2023
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Ingår i: Journal of Geometric Analysis. - 1050-6926. ; 33:7
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Tidskriftsartikel (refereegranskat)abstract
- We study norm-estimates for the (Formula presented.)-equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen–Macaulay and whose underlying reduced space is smooth, the (Formula presented.)-equation for (0, 1)-forms can be solved with Lp -estimates.
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| 4. |
- Andersson, Mats, 1957, et al.
(författare)
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Norm Estimates for the (partial derivative)over-bar-Equation on a Non-reduced Space
- 2023
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Ingår i: Journal of Geometric Analysis. - 1050-6926. ; 33:7
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Tidskriftsartikel (refereegranskat)abstract
- We study norm-estimates for the (partial derivative) over bar -equation on non-reduced analytic spaces. Our main result is that on a non-reduced analytic space, which is Cohen-Macaulay and whose underlying reduced space is smooth, the (partial derivative) over bar -equation for (0, 1)-forms can be solved with L-p-estimates.
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| 5. |
- Andersson, Mats, 1957, et al.
(författare)
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The dbar-equation on a non-reduced analytic space
- 2017
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Ingår i: [Preprint].
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of dbar-equation on X and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves A_q^X of (0,q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf O_X. Our construction is based on intrinsic semi-global Koppelman formulas on X.
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| 6. |
- Andersson, Mats, 1957, et al.
(författare)
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The ∂¯ -equation on a non-reduced analytic space
- 2019
-
Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 374:1-2, s. 553-599
-
Tidskriftsartikel (refereegranskat)abstract
- Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of ∂¯ -equation on X and prove a Dolbeault–Grothendieck lemma. We obtain fine sheaves AXq of (0, q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf OX. Our construction is based on intrinsic semi-global Koppelman formulas on X.
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| 7. |
- Andersson, Mats, et al.
(författare)
-
The (partial derivative)over-bar-equation on a non-reduced analytic space
- 2019
-
Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 374:1-2, s. 553-599
-
Tidskriftsartikel (refereegranskat)abstract
- Let X be a, possibly non-reduced, analytic space of pure dimension. We introduce a notion of -equation on X and prove a Dolbeault-Grothendieck lemma. We obtain fine sheaves AXq of (0,q)-currents, so that the associated Dolbeault complex yields a resolution of the structure sheaf OX. Our construction is based on intrinsic semi-global Koppelman formulas on X.
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| 8. |
- Lärkäng, Richard, 1985
(författare)
-
A comparison formula for residue currents
- 2012
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Given two ideals $\mathcal{I}$ and $\mathcal{J}$ of holomorphic functions such that $\mathcal{I} \subseteq \mathcal{J}$, we describe a comparison formula relating the Andersson-Wulcan currents of $\mathcal{I}$ and $\mathcal{J}$. More generally, this comparison formula holds for residue currents associated to two generically exact complexes of vector bundles, together with a morphism between the complexes. We then show various applications of the comparison formula including generalizing the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals, proving that there exists a natural current $R^\mathcal{J}_Z$ on a singular variety $Z$ such that $\ann R^\mathcal{J}_Z = \mathcal{J}$, and giving an analytic proof of a theorem of Hickel related to the Jacobian determinant of a holomorphic mapping by means of residue currents.
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| 9. |
- Lärkäng, Richard, 1985
(författare)
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A comparison formula for residue currents
- 2019
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Ingår i: Mathematica Scandinavica. - : Det Kgl. Bibliotek/Royal Danish Library. - 0025-5521 .- 1903-1807. ; 125:1, s. 39-66
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Tidskriftsartikel (refereegranskat)abstract
- Given two ideals I and J of holomorphic functions such that I⊆J, we describe a comparison formula relating the Andersson-Wulcan currents of I and J. More generally, this comparison formula holds for residue currents associated to two generically exact Hermitian complexes together with a morphism between the complexes. One application of the comparison formula is a generalization of the transformation law for Coleff-Herrera products to Andersson-Wulcan currents of Cohen-Macaulay ideals. We also use it to give an analytic proof by means of residue currents of theorems of Hickel, Vasconcelos and Wiebe related to the Jacobian ideal of a holomorphic mapping.
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| 10. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Chern currents of coherent sheaves
- 2022
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Ingår i: Epijournal de Geometrie Algebrique. - : Centre pour la Communication Scientifique Directe (CCSD). - 2491-6765. ; 6
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Tidskriftsartikel (refereegranskat)abstract
- Given a finite locally free resolution of a coherent analytic sheaf F, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of F, that represents the Chern class of F and has support on the support of F . If the connections are (1,0)-connections and F has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of F . The proof of this goes through a generalized Poincaré–Lelong formula, previously obtained by the authors, and a result that relates the Chern current to the residue current associated with the locally free resolution.
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| 11. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Chern Forms of Hermitian Metrics with Analytic Singularities on Vector Bundles
- 2022
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 71:5, s. 153-189
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Tidskriftsartikel (refereegranskat)abstract
- We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular Hermitian metric h with analytic singularities on a holomorphic vector bundle E. The currents are constructed as pushforwards of generalized Monge-Ampere products on the projectivization of E. The Chern and Segre currents represent the Chern and Segre classes of E, respectively, and coincide with the Chern and Segre forms of E and h where h is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined.
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| 12. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Computing residue currents of monomial ideals using comparison formulas
- 2012
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Given a free resolution of an ideal $\mathfrak{a}$ of holomorphic functions, one can construct a vector-valued residue current, $R$, which coincides with the classical Coleff-Herrera product if $\mathfrak{a}$ is a complete intersection ideal and whose annihilator ideal is precisely $\mathfrak{a}$. We give a complete description of $R$ in the case when $\mathfrak{a}$ is an Artinian monomial ideal and the resolution is the hull resolution (or a more general cellular resolution), extending previous results by the second author. The main ingredient in the proof is a comparison formula for residue currents due to the first author.
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| 13. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Computing residue currents of monomial ideals using comparison formulas
- 2014
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Ingår i: Bulletin des Sciences Mathématiques. - : Elsevier BV. - 0007-4497. ; 138:3, s. 376-392
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Tidskriftsartikel (refereegranskat)abstract
- Given a free resolution of an ideal a of holomorphic functions, one can construct a vector-valued residue current R, which coincides with the classical Coleff–Herrera product if a is a complete intersection ideal and whose annihilator ideal is precisely a. We give a complete description of R in the case when a is an Artinian monomial ideal and the resolution is the hull resolution (or a more general cellular resolution). The main ingredient in the proof is a comparison formula for residue currents due to the first author. By means of this description, we obtain in the monomial case a current version of a factorization of the fundamental cycle of a due to Lejeune-Jalabert.
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| 14. |
- Lärkäng, Richard, 1985, et al.
(författare)
-
Elementary construction of residue currents associated to Cohen-Macaulay ideals
- 2016
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- For a Cohen-Macaulay ideal of holomorphic functions, we construct by elementary means residue currents whose annihilator is precisely the given ideal. We give two proofs that the currents have the prescribed annihilator, one using the theory of linkage, and another using an explicit division formula involving these residue currents to express the ideal membership.
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| 15. |
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| 16. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Elementary construction of residue currents associated to Cohen-Macaulay ideals
- 2018
-
Ingår i: Annales de lInstitut Fourier. - : Cellule MathDoc/CEDRAM. - 0373-0956 .- 1777-5310. ; 68:1, s. 377-391
-
Tidskriftsartikel (refereegranskat)abstract
- For a Cohen-Macaulay ideal of holomorphic functions, we construct by elementary means residue currents whose annihilator is precisely the given ideal. We give two proofs that the currents have the prescribed annihilator, one using the theory of linkage, and another using an explicit division formula involving these residue currents to express the ideal membership.
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| 17. |
- Lärkäng, Richard, 1985
(författare)
-
Explicit versions of the local duality theorem in C^n
- 2015
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider versions of the local duality theorem in C^n. We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff-Herrera and Andersson-Wulcan, and we give several different proofs of non-degeneracy of the pairings. One of the proofs of non-degeneracy uses the theory of linkage, and conversely, we can use the non-degeneracy to obtain results about linkage for modules. We also discuss a variant of such pairings based on residues considered by Passare, Lejeune-Jalabert and Lundqvist.
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| 18. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Extending holomorphic maps from Stein manifolds into affine toric varieties
- 2016
-
Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 144:11, s. 4613-4626
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Tidskriftsartikel (refereegranskat)abstract
- A complex manifold Y is said to have the interpolation property if a holomorphic map to Y from a subvariety S of a reduced Stein space X has a holomorphic extension to X if it has a continuous extension. Taking S to be a contractible submanifold of X = C^n gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstneric, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular targets. The targets that we study are affine toric varieties, not necessarily normal. We prove that every affine toric variety satisfies a weakening of the interpolation property that is much stronger than the convex interpolation property, but the full interpolation property fails for most affine toric varieties, even for a source as simple as the product of two annuli embedded in C^4.
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| 19. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Koppelman formulas on affine cones over smooth projective complete intersections
- 2015
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Annan publikation (övrigt vetenskapligt/konstnärligt)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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| 20. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Koppelman Formulas on Affine Cones Over Smooth Projective Complete Intersections
- 2018
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Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 67:2, s. 753-780
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Tidskriftsartikel (refereegranskat)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 partial derivative-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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| 21. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Koppelman formulas on the A_1-singularity
- 2016
-
Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 437:1, s. 214-240
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Tidskriftsartikel (refereegranskat)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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| 22. |
- Lärkäng, Richard, 1985, et al.
(författare)
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On a mixed Monge-Ampere operator for quasiplurisubharmonic functions with analytic singularities
- 2020
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Ingår i: Bulletin of the London Mathematical Society. - : Wiley. - 0024-6093 .- 1469-2120. ; 52:1, s. 77-93
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Tidskriftsartikel (refereegranskat)abstract
- We consider mixed Monge-Ampere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one-parameter limits of mixed Monge-Ampere products of smooth functions, generalizing results of Andersson, Blocki and the last author in the case of non-mixed Monge-Ampere products. Connections to the theory of residue currents, going back to Coleff-Herrera, Passare and others, play an important role in the proof. As a consequence we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.
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| 23. |
- Lärkäng, Richard, 1985
(författare)
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On the duality theorem on an analytic variety
- 2010
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The duality theorem for Coleff-Herrera products on a complex manifold says that if $f = (f_1,\dots,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product $\mu^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf $\O_Z$.
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| 24. |
- Lärkäng, Richard, 1985
(författare)
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On the duality theorem on an analytic variety
- 2013
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 355:1, s. 215-234
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Tidskriftsartikel (refereegranskat)abstract
- The duality theorem for Coleff-Herrera products on a complex manifold says that if $f = (f_1,\dots,f_p)$ defines a complete intersection, then the annihilator of the Coleff-Herrera product $\mu^f$ equals (locally) the ideal generated by $f$. This does not hold unrestrictedly on an analytic variety $Z$. We give necessary, and in many cases sufficient conditions for when the duality theorem holds. These conditions are related to how the zero set of $f$ intersects certain singularity subvarieties of the sheaf $\Ok_Z$.
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| 25. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Residue currents and fundamental cycles
- 2015
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We give a factorization of the fundamental cycle of an analytic space in terms of certain differential forms and residue currents associated with a locally free resolution of its structure sheaf. Our result can be seen as a generalization of the classical Poincaré-Lelong formula. It is also a current version of a result by Lejeune-Jalabert, who similarly expressed the fundamental class of a Cohen-Macaulay analytic space in terms of differential forms and cohomological residues.
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| 26. |
- Lärkäng, Richard, 1985, et al.
(författare)
-
Residue currents and fundamental cycles
- 2018
-
Ingår i: Indiana University Mathematics Journal. - : Indiana University Mathematics Journal. - 0022-2518. ; 67:3, s. 1085-1114
-
Tidskriftsartikel (refereegranskat)abstract
- We give a factorization of the fundamental cycle of an analytic space in terms of certain differential forms and residue currents associated with a locally free resolution of its structure sheaf. Our result can be seen as a generalization of the classical Poincare-Lelong formula. It is also a current version of a result by Lejeune-Jalabert, who similarly expressed the fundamental class of a Cohen-Macaulay analytic space in terms of differential forms and cohomological residues.
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| 27. |
- Lärkäng, Richard, 1985
(författare)
-
Residue currents associated with weakly holomorphic functions
- 2009
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We construct Coleff-Herrera products and Bochner-Martinelli type residue currents associated with a tuple $f$ of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case, as the transformation law, the Poincar\'e-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli type residue current associated with $f$ when $f$ defines a complete intersection.
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| 28. |
- Lärkäng, Richard, 1985
(författare)
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Residue currents associated with weakly holomorphic functions
- 2012
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Ingår i: Arkiv för Matematik. - : International Press of Boston. - 0004-2080 .- 1871-2487. ; 50:1, s. 135-164
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Tidskriftsartikel (refereegranskat)abstract
- We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple f of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with f when f defines a complete intersection.
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| 29. |
- Lärkäng, Richard, 1985
(författare)
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Residue currents on analytic spaces
- 2010
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns residue currents on analytic spaces. In the first paper, we construct Coleff-Herrera products and Bochner-Martinelli type currents associated with a weakly holomorphic mapping, and show that these currents satisfy well-known properties from the strongly holomorphic case. This includes the transformation law, the Poincaré-Lelong formula and the equivalence of the Coleff-Herrera product and the Bochner-Martinelli current associated with a complete intersection of weakly holomorphic functions. In the second paper, we discuss the duality theorem on singular varieties. In the case of a complex manifold, the duality theorem, proven by Dickenstein-Sessa and Passare, says that the annihilator of the Coleff-Herrera product associated with a complete intersection $f$ equals the ideal generated by $f$. We give sufficient and in many cases necessary conditions in terms of certain singularity subvarieties of the sheaf $\mathcal{O}_Z$ for when the duality theorem holds on a singular variety $Z$.
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| 30. |
- Lärkäng, Richard, 1985
(författare)
-
Residue currents on singular varieties
- 2012
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis concerns various aspects of the theory of residue currents. Particularly, we study residue currents on singular varieties and duality theorems for such currents. On a singular variety, there are various notions of holomorphic functions. In Paper I, we study how to extend the definition of Coleff-Herrera products and Bochner-Martinelli type residue currents from the case of strongly holomorphic functions to weakly holomorphic functions, and investigate how various properties known in the strongly holomorphic case transform into the weakly holomorphic case. The duality theorem for Coleff-Herrera products on a complex manifold is one of the key properties of the Coleff-Herrera product. On a singular variety, the duality theorem for Coleff-Herrera products is in general false. In Paper II, we discuss necessary and sufficient conditions for when the duality theorem holds, and in particular we show that on any singular variety, one can find examples where the duality principle fails. Another important property of the Coleff-Herrera product is the transformation law. In Paper III, we describe a comparison formula for Andersson-Wulcan currents, generalizing the transformation law. Applications of this formula include giving a proof by means of residue currents of a theorem of Hickel related to the Jacobian of a holomorphic mapping, and constructing a current on a singular variety satisfying the duality principle. The failure of the duality theorem for Coleff-Herrera products leads to the search for an alternative. In Paper IV, we elaborate on the construction in Paper III, of a current satisfying the duality principle for an arbitrary ideal. In particular, using the comparison formula, we explain how we can view this construction as an intrinsic construction on the variety, generalizing the construction of Andersson and Wulcan.
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| 31. |
- Lärkäng, Richard, 1985
(författare)
-
Residue currents with prescribed annihilator ideals on singular varieties
- 2012
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Given an ideal $\mathcal{J}$ on a complex manifold, Andersson and Wulcan constructed a current $R^\mathcal{J}$ such that $\ann R^\mathcal{J} = \mathcal{J}$, generalizing the duality theorem for Coleff-Herrera products. We describe a way to generalize this construction to ideals on singular varieties.
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| 32. |
- Lärkäng, Richard, 1985
(författare)
-
Residue currents with prescribed annihilator ideals on singular varieties
- 2015
-
Ingår i: Mathematische Zeitschrift. - : Springer Science and Business Media LLC. - 0025-5874 .- 1432-1823. ; 279:1, s. 333-358
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Tidskriftsartikel (refereegranskat)abstract
- Given an ideal J on a complex manifold, Andersson and Wulcan constructed explicitly a current R^J such that the annihilator of R^J is J, generalizing the duality theorem for Coleff–Herrera products. We describe a way to generalize this construction to ideals on singular varieties.
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| 33. |
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| 34. |
- Lärkäng, Richard, 1985, et al.
(författare)
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Various approaches to products of residue currents
- 2013
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Ingår i: Journal of Functional Analysis. - : Elsevier BV. - 0022-1236 .- 1096-0783. ; 264:1, s. 118-138
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Tidskriftsartikel (refereegranskat)abstract
- We describe various approaches to Coleff-Herrera productsof residue currents Rj (of Cauchy-Fantappiè-Leray type) associatedto holomorphic mappings fj. More precisely, we study towhich extent (exterior) products of natural regularizations of theindividual currents Rj yield regularizations of the correspondingColeff-Herrera products. Our results hold globally on an arbitrarypure-dimensional complex space.
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