arXiv subjects

Mats Andersson

Publications and source records attributed to Mats Andersson.

48 records · Page 3Linked to original sources

arXiv2009

Decomposition of residue currents

Given a submodule $J\subset \mathcal O_0^{\oplus r}$ and a free resolution of $J$ one can define a certain vector valued residue current whose annihilator is $J$. We make a decomposition of the current with respect to Ass$(J)$ that correspond to a primary decomposition of $J$. As a tool we introduce a class of currents that includes usual residue and principal value currents; in particular these currents admit a certain type of restriction to analytic varieties and more generally to constructible sets.

math.CV↗

arXiv2009

On the Briancon-Skoda theorem on a singular variety

Let $Z$ be a germ of a reduced analytic space of pure dimension. We provide an analytic proof of the uniform Briancon-Skoda theorem for the local ring $\mathcal{O}_Z$; a result which was previously proved by Huneke by algebraic methods. For ideals with few generators we also get much sharper results.

math.CV↗

arXiv2009

A residue criterion for strong holomorphicity

We give a local criterion in terms of a residue current for strong holomorphicity of a meromorphic function on an arbitrary pure-dimensional analytic variety. This generalizes a result by A Tsikh for the case of a reduced complete intersection.

math.CV↗

arXiv2008

Koppelman formulas and the $\dbar$-equation on an analytic space

Let $X$ be an analytic space of pure dimension. We introduce a formalism to generate intrinsic weighted Koppelman formulas on $X$ that provide solutions to the $\dbar$-equation. We prove that if $ϕ$ is a smooth $(0,q+1)$-form on a Stein space $X$ with $\dbarϕ=0$, then there is a smooth $(0,q)$-form $ψ$ on $X_{reg}$ with at most polynomial growth at $X_{sing}$ such that $\dbarψ=ϕ$. The integral formulas also give other new existence results for the $\dbar$-equation and Hartogs theorems, as well as new proofs of various known results.

math.CV↗

arXiv2007

Uniqueness and factorization of Coleff-Herrera currents

We prove a uniqueness result for Coleff-Herrera currents which in particular means that if $f=(f_1,..., f_m)$ defines a complete intersection, then the classical Coleff-Herrera product associated to $f$ is the unique Coleff-Herrera current that is cohomologous to 1 with respect to the operator $δ_f-\dbar$, where $δ_f$ is interior multiplication with $f$. From the uniqueness result we deduce that any Coleff-Herrera current on a variety $Z$ is a finite sum of products of residue currents with support on $Z$ and holomorphic forms.

math.CV↗

arXiv2007

Residue currents with prescribed annihilator ideals

Given a coherent ideal sheaf $J$ we construct locally a vector-valued residue current $R$ whose annihilator is precisely the given sheaf. In case $J$ is a complete intersection, $R$ is just the classical Coleff-Herrera product. By means of these currents we can extend various results, previously known for a complete intersection, to general ideal sheaves. Combining with integral formulas we obtain a residue version of the Ehrenpreis-Palamodov fundamental principle.

math.CV↗

arXiv2005

Explicit versions of the Briançon-Skoda theorem with variations

We give new a proof of the general Briançon-Skoda theorem about ideals of holomorphic functions by means of multivariable residue calculus. The method gives new variants of this theorem for products of ideals. Moreover, we obtain a related result for the ideal generated by the the subdeterminants of a matrix-valued generically surjective holomorphic function, generalizing the duality theorem for a complete intersection. We also provide explicit versions of the various results, including the general Briançon-Skoda theorem, with integral representation formulas.

math.CV↗

arXiv2005

Integral representation with weights II, division and interpolation

Let $f$ be a $r\times m$-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic $ψ$ such that $ϕ=fψ$, provided that $ϕ$ is holomorphic and annihilates a certain residue current with support on the set where $f$ is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previously known for the case $r=1$.

math.CV↗

arXiv2005

Residue currents of holomorphic morphisms

Given a generically surjective holomorphic vector bundle morphism $f\colon E\to Q$, $E$ and $Q$ Hermitian bundles, we construct a current $R^f$ with values in $\Hom(Q,H)$, where $H$ is a certain derived bundle, and with support on the set $Z$ where $f$ is not surjective. The main property is that if $ϕ$ is a holomorphic section of $Q$, and $R^fϕ=0$, then locally $fψ=ϕ$ has a holomorphic solution $ψ$. In the generic case also the converse holds. This gives a generalization of the corresponding theorem for a complete intersection, due to Dickenstein-Sessa and Passare. We also present results for polynomial mappings, related to M Noether's theorem and the effective Nullstellensatz. The construction of the current is based on a generalization of the Koszul complex. By means of this complex one can also obtain new global estimates of solutions to $fψ=ϕ$, and as an example we give new results related to the $H^p$-corona problem.

math.CV↗

arXiv2005

Operators with smooth functional calculi

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, that contains all operators of Helffer-Sjöstrand type and is closed under the action of smooth proper mappings. Moreover, the class is closed under tensor product of commuting operators. In general an operator in this class has no resolvent in the usual sense so the spectrum must be defined in terms of the functional calculus. We also consider invariant subspaces and spectral decompositions.

math.SP↗

arXiv2005

The membership problem for polynomial ideals in terms of residue currents

We find a relation between the vanishing of a globally defined residue current on $¶^n$ and solution of the membership problem with control of the polynomial degrees. Several classical results appear as special cases, such as Max Nöther's theorem, and we also obtain a generalization of that theorem. There are also connections to effective versions of the Nullstellensatz. We also provide explicit integral representations of the solutions.

math.CV↗

arXiv2003

Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula

We define a smooth functional calculus for a non-commuting tuple of (unbounded) operators $A_j$ on a Banach space with real spectra and resolvents with temperate growth, by means of an iterated Cauchy formula. The construction is also extended to tuples of more general operators allowing smooth functional calculii. We also discuss the relation to the case with commuting operators.

math.SP↗