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Elizabeth Wulcan

Publications and source records attributed to Elizabeth Wulcan.

At least 37 records · Page 2Linked to original sources
arXiv2014

Green functions, Segre numbers, and King's formula

Let $\mathcal J$ be a coherent ideal sheaf on a complex manifold $X$ with zero set $Z$, and let $G$ be a plurisubharmonic function such that $G=\log|f|+\mathcal O(1)$ locally at $Z$, where $f$ is a tuple of holomorphic functions that defines $\mathcal J$. We give a meaning to the Monge-Ampère products $(dd^c G)^k$ for $k=0,1,2,...$, and prove that the Lelong numbers of the currents $M_k^{\mathcal J}:=\mathbf 1_Z(dd^c G)^k$ at $x$ coincide with the so-called Segre numbers of $\mathcal J$ at $x$, introduced independently by Tworzewski, Gaffney-Gassler, and Achilles-Manaresi. More generally, we show that $M_k^{\mathcal J}$ satisfy a certain generalization of the classical King formula.

math.CV↗
arXiv2014

Global effective versions of the Brian\ccon-Skoda-Huneke theorem

We prove global effective versions of the Brian\ccon-Skoda-Huneke theorem. Our results extend, to singular varieties, a result of Hickel on the membership problem in polynomial ideals in $\mathbf C^n$, and a related theorem of Ein and Lazarsfeld for smooth projective varieties. The proofs rely on known geometric estimates and new results on multivariable residue calculus.

math.CV↗
arXiv2014

Segre numbers, a generalized King formula, and local intersections

Let $\mathcal J$ be an ideal sheaf on a reduced analytic space $X$ with zero set $Z$. We show that the Lelong numbers of the restrictions to $Z$ of certain generalized Monge-Ampère products $(dd^c\log|f|^2)^k$, where $f$ is a tuple of generators of $\mathcal J$, coincide with the so-called Segre numbers of $\mathcal J$, introduced independently by Tworzewski and Gaffney-Gassler. More generally we show that these currents satisfy a generalization of the classical King formula that takes into account fixed and moving components of Vogel cycles associated with $\mathcal J$. A basic tool is a new calculus for products of positive currents of Bochner-Martinelli type. We also discuss connections to intersection theory.

math.CV↗
arXiv2013

Computing residue currents of monomial ideals using comparison formulas

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. By means of this description we obtain in the monomial case a current version of a factorization of the fundamental cycle of $\mathfrak{a}$ due to Lejeune-Jalabert.

math.CV↗
arXiv2013

Adjunction for the Grauert-Riemenschneider canonical sheaf and extension of L2-cohomology classes

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.

math.CV↗
arXiv2013

Stabilization of monomial maps in higher codimension

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees $λ_k$ of $f$ satisfy $λ_k^2>λ_{k-1}λ_{k+1}$. On the other hand, we give examples of monomial maps $f$, where this condition is not satisfied and where the degree sequences $°_k(f^n)$ do not satisfy any linear recurrence. It follows that such an $f$ is not $k$-stable on any toric model with at worst quotient singularities.

math.DS↗
arXiv2012

On the effective membership problem for polynomial ideals

We discuss the possibility of representing elements in polynomial ideals in $\C^N$ with optimal degree bounds. Classical theorems due to Macaulay and Max Noether say that such a representation is possible under certain conditions on the variety of the associated homogeneous ideal. We present some variants of these results, as well as generalizations to subvarieties of $\C^N$.

math.CV↗
arXiv2012

Canonical heights for plane polynomial maps of small topological degree

We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller than the first dynamical degree. The proof uses the existence, proved by Favre and the first author, of certain compactifications of the plane adapted to the dynamics.

math.NT↗
arXiv2011

Degree growth of monomial maps and McMullen's polytope algebra

We compute all dynamical degrees of monomial maps by interpreting them as mixed volumes of polytopes. By exploiting further the isomorphism between the polytope algebra of P. McMullen and the universal cohomology of complete toric varieties, we construct invariant positive cohomology classes when the dynamical degrees have no resonance.

math.DS↗
arXiv2010

Stabilization of monomial maps

A monomial (or equivariant) selfmap of a toric variety is called stable if its action on the Picard group commutes with iteration. Generalizing work of Favre to higher dimensions, we show that under suitable conditions, a monomial map can be made stable by refining the underlying fan. In general, the resulting toric variety has quotient singularities; in dimension two we give criteria for when it can be chosen smooth, as well as examples when it cannot.

math.DS↗
arXiv2010

Some variants of Macaulay's and Max Noether's theorems

We use residue currents on toric varieties to obtain bounds on the support of solutions to polynomial ideal membership problems. Our bounds depend on the Newton polytopes of the polynomial systems and are therefore well adjusted to sparse systems of polynomials. We present variants of classical results due to Macaulay and Max Noether.

math.CV↗
arXiv2010

Sparse effective membership problems via residue currents

We use residue currents on toric varieties to obtain bounds on the degrees of solutions to polynomial ideal membership problems. Our bounds depend on (the volume of) the Newton polytope of the polynomial system and are therefore well adjusted to sparse polynomial systems. We present sparse versions of Max Nöther's $AF+BG$ Theorem, Macaulay's Theorem, and Kollár's Effective Nullstellensatz, as well as recent results by Hickel and Andersson-Götmark.

math.CV↗
arXiv2009

On weighted Bochner-Martinelli residue currents

We study the weighted Bochner-Martinelli residue current R^p(f) associated with a sequence f=(f_1,...,f_m) of holomorphic germs at the origin in C^n, whose common zero set equals the origin, and p=(p_1,..., p_m)\in N^n. Our main results are a description of R^p(f) and its annihilator ideal in terms of the Rees valuations of the ideal generated by (f_1^{p_1},..., f_m^{p_m}) and an explicit description of R^p(f) when f is monomial. For a monomial sequence f we show that R^p(f) is independent of p if and only if f is a regular sequence.

math.CV↗
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↗
arXiv2008

On Bochner-Martinelli residue currents and their annihilator ideals

We study the residue current R^f of Bochner-Martinelli type associated with a tuple f=(f_1,...,f_m) of holomorphic germs at the origin in C^n, whose common zero set equals the origin. Our main results are a geometric description of R^f in terms of the Rees valuations associated with the ideal (f) generated by f and a characterization of when the annihilator ideal of R^f equals (f).

math.CV↗
arXiv2008

Residue currents constructed from resolutions of monomial ideals

Given a free resolution of an ideal $J$ of holomorphic functions, one can construct a vector valued residue current $R$, whose annihilator is precisely $J$. In this paper we compute $R$ in case $J$ is a monomial ideal and the resolution is a cellular resolution in the sense of Bayer and Sturmfels. A description of $R$ is given in terms of the underlying polyhedral cell complex and it is related to irreducible decompositions of $J$.

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↗