arXiv subjects

Elizabeth Wulcan

Publications and source records attributed to Elizabeth Wulcan.

At least 19 recordsLinked to original sources

arXiv2024

Global Chern currents of coherent sheaves and Baum Bott currents

We provide global extensions of previous results about representations of characteristic classes of coherent analytic sheaves and of Baum-Bott residues of holomorphic foliations. We show in the first case that they can be represented by currents with support on the support of the given coherent analytic sheaf, and in the second case, by currents with support on the singular set of the foliation. In previous works, we have constructed such representatives provided global resolutions of the appropriate sheaves existed. In this article, we show that the definition of Chern classes of Green and the associated techniques, which work on arbitrary complex manifolds without any assumption on the existence of global resolutions, may be combined with our previous constructions to yield the desired representatives. We also prove a transgression formula for such representatives, which is new even in the case when global resolutions exist. More precisely, the representatives depend on local resolutions of the sheaf, and on choices of metrics and connections on these bundles, i.e., the currents for two different choices differ by a current of the form $dN$, where $N$ is an explicit current, which in the first case above has support on the support of the given coherent analytic sheaf, and in the second case above has support on the singular set of the foliation.

math.CV↗

arXiv2023

Baum-Bott residue currents

Let $\mathscr{F}$ be a holomorphic foliation of rank $\kappa$ on a complex manifold $M$ of dimension $n$, let $Z$ be a compact connected component of the singular set of $\mathscr{F}$, and let $\Phi \in \mathbb C[z_1,\ldots,z_n]$ be a homogeneous symmetric polynomial of degree $\ell$ with $n-\kappa < \ell \leq n$. Given a locally free resolution of the normal sheaf of $\mathscr{F}$, equipped with Hermitian metrics and certain smooth connections, we construct an explicit current $R^\Phi_Z$ with support on $Z$ that represents the Baum-Bott residue $\text{res}^\Phi(\mathscr{F}; Z)\in H_{2n-2\ell}(Z, \mathbb C)$ and is obtained as the limit of certain smooth representatives of $\text{res}^\Phi(\mathscr{F}; Z)$. If the connections are $(1,0)$-connections and $\text{codim} Z\geq \ell$, then $R^\Phi_Z$ is independent of the choice of metrics and connections. When $\mathscr{F}$ has rank one we give a more precise description of $R^\Phi_Z$ in terms of so-called residue currents of Bochner-Martinelli type. In particular, when the singularities are isolated, we recover the classical expression of Baum-Bott residues in terms of Grothendieck residues.

math.CV↗

arXiv2021

Non-pluripolar energy and the complex Monge-Amp\`ere operator

Given a domain $\Omega\subset \mathbf C^n$ we introduce a class of plurisubharmonic (psh) functions $\mathcal G(\Omega)$ and Monge-Amp\`ere operators $u\mapsto [dd^c u]^p$, $p\leq n$, on $\mathcal G(\Omega)$ that extend the Bedford-Taylor-Demailly Monge-Amp\`ere operators. Here $[dd^c u]^p$ is a closed positive current of bidegree $(p,p)$ that dominates the non-pluripolar Monge-Amp\`ere current $\langle dd^c u\rangle^p$. We prove that $[dd^c u]^p$ is the limit of Monge-Amp\`ere currents of certain natural regularizations of $u$. On a compact K\"ahler manifold $(X, \omega)$ we introduce a notion of non-pluripolar energy and a corresponding finite energy class $\mathcal G(X, \omega)\subset \text{PSH}(X, \omega)$ that is a global version of $\mathcal G(\Omega)$. From the local construction we get global Monge-Amp\`ere currents $[dd^c \varphi + \omega]^p$ for $\varphi\in \mathcal G(X,\omega)$ that only depend on the current $dd^c \varphi+ \omega$. The limits of Monge-Amp\`ere currents of certain natural regularizations of $\varphi$ can be expressed in terms of $[dd^c \varphi + \omega]^j$ for $j\leq p$. We get a mass formula involving the currents $[dd^c \varphi+\omega]^p$ that describes the loss of mass of the non-pluripolar Monge-Amp\`ere measure $\langle dd^c \varphi+\omega\rangle^n$. The class $\mathcal G(X, \omega)$ includes $\omega$-psh functions with analytic singularities and the class $\mathcal E(X, \omega)$ of $\omega$-psh functions of finite energy and certain other convex energy classes, although it is not convex itself.

math.CV↗

arXiv2021

Chern currents of coherent sheaves

Given a finite locally free resolution of a coherent analytic sheaf $\mathcal F$, equipped with Hermitian metrics and connections, we construct an explicit current, obtained as the limit of certain smooth Chern forms of $\mathcal F$, that represents the Chern class of $\mathcal F$ and has support on the support of $\mathcal F$. If the connections are $(1,0)$-connections and $\mathcal F$ has pure dimension, then the first nontrivial component of this Chern current coincides with (a constant times) the fundamental cycle of $\mathcal F$. The proof of this goes through a generalized Poincar\'e-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.

math.CV↗

arXiv2020

Global representation of Segre numbers by Monge-Ampère products

On a reduced analytic space $X$ we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding quotient $\mathcal{B}(X)$ that we think of as an analogue of the Chow group and a refinement of de Rham cohomology. This group allows us to study both global and local intersection theoretic properties. We provide many $\mathcal{B}$-analogues of classical intersection theoretic constructions: For an analytic subspace $V\subset X$ we define a $\mathcal{B}$-Segre class, which is an element of $\mathcal{B}(X)$ with support in $V$. It satisfies a global King formula and, in particular, its multiplicities at each point coincide with the Segre numbers of $V$. When $V$ is cut out by a section of a vector bundle we interpret this class as a Monge-Ampère-type product. For regular embeddings we construct a $\mathcal{B}$-analogue of the Gysin morphism.

math.CV↗

arXiv2020

On non-proper intersections and local intersection numbers

Given pure-dimensional (generalized) cycles $\mu_1$ and $\mu_2$ on a complex manifold $Y$ we introduce a product $\mu_1\diamond_{Y} \mu_2$ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. % If $Y$ is projective, then given a very ample line bundle $L\to Y$ we define a product $\mu_1\bl \mu_2$ whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $\mu_1$ and $\mu_2$ are effective, this product satisfies a B\'ezout inequality. If $i\colon Y\to \Pk^N$ is an embedding such that $i^*\Ok(1)=L$, then $\mu_1\bl \mu_2$ can be expressed as a mean value of St\"uckrad-Vogel cycles on $\Pk^N$. There are quite explicit relations between $\di_Y$ and $\bl$.

math.CV↗

arXiv2019

On a mixed Monge-Ampère operator for quasiplurisubharmonic functions with analytic singularities

We consider mixed Monge-Ampère products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Ampère products of smooth functions, generalizing results of Andersson, Błocki and the last author in the case of non-mixed Monge-Ampère 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.

math.CV↗

arXiv2019

Nonproper intersection products and generalized cycles

In this article we develop intersection theory in terms of the $\mathcal{B}$-group of a reduced analytic space. This group was introduced in a previous work as an analogue of the Chow group; it is generated by currents that are direct images of Chern forms and it contains all usual cycles. However, contrary to Chow classes, the $\mathcal{B}$-classes have well-defined multiplicities at each point. We focus on a $\mathcal{B}$-analogue of the intersection theory based on the Stückrad-Vogel procedure and the join construction in projective space. Our approach provides global $\mathcal{B}$-classes which satisfy a Bézout theorem and have the expected local intersection numbers. An essential feature is that we take averages, over various auxiliary choices, by integration. We also introduce $\mathcal{B}$-analogues of more classical constructions of intersections using the Gysin map of the diagonal. These constructions are connected via a $\mathcal{B}$-variant of van Gastel's formulas. Furthermore, we prove that our intersections coincide with the classical ones on cohomology level.

math.AG↗

arXiv2019

A note on the singularities of residue currents of integrally closed ideals

Given a free resolution of an ideal $\mathfrak a$ of holomorpic functions there is an associated residue current $R$ that coincides with the classical Coleff-Herrera product if $\mathfrak a$ is a complete intersection ideal and whose annihilator ideal equals $\mathfrak a$. In the case when $\mathfrak a$ is an Artinian monomial ideal, we show that the singularities of $R$ are small in a certain sense if and only if $\mathfrak a$ is integrally closed.

math.CV↗

arXiv2018

Residue currents and cycles of complexes of vector bundles

We give a factorization of the cycle of a bounded complex of vector bundles in terms of certain associated differential forms and residue currents. This is a generalization of previous results in the case when the complex is a locally free resolution of the structure sheaf of an analytic space and it can be seen as a generalization of the classical Poincar\'e-Lelong formula.

math.CV↗

arXiv2018

Residue currents and fundamental cycles

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.

math.CV↗

arXiv2018

Estimates for the $\bar{\partial}$-equation on canonical surfaces

We study the solvability in $L^p$ of the $\bar\partial$-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 $\bar\partial_s$ and $\bar\partial_w$ of $\bar\partial$. For $\bar\partial_s$ we have solvability, whereas for $\bar\partial_w$ 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 $\bar\partial$ introduced by the first and fourth author, and we study mapping properties of these operators at the singularity.

math.CV↗

arXiv2018

Chern forms of hermitian metrics with analytic singularities on vector bundles

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-Amp\`ere 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.

math.CV↗

arXiv2018

Direct images of semi-meromorphic currents

We introduce a calculus for the class $ASM(X)$ of direct images of semi-meromorphic currents on a reduded analytic space $X$, that extends the classical calculus due to Coleff, Herrera and Passare. Our main result is that each element in this class acts as a kind of multiplication on the sheaf $\PM_X$ of pseudomeromorphic currents on $X$. We also prove that $ASM(X)$ as well as $\PM_X$ and certain subsheaves are closed under the action of holomorphic differential operators and interior multiplication by holomorphic vector fields.

math.CV↗

arXiv2017

On a Monge-Ampère operator for plurisubharmonic functions with analytic singularities

We study continuity properties of generalized Monge-Ampère operators for plurisubharmonic functions with analytic singularities. In particular, we prove continuity for a natural class of decreasing approximating sequences. We also prove a formula for the total mass of the Monge-Ampère measure of such a function on a compact Kähler manifold.

math.CV↗

arXiv2017

Regularity of pseudomeromorphic currents

Let $X$ be a (reduced) pure-dimensional analytic space. We prove that direct images of principal value and residue currents on $X$ are smooth outside sets that are small in a certain sense. We also prove that the sheaf of such currents, provided that $X$ is smooth, is a stalkwise injective $\Ok_X$-module.

math.CV↗

arXiv2016

Explicit Serre duality on complex spaces

In this paper we use recently developed calculus of residue currents together with integral formulas to give a new explicit analytic realization, as well as 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 concrete fine sheaves $\mathscr{B}_X^{n,q}$ of certain currents on $X$ of bidegree $(n,q)$, such that the Dolbeault complex $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ becomes, in a certain sense, a dualizing complex. In particular, if $X$ is Cohen-Macaulay (e.g., Gorenstein or a complete intersection) then $(\mathscr{B}_X^{n,\bullet},\,\bar{\partial})$ is an explicit fine resolution of the Grothendieck dualizing sheaf.

math.CV↗

arXiv2016

On a representation of the fundamental class of an ideal due to Lejeune-Jalabert

Lejeune-Jalabert showed that the fundamental class of a Cohen-Macaulay ideal $\mathfrak a\subset \mathcal O_0$ admits a representation as a residue, constructed from a free resolution of $\mathfrak a$, multiplied by a certain differential form coming from the resolution. We give an explicit description of this differential form in the case where the free resolution is the Scarf resolution of a generic monomial ideal. As a consequence we get a new proof and a refinement of Lejeune-Jalabert's result in this case.

math.AG↗