# Chern currents of coherent sheaves

Richard Lärkäng  Department of Mathematics, Chalmers University of Technology
and the University of Gothenburg, S-412 96 Gothenburg, Sweden
<EMAIL_ADDRESS>and  Elizabeth Wulcan  Department of Mathematics, Chalmers
University of Technology and the University of Gothenburg, S-412 96
Gothenburg, Sweden<EMAIL_ADDRESS>
  *     * scAbstract. 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 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.

scKeywords. Chern classes; coherent sheaves; residue currents

sc2020 Mathematics Subject Classification. 32A27; 14C17; 32C30; 14F06; 53C05

  *
cJanuary 19, 2022Received by the Editors on November 3, 2021.
Accepted on March 16, 2022.

Department of Mathematics, Chalmers University of Technology and the
University of Gothenburg, S-412 96 Gothenburg, Sweden

sce-mail<EMAIL_ADDRESS>
Department of Mathematics, Chalmers University of Technology and the
University of Gothenburg, S-412 96 Gothenburg, Sweden

sce-mail<EMAIL_ADDRESS>
The first author was partially supported by the Swedish Research Council
(2017-04908) and the second author was partially supported by the Swedish
Research Council (2017-03905) and the Göran Gustafsson Foundation for Research
in Natural Sciences and Medicine.

© by the author(s) This work is licensed under
http://creativecommons.org/licenses/by-sa/4.0/

###### Contents

  1. 1 Introduction
  2. 2 Currents associated with complexes of vector bundles
  3. 3 Chern forms and Chern characters
  4. 4 Connections compatible with a complex
  5. 5 The Chern current $c^{\mathrm{Res}}(E,D)$
  6. 6 An explicit description of Chern currents of low degrees
  7. 7 An example

## 1\. Introduction

Let $X$ be a complex manifold and let $\mathcal{F}$ be a coherent analytic
sheaf on $X$ of positive codimension, i.e. such that the support
$\operatorname{supp}\mathcal{F}$ has positive codimension. Assume that
$\mathcal{F}$ has a locally free resolution of the form

(1.1)
$0\to{\mathcal{O}}(E_{N})\xrightarrow[]{\varphi_{N}}\cdots\xrightarrow[]{\varphi_{1}}{\mathcal{O}}(E_{0})\to\mathcal{F}\to
0,$

where $E_{k}$ are holomorphic vector bundles on $X$, and
${\mathcal{O}}(E_{k})$ denote the corresponding locally free sheaves. Then the
(total) Chern class of $\mathcal{F}$ equals

(1.2) $c(\mathcal{F})=\prodop\displaylimits_{k=0}^{N}c(E_{k})^{(-1)^{k}},$

where $c(E_{k})$ is the (total) Chern class of $E_{k}$, see Section 3.1. In
this paper we construct explicit representatives of the nontrivial part of
$c(\mathcal{F})$ with support on $\operatorname{supp}\mathcal{F}$.

Let us briefly describe our construction; the representatives of
$c(\mathcal{F})$ will be currents obtained as limits of certain Chern forms.
Assume that $(E,\varphi)$ is a locally free resolution of $\mathcal{F}$ of the
form (1.1). Moreover assume that each vector bundle $E_{k}$ is equipped with a
Hermitian metric and a connection $D_{k}$ (that is not necessarily the Chern
connection of the metric). Let $\sigma_{k}$ be the minimal inverse of
$\varphi_{k}$, see Section 2.3, let $\chi:\mathbb R_{\geq 0}\to\mathbb R_{\geq
0}$ be a smooth cut-off function such that $\chi(t)\equiv 0$ for $t\ll 1$ and
$\chi(t)\equiv 1$ for $t\gg 1$, let $s$ be a (generically non-vanishing)
holomorphic section of a vector bundle such that
$\\{s=0\\}\supseteq\operatorname{supp}\mathcal{F}$, let
$\chi_{\epsilon}=\chi(|s|^{2}/\epsilon)$, and let $\widehat{D}^{\epsilon}_{k}$
be the connection

(1.3)
$\widehat{D}_{k}^{\epsilon}=-\chi_{\epsilon}\sigma_{k}D\varphi_{k}+D_{k};$

here $D$ is a connection on $\operatorname{End}E$, where
$E=\bigoplusop\displaylimits E_{k}$, induced by the $D_{k}$, see Section 2.2.
Then clearly the Chern form

(1.4)
$c(E,\widehat{D}^{\epsilon}):=\prodop\displaylimits_{k=0}^{N}c(E_{k},\widehat{D}^{\epsilon}_{k})^{(-1)^{k}}$

is a representative for $c(\mathcal{F})$. Throughout this paper we consider
Chern classes as de Rham cohomology classes of smooth forms or currents. Our
first main result asserts that the limit of this form is a current with the
desired properties.

######  Theorem 1.1.

Assume that $\mathcal{F}$ is a coherent analytic sheaf of positive codimension
that admits a locally free resolution of the form (1.1). Moreover, assume that
each $E_{k}$ is equipped with a Hermitian metric and a connection $D_{k}$. Let
$c(E,\widehat{D}^{\epsilon})$ be the Chern form of $\mathcal{F}$ defined by
(1.4), and let $c_{\ell}(E,\widehat{D}^{\epsilon})$ denote the component of
degree $2\ell$. Let $\ell_{1},\dots,\ell_{m}\in\mathbb N_{>0}$. Then

(1.5) $c_{\ell_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge
c_{\ell_{m}}^{\mathrm{Res}}(E,D):=\lim_{\epsilon\to
0}c_{\ell_{1}}(E,\widehat{D}^{\epsilon})\wedge\dots\wedge
c_{\ell_{m}}(E,\widehat{D}^{\epsilon})$

is a well-defined closed current, independent of $\chi_{\epsilon}$, that
represents $c_{\ell_{1}}(\mathcal{F})\wedge\dots\wedge
c_{\ell_{m}}(\mathcal{F})$ and has support on
$\operatorname{supp}\mathcal{F}$.

The Chern currents (1.5) are pseudomeromorphic in the sense of [AW10], see
Theorem 5.1, which means that they have a geometric nature similar to closed
positive (or normal) currents, see Section 2.1. We let

$c^{\mathrm{Res}}(E,D)=1+c_{1}^{\mathrm{Res}}(E,D)+c_{2}^{\mathrm{Res}}(E,D)+\cdots.$

The first nontrivial component of $c^{\mathrm{Res}}(E,D)$ is (the current of
integration along) a cycle, see Theorem 1.4 below. We do not know whether
Chern currents of higher degree are of order $0$ in general.

######  Remark 1.2.

If all the connections $D_{k}$ are $(1,0)$-connections, i.e. the $(0,1)$-part
of each $D_{k}$ equals ${\bar{\partial}}$, then so are the connections
$\widehat{D}_{k}^{\epsilon}$. However, even if the $D_{k}$ are Chern
connections, the $\widehat{D}_{k}^{\epsilon}$ are not Chern connections in
general. Thus, it might be the case that the involved forms and currents in
(1.5) contain terms of bidegree $(\ell+m,\ell-m)$ with $m>0$ (but only when
$\ell>\operatorname{codim}\mathcal{F}$ by Theorem 1.4 below).

Our construction of Chern currents is inspired by the paper [BB72] by Baum and
Bott, where singular holomorphic foliations are studied by expressing
characteristic classes associated to a foliation as certain cohomological
residues, more precisely as push-forwards of cohomology classes living in the
singular set of the foliation. A key point in the proofs in [BB72] are the
concepts of connections compatible with and fitted to a complex of vector
bundles. One may check that their constructions of fitted connections (with
some minor adaptations) correspond to connections of the form (1.3). For the
results in [BB72], it was sufficient to consider Chern forms associated to
connections (1.3) for $\epsilon$ small enough, but fixed, while in the present
paper, we study the limit of such forms when $\epsilon\to 0$.

######  Example 1.3.

Let us compute $c^{\mathrm{Res}}(E,D)$ when $\mathcal{F}$ is the structure
sheaf ${\mathcal{O}}_{Z}$ of a divisor $Z\subset X$, defined by a holomorphic
section $s$ of a holomorphic line bundle $L$ over $X$, and $(E,\varphi)$ is
the locally free resolution

(1.6) $0\to{\mathcal{O}}(L^{*})\stackrel{{\scriptstyle
s}}{{\to}}{\mathcal{O}}\to{\mathcal{O}}_{Z}\to 0.$

Assume that $L$ is equipped with a connection $D_{L}$; equip $E_{1}=L^{*}$
with the induced dual connection $D_{L^{*}}$, and $E_{0}$ with the trivial
connection. The minimal inverse of $s$ is $1/s$ and $Ds=D_{L}s$, so
$\widehat{D}^{\epsilon}_{1}=\chi_{\epsilon}(D_{L}s/s)+D_{L^{*}}$, and
$\widehat{D}^{\epsilon}_{0}$ is the trivial connection. The curvature form of
$\widehat{D}_{1}^{\epsilon}$ equals
$\widehat{\Theta}^{\epsilon}_{1}=d\chi_{\epsilon}\wedge(D_{L}s/s)-(1-\chi_{\epsilon}){}_{L}$,
where L is the curvature form of $D_{L}$ (which equals minus the curvature
form of $D_{L^{*}}$). By an appropriate formulation of the Poincaré-Lelong
formula,

(1.7) $\lim_{\epsilon\to 0}d\chi_{\epsilon}\wedge\frac{D_{L}s}{s}=2\pi i[Z],$

where $[Z]$ is the current of integration along (the cycle of) $Z$. Note that

$c(E,\widehat{D}^{\epsilon})=c(E_{1},\widehat{D}_{1}^{\epsilon})^{-1}=1-c_{1}(E_{1},\widehat{D}_{1}^{\epsilon})+c_{1}(E_{1},\widehat{D}_{1}^{\epsilon})^{2}-\cdots.$

Thus, since L is smooth,

(1.8) $c^{\mathrm{Res}}_{1}(E,D)=\lim_{\epsilon\to
0}-c_{1}(E_{1},\widehat{D}^{\epsilon}_{1})=-\lim_{\epsilon\to
0}\frac{i}{2\pi}\widehat{\Theta}_{1}^{\epsilon}=[Z].$

Thus, the first Chern current coincides with $[Z]$, which can be seen as a
canonical representative of $c_{1}({\mathcal{O}}_{Z})$ with support on
$\operatorname{supp}{\mathcal{O}}_{Z}$.

By further calculations of terms of all degrees, one can show that

$c^{\mathrm{Res}}(E,D)=1+[Z](1+c_{1}(L,D)+\dots+c_{1}^{n-1}(L,D)).$

Our next result is an explicit description of the first nontrivial Chern
current $c_{p}^{\mathrm{Res}}(E,D)$ in the case when $\mathcal{F}$ has pure
codimension $p$, i.e. $\operatorname{supp}\mathcal{F}$ has pure dimension
$\dim X-p$, that generalizes (1.8). Recall that the _(fundamental) cycle_ of
$\mathcal{F}$ is the cycle

(1.9) $[\mathcal{F}]=\sumop\displaylimits_{i}m_{i}[Z_{i}]$

(considered as a current of integration), where $Z_{i}$ are the irreducible
components of $\operatorname{supp}\mathcal{F}$, and $m_{i}$ is the _geometric
multiplicity_ of $Z_{i}$ in $\mathcal{F}$, see _e.g._ [Ful98, Chapter 1.5].

######  Theorem 1.4.

Assume that $\mathcal{F}$ is a coherent analytic sheaf of pure codimension
$p>0$ that admits a locally free resolution $(E,\varphi)$ of the form (1.1).
Moreover, assume that each $E_{k}$ is equipped with a Hermitian metric and a
$(1,0)$-connection $D_{k}$. Then

$c_{p}^{\mathrm{Res}}(E,D)=(-1)^{p-1}(p-1)![\mathcal{F}].$

Moreover

(1.10) $c_{\ell}^{\mathrm{Res}}(E,D)=0\quad\text{ for }0<\ell<p$

and

(1.11) $c_{\ell_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge
c_{\ell_{m}}^{\mathrm{Res}}(E,D)=0\quad\text{ for }m\geq 2\text{ and
}0<\ell_{1}+\dots+\ell_{m}\leq p.$

Here the Chern currents on the left hand sides are defined by (1.5).

In case $\mathcal{F}$ has codimension $p$, but not necessarily _pure_
codimension $p$, then Theorem 1.4 still holds if we replace the first equation
by

(1.12) $c_{p}^{\mathrm{Res}}(E,D)=(-1)^{p-1}(p-1)![\mathcal{F}]_{p},$

where $[\mathcal{F}]_{p}$ denotes the part of $[\mathcal{F}]$ of codimension
$p$, i.e. in (1.9), one only sums over the components $Z_{i}$ of codimension
$p$.

In particular, $c_{p}^{\mathrm{Res}}(E,D)$ is independent of the choice of
Hermitian metrics and $(1,0)$-connections on $(E,\varphi)$. Moreover, it
follows that on cohomology level

(1.13) $c_{p}(\mathcal{F})=(-1)^{p-1}(p-1)![\mathcal{F}],$

where now the right hand side should be interpreted as a de Rham class. When
$\mathcal{F}$ is the pushforward of a vector bundle from a subvariety, that
(1.13) holds is a well-known consequence of the Grothendieck-Riemann-Roch
theorem, _cf._ [Ful98, Examples 15.2.16 and 15.1.2].

The proof of Theorem 1.4 relies on a generalization of the Poincaré-Lelong
formula. Given a complex (1.1) equipped with Hermitian metrics, Andersson and
the second author defined in [AW07] an associated so-called residue current
$R^{E}=R=\sumop\displaylimits R_{k}$ with support on
$\operatorname{supp}\mathcal{F}$, where $R_{k}$ is a
$\operatorname{Hom}(E_{0},E_{k})$-valued $(0,k)$-current for $k=0,\dots,N$,
see Section 2.3. The construction involves the minimal inverses $\sigma_{k}$
of $\varphi_{k}$. If $(E,\varphi)$ is the complex (1.6), then $R^{E}$
coincides with the residue current ${\bar{\partial}}(1/s)=\lim_{\epsilon\to
0}{\bar{\partial}}\chi_{\epsilon}(1/s)$; more generally if $(E,\varphi)$ is
the Koszul complex of a complete intersection, then $R^{E}$ coincides with the
classical Coleff-Herrera residue current, [CH78]. Using residue currents, we
can write the Poincaré-Lelong formula (1.7) as

${\bar{\partial}}\frac{1}{s}\wedge D_{L}s=2\pi i[Z];$

indeed, the left hand side in (1.7) equals
$\lim{\bar{\partial}}\chi_{\epsilon}\wedge(D_{L}s)/s$. Given a
$\operatorname{Hom}(E_{\ell},E_{\ell})$-valued current $\alpha$, let
$\operatorname{tr}\alpha$ denote the trace of $\alpha$. In [LW18, LW21] we
proved the following generalization of the Poincaré-Lelong formula:

_Assume that $R^{E}$ is the residue current associated with a finite locally
free resolution $(E,\varphi)$ of a coherent analytic sheaf $\mathcal{F}$ of
pure codimension $p$. Moreover assume that $D$ is a connection on
$\operatorname{End}E$ induced by arbitrary $(1,0)$-connections on $E_{k}$.
Then_

(1.14) $\frac{1}{(2\pi i)^{p}p!}\operatorname{tr}(D\varphi_{1}\cdots
D\varphi_{p}R_{p})=[\mathcal{F}].$

If $\mathcal{F}$ has codimension $p$, but not necessarily pure codimenion,
(1.14) still holds if we replace $[\mathcal{F}]$ by $[\mathcal{F}]_{p}$, _cf._
[LW18, Theorem 1.5]. In view of this, Theorem 1.4, as well as (1.12), are
direct consequences of the following explicit description of (the components
of low degree of) $c^{\mathrm{Res}}(E,D)$ in terms of $R^{E}$.

######  Theorem 1.5.

Assume that $\mathcal{F}$ is a coherent analytic sheaf of codimension $p>0$
that admits a locally free resolution $(E,\varphi)$ of the form (1.1).
Moreover, assume that each $E_{k}$ is equipped with a Hermitian metric and a
$(1,0)$-connection $D_{k}$. Let $R$ be the associated residue current and $D$
the connection on $\operatorname{End}E$ induced by the $D_{k}$. Then

$c_{p}^{\mathrm{Res}}(E,D)=\frac{(-1)^{p-1}}{(2\pi
i)^{p}p}\operatorname{tr}(D\varphi_{1}\cdots D\varphi_{p}R_{p}).$

Moreover (1.10) and (1.11) hold.

In fact, we formulate and prove our results in a slightly more general
setting. We consider the Chern class $c(E)$ of a generically exact complex of
vector bundles $(E,\varphi)$ that is not necessarily a locally free resolution
of a coherent sheaf. Theorem 5.1 below asserts that
$c(E,\widehat{D}^{\epsilon})$ as well as products of such currents have well-
defined limits when $\epsilon\to 0$ and represent the corresponding (products
of) Chern classes. In particular, Theorem 1.1 follows. In Theorem 6.1, if
$(E,\varphi)$ is exact outside a variety of codimension $p$, we give an
explicit description of $c_{p}^{\mathrm{Res}}(E,D):=\lim_{\epsilon\to
0}c_{p}(E,\widehat{D}^{\epsilon})$ terms of residue currents that generalizes
Theorem 1.5. From this and a more general version of the Poincaré-Lelong
formula (1.14) it follows that if the cohomology groups are of pure
codimension $p$, then $c_{p}^{\mathrm{Res}}(E,D)=(-1)^{p-1}(p-1)![E]$, where
$[E]$ is the cycle of $(E,\varphi)$, see Corollary 6.7 and (2.19).

Our results could alternatively be formulated in term of the Chern character
$\operatorname{ch}(E)$ of $E$. From Theorem 1.1, for $\ell>0$, we obtain a
current $\operatorname{ch}^{\mathrm{Res}}_{\ell}(E,D)$ that represents the
$\ell^{\mathrm{th}}$ graded piece $\operatorname{ch}_{\ell}(E)$ of the Chern
character, see Section 6. Theorems 1.4 and 1.5 are then equivalent to

$\displaystyle\operatorname{ch}_{p}^{\mathrm{Res}}(E,D)$
$\displaystyle=\frac{1}{(2\pi i)^{p}p!}\operatorname{tr}(D\varphi_{1}\cdots
D\varphi_{p}R_{p})=[\mathcal{F}],$
$\displaystyle\operatorname{ch}_{\ell}^{\mathrm{Res}}(E,D)$
$\displaystyle=0\quad\text{ for }\ell<p,\ \text{and}$
$\displaystyle\operatorname{ch}_{\ell_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge\operatorname{ch}_{\ell_{m}}^{\mathrm{Res}}(E,D)$
$\displaystyle=0\quad\text{ for }m\geq 2\ \text{and}\
\ell_{1}+\dots+\ell_{m}\leq p,$

see Theorem 6.2 and Remark 6.3.

We refer to the currents in Theorem 1.1 as Chern currents, in analogy with the
usual Chern forms representing Chern classes. In works of Bismut, Gillet, and
Soulé [BGS90a, BGS90b] appears the similarly named concept of Bott-Chern
currents, that are certain explicit $dd^{c}$-potentials in a transgresssion
formula in a Grothendieck-Riemann-Roch theorem, and not directly related to
our currents.

There are some similarities between our results and results by Harvey and
Lawson. In [HL93] they study characteristic classes of morphisms
$\varphi:E_{0}\to E_{1}$ of vector bundles, and only in very special
situations there is overlap between their results and ours. We remark that the
connection (1.3) that plays a crucial role in our work essentially appears and
is important in [HL93], see, in particular, [HL93, Section I.4].

Chern classes of coherent sheaves, without the assumption of the existence of
a global locally free resolution, were studied in the thesis of Green,
[Gre80], as well as in various recent papers, including [Gri10, Hos20a,
Hos20b, Qia16, BSW21, Wu20]. Several of these papers also concern classes in
finer cohomology theories than de Rham cohomology, as for example (rational or
complex) Bott-Chern or Deligne cohomology.

In the present paper, our focus has been to find explicit representatives of
Chern classes of a coherent sheaf with support on the support of the sheaf, a
type of result which as far as we can tell, none of the above mentioned works
seems to consider. By incorporating the construction of residue currents
associated with a twisted resolution from [JL21], it might be possible to
extend our results to arbitrary coherent sheaves, without any assumptions
about the existence of a global locally free resolution. We plan to explore
this in future work. The currents we study provide representatives of the
Chern classes in de Rham cohomology. Our methods unfortunately do not seem to
yield representatives in the finer cohomology theories mentioned above, since
for example Chern classes in complex Bott-Chern cohomology as in [Qia16,
BSW21], are naturally obtained from Chern forms of the Chern connection of a
hermitian metric, while our construction, building on the techniques in
[BB72], involve Chern forms of connections that are not Chern connections of a
hermitian metric.

The paper is organized as follows. In Section 2 we give some necessary
background on (residue) currents. In Section 3 we describe Chern forms and
Chern characters, and in Section 4 we discuss compatible connections. The
proofs of (the generalized versions of) Theorems 1.1 and 1.5 occupy Sections 5
and 6, respectively. Finally in Section 7 we compute $c^{\mathrm{Res}}(E,D)$
for an explicit choice of a locally free resolution $(E,\varphi)$ of a
coherent sheaf $\mathcal{F}$. In particular, we compute
$c^{\mathrm{Res}}_{\ell}(E,D)$ for $\ell>\operatorname{codim}\mathcal{F}$ in
this case.

### Acknowledgements

This paper is very much inspired by an ongoing joint project with Lucas
Kaufmann, which aims to understand Baum-Bott residues in terms of (residue)
currents. We are greatly indebted to him for many valuable discussions on this
topic. We would also like to thank Dennis Eriksson for many important
discussions and helpful comments on a previous version of this paper.

## 2\. Currents associated with complexes of vector bundles

We say that a function $\chi:\mathbb R_{\geq 0}\to\mathbb R_{\geq 0}$ is a
_smooth approximand of the characteristic function_ $\chi_{[1,\infty)}$ of the
interval $[1,\infty)$ and write

$\chi\sim\chi_{[1,\infty)}$

if $\chi$ is smooth and $\chi(t)\equiv 0$ for $t\ll 1$ and $\chi(t)\equiv 1$
for $t\gg 1$. Note that if $\chi\sim\chi_{[1,\infty)}$ and
$\hat{\chi}=\chi^{\ell}$, then $\hat{\chi}\sim\chi_{[1,\infty)}$ and

(2.1) $d\hat{\chi}=\ell\chi^{\ell-1}d\chi.$

### 2.1. Pseudomeromorphic currents

Let $f$ be a (generically nonvanishing) holomorphic function on a (connected)
complex manifold $X$. Herrera and Lieberman [HL71], proved that the _principal
value_

$\lim_{\epsilon\to 0}\intop\nolimits_{|f|^{2}>\epsilon}\frac{\xi}{f}$

exists for test forms $\xi$ and defines a current $[1/f]$. It follows that
${\bar{\partial}}[1/f]$ is a current with support on the zero set $Z(f)$ of
$f$; such a current is called a _residue current_. Assume that
$\chi\sim\chi_{[1,\infty)}$ and that $F$ is a (generically nonvanishing)
section of a Hermitian vector bundle such that $Z(f)\subseteq\\{F=0\\}$. Then

(2.2) $[1/f]=\lim_{\epsilon\to
0}\frac{\chi(|F|^{2}/\epsilon)}{f}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{ and
}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
{\bar{\partial}}[1/f]=\lim_{\epsilon\to
0}\frac{{\bar{\partial}}\chi(|F|^{2}/\epsilon)}{f},$

see _e.g._ [AW18]. In particular, the limits are independent of $\chi$ and
$F$.

In the literature there are various generalizations of residue currents and
principal value currents. In particular, Coleff and Herrera [CH78] introduced
products like

(2.3)
$[1/f_{1}]\cdots[1/f_{r}]{\bar{\partial}}[1/f_{r+1}]\wedge\cdots\wedge{\bar{\partial}}[1/f_{m}].$

In order to obtain a coherent approach to questions about residue and
principal value currents was introduced in [AW10] the sheaf
${\mathcal{PM}}_{X}$ of pseudomeromorphic currents on $X$, consisting of
direct images under holomorphic mappings of products of test forms and
currents like (2.3). See _e.g._ [AW18, Section 2.1] for a precise definition;
in particular it follows from the definition that ${\mathcal{PM}}$ is closed
under push-forwards of modifications. Also, we refer to [AW18] for the results
mentioned in this subsection. The sheaf ${\mathcal{PM}}_{X}$ is closed under
${\bar{\partial}}$ and under multiplication by smooth forms. Pseudomeromorphic
currents have a geometric nature, similar to closed positive (or normal)
currents. For example, the dimension principle states that if the
pseudomeromorphic current $\mu$ has bidegree $(*,p)$ and support on a variety
of codimension strictly larger than $p$, then $\mu$ vanishes.

The sheaf ${\mathcal{PM}}_{X}$ admits natural restrictions to constructible
subsets. In particular, if $W$ is a subvariety of the open subset
$\mathcal{U}\subseteq X$, and $F$ is a section of a vector bundle such that
$\\{F=0\\}=W$, then the restriction to $\mathcal{U}\setminus W$ of a
pseudomeromorphic current $\mu$ on $\mathcal{U}$ is the pseudomeromorphic
current

$\textbf{1}_{\mathcal{U}\setminus W}\mu:=\lim_{\epsilon\to
0}\chi(|F|^{2}/\epsilon)\mu|_{\mathcal{U}},$

where $\chi\sim\chi_{[1,\infty)}$ as above. This definition is independent of
the choice of $F$ and $\chi$.

A pseudomeromorphic current $\mu$ on $X$ is said to have the _standard
extension property_ (SEP) if $\textbf{1}_{\mathcal{U}\setminus
W}\mu=\mu|_{\mathcal{U}}$ for any subvariety $W\subseteq\mathcal{U}$ of
positive codimension, where $\mathcal{U}\subseteq X$ is any open subset. By
definition, it follows that if $\mu$ has the SEP and $F\nequiv 0$ is any
holomorphic section of a vector bundle, then

(2.4) $\lim_{\epsilon\to 0}\chi(|F|^{2}/\epsilon)\mu=\mu.$

### 2.2. Superstructure and connections on a complex of vector bundles

Let $(E,\varphi)$ be a complex

(2.5)
$0\xrightarrow[]{}E_{N}\xrightarrow[]{\varphi_{N}}E_{N-1}\xrightarrow[]{\varphi_{N-1}}\cdots\xrightarrow[]{\varphi_{2}}E_{1}\xrightarrow[]{\varphi_{1}}E_{0}\xrightarrow[]{}0,$

of vector bundles over $X$. As in [AW07], see also [LW18], we will consider
the complex $(E,\varphi)$ to be equipped with a so-called superstructure, i.e.
a $\mathbb{Z}_{2}$-grading, which splits $E:=\oplus E_{k}$ into odd and even
parts $E^{+}$ and $E^{-}$, where $E^{+}=\oplus E_{2k}$ and $E^{-}=\oplus
E_{2k+1}$. Also $\operatorname{End}E$ gets a superstructure by letting the
even part be the endomorphisms preserving the degree, and the odd part the
endomorphisms switching degrees.

This superstructure affects how form- and current-valued endomorphisms act.
Assume that $\alpha=\omega\otimes\gamma$ is a section of
${\mathcal{E}}^{\bullet}(\operatorname{End}E)$, where $\gamma$ is a
holomorphic section of $\operatorname{Hom}(E_{\ell},E_{k})$, and $\omega$ is a
smooth form of degree $m$. Then we let $\deg_{f}\alpha=m$ and
$\deg_{e}\alpha=k-\ell$ denote the _form_ and _endomorphism_ degrees,
respectively, of $\alpha$. The _total_ degree is
$\deg\alpha=\deg_{f}\alpha+\deg_{e}\alpha$. If $\beta$ is a form-valued
section of $E$, i.e. $\beta=\eta\otimes\xi$, where $\eta$ is a scalar form,
and $\xi$ is a section of $E$, both homogeneous in degree, then the action of
$\alpha$ on $\beta$ is defined by

(2.6)
$\alpha(\beta):=(-1)^{(\deg_{e}\alpha)(\deg_{f}\beta)}\omega\wedge\eta\otimes\gamma(\xi).$

If furthermore, $\alpha^{\prime}=\omega^{\prime}\otimes\gamma^{\prime}$, where
$\gamma^{\prime}$ is a holomorphic section of $\operatorname{End}E$, and
$\omega^{\prime}$ is a smooth form, both homogeneous in degree, then we define

$\alpha\alpha^{\prime}:=(-1)^{(\deg_{e}\alpha)(\deg_{f}\alpha^{\prime})}\omega\wedge\omega^{\prime}\otimes\gamma\circ\gamma^{\prime}.$

For an $(m\times n)$-matrix $A$ and an $(n\times m)$-matrix $B$, we have that
$\operatorname{tr}(AB)=\operatorname{tr}(BA)$, while for the morphisms
$\alpha$ and $\alpha^{\prime}$ above, we get such an equality with a sign due
to the superstructure,

(2.7)
$\operatorname{tr}(\alpha\alpha^{\prime})=(-1)^{(\deg\alpha)(\deg\alpha^{\prime})-(\deg_{e}\alpha)(\deg_{e}\alpha^{\prime})}\operatorname{tr}(\alpha^{\prime}\alpha),$

see [LW18, Equation (2.14)].

Note that ${\bar{\partial}}$ extends in a way that respects the superstructure
to act on $\operatorname{End}E$-valued morphisms. In particular,

(2.8)
${\bar{\partial}}(\alpha\alpha^{\prime})={\bar{\partial}}\alpha\alpha^{\prime}+(-1)^{\deg\alpha}\alpha{\bar{\partial}}\alpha^{\prime}.$

We will consider the situation when $(E,\varphi)$ is equipped with a
connection $D=D_{E}=(D_{0},\ldots,D_{N})$, where $D_{k}$ is a connection on
$E_{k}$. Then there is an induced connection $\oplus D_{k}$ on $E$, that we
also denote by $D_{E}$. This in turn induces a connection
$D_{\operatorname{End}}$ on $\operatorname{End}E$ that takes the
superstructure into account, defined by

(2.9)
$D_{\operatorname{End}}\alpha:=D_{E}\circ\alpha-(-1)^{\deg\alpha}\alpha\circ
D_{E},$

if $\alpha$ is a $\operatorname{End}E$-valued form. It satisfies the following
Leibniz rule, [LW18, Equation (2.4)], _cf._ (2.8)

(2.10)
$D_{\operatorname{End}}(\alpha\alpha^{\prime})=D_{\operatorname{End}}\alpha\alpha^{\prime}+(-1)^{\deg\alpha}\alpha
D_{\operatorname{End}}\alpha^{\prime}.$

To simplify notation, we will sometimes drop the subscript
$\operatorname{End}$ and simply denote this connection by $D$. If k denotes
the curvature form of $D_{k}$, and $\alpha:E_{k}\to E_{\ell}$, then, by (2.9),

(2.11)
$DD\alpha={}_{\ell}\alpha+(-1)^{\deg\alpha+\deg\alpha+1}\alpha{}_{k}={}_{\ell}\alpha-\alpha{}_{k}.$

The above formulas hold also when $\alpha$ and $\alpha^{\prime}$ are current-
valued instead of form-valued, as long as the involved products of currents
are well-defined.

We let $D_{k}^{\prime}$ and $D_{k}^{\prime\prime}$ denote the $(1,0)$\- and
$(0,1)$-parts of $D_{k}$, respectively, and we let
$D^{\prime}=(D_{k}^{\prime})$ and $D^{\prime\prime}=(D^{\prime\prime}_{k})$
denote the corresponding $(1,0)$\- and $(0,1)$-parts of $D_{E}=(D_{k})$. We
say that $D_{E}$ is a _$(1,0)$ -connection_ if each $D_{k}$ is a
$(1,0)$-connection, i.e. $D^{\prime\prime}_{k}={\bar{\partial}}$. We will use
the following consequence of (2.11): assume that $D_{E}$ is a
$(1,0)$-connection, and $\alpha:E_{k}\to E_{\ell}$ is a holomorphic (or more
generally a ${\bar{\partial}}$-closed form-valued) morphism. Then

(2.12)
${\bar{\partial}}D\alpha=({}_{\ell})_{(1,1)}\alpha-\alpha({}_{k})_{(1,1)},$

where $(\cdot)_{(1,1)}$ denotes the component of bidegree $(1,1)$.

Since $(E,\varphi)$ is a complex and $\varphi_{k}$ has odd degree, it follows
from (2.10) that

(2.13) $\varphi_{k-1}D\varphi_{k}=D\varphi_{k-1}\varphi_{k}.$

### 2.3. Residue currents associated to a complex

Let us briefly recall the construction in [AW07]. Assume that we have a
generically exact complex $(E,\varphi)$ of vector bundles over a complex
manifold $X$ of the form (2.5), and assume that each $E_{k}$ is equipped with
some Hermitian metric. If $Z_{k}$ is the analytic set where $\varphi_{k}$ has
lower rank than its generic rank, then outside of $Z_{k}$ the minimal (or
Moore-Penrose) inverse $\sigma_{k}:E_{k-1}\to E_{k}$ of $\varphi_{k}$ is
determined by the following properties:
$\varphi_{k}\sigma_{k}\varphi_{k}=\varphi_{k}$,
$\operatorname{im}\sigma_{k}\perp\operatorname{im}\varphi_{k+1}$, and
$\sigma_{k+1}\sigma_{k}=0$. One can verify that $\sigma_{k}$ is smooth outside
of $Z_{k}$. Since $\sigma_{k}\sigma_{k-1}=0$ and $\sigma_{k}$ has odd degree,
by (2.8),

(2.14)
$\sigma_{k}{\bar{\partial}}\sigma_{k-1}={\bar{\partial}}\sigma_{k}\sigma_{k-1}.$

Let $Z$ be the set where $(E,\varphi)$ is not pointwise exact. It follows from
the definition of $\sigma_{k}$ that

(2.15)
$\varphi_{k}\sigma_{k}+\sigma_{k-1}\varphi_{k-1}=\operatorname{Id}_{E_{k-1}}$

outside $Z$, or more generally outside $Z_{k}\cup Z_{k-1}$. Applying (2.8) to
(2.15), we obtain that outside $Z$

(2.16)
$\varphi_{k}{\bar{\partial}}\sigma_{k}={\bar{\partial}}\sigma_{k-1}\varphi_{k-1}$

and furthermore applying (2.10) to this equality, we get that

(2.17)
$D\varphi_{k}{\bar{\partial}}\sigma_{k}=D{\bar{\partial}}\sigma_{k-1}\varphi_{k-1}+{\bar{\partial}}\sigma_{k-1}D\varphi_{k-1}+\varphi_{k}D{\bar{\partial}}\sigma_{k}.$

######  Lemma 2.1.

Let $X$, $(E,\varphi)$, $Z$, and $\sigma_{k}$ be as above. Assume that for
each $j=1,\ldots,m$, $s_{j}$ is an entry of $\sigma_{k}$,
$\partial\sigma_{k}$, or ${\bar{\partial}}\sigma_{k}$ for some $k$ in some
local trivialization, and let $s=s_{1}\cdots s_{m}$. Assume that
$\chi\sim\chi_{[1,\infty)}$ and that $F$ is a (generically nonvanishing)
holomorphic section of a vector bundle over $X$ such that $Z\subset\\{F=0\\}$.
Then the limits

$\lim_{\epsilon\to
0}\chi(|F|^{2}/\epsilon)s\qquad\text{and}\qquad\lim_{\epsilon\to
0}{\bar{\partial}}\chi(|F|^{2}/\epsilon)\wedge s$

exist and define pseudomeromorphic currents on $X$ that are independent of the
choices of $\chi$ and $F$; the support of the second current is contained in
$Z$. Furthermore,

$\lim_{\epsilon\to 0}\partial\chi(|F|^{2}/\epsilon)\wedge s=0.$

###### Proof.

By Hironaka’s theorem there is a holomorphic modification $\pi:\tilde{X}\to
X$, such that for each $k$, $\pi^{*}\sigma_{k}$ is locally of the form
$(1/\gamma_{k})\tilde{\sigma}_{k}$, where $\gamma_{k}$ is holomorphic with
$Z(\gamma_{k})\subset\tilde{Z}:=\pi^{-1}Z$, and $\tilde{\sigma}_{k}$ is
smooth, see [AW07, Section 2]. Now, where
$\chi(|\pi^{*}F|^{2}/\epsilon)\nequiv 0$,
${\bar{\partial}}\pi^{*}\sigma_{k}=(1/\gamma_{k}){\bar{\partial}}\tilde{\sigma}_{k}$
and
$\partial\pi^{*}\sigma_{k}=\partial(1/\gamma_{k})\tilde{\sigma}_{k}+(1/\gamma_{k})\partial\tilde{\sigma}_{k}$.
Since each holomorphic derivative ${\partial/\partial z_{i}}(1/\gamma_{k})$ is
a meromorphic function with poles contained in $\tilde{Z}$ it follows that
$\pi^{*}s_{j}$ equals (a sum of terms of the form) $(1/g_{j})\tilde{s}_{j}$,
where $g_{j}$ is holomorphic with $Z(g_{j})\subset\tilde{Z}$, and
$\tilde{s}_{j}$ is smooth. Thus $\pi^{*}s$ equals (a sum of terms of the form)
$(1/g)\tilde{s}$, where $g$ is holomorphic with $Z(g)\subset\tilde{Z}$, and
$\tilde{s}$ is smooth. In view of (2.2),

$\lim_{\epsilon\to
0}\chi(|\pi^{*}F|^{2}/\epsilon)\pi^{*}s\quad\text{and}\quad\lim_{\epsilon\to
0}{\bar{\partial}}\chi(|\pi^{*}F|^{2}/\epsilon)\wedge\pi^{*}s$

are well-defined pseudomeromorphic currents on $\tilde{X}$ independent of
$\chi$ and $F$; the second current has support on $\tilde{Z}$. Since
${\mathcal{PM}}$ is closed under push-forwards of modifications, _cf._ Section
2.1, this proves the first part of the lemma.

As proved above, the limit

$\mu:=\lim_{\epsilon\to 0}\chi(|F|^{2}/\epsilon)s$

exists. This current is in fact a so-called almost semi-meromorphic current,
_cf._ [AW18, Section 4], and in particular, it has the SEP. By [AW18, Theorem
3.7], $\partial\mu$ also has the SEP. Thus,

$\lim_{\epsilon\to 0}\partial\chi(|F|^{2}/\epsilon)\wedge\mu=\lim_{\epsilon\to
0}\partial(\chi(|F|^{2}/\epsilon)\wedge\mu)-\lim_{\epsilon\to
0}\chi(|F|^{2}/\epsilon)\partial\mu=\partial\mu-\partial\mu=0,$

which proves the last part of the lemma. Here, in the second equality, we have
used that the two limits exist and are both equal to $\partial\mu$ by (2.4). ∎

In particular

(2.18) $R^{\ell}_{k}:=\lim_{\epsilon\to
0}{\bar{\partial}}\chi(|F|^{2}/\epsilon)\wedge\sigma_{k}{\bar{\partial}}\sigma_{k-1}\cdots{\bar{\partial}}\sigma_{\ell+1}$

is a $\operatorname{Hom}(E_{\ell},E_{k})$-valued pseudomeromorphic current of
bidegree $(0,k-\ell)$ with support on $Z$; in fact, it follows from the proof
that the support is contained in $Z_{\ell+1}\cup\cdots\cup Z_{k}$. If
$\ell=k-1$, then the right hand side of (2.18) should be interpreted as
$\lim_{\epsilon\to 0}{\bar{\partial}}\chi(|F|^{2}/\epsilon)\wedge\sigma_{k}$.
The residue current $R^{E}=R:=\sumop\displaylimits R_{k}^{\ell}$ associated
with $(E,\varphi)$ was introduced in [AW07], _cf._ the introduction. Assume
that $(E,\varphi)$ is a locally free resolution of a coherent analytic sheaf
$\mathcal{F}$. Then $R_{k}^{\ell}$ vanishes for $\ell>0$ by [AW07, Theorem
3.1]. In this case $R=\sumop\displaylimits R_{k}$, where $R_{k}=R_{k}^{0}$.

Given a complex $(E,\varphi)$ of vector bundles of the form (2.5), following
[LW21], we define the _cycle_

(2.19) $[E]=\sumop\displaylimits_{k=0}^{N}(-1)^{k}[\mathcal{H}_{k}(E)],$

where $\mathcal{H}_{k}$ is the homology sheaf of $(E,\varphi)$ at level $k$.
Note that if $(E,\varphi)$ is a locally free resolution of a coherent analytic
sheaf $\mathcal{F}$, then $[E]=[\mathcal{F}]$. In [LW21] we prove the
following generalization of (1.14).

######  Theorem 2.2.

Let $(E,\varphi)$ be a complex of Hermitian vector bundles of the form (2.5)
such that $\mathcal{H}_{k}(E)$ has pure codimension $p>0$ or vanishes, for
$k=0,\dots,N$, and let $D$ be an arbitrary $(1,0)$-connection on
$(E,\varphi)$. Then,

$\frac{1}{(2\pi
i)^{p}p!}\sumop\displaylimits_{k=0}^{N-p}(-1)^{k}\operatorname{tr}(D\varphi_{k+1}\cdots
D\varphi_{k+p}R^{k}_{k+p})=[E].$

## 3\. Chern forms and Chern characters

### 3.1. Chern classes and forms

Assume that $E$ is a holomorphic vector bundle of rank $r$ equipped with a
connection $D$. Then recall that the (total) Chern form
$c(E,D)=1+c_{1}(E,D)+\cdots+c_{r}(E,D)$ is defined by

$\sumop\displaylimits_{\ell=0}^{r}c_{\ell}(E,D)t^{\ell}=\det\left(I+\frac{i}{2\pi}\Theta
t\right),$

where is the curvature matrix of $D$ in a local trivialization; in particular,
$c_{\ell}(E,D)$ is a form of degree $2\ell$. The de Rham cohomology class of
$c(E,D)$ is the _(total) Chern class_ $c(E)=\sumop\displaylimits c_{\ell}(E)$
of the vector bundle $E$.

If $(E,\varphi)$ is a complex of vector bundles of the form (2.5) that is not
necessarily a locally free resolution of a coherent analytic sheaf, in line
with the Chern theory of virtual bundles as in _e.g._ [BB72, Section 4] or
[Suw98, Section II.8.C], we let

$c(E)=\prodop\displaylimits_{k=0}^{N}c(E_{k})^{(-1)^{k}}.$

Moreover, if $(E,\varphi)$ is equipped with a connection $D=(D_{k})$, _cf._
Section 2.2, we let

(3.1) $c(E,D)=\prodop\displaylimits_{k=0}^{N}c(E_{k},D_{k})^{(-1)^{k}}$

and we let $c_{\ell}(E,D)=c(E,D)_{\ell}$ be the component of degree $2\ell$.

Consider now a coherent analytic sheaf $\mathcal{F}$ with a locally free
resolution (1.1). We define the Chern class of $\mathcal{F}$ by (1.2), i.e.
$c(\mathcal{F})=c(E)$, and if $(E,\varphi)$ is equipped with a connection $D$,
then this class may be represented by (3.1). This definition of Chern classes
of coherent sheaves may be motivated in terms of K-theory. However, it is
typically considered only on manifolds with the so-called resolution property.
Recall that a complex manifold $X$ is said to have the _resolution property_
if any coherent analytic sheaf $\mathcal{F}$ on $X$ has a finite locally free
resolution (1.1). For such manifolds, the definition (1.2) is the unique
extension of the definition of Chern classes from locally free sheaves to
coherent analytic sheaves that satisfies the following _Whitney formula_ : if
$0\to\mathcal{F}^{\prime}\to\mathcal{F}\to\mathcal{F}^{\prime\prime}\to 0$ is
a short exact sequence of sheaves, then
$c(\mathcal{F})=c(\mathcal{F}^{\prime\prime})c(\mathcal{F}^{\prime\prime})$,
_cf._ [BS58, Théorème 2] or [EH16, Chapter 14.2].

In this paper, we define Chern classes of coherent sheaves by (1.2) also on
manifolds which do not have the resolution property, but then necessarily only
for coherent sheaves with a locally free resolution (1.1). Note that if we are
on a manifold for which the resolution property does not hold, it is not
immediate that the de Rham cohomology class of (1.2) is well-defined, i.e.
independent of the resolution. However, that it is well-defined follows from a
construction of Chern classes of arbitrary coherent analytic sheaves on
arbitrary complex manifolds by Green, [Gre80], see also [TT86], since in case
one has a global locally free resolution of finite length, the definition in
[Gre80] coincides with the one in (1.2).

### 3.2. The Chern character (form) of a vector bundle

Assume that $E$ is a holomorphic vector bundle of rank $r$. Then formally we
can write

$1+c_{1}(E)t+\cdots+c_{r}(E)t^{r}=\prodop\displaylimits_{i=1}^{r}(1+\alpha_{i}t),$

where $\alpha_{i}$ are the so-called _Chern roots_ of $E$, see _e.g._ [Ful98,
Remark 3.2.3]. In particular, this means that
$c_{\ell}(E)=e_{\ell}(\alpha_{1},\ldots,\alpha_{r})$, where $e_{\ell}$ is the
$\ell^{\mathrm{th}}$ _elementary symmetric polynomial_

$e_{\ell}(x)=e_{\ell}(x_{1},\ldots,x_{r})=\sumop\displaylimits_{1\leq
i_{1}<\dots<i_{\ell}\leq r}x_{i_{1}}\cdots x_{i_{\ell}}.$

The _Chern character_ of $E$ may formally be defined as the symmetric
polynomial $\operatorname{ch}(E)=\sumop\displaylimits_{i=1}^{r}e^{\alpha_{i}}$
in the Chern roots, see _e.g._ [Ful98, Example 3.2.3]. In particular, the
$\ell^{\mathrm{th}}$ graded piece is

(3.2)
$\operatorname{ch}_{\ell}(E)=\frac{1}{\ell!}p_{\ell}(\alpha_{1},\ldots,\alpha_{r}),$

where $p_{\ell}$ is the $\ell^{\mathrm{th}}$ _power sum polynomial_

$p_{\ell}(x)=p_{\ell}(x_{1},\ldots,x_{r})=\sumop\displaylimits_{i=1}^{r}x_{i}^{\ell}.$

Since any symmetric polynomial in $x_{i}$ may be expressed as a unique
polynomial in $e_{j}(x)$, there are polynomials
$Q_{\ell}(t_{1},\ldots,t_{\ell})$, $\ell\geq 1$, such that
$p_{\ell}(x)=Q_{\ell}(e_{1}(x),\ldots,e_{\ell}(x))$; these are sometimes
called _Hirzebruch–Newton polynomials_. If $t_{j}$ is given weight $j$, then
$Q_{\ell}(t_{1},\ldots,t_{\ell})$ is a weighted homogenous polynomial of
degree $\ell$. Written out explicitly, Definition (3.2) should be read as

$\operatorname{ch}_{\ell}(E)=\frac{1}{\ell!}Q_{\ell}\big{(}c_{1}(E),\ldots,c_{\ell}(E)\big{)}.$

If $E$ is equipped with a connection $D$, one can analogously define _Chern
character forms_

(3.3)
$\operatorname{ch}_{\ell}(E,D)=\frac{1}{\ell!}Q_{\ell}\big{(}c_{1}(E,D),\ldots,c_{\ell}(E,D)\big{)}$

and $\operatorname{ch}(E,D)=\sumop\displaylimits\operatorname{ch}_{\ell}(E,D)$
representing the Chern character. If is the curvature corresponding to $D$ (in
a local trivialization), then

(3.4)
$\operatorname{ch}_{\ell}(E,D)=\frac{1}{\ell!}\operatorname{tr}\left(\frac{i}{2\pi}\Theta\right)^{\ell},$

_cf._ _e.g._ [Tu17, §B.4-6].

The polynomials $Q_{\ell}$ may be computed recursively through Newton’s
identities,

(3.5) $p_{\ell}(x)=(-1)^{\ell-1}\ell
e_{\ell}(x)+\sumop\displaylimits_{i=1}^{\ell-1}(-1)^{\ell-i-1}e_{\ell-i}(x)p_{i}(x),\quad\ell\geq
1.$

In particular, it follows that the $Q_{\ell}$ are independent of $r$.
Moreover, $\operatorname{ch}_{\ell}(E,D)$ is of the form

(3.6)
$\operatorname{ch}_{\ell}(E,D)=\frac{(-1)^{\ell-1}}{(\ell-1)!}c_{\ell}(E,D)+\widetilde{Q}_{\ell}\big{(}c_{1}(E,D),\ldots,c_{\ell-1}(E,D)\big{)},$

where $\widetilde{Q}_{\ell}$ is a weighted homogeneous polynomial of degree
$\ell$, and conversely,

(3.7)
$c_{\ell}(E,D)=(-1)^{\ell-1}(\ell-1)!\operatorname{ch}_{\ell}(E,D)+\widehat{Q}_{\ell}\big{(}\operatorname{ch}_{1}(E,D),\ldots,\operatorname{ch}_{\ell-1}(E,D)\big{)},$

where $\widehat{Q}_{\ell}$ is a weighted homogeneous polynomial of degree
$\ell$.

######  Example 3.1.

We obtain from (3.5) that $p_{1}=e_{1}\text{ and }p_{2}=e_{1}^{2}-2e_{2}$.
Thus, $Q_{1}(t_{1})=t_{1}$ and $Q_{2}(t_{1},t_{2})=t_{1}^{2}-2t_{2}$, so

(3.8) $\operatorname{ch}_{1}(E,D)=c_{1}(E,D)\quad\text{ and
}\quad\operatorname{ch}_{2}(E,D)=\frac{1}{2}(c_{1}(E,D)^{2}-2c_{2}(E,D)).$

We have the following (formal) relationship between $e_{\ell}(x)$ and
$p_{\ell}(x)$, and thus $Q_{\ell}(e_{1},\ldots,e_{\ell})$:

(3.9) $\ln\left(\sumop\displaylimits_{\ell\geq
0}e_{\ell}(x)t^{\ell}\right)=\sumop\displaylimits_{\ell\geq
1}\frac{(-1)^{\ell-1}}{\ell}p_{\ell}(x)t^{\ell}=\sumop\displaylimits_{\ell\geq
1}\frac{(-1)^{\ell-1}}{\ell}Q_{\ell}(e_{1},\ldots,e_{\ell})t^{\ell}.$

This follows _e.g._ by integrating [Mac95, Chapter I, Equation (2.10’)] with
respect to $t$. Since $e_{1},\ldots,e_{r}$ are algebraically independent,
(3.9) holds if we replace the $e_{\ell}$ by $a_{\ell}$ in any commutative
ring. In particular, if we apply (3.9) to $e_{\ell}=c_{\ell}(E,D)$ and take
the components of degree $2\ell$ (the coefficents of $t^{\ell}$) we get

(3.10)
$\ln\big{(}c(E,D)\big{)}_{\ell}=(-1)^{\ell-1}(\ell-1)!\operatorname{ch}_{\ell}(E,D);$

here $()_{\ell}$ denotes the part of form degree $2\ell$.

### 3.3. The Chern character of a complex of vector bundles

Let $(E,\varphi)$ be a complex of vector bundles of the form (2.5). Then the
Chern character can be defined as

$\operatorname{ch}(E)=\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{ch}(E_{k}),$

_cf._ , _e.g._ , [EH16, Chapter 14.2.1] and [Kar08, Chapter V.3].

If $(E,\varphi)$ is equipped with a connection $D=(D_{k})$, for $\ell\geq 1$
we define a _Chern character form_ $\operatorname{ch}_{\ell}(E,D)$ through
(3.3). Then $\operatorname{ch}_{\ell}(E,D)$ inherits properties from the
vector bundle case. In particular (3.6) and (3.7) hold. Also (3.10) holds and,
using that

$\ln\big{(}c(E,D)\big{)}=\ln\left(\prodop\displaylimits_{k=0}^{N}c(E_{k},D_{k})^{(-1)^{k}}\right)=\sumop\displaylimits_{k=0}^{N}(-1)^{k}\ln\big{(}c(E_{k},D_{k})\big{)},$

we get that

(3.11)
$\operatorname{ch}_{\ell}(E,D)=\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{ch}_{\ell}(E_{k},D_{k}).$

In particular, $\operatorname{ch}_{\ell}(E,D)$ represents
$\operatorname{ch}_{\ell}(E)$.

Let k denote the curvature matrix of $D_{k}$ (in some local trivialization)
and define111To be consistent with (3.2) we should have a factor
$(i/2\pi)^{\ell}$ in the definition of $p_{\ell}(E,D)$, _cf._ (3.13). However,
the normalization (3.12) is more convenient to work with.

(3.12)
$p_{\ell}(E,D)=\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}{}_{k}^{\ell}.$

In view of (3.4) and (3.11), for $\ell\geq 1$,

(3.13)
$\operatorname{ch}_{\ell}(E,D)=\frac{i^{\ell}}{(2\pi)^{\ell}\ell!}p_{\ell}(E,D).$

Assume that $D=(D_{k})$ is a $(1,0)$-connection. Let $()_{(q,r)}$ denote the
part of bidegree $(q,r)$ of a form. Since the curvature matrices k (in local
trivializations) consist of forms of bidegree $(2,0)$ and $(1,1)$, it follows
that

(3.14)
$p_{(\ell,\ell)}(E,D)=\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}({}_{k})_{(1,1)}^{\ell}$

and by (3.13) that

(3.15)
$\operatorname{ch}_{(\ell,\ell)}(E,D)=\frac{i^{\ell}}{(2\pi)^{\ell}\ell!}p_{(\ell,\ell)}(E,D).$

## 4\. Connections compatible with a complex

Assume that $(E,\varphi)$ is a complex of vector bundles of the form

(4.1) $0\to
E_{N}\xrightarrow[]{\varphi_{N}}\cdots\xrightarrow[]{\varphi_{1}}E_{0}\xrightarrow[]{\varphi_{0}}E_{-1}\to
0.$

Moreover assume that each $E_{k}$ is equipped with a connection $D_{k}$. Then,
following [BB72], we say that the connection $D=(D_{-1},\ldots,D_{N})$ on
$(E,\varphi)$ is _compatible_ with $(E,\varphi)$ if

(4.2) $D_{k-1}\circ\varphi_{k}=-\varphi_{k}\circ D_{k}$

for $k=0,\ldots,N$. In terms of the induced connection
$D=D_{\operatorname{End}}$ on $\operatorname{End}E$, this can succinctly be
written as $D\varphi_{k}=0$.

Note that in contrast to above, (4.1) starts at level $-1$. The typical
situation we consider is when we start with a complex (2.5) that is pointwise
exact outside an analytic variety $Z$ and then restrict to $X\setminus Z$;
then $E_{-1}=0$.

######  Remark 4.1.

By [BB72, Lemma 4.17], given a complex $(E,\varphi)$ of vector bundles of the
form (4.1) one can always extend a given connection $D_{-1}$ on $E_{-1}$ to a
connection $D=(D_{k})$ that is compatible with $(E,\varphi)$ where it is
pointwise exact. In fact, Lemma 4.4 below gives an explicit formula for such a
connection, see Remark 4.5.

######  Remark 4.2.

In [BB72], the condition of being compatible is stated without the minus sign
in (4.2); our condition on $D$ is actually the same, but we need to introduce
the minus sign since we use the conventions of the superstructure. Indeed, if
$\xi$ is a section of $E_{k}$ of form-degree $0$, then
$D_{k-1}\circ\varphi_{k}\xi$ is defined in the same way with or without the
superstructure, while the action of $\varphi_{k}$ on $D_{k}\xi$ changes sign
depending on whether the superstructure is used or not since $D_{k}\xi$ has
form-degree $1$, _cf._ (2.6).

Compatible connections satisfy the following Whitney formula, [BB72, Lemma
4.22], _cf._ Section 3.1.

######  Lemma 4.3.

Assume that $(E,\varphi)$ is an exact complex of vector bundles of the form
(4.1) that is equipped with a connection $D=(D_{k})$ that is compatible with
$(E,\varphi)$. Then

$c(E_{-1},D_{-1})=\prodop\displaylimits_{k=0}^{N}c(E_{k},D_{k})^{(-1)^{k}}.$

### 4.1. The connection $\widehat{D}^{\epsilon}$

We will consider a specific situation and choice of compatible connection. As
in previous sections, let $(E,\varphi)$ be a complex of vector bundles of the
form (2.5) that is pointwise exact outside the analytic set $Z$. Moreover, let
$\chi$ be a smooth approximand of $\chi_{[1,\infty)}$, let $F$ be a
(generically nonvanishing) section of a vector bundle such that
$Z\subseteq\\{F=0\\}$, and let $\chi_{\epsilon}=\chi(|F|^{2}/\epsilon)$. Then
$\chi_{\epsilon}\equiv 0$ in a neighborhood of $Z$. Consider now a fixed
choice of connection $D=(D_{k})$ on $(E,\varphi)$, and for $\epsilon>0$,
define a new connection $\widehat{D}^{\epsilon}=(\widehat{D}^{\epsilon}_{k})$
on $(E,\varphi)$ through

(4.3)
$\widehat{D}_{k}^{\epsilon}=-\chi_{\epsilon}\sigma_{k}D\varphi_{k}+D_{k}.$

Note that if $D$ is a $(1,0)$-connection, then so is $\widehat{D}^{\epsilon}$.

######  Lemma 4.4.

The connection $\widehat{D}^{\epsilon}$ is compatible with $(E,\varphi)$ where
$\chi_{\epsilon}\equiv 1$.

###### Proof.

Using (2.9), (2.13) and (2.15) we obtain that

$\displaystyle\widehat{D}^{\epsilon}\varphi_{k}$
$\displaystyle=\widehat{D}_{k-1}^{\epsilon}\circ\varphi_{k}+\varphi_{k}\circ\widehat{D}_{k}^{\epsilon}$
$\displaystyle=\chi_{\epsilon}\left(-\sigma_{k-1}D\varphi_{k-1}\varphi_{k}-\varphi_{k}\sigma_{k}D\varphi_{k}\right)+D_{k-1}\circ\varphi_{k}+\varphi_{k}\circ
D_{k}$
$\displaystyle=-\chi_{\epsilon}(\sigma_{k-1}\varphi_{k-1}+\varphi_{k}\sigma_{k})D\varphi_{k}+D_{k-1}\circ\varphi_{k}+\varphi_{k}\circ
D_{k}$ $\displaystyle=(1-\chi_{\epsilon})D\varphi_{k}.$

In particular, $\widehat{D}^{\epsilon}$ is compatible with the complex where
$\chi_{\epsilon}\equiv 1$. ∎

######  Remark 4.5.

Assume that $(E,\varphi)$ is a pointwise exact complex of vector bundles
equipped with some connection $D=(D_{k})$. Then, as in the proof above, it
follows that the connection $\widetilde{D}$ defined by

$\widetilde{D}_{k}=-\sigma_{k}D\varphi_{k}+D_{k}$

is compatible with $(E,\varphi)$. Moreover, $\widetilde{D}_{-1}=D_{-1}$, _cf._
Remark 4.1.

Assume that $\theta_{k}$ is a connection matrix for $D_{k}$ in a local
trivialization, i.e. $D_{k}\alpha=d\alpha+\theta_{k}\wedge\alpha$. Then the
connection matrix for $\widehat{D}_{k}^{\epsilon}$ is

$\hat{\theta}^{\epsilon}_{k}=-\chi_{\epsilon}\sigma_{k}D\varphi_{k}+\theta_{k}$

and thus the curvature matrix of $\widehat{D}_{k}^{\epsilon}$ equals

(4.4)
$\widehat{\Theta}_{k}^{\epsilon}=d\hat{\theta}^{\epsilon}_{k}+(\hat{\theta}_{k}^{\epsilon})^{2}=-d(\chi_{\epsilon}\sigma_{k}D\varphi_{k})+\chi_{\epsilon}^{2}\sigma_{k}D\varphi_{k}\sigma_{k}D\varphi_{k}-\chi_{\epsilon}(\theta_{k}\sigma_{k}D\varphi_{k}+\sigma_{k}D\varphi_{k}\theta_{k})+{}_{k},$

where k is the curvature matrix of $D_{k}$.

## 5\. The Chern current $c^{\mathrm{Res}}(E,D)$

In this section we prove that the limits as $\epsilon\to 0$ of products of
Chern forms $c_{\ell}(E,\widehat{D}^{\epsilon})$, where
$\widehat{D}^{\epsilon}$ is the connection from the previous section, give the
desired currents in (1.5). More generally, we prove the following
generalization of Theorem 1.1.

######  Theorem 5.1.

Assume that $(E,\varphi)$ is a complex of Hermitian vector bundles of the form
(2.5) that is pointwise exact outside a subvariety $Z$ of positive
codimension. Moreover assume that $D=(D_{k})$ is a connection on $(E,\varphi)$
and let $\widehat{D}^{\epsilon}$ be the connection defined by (4.3). Then, for
$\ell_{1},\ldots,\ell_{m}\in\mathbb N_{>0}$,

(5.1) $c_{\ell_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge
c_{\ell_{m}}^{\mathrm{Res}}(E,D)=\lim_{\epsilon\to
0}c_{\ell_{1}}(E,\widehat{D}^{\epsilon})\wedge\dots\wedge
c_{\ell_{m}}(E,\widehat{D}^{\epsilon}),$

where the right side is defined by (3.1), is a well-defined closed
pseudomeromorphic current that is independent of the choice of
$\chi_{\epsilon}$, has support on $Z$, and represents
$c_{\ell_{1}}(E)\wedge\cdots\wedge c_{\ell_{m}}(E)$.

Theorem 1.1 is an immediate consequence of Theorem 5.1.

###### Proof.

Let

(5.2) $M_{\epsilon}=c_{\ell_{1}}(E,\widehat{D}^{\epsilon})\wedge\dots\wedge
c_{\ell_{m}}(E,\widehat{D}^{\epsilon}).$

We first prove that $\lim_{\epsilon\to 0}M_{\epsilon}$ exists and is a
pseudomeromorphic current. This is a local statement and we may therefore work
in a local trivialization where $D_{k}$ is determined by the connection matrix
$\theta_{k}$. By (4.4), $\widehat{\Theta}_{k}^{\epsilon}$ is a (form-valued)
matrix of the form

$\widehat{\Theta}_{k}^{\epsilon}=\alpha_{k}+\chi_{\epsilon}\beta_{k}^{\prime}+\chi_{\epsilon}^{2}\beta_{k}^{\prime\prime}+d\chi_{\epsilon}\wedge\beta_{k}^{\prime\prime\prime},$

where $\alpha_{k}={}_{k}$ is smooth and $\beta_{k}^{\prime}$,
$\beta_{k}^{\prime\prime}$ and $\beta_{k}^{\prime\prime\prime}$ are
polynomials in $\sigma_{k}$, $D\varphi_{k}$, $\theta_{k}$ and exterior
derivatives of such factors. In particular $\alpha_{k}$, $\beta_{k}^{\prime}$,
$\beta_{k}^{\prime\prime}$, and $\beta_{k}^{\prime\prime\prime}$ are
independent of $\epsilon$.

Since $M_{\epsilon}$ is a polynomial in the entries of
$\widehat{\Theta}_{0}^{\epsilon},\dots,\widehat{\Theta}_{N}^{\epsilon}$, see
Section 3.1, we can write

$M_{\epsilon}=A+\sumop\displaylimits_{j\geq
1}\chi_{\epsilon}^{j}B^{\prime}_{j}+\sumop\displaylimits_{j\geq
1}\chi_{\epsilon}^{j-1}d\chi_{\epsilon}\wedge B^{\prime\prime}_{j},$

where $A$, $B^{\prime}_{j}$, and $B^{\prime\prime}_{j}$ are independent of
$\epsilon$, $A$ is smooth, and $B^{\prime}_{j}$ and $B^{\prime\prime}_{j}$ are
polynomials in entries of $\sigma_{k}$, $D\varphi_{k}$, $\theta_{k}$ and
exterior derivatives of such factors. Let
$\hat{\chi}_{\epsilon}=\hat{\chi}(|F|^{2}/\epsilon)$, where
$\hat{\chi}=\chi^{j}\sim\chi_{[1,\infty)}$, _cf._ Section 2. Then by Lemma
2.1, the limits of

$\chi^{j}_{\epsilon}B_{j}^{\prime}=\hat{\chi}_{\epsilon}B_{j}^{\prime}\qquad\text{and}\qquad\chi^{j-1}_{\epsilon}d\chi_{\epsilon}\wedge
B_{j}^{\prime\prime}=d\hat{\chi}_{\epsilon}\wedge B_{j}^{\prime\prime}/j$

as $\epsilon\to 0$ exist and are pseudomeromorphic currents that are
independent of $\chi_{\epsilon}$. It follows that the limit (5.1) exists and
is a pseudomeromorphic current that is independent of $\chi_{\epsilon}$.

By Lemma 4.4, $\widehat{D}^{\epsilon}$ is compatible with $(E,\varphi)$ where
$\chi_{\epsilon}\equiv 1$ and therefore, by Lemma 4.3,
$c(E,\widehat{D}^{\epsilon})=0$ there. It follows that $M_{\epsilon}$ has
support where $\chi_{\epsilon}\nequiv 1$. Note that the $\sigma_{k}$ are
smooth outside of $Z$. By Lemma 2.1, the limit (5.1) is independent of the
choice of $\chi_{\epsilon}$. In particular, we may assume that the section $F$
defining $\chi_{\epsilon}=\chi(|F|^{2}/\epsilon)$ is locally defined such that
$\\{F=0\\}=Z$. It then follows that the limit (5.1) has support on $Z$. That
(5.1) represents $c_{\ell_{1}}(E)\wedge\cdots\wedge c_{\ell_{m}}(E)$ follows
by Poincaré duality, since the forms on the right hand side of (5.1) represent
this class for all $\epsilon>0$. Also (5.1) is closed since the forms on the
right hand side are for all $\epsilon>0$. ∎

######  Remark 5.2.

Assume that $D=(D_{k})$ in Theorem 5.1 is a $(1,0)$-connection. Then
$\widehat{\Theta}_{k}^{\epsilon}$ only has components of bidegree $(2,0)$ and
$(1,1)$, _cf._ (4.4). It follows that (5.2) and consequently (5.1) consist of
components of bidegree $(\ell+q,\ell-q)$ with $q\geq 0$, where
$\ell=\ell_{1}+\cdots+\ell_{m}$.

## 6\. An explicit description of Chern currents of low degrees

In this section we study the Chern current $c^{\mathrm{Res}}(E,D)$ of a
complex $(E,\varphi)$ that is equipped with a $(1,0)$-connection $D$. Our main
result is the following generalization of Theorem 1.5 that is an explicit
description of $c_{p}^{\mathrm{Res}}(E,D)$ in terms of the residue current $R$
associated with $(E,\varphi)$.

######  Theorem 6.1.

Assume that $(E,\varphi)$ is a complex of Hermitian vector bundles of the form
(2.5) that is pointwise exact outside a subvariety $Z$ of codimension $p$, and
let $R$ be the corresponding residue current. Moreover, assume that
$D=(D_{k})$ is a $(1,0)$-connection on $(E,\varphi)$ and let
$c^{\mathrm{Res}}(E,D)$ be the corresponding Chern current. Then

(6.1) $c_{p}^{\mathrm{Res}}(E,D)=\frac{(-1)^{p-1}}{(2\pi
i)^{p}p}\sumop\displaylimits_{k=0}^{N-p}(-1)^{k}\operatorname{tr}(D\varphi_{k+1}\cdots
D\varphi_{k+p}R^{k}_{k+p}).$

Moreover

(6.2) $c_{\ell}^{\mathrm{Res}}(E,D)=0\quad\text{ for }0<\ell<p$

and

(6.3) $c_{\ell_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge
c_{\ell_{m}}^{\mathrm{Res}}(E,D)=0\quad\text{ for }m\geq 2\text{ and
}0<\ell_{1}+\dots+\ell_{m}\leq p.$

In fact, Theorem 6.1 follows from the following formulation in terms of the
Chern character (forms). For $(E,\varphi)$ and $D$ as in the theorem and for
$\ell_{1},\dots,\ell_{m}\geq 1$ we let

(6.4)
$\operatorname{ch}^{\mathrm{Res}}_{\ell_{1}}(E,D)\wedge\dots\wedge\operatorname{ch}^{\mathrm{Res}}_{\ell_{m}}(E,D):=\lim_{\epsilon\to
0}\operatorname{ch}_{\ell_{1}}(E,\widehat{D}^{\epsilon})\wedge\dots\wedge\operatorname{ch}_{\ell_{m}}(E,\widehat{D}^{\epsilon}),$

where $\widehat{D}^{\epsilon}$ is the connection defined by (4.3). By Theorem
5.1 this is a well-defined current with support on $Z$ that represents
$\operatorname{ch}_{\ell_{1}}(E)\wedge\dots\wedge\operatorname{ch}_{\ell_{m}}(E)$.

######  Theorem 6.2.

Assume that $(E,\varphi)$, $D$, $R$, and $p$ are as in Theorem 6.1. For
$\ell\geq 1$, let $\operatorname{ch}_{\ell}^{\mathrm{Res}}(E,D)$ be the
corresponding Chern character current (6.4). Then

(6.5) $\operatorname{ch}_{p}^{\mathrm{Res}}(E,D)=\frac{1}{(2\pi
i)^{p}p!}\sumop\displaylimits_{k=0}^{N-p}(-1)^{k}\operatorname{tr}(D\varphi_{k+1}\cdots
D\varphi_{k+p}R^{k}_{k+p}).$

Moreover

(6.6) $\operatorname{ch}_{\ell}^{\mathrm{Res}}(E,D)=0\quad\text{ for }\ell<p$

and

(6.7)
$\operatorname{ch}_{\ell_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge\operatorname{ch}_{\ell_{m}}^{\mathrm{Res}}(E,D)=0\quad\text{
for }m\geq 2\text{ and }\ell_{1}+\dots+\ell_{m}\leq p.$

###### Proof of Theorem 6.1.

Since $\widehat{Q}_{\ell}$ in (3.7) is a polynomial of weighted degree $\ell$
we get that

(6.8)
$\widehat{Q}_{\ell}\big{(}\operatorname{ch}_{1}^{\mathrm{Res}}(E,D),\ldots,\operatorname{ch}_{\ell-1}^{\mathrm{Res}}(E,D)\big{)}$

is a sum of terms

(6.9)
$\operatorname{ch}_{\lambda_{1}}^{\mathrm{Res}}(E,D)\wedge\dots\wedge\operatorname{ch}_{\lambda_{s}}^{\mathrm{Res}}(E,D),\text{
where }s\geq 2\text{ and }\lambda_{1}+\cdots+\lambda_{s}=\ell.$

Thus (6.8) vanishes by (6.7) for $\ell\leq p$. Now (6.1) and (6.2) follow by
(3.7), (6.5), and (6.6). Also, the left hand side of (6.3) is a sum of terms
of the form (6.9) and thus it vanishes. ∎

######  Remark 6.3.

Taking Theorem 6.1 for granted, by similar arguments as in the proof above,
using (3.6), we get Theorem 6.2. Thus Theorems 6.1 and 6.2 are equivalent.

Recall from Section 2.3 that if $(E,\varphi)$ is a locally free resolution of
a sheaf $\mathcal{F}$ of codimension $p$, then $R^{\ell}_{k}=0$ for $\ell>0$.
Thus the only nonvanishing residue current in (6.1) is $R_{p}=R_{p}^{0}$, and
hence Theorem 1.5 follows. It may be noted that our proof of Theorem 6.1 does
not become simpler in the situation of Theorem 1.5.

To organize the proof of Theorem 6.2 we will introduce a certain class
$O_{Z,\ell,\epsilon}$ of forms depending on $\epsilon>0$ that in the limit are
pseudomeromorphic currents with support on $Z$ that vanish if
$\ell\leq\operatorname{codim}Z$. Throughout this section, let $(E,\varphi)$ be
fixed as the complex from Theorems 6.1 and 6.2 and let $\sigma_{k}$ be the
minimal inverse of $\varphi_{k}$ as in Section 2.3. Let
$\mathcal{E}_{q,\epsilon}$ denote smooth forms of bidegree $(*,q)$ that can be
written as polynomials in $\chi_{\epsilon}$,
${\bar{\partial}}\chi_{\epsilon}$, entries of $\sigma_{k}$,
$\partial\sigma_{k}$ or ${\bar{\partial}}\sigma_{k}$ in some local
trivialization for $k=1,\dots,N$, and smooth forms independent of $\epsilon$.
Here $\chi_{\epsilon}=\chi(|F|^{2}/\epsilon)$, where $\chi$ is a smooth
approximand of $\chi_{[1,\infty)}$ and $F$ is a generically non-vanishing
section of a holomorphic vector bundle such that $Z=\\{F=0\\}$. We say that
$\psi_{\epsilon}\in\mathcal{E}_{\epsilon}:=\oplus\mathcal{E}_{q,\epsilon}$ is
in $O_{Z,\ell,\epsilon}$ if $\psi_{\epsilon}$ is a sum of terms of the form
$a\wedge b_{\epsilon}$, where $a$ is a smooth form that is independent of
$\epsilon$, and $b_{\epsilon}$ is in $\mathcal{E}_{q,\epsilon}$, where
$q<\ell$, and vanishes where $\chi_{\epsilon}\equiv 1$. In particular, if
$\psi_{\epsilon}\in\mathcal{E}_{q,\epsilon}$ vanishes where
$\chi_{\epsilon}\equiv 1$, then $\psi_{\epsilon}\in O_{Z,\ell,\epsilon}$ for
any $\ell>q$. Note that

(6.10) $\mathcal{E}_{q,\epsilon}\wedge O_{Z,\ell,\epsilon}\subseteq
O_{Z,\ell+q,\epsilon}.$

######  Lemma 6.4.

Assume that $Z$ has codimension $p$, and let $\psi_{\epsilon}$ be a form in
$O_{Z,\ell,\epsilon}$ with $\ell\leq p$. Then $\lim_{\epsilon\to
0}\psi_{\epsilon}=0.$

###### Proof.

Consider a term $a\wedge b_{\epsilon}$ of $\psi_{\epsilon}$ as above. Then
$b_{\epsilon}\in\mathcal{E}_{q,\epsilon}$, where $q<\ell\leq p$. By Lemma 2.1,
the limit $b:=\lim_{\epsilon\to 0}b_{\epsilon}$ exists and is a
pseudomeromorphic current of bidegree $(*,q)$. Since $b_{\epsilon}\equiv 0$
where $\chi_{\epsilon}\equiv 1$, $b$ has support on $Z$ and thus $b=0$ by the
dimension principle, see Section 2.1. Since $a$ is smooth and independent of
$\epsilon$, it follows that $\lim_{\epsilon\to 0}(a\wedge
b_{\epsilon})=a\wedge b=0$. ∎

Throughout this section, let $D=(D_{k})$ be a $(1,0)$-connection on
$(E,\varphi)$, let $\chi\sim\chi_{[1,\infty)}$, let $\chi_{\epsilon}$ and
$\widehat{D}^{\epsilon}$ be defined as in Section 4, and let
$c(E,\widehat{D}^{\epsilon})=\sumop\displaylimits
c_{\ell}(E,\widehat{D}^{\epsilon})$ be the corresponding Chern form defined by
(3.1). Since the limits in Theorem 5.1 are independent of the choice of
$\chi_{\epsilon}$, and the results in this section are local statements, we
may assume locally that the section $F$ in the definition of
$\chi_{\epsilon}=\chi(|F|^{2}/\epsilon)$ is such that $\\{F=0\\}=Z$.

######  Lemma 6.5.

For $\ell\geq 1$ and $\epsilon>0$, we have

(6.11) $\operatorname{ch}_{\ell}(E,\widehat{D}^{\epsilon})=\frac{1}{(2\pi
i)^{\ell}\ell!}{\bar{\partial}}\chi_{\epsilon}^{\ell}\wedge\sumop\displaylimits_{k=1}^{N}(-1)^{k}\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{\ell-1}\big{)}+O_{Z,\ell,\epsilon}.$

###### Proof.

We may work in a local trivialization; let $\widehat{\Theta}_{k}^{\epsilon}$
be the curvature matrix of $\widehat{D}^{\epsilon}_{k}$. By Remark 5.2, since
the $D_{k}$ are $(1,0)$-connections,
$\operatorname{ch}_{\ell}(E,\widehat{D}^{\epsilon})$ consists of components of
bidegree $(\ell+q,\ell-q)$ with $q\geq 0$. From the proof of Theorem 5.1 it
follows that $c(E,\widehat{D}^{\epsilon})$ is in $\mathcal{E}_{\epsilon}$ and
vanishes where $\chi_{\epsilon}\equiv 1$, and consequently, the same holds for
$\operatorname{ch}(E,\widehat{D}^{\epsilon})$. It follows that
$\operatorname{ch}_{(\ell+q,\ell-q)}(E,\widehat{D}^{\epsilon})\in
O_{Z,\ell,\epsilon}$ for $q>0$, so

(6.12)
$\operatorname{ch}_{\ell}(E,\widehat{D}^{\epsilon})=\operatorname{ch}_{(\ell,\ell)}(E,\widehat{D}^{\epsilon})+O_{Z,\ell,\epsilon}.$

Since $\widehat{D}^{\epsilon}$ is a $(1,0)$-connection, by (3.15),

(6.13)
$\operatorname{ch}_{(\ell,\ell)}(E,\widehat{D}^{\epsilon})=\frac{i^{\ell}}{(2\pi)^{\ell}\ell!}p_{(\ell,\ell)}(E,\widehat{D}^{\epsilon}),$

where $p_{(\ell,\ell)}$ is given by (3.14). To prove the lemma it thus
suffices to show that

(6.14)
$p_{(\ell,\ell)}(E,\widehat{D}^{\epsilon})=(-1)^{\ell}{\bar{\partial}}\chi_{\epsilon}^{\ell}\wedge\sumop\displaylimits_{k=1}^{N}(-1)^{k}\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{\ell-1}\big{)}+O_{Z,\ell,\epsilon}.$

To prove (6.14), first note in view of (4.4) that since $D$ is a
$(1,0)$-connection,

(6.15)
$(\widehat{\Theta}_{k}^{\epsilon})_{(1,1)}=-{\bar{\partial}}(\chi_{\epsilon}\sigma_{k}D\varphi_{k})+({}_{k})_{(1,1)},$

where k is the curvature matrix of $D_{k}$. We make the following
decomposition:

(6.16)
$\begin{split}-{\bar{\partial}}(\chi_{\epsilon}\sigma_{k}D\varphi_{k})+({}_{k})_{(1,1)}&=-{\bar{\partial}}\chi_{\epsilon}\wedge\sigma_{k}D\varphi_{k}-\chi_{\epsilon}\big{(}{\bar{\partial}}(\sigma_{k}D\varphi_{k})-({}_{k})_{(1,1)}\big{)}+(1-\chi_{\epsilon})({}_{k})_{(1,1)}\\\
&=:\alpha_{k}+\beta_{k}+\gamma_{k}.\end{split}$

Let us consider

$\operatorname{tr}(\widehat{\Theta}_{k}^{\epsilon})^{\ell}_{(1,1)}=\operatorname{tr}(\alpha_{k}+\beta_{k}+\gamma_{k})^{\ell}$

and expand the product. Note that $\gamma_{k}\in O_{Z,1,\epsilon}$, and thus
by (6.10) all terms with a factor $\gamma_{k}$ are in $O_{Z,\ell,\epsilon}$.
Next, note that since $({\bar{\partial}}\chi_{\epsilon})^{2}=0$,
$\alpha_{k}^{2}=0$, and since $\alpha_{k}$ and $\beta_{k}$ have total degree
$4$ and endomorphism degree $2$, all terms containing one $\alpha_{k}$ and the
remaining $\ell-1$ factors being $\beta_{k}$ are all equal to
$\operatorname{tr}(\alpha_{k}\wedge\beta_{k}^{\ell-1})$ by (2.7). To conclude,

(6.17)
$\operatorname{tr}(\widehat{\Theta}_{k}^{\epsilon})^{\ell}_{(1,1)}=\ell\operatorname{tr}(\alpha_{k}\wedge\beta_{k}^{\ell-1})+\operatorname{tr}\beta_{k}^{\ell}+O_{Z,\ell,\epsilon}.$

We have that

$\displaystyle\ell\operatorname{tr}(\alpha_{k}\wedge\beta_{k}^{\ell-1})$
$\displaystyle=(-1)^{\ell}{\bar{\partial}}\chi_{\epsilon}^{\ell}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}\big{(}{\bar{\partial}}(\sigma_{k}D\varphi_{k})-({}_{k})_{(1,1)}\big{)}^{\ell-1}\big{)}$
$\displaystyle=(-1)^{\ell}{\bar{\partial}}\chi_{\epsilon}^{\ell}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}\big{(}{\bar{\partial}}(\sigma_{k}D\varphi_{k})\big{)}^{\ell-1}\big{)}+O_{Z,\ell,\epsilon},$

since
$\ell\chi_{\epsilon}^{\ell-1}{\bar{\partial}}\chi_{\epsilon}={\bar{\partial}}\chi_{\epsilon}^{\ell}$,
_cf._ (2.1), and in the middle expression all terms having a factor
$({}_{k})_{(1,1)}$ also contain a factor
${\bar{\partial}}\chi_{\epsilon}^{\ell}$, and thus are in
$O_{Z,\ell,\epsilon}$. Moreover, by (2.8) and (2.12),

${\bar{\partial}}(\sigma_{k}D\varphi_{k})={\bar{\partial}}\sigma_{k}D\varphi_{k}-\sigma_{k}{\bar{\partial}}(D\varphi_{k})={\bar{\partial}}\sigma_{k}D\varphi_{k}-\sigma_{k}({}_{k-1})_{(1,1)}\varphi_{k}+\sigma_{k}\varphi_{k}({}_{k})_{(1,1)},$

and hence

(6.18)
$\ell\operatorname{tr}(\alpha_{k}\wedge\beta_{k}^{\ell-1})=(-1)^{\ell}{\bar{\partial}}\chi_{\epsilon}^{\ell}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}\big{(}{\bar{\partial}}\sigma_{k}D\varphi_{k}\big{)}^{\ell-1}\big{)}+O_{Z,\ell,\epsilon},$

since all terms containing a factor $({}_{k})_{(1,1)}$ or $({}_{k-1})_{(1,1)}$
also contain a factor ${\bar{\partial}}\chi_{\epsilon}^{\ell}$. Note that
$\alpha_{0}=0$ since $\varphi_{0}=0$. It thus follows from (6.17) and (6.18)
that

$\displaystyle p_{(\ell,\ell)}(E,\widehat{D}^{\epsilon})$
$\displaystyle=\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}(\widehat{\Theta}_{k}^{\epsilon})_{(1,1)}^{\ell}$
$\displaystyle=(-1)^{\ell}{\bar{\partial}}\chi_{\epsilon}^{\ell}\wedge\sumop\displaylimits_{k=1}^{N}(-1)^{k}\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}\big{(}{\bar{\partial}}\sigma_{k}D\varphi_{k}\big{)}^{\ell-1}\big{)}+\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}\beta_{k}^{\ell}+O_{Z,\ell,\epsilon}.$

Thus, to prove (6.14) it suffices to show that
$\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}\beta_{k}^{\ell}$
vanishes for $\ell\geq 1$. Outside $Z$, let $\widetilde{D}$ be the connection
on $(E,\varphi)$ defined by

(6.19) $\widetilde{D}_{k}:=-\sigma_{k}D\varphi_{k}+D_{k}$

and let $\tilde{c}=c(E,\widetilde{D})$ be the corresponding Chern form defined
by (3.1). If follows from Lemma 4.4 that $\widetilde{D}$ is compatible with
$(E,\varphi)$ and thus by Lemma 4.3, ${\tilde{c}}_{j}$ vanishes for $j\geq 1$.
For $\ell\geq 1$, let $\tilde{p}_{\ell}:=p_{(\ell,\ell)}(E,\widetilde{D})$,
where $p_{(\ell,\ell)}$ is given by (3.14). By (3.15) and (3.6),
$\tilde{p}_{\ell}$ is a polynomial in
$\tilde{c}_{(1,1)},\dots,\tilde{c}_{(\ell,\ell)}$. Since $\tilde{c}_{(j,j)}$
vanishes for any $j\geq 1$, $\tilde{p}_{\ell}=0$. Note that
$\beta_{k}=\chi_{\epsilon}(\widetilde{\Theta}_{k})_{(1,1)}$, where
$\widetilde{\Theta}_{k}$ is the curvature matrix corresponding to
$\widetilde{D}_{k}$. Thus

$\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}\beta_{k}^{\ell}=\chi_{\epsilon}^{\ell}\sumop\displaylimits_{k=0}^{N}(-1)^{k}\operatorname{tr}(\widetilde{\Theta}_{k})_{(1,1)}^{\ell}=\chi_{\epsilon}^{\ell}\tilde{p}_{\ell}=0$

for $\epsilon>0$. This concludes the proof of (6.14). ∎

######  Lemma 6.6.

For $\ell\geq 1$ and $\epsilon>0$, we have

(6.20)
${\bar{\partial}}\chi_{\epsilon}\wedge\sumop\displaylimits_{k=1}^{N}(-1)^{k}\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{\ell-1}\big{)}=\\\
(-1)^{\ell}\sumop\displaylimits_{k=0}^{N-\ell}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+\ell}{\bar{\partial}}\sigma_{k+\ell-1}\cdots{\bar{\partial}}\sigma_{k+1}D\varphi_{k+1}\cdots
D\varphi_{k+\ell}\big{)}+O_{Z,\ell,\epsilon}+{\bar{\partial}}O_{Z,\ell,\epsilon}.$

If $\ell>N$, the sum on the right hand side should be interpreted as $0$.

Here ${\bar{\partial}}O_{Z,\ell,\epsilon}$ means forms of the form
${\bar{\partial}}\psi_{\epsilon}$, where $\psi_{\epsilon}\in
O_{Z,\ell,\epsilon}$.

###### Proof.

For $\ell=1$ the sums differ only by a shift in the indices, so we may assume
$\ell\geq 2$. For fixed $k\in\mathbb Z$ and $m,r,s\geq 0$, let

$\rho_{k,m}^{r,s}={\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m+1}{\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}\varphi_{k+m+1}\big{)}.$

If $m=0$, then the factor
${\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}$ should be
interpreted as $1$. Moreover since $(E,\varphi)$ starts at level $0$ and ends
at level $N$, we interpret $\varphi_{j}$ and $\sigma_{j}$ as $0$ if $j>N$ or
$j<1$, and consequently we interpret $\rho_{k,m}^{r,s}$ as $0$ if $k+m\geq N$
or $k\leq 0$.

We claim that

(6.21)
${\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}({\bar{\partial}}\sigma_{k-1}D\varphi_{k-1})^{r}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s+1}D\varphi_{k}\cdots
D\varphi_{k+m}\big{)}=\\\
{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}({\bar{\partial}}\sigma_{k-1}D\varphi_{k-1})^{r+1}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}\big{)}\\\
+\rho_{k-1,m}^{r,s}+\rho_{k,m}^{r,s}+O_{Z,r+s+m+2,\epsilon}+{\bar{\partial}}O_{Z,r+s+m+2,\epsilon}.$

Let us take (6.21) for granted and let
$\rho_{k,m}^{t}=\sumop\displaylimits_{r=0}^{t}\rho_{k,m}^{r,t-r}$. Then by
inductively applying (6.21) to $r=0,\ldots,t$ with $s=t-r$, we get

(6.22)
${\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{t+1}D\varphi_{k}\cdots
D\varphi_{k+m}\big{)}=\\\
{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k-1}(D\varphi_{k-1}{\bar{\partial}}\sigma_{k-1})^{t}D\varphi_{k-1}\cdots
D\varphi_{k+m}\big{)}\\\
+\rho_{k-1,m}^{t}+\rho_{k,m}^{t}+O_{Z,t+m+2,\epsilon}+{\bar{\partial}}O_{Z,t+m+2,\epsilon}.$

It follows that, for fixed $m$ and $t$,

$\sumop\displaylimits_{k=1}^{N-m}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{t+1}D\varphi_{k}\cdots
D\varphi_{k+m}\big{)}=\\\
\sumop\displaylimits_{k=1}^{N-m}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k-1}(D\varphi_{k-1}{\bar{\partial}}\sigma_{k-1})^{t}D\varphi_{k-1}\cdots
D\varphi_{k+m}\big{)}\\\
-\rho_{0,m}^{t}+(-1)^{N-m}\rho_{N-m,m}^{t}+O_{Z,t+m+2,\epsilon}+{\bar{\partial}}O_{Z,t+m+2,\epsilon}.$

Thus, since $\rho_{0,m}^{r,s}$ and $\rho_{N-m,m}^{r,s}$ vanish,

(6.23)
$\sumop\displaylimits_{k=1}^{N-m}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{t+1}D\varphi_{k}\cdots
D\varphi_{k+m}\big{)}=\\\
-\sumop\displaylimits_{k=1}^{N-m-1}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+m+1}{\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{t}D\varphi_{k}\cdots
D\varphi_{k+m+1}\big{)}\\\
+O_{Z,t+m+2,\epsilon}+{\bar{\partial}}O_{Z,t+m+2,\epsilon}.$

Assume that $2\leq\ell\leq N$. By inductively applying (6.23) to
$m=0,\ldots,\ell-2$ with $t=\ell-2-m$, we get

$\displaystyle\sumop\displaylimits_{k=1}^{N}(-1)^{k}$
$\displaystyle{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{\ell-1}D\varphi_{k}\big{)}$
$\displaystyle=-\sumop\displaylimits_{k=1}^{N-1}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+1}{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{\ell-2}D\varphi_{k}D\varphi_{k+1}\big{)}+O_{Z,\ell,\epsilon}+{\bar{\partial}}O_{Z,\ell,\epsilon}$
$\displaystyle=\cdots$
$\displaystyle=(-1)^{\ell-1}\sumop\displaylimits_{k=1}^{N-\ell+1}(-1)^{k}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{k+\ell-1}{\bar{\partial}}\sigma_{k+\ell-2}\cdots{\bar{\partial}}\sigma_{k}D\varphi_{k}\cdots
D\varphi_{k+\ell-1}\big{)}+O_{Z,\ell,\epsilon}+{\bar{\partial}}O_{Z,\ell,\epsilon},$

which after a shift in indices is exactly (6.20). If $\ell>N$ and we perform
the same induction, after $N-1$ steps we end up with

$(-1)^{N}{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}\big{(}\sigma_{N}{\bar{\partial}}\sigma_{N-1}\cdots{\bar{\partial}}\sigma_{1}(D\varphi_{1}{\bar{\partial}}\sigma_{1})^{\ell-N}D\varphi_{1}\cdots
D\varphi_{N}\big{)}+O_{Z,\ell,\epsilon}+{\bar{\partial}}O_{Z,\ell,\epsilon},$

which by (6.22) equals
$\rho_{0,N-1}^{\ell-N-1}+\rho_{1,N-1}^{\ell-N-1}+O_{Z,\ell,\epsilon}=O_{Z,\ell,\epsilon}$;
thus (6.20) holds also in this case.

It remains to prove (6.21). To do this let us replace the first factor
$D\varphi_{k}{\bar{\partial}}\sigma_{k}$ in the left hand side of (6.21) by
the right hand side of (2.17); we then get three terms. The term corresponding
to the second term ${\bar{\partial}}\sigma_{k-1}D\varphi_{k-1}$ in (2.17) is
precisely the first term in the right hand side of (6.21). Next, by (2.13) and
(2.16),

$\varphi_{k-1}D\varphi_{k}{\bar{\partial}}\sigma_{k}=D\varphi_{k-1}\varphi_{k}{\bar{\partial}}\sigma_{k}=D\varphi_{k-1}{\bar{\partial}}\sigma_{k-1}\varphi_{k-1}.$

Applying this repeatedly we get

$\varphi_{k-1}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}=(D\varphi_{k-1}{\bar{\partial}}\sigma_{k-1})^{s}\varphi_{k-1}.$

Using this and (2.13) (to “move” the $D$), we get

$\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}({\bar{\partial}}\sigma_{k-1}D\varphi_{k-1})^{r}D{\bar{\partial}}\sigma_{k-1}\varphi_{k-1}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}=\\\
\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}({\bar{\partial}}\sigma_{k-1}D\varphi_{k-1})^{r}D{\bar{\partial}}\sigma_{k-1}(D\varphi_{k-1}{\bar{\partial}}\sigma_{k-1})^{s}D\varphi_{k-1}\cdots
D\varphi_{k+m-1}\varphi_{k+m}.$

It follows that the term corresponding to the first term in (2.17) equals
$\rho_{k-1,m}^{r,s}$.

Finally we consider the term corresponding to the last term in (2.17). As
above, using (2.13) and (2.16), we get that

$({\bar{\partial}}\sigma_{k-1}D\varphi_{k-1})^{r}\varphi_{k}=\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}$

and thus, using this and (2.14) (to “move” the ${\bar{\partial}}$),

(6.24)
$\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k}({\bar{\partial}}\sigma_{k-1}D\varphi_{k-1})^{r}\varphi_{k}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}=\\\
{\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}\sigma_{k}\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}.$

In view of (2.15) we can replace the factor $\sigma_{k}\varphi_{k}$ by
$\operatorname{Id}_{E_{k}}-\varphi_{k+1}\sigma_{k+1}$:

(6.25)
${\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}\sigma_{k}\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}=\\\
{\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}+\\\
(-{\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}\varphi_{k+1}\sigma_{k+1}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m})=:\xi+\delta.$

By repeatedly using (2.16) we get that

${\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}\varphi_{k+1}\sigma_{k+1}={\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+2}\varphi_{k+2}{\bar{\partial}}\sigma_{k+2}\sigma_{k+1}=\dots=\varphi_{k+m+1}{\bar{\partial}}\sigma_{k+m+1}\cdots{\bar{\partial}}\sigma_{k+2}\sigma_{k+1}.$

It follows, using (2.14), that $\delta$ in (6.25) equals

$\delta=-\varphi_{k+m+1}\sigma_{k+m+1}{\bar{\partial}}\sigma_{k+m}\cdots{\bar{\partial}}\sigma_{k+1}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}=:-\varphi_{k+m+1}\beta.$

Note that $\deg\beta=4\kappa+2$ and $\deg_{e}\beta=2\kappa+1$, where
$\kappa=m+r+s+1$, and since

$\deg\varphi_{k+m+1}=\deg_{e}\varphi_{k+m+1}=1,$

we get, in view of (2.7), that
$\operatorname{tr}(\varphi_{k+m+1}\beta)=-\operatorname{tr}(\beta\varphi_{k+m+1})$
and it follows that this terms equals $\rho_{k,m}^{r,s}$.

It remains to consider $\xi$ in (6.25). Let

$\eta:=\sigma_{k+m}{\bar{\partial}}\sigma_{k+m-1}\cdots{\bar{\partial}}\sigma_{k+1}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{r}D{\bar{\partial}}\sigma_{k}(D\varphi_{k}{\bar{\partial}}\sigma_{k})^{s}D\varphi_{k}\cdots
D\varphi_{k+m}$

Then, by (2.8), ${\bar{\partial}}\eta=\xi+\xi^{\prime}$, where $\xi^{\prime}$
consists of a sum of terms with a factor ${\bar{\partial}}D\varphi_{j}$ or
${\bar{\partial}}D{\bar{\partial}}\sigma_{j}$. Let $q=m+r+s+2$. Then, note
that ${\bar{\partial}}\chi_{\epsilon}\wedge\eta$ is in $O_{Z,q,\epsilon}$, and
thus
${\bar{\partial}}\chi_{\epsilon}\wedge{\bar{\partial}}\eta=-{\bar{\partial}}({\bar{\partial}}\chi_{\epsilon}\wedge\eta)\in{\bar{\partial}}O_{Z,q,\epsilon}$.
Moreover, by (2.12), each term in $\xi^{\prime}$ has a factor that is a smooth
$(1,1)$-form. Therefore ${\bar{\partial}}\chi_{\epsilon}\wedge\xi^{\prime}\in
O_{Z,q,\epsilon}$, and hence
$\operatorname{tr}{\bar{\partial}}\chi_{\epsilon}\wedge\xi=-\operatorname{tr}({\bar{\partial}}\chi_{\epsilon}\wedge\xi^{\prime})+\operatorname{tr}({\bar{\partial}}\chi_{\epsilon}\wedge{\bar{\partial}}\eta)\in
O_{Z,q,\epsilon}+{\bar{\partial}}O_{Z,q,\epsilon}$. This concludes the proof
of (6.21). ∎

###### Proof of Theorem 6.2.

We first prove (6.5). Since $Z$ has codimension $p$ and
$\chi^{p}\sim\chi_{[1,\infty)}$, by Lemmas 6.4, 6.5, and 6.6, and by (2.18),
we have

$\displaystyle\operatorname{ch}_{p}^{\mathrm{Res}}(E,D)$
$\displaystyle=\lim_{\epsilon\to
0}\operatorname{ch}_{p}(E,\widehat{D}^{\epsilon})$
$\displaystyle=\frac{1}{(2\pi i)^{p}p!}\lim_{\epsilon\to
0}{\bar{\partial}}\chi_{\epsilon}^{p}\wedge\sumop\displaylimits_{k=1}^{N}(-1)^{k}\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}({\bar{\partial}}\sigma_{k}D\varphi_{k})^{p-1}\big{)}$
$\displaystyle=\frac{(-1)^{p}}{(2\pi
i)^{p}p!}\sumop\displaylimits_{k=0}^{N-p}(-1)^{k}\lim_{\epsilon\to
0}{\bar{\partial}}\chi_{\epsilon}^{p}\wedge\operatorname{tr}\big{(}\sigma_{k+p}{\bar{\partial}}\sigma_{k+p-1}\cdots{\bar{\partial}}\sigma_{k+1}D\varphi_{k+1}\cdots
D\varphi_{k+p}\big{)}$ $\displaystyle=\frac{(-1)^{p}}{(2\pi
i)^{p}p!}\sumop\displaylimits_{k=0}^{N-p}(-1)^{k}\operatorname{tr}\big{(}R^{k}_{k+p}D\varphi_{k+1}\cdots
D\varphi_{k+p}\big{)}.$

Since $D\varphi_{k+1}\cdots D\varphi_{k+p}$ and $R^{k}_{k+p}$ both have total
degree $2p$ and endomorphism degree $p$, it follows from (2.7) that

$\operatorname{tr}\big{(}R^{k}_{k+p}D\varphi_{k+1}\cdots
D\varphi_{k+p}\big{)}=(-1)^{p}\operatorname{tr}\big{(}D\varphi_{k+1}\cdots
D\varphi_{k+p}R^{k}_{k+p}\big{)},$

and thus (6.5) follows.

Next, by Theorem 5.1 and Remark 5.2,
$\operatorname{ch}_{\ell}^{\mathrm{Res}}(E,D)$ is a pseudomeromorphic current
with support on $Z$ and with components of bidegree $(\ell+q,\ell-q)$ where
$q\geq 0$. Therefore it vanishes by the dimension principle when $\ell<p$, see
Section 2.1. This proves (6.6).

It remains to prove (6.7). From the beginning of the proof of Lemma 6.5 and
(6.10) it follows that

(6.26)
$\operatorname{ch}_{\ell_{1}}(E,\widehat{D}^{\epsilon})\wedge\dots\wedge\operatorname{ch}_{\ell_{m}}(E,\widehat{D}^{\epsilon})=C{p}_{(\ell_{1},\ell_{1})}(E,\widehat{D}^{\epsilon})\wedge\cdots\wedge{p}_{(\ell_{m},\ell_{m})}(E,\widehat{D}^{\epsilon})+O_{Z,\ell_{1}+\cdots+\ell_{m},\epsilon},$

for some appropriate constant $C$. By (6.14), the fact that
$({\bar{\partial}}\chi_{\epsilon})^{2}=0$, and (6.10), it follows that

${p}_{(\ell_{1},\ell_{1})}(E,\widehat{D}^{\epsilon})\wedge\cdots\wedge{p}_{(\ell_{m},\ell_{m})}(E,\widehat{D}^{\epsilon})\in
O_{Z,p,\epsilon}$

if $m\geq 2$ and $\ell_{1}+\cdots+\ell_{m}\leq p$. Thus the limit of (6.26)
vanishes in this case, which proves (6.7). ∎

Assume that $(E,\varphi)$ is a complex of Hermitian vector bundles of the form
(2.5) such that $\mathcal{H}_{k}(E)$ has pure codimension $p$ or vanishes for
$k=0,\dots,N$, and let $Z=\cup\operatorname{supp}\mathcal{H}_{k}(E)$. Then
$(E,\varphi)$ is pointwise exact outside $Z$, which has codimension $p$. Now,
by combining Theorem 2.2 and Theorem 6.1, we obtain the following
generalization of Theorem 1.4.

######  Corollary 6.7.

Assume that $(E,\varphi)$ is a complex of Hermitian vector bundles of the form
(2.5) such that $\mathcal{H}_{k}(E)$ has pure codimension $p$ or vanishes for
$k=0,\dots,N$. Moreover, assume that $D=(D_{k})$ is a $(1,0)$-connection on
$(E,\varphi)$. Then

$c_{p}^{\mathrm{Res}}(E,D)=(-1)^{p-1}(p-1)![E].$

Moreover (1.10) and (1.11) hold.

Here $[E]$ is the cycle of $(E,\varphi)$ defined by (2.19). In particular, it
follows from Corollary 6.7 that $c_{p}^{\mathrm{Res}}(E,D)$ is independent of
the choice of Hermitian metric and $(1,0)$-connection $D$ on $(E,\varphi)$.

By equipping a complex of vector bundles $(E,\varphi)$ with Hermitian metrics
and $(1,0)$-connections and taking cohomology we get the following
generalization of (1.13).

######  Corollary 6.8.

Assume that $(E,\varphi)$ is a complex of vector bundles of the form (2.5)
such that $\mathcal{H}_{k}(E)$ has pure codimension $p$ or vanishes for
$k=0,\dots,N$. Then

$\big{[}(-1)^{p-1}(p-1)![E]\big{]}=c_{p}(E).$

## 7\. An example

We will compute (products of) Chern currents $c^{\mathrm{Res}}(E,D)$ for an
explicit choice of $(E,\varphi)$ and $D$. Let
$\mathcal{J}\subseteq{\mathcal{O}}_{\mathbb P^{2}_{[t,x,y]}}$ be defined by
$\mathcal{J}=\mathcal{J}(y^{k},x^{\ell}y^{m})$, where $m<k$, and let
${\mathcal{O}}_{Z}:={\mathcal{O}}_{\mathbb P^{2}}/\mathcal{J}$. Then $Z$ has
pure dimension $1$, since $Z_{\text{red}}=\\{y=0\\}$, which is irreducible.
However, note that $\mathcal{J}$ has an embedded prime $\mathcal{J}(x,y)$ of
dimension $0$. Now $\mathcal{F}={\mathcal{O}}_{Z}$ has a locally free
resolution of the form

(7.1)
$0\to{\mathcal{O}}(-k-\ell)\xrightarrow[]{\varphi_{2}}{\mathcal{O}}(-k)\oplus{\mathcal{O}}(-\ell-m)\xrightarrow[]{\varphi_{1}}{\mathcal{O}}_{\mathbb
P^{2}}\to\mathcal{F}\to 0,$

where in the trivialization in the coordinate chart $\mathbb C^{2}=\mathbb
C^{2}_{(x,y)}$

(7.2) $\varphi_{2}=\left[\begin{array}[]{c}-x^{\ell}\\\
y^{k-m}\end{array}\right]\text{ and
}\varphi_{1}=\left[\begin{array}[]{cc}y^{k}&x^{\ell}y^{m}\end{array}\right].$

Let us start by computing the Chern class of $\mathcal{F}$. Let $\omega$
denote $c_{1}({\mathcal{O}}(1))$. Then

$\displaystyle c(E_{0})$ $\displaystyle=c({\mathcal{O}}_{\mathbb P^{2}})=1$
$\displaystyle c(E_{1})$
$\displaystyle=c\big{(}{\mathcal{O}}(-k)\oplus{\mathcal{O}}(-\ell-m)\big{)}=1-(k+\ell+m)\omega+k(\ell+m)\omega^{2}$
$\displaystyle c(E_{2})$
$\displaystyle=c\big{(}{\mathcal{O}}(-k-\ell)\big{)}=1-(k+\ell)\omega,$

see _e.g._ [Ful98, Chapter 3]. Moreover,

(7.3) $c(E_{1})^{-1}=1-c_{1}(E_{1})+c_{1}(E_{1})^{2}-c_{2}(E_{1});$

here the sum ends in degree $2$, since we are in dimension $2$. Thus, by
(1.2),

$\displaystyle c(\mathcal{F})=c(E_{0})c(E_{1})^{-1}c(E_{2})$
$\displaystyle=1-c_{1}(E_{1})+c_{1}(E_{2})+c_{1}(E_{1})^{2}-c_{2}(E_{1})-c_{1}(E_{1})c_{1}(E_{2})$
$\displaystyle=1+m\omega+\big{(}m^{2}+\ell(m-k)\big{)}\omega^{2}.$

In particular,

(7.4) $c_{1}(\mathcal{F})=m\omega\quad\text{ and }\quad
c_{2}(\mathcal{F})=\big{(}m^{2}+\ell(m-k)\big{)}\omega^{2}.$

### 7.1. Chern currents

Assume that each $E_{k}$ in (7.1) is equipped with the metric induced by the
standard metric on $\mathcal{O}(1)\to\mathbb P^{2}$, in turn induced by the
standard metric on $\mathbb C^{3}$, and let $D_{k}$ the corresponding Chern
connection. Let $D=(D_{k})$, let $\chi\sim\chi_{[1,\infty)}$, and let
$\chi_{\epsilon}$ and $\widehat{D}^{\epsilon}$ be as in Section 4. By Theorem
1.4,

$c_{1}^{\mathrm{Res}}(E,D)=[\mathcal{F}]=[Z]=m[y=0],$

which clearly is a representative of $c_{1}(\mathcal{F})$, see (7.4), with
support on $\operatorname{supp}\mathcal{F}=Z_{\text{red}}=\\{y=0\\}$. We want
to compute $c^{\mathrm{Res}}_{2}(E,D)$ and $(c^{\mathrm{Res}}_{1})^{2}(E,D)$.
Note that these currents are not covered by Theorem 1.5, since $p=1$ in this
case.

Let $\hat{p}_{\ell}=p_{(\ell,\ell)}(E,\widehat{D}^{\epsilon})$, where
$p_{(\ell,\ell)}$ is given by (3.14). For degree reasons,
$\operatorname{ch}_{2}(E,\widehat{D}^{\epsilon})=\operatorname{ch}_{(2,2)}(E,\widehat{D}^{\epsilon})$
and
$\operatorname{ch}_{1}^{2}(E,\widehat{D}^{\epsilon})=\operatorname{ch}_{(1,1)}^{2}(E,\widehat{D}^{\epsilon})$,
_cf._ Remark 5.2. It follows in view of (3.8) and (3.15) that

(7.5)
$c_{2}(E,\widehat{D}^{\epsilon})=\left(\frac{i}{2\pi}\right)^{2}\frac{1}{2}(\hat{p}_{1}^{2}-\hat{p}_{2})\quad\text{
and }\quad
c_{1}^{2}(E,\widehat{D}^{\epsilon})=\left(\frac{i}{2\pi}\right)^{2}\hat{p}_{1}^{2}.$

Thus, to compute $c_{2}^{\mathrm{Res}}(E,D)$ and
$(c_{1}^{\mathrm{Res}})^{2}(E,D)$, it suffices to calculate the limits of
$\hat{p}_{1}^{2}$ and $\hat{p}_{2}$ as $\epsilon\to 0$.

Note first that only two of the standard coordinate charts of $\mathbb P^{2}$
intersect $Z$. In $\mathbb C^{2}_{(t,y)}$, we have that
$\varphi_{1}=y^{m}\left[\begin{array}[]{cc}y^{k-m}&1\end{array}\right]$, so
$\sigma_{1}=(1/y^{m})\sigma_{1}^{\prime}$, where $\sigma_{1}^{\prime}$ is
smooth. By using (4.4) one can check that the limits of $\hat{p}_{1}^{2}$ and
$\hat{p}_{2}$ put no mass at $\\{t=y=0\\}$. Thus it is enough to compute the
limits in the coordinate chart $\mathbb C^{2}_{(x,y)}$ where $\varphi_{j}$ are
given by (7.2). Note that $\varphi_{1}=y^{m}\varphi_{1}^{\prime}$, where
$\varphi_{1}^{\prime}=[y^{k-m}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ x^{\ell}]$ has rank $1$ outside of the origin. Then
$\sigma_{1}=(1/y^{m})\sigma_{1}^{\prime}$, where $\sigma_{1}^{\prime}$ is
smooth outside the origin, and

(7.6)
$\sigma_{1}^{\prime}\varphi_{1}^{\prime}|_{\\{y=0\\}}=\left[\begin{array}[]{cc}0&0\\\
0&1\end{array}\right]$

when $x\neq 0$ and $\varphi_{1}^{\prime}\sigma_{1}^{\prime}=1$ outside the
origin. Also note that $\sigma_{2}$ is smooth outside the origin, since
$\varphi_{2}$ has constant rank there. Let $O^{\prime}_{Z,\ell,\epsilon}$ be
defined as in the beginning of Section 6 but with $\sigma_{1}$ replaced by
$\sigma_{1}^{\prime}$, and let $O_{\epsilon}=O^{\prime}_{Z,2,\epsilon}$. Then
$\psi_{\epsilon}\in O_{\epsilon}$ is smooth outside the origin and, by
arguments as in the proof of Lemma 6.4, $\lim_{\epsilon\to
0}\psi_{\epsilon}=0$.

Next, let $\hat{\omega}=(2\pi/i)\omega$, where $\omega$ now denotes the
Fubini-Study form. Then

(7.7) ${}_{1}=\left[\begin{array}[]{cc}-k&0\\\
0&-(\ell+m)\end{array}\right]\hat{\omega},\quad{}_{2}=-(k+\ell)\hat{\omega}.$

In particular, k is of bidegree $(1,1)$. Let
$\widetilde{D}=(\widetilde{D}_{k})$ be the connection on $\mathbb
P^{2}\setminus Z$ defined by (6.19) and let $\widetilde{\Theta}_{k}$ be the
corresponding curvature forms. Then a computation, _cf._ (6.15) and (6.16),
yields

(7.8)
$(\widehat{\Theta}^{\epsilon}_{k})_{(1,1)}=-{\bar{\partial}}\chi_{\epsilon}\wedge\sigma_{k}D\varphi_{k}+\chi_{\epsilon}(\widetilde{\Theta}_{k})_{(1,1)}+(1-\chi_{\epsilon}){}_{k}.$

Let us start by computing $\hat{p}_{1}^{2}$. Recall from the proof of Lemma
6.5 that
$\tilde{p}_{j}=-\operatorname{tr}(\widetilde{\Theta}_{1})_{(1,1)}^{j}+\operatorname{tr}(\widetilde{\Theta}_{2})_{(1,1)}^{j}$
vanishes where $\chi_{\epsilon}\nequiv 0$ for $j=1,2$. Moreover, note in view
of (7.7) that
$-\operatorname{tr}{}_{1}+\operatorname{tr}{}_{2}=m\hat{\omega}$. It follows
that

$\hat{p}_{1}=-\operatorname{tr}(\widehat{\Theta}^{\epsilon}_{1})_{(1,1)}+\operatorname{tr}(\widehat{\Theta}^{\epsilon}_{2})_{(1,1)}=\\\
{\bar{\partial}}\chi_{\epsilon}\wedge\big{(}\operatorname{tr}(\sigma_{1}D\varphi_{1})-\operatorname{tr}(\sigma_{2}D\varphi_{2})\big{)}+(1-\chi_{\epsilon})m\hat{\omega}.$

Note that
$(1-\chi_{\epsilon}){\bar{\partial}}\chi_{\epsilon}=(1/2){\bar{\partial}}\tilde{\chi}_{\epsilon}$,
where $\tilde{\chi}=2(\chi-\chi^{2}/2)\sim\chi_{[1,\infty)}$. Using this and
that $({\bar{\partial}}\chi_{\epsilon})^{2}=0$, we get

$\displaystyle\hat{p}_{1}^{2}$
$\displaystyle=2m\hat{\omega}\wedge(1-\chi_{\epsilon}){\bar{\partial}}\chi_{\epsilon}\wedge\big{(}\operatorname{tr}(\sigma_{1}D\varphi_{1})-\operatorname{tr}(\sigma_{2}D\varphi_{2})\big{)}+(1-\chi_{\epsilon})^{2}m^{2}\hat{\omega}^{2}$
$\displaystyle=m\hat{\omega}\wedge{\bar{\partial}}\tilde{\chi}_{\epsilon}\wedge\operatorname{tr}(\sigma_{1}D\varphi_{1})+O_{\epsilon}.$

Note that

(7.9)
$\sigma_{1}D\varphi_{1}=-\frac{Dy^{m}}{y^{m}}\sigma_{1}^{\prime}\varphi_{1}^{\prime}+\theta_{1}\sigma_{1}^{\prime}\varphi_{1}^{\prime}+\sigma_{1}^{\prime}D\varphi_{1}^{\prime}.$

Therefore, in view of the Poincaré-Lelong formula, _cf._ (1.7) and (7.6),
since $\sigma_{1}^{\prime}\varphi_{1}^{\prime}$ is smooth outside the origin,

(7.10)
${\bar{\partial}}\tilde{\chi}_{\epsilon}\wedge\sigma_{1}D\varphi_{1}=-{\bar{\partial}}\tilde{\chi}_{\epsilon}\wedge\frac{Dy^{m}}{y^{m}}\sigma_{1}^{\prime}\varphi_{1}^{\prime}+O_{\epsilon}\xrightarrow[\epsilon\to
0]{}\frac{2\pi}{i}m[y=0]\left[\begin{array}[]{cc}0&0\\\ 0&1\end{array}\right]$

outside the origin. Since the limit is a pseudomeromorphic $(1,1)$-current,
(7.10) holds everywhere by the dimension principle. It follows that

(7.11) $\lim_{\epsilon\to 0}\hat{p}_{1}^{2}=\lim_{\epsilon\to
0}m\hat{\omega}\wedge{\bar{\partial}}\tilde{\chi}_{\epsilon}\wedge\operatorname{tr}(\sigma_{1}D\varphi_{1})=\frac{2\pi}{i}m^{2}\hat{\omega}\wedge[y=0].$

Let us next consider
$\hat{p}_{2}=-\operatorname{tr}(\widehat{\Theta}_{1})^{2}_{(1,1)}+\operatorname{tr}(\widehat{\Theta}_{2})_{(1,1)}^{2}$.
A computation using (2.7), _cf._ (7.8), yields

(7.12)
$\displaystyle\operatorname{tr}(\widehat{\Theta}^{\epsilon}_{k})_{(1,1)}^{2}$
$\displaystyle=\operatorname{tr}\big{(}-{\bar{\partial}}\chi_{\epsilon}\wedge\sigma_{k}D\varphi_{k}-\chi_{\epsilon}{\bar{\partial}}(\sigma_{k}D\varphi_{k})+{}_{k}\big{)}^{2}$
$\displaystyle={\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}{\bar{\partial}}(\sigma_{k}D\varphi_{k})\big{)}-2{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}(\sigma_{k}D\varphi_{k}{}_{k})+\operatorname{tr}\big{(}\chi_{\epsilon}(\widetilde{\Theta}_{k})_{(1,1)}+(1-\chi_{\epsilon}){}_{k}\big{)}^{2}.$

Again using that
$\tilde{p}_{j}=-\operatorname{tr}(\widetilde{\Theta}_{1})_{(1,1)}^{j}+\operatorname{tr}(\widetilde{\Theta}_{2})_{(1,1)}^{j}$
vanishes where $\chi_{\epsilon}\nequiv 0$, we get

(7.13)
$-\big{(}\chi_{\epsilon}(\widetilde{\Theta}_{1})_{(1,1)}+(1-\chi_{\epsilon}){}_{1}\big{)}^{2}+\big{(}\chi_{\epsilon}(\widetilde{\Theta}_{2})_{(1,1)}+(1-\chi_{\epsilon}){}_{2}\big{)}^{2}=O_{\epsilon}\to
0.$

Note that ${\bar{\partial}}\chi_{\epsilon}\wedge\sigma_{2}D\varphi_{2}{}_{2}$
is in $O_{\epsilon}$. Therefore, in view of (7.7) and (7.10),

(7.14)
$2{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}(\sigma_{1}D\varphi_{1}{}_{1})-2{\bar{\partial}}\chi_{\epsilon}\wedge\operatorname{tr}(\sigma_{2}D\varphi_{2}{}_{2})=\\\
-2{\bar{\partial}}\chi_{\epsilon}\wedge\frac{Dy^{m}}{y^{m}}\operatorname{tr}(\sigma_{1}^{\prime}\varphi_{1}^{\prime}{}_{1})+O_{\epsilon}\xrightarrow[\epsilon\to
0]{}-\frac{2\pi}{i}2m(\ell+m)\hat{\omega}\wedge[y=0].$

Let us next consider the contribution from the first term

(7.15)
${\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}{\bar{\partial}}(\sigma_{k}D\varphi_{k})\big{)}={\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}{\bar{\partial}}\sigma_{k}D\varphi_{k}\big{)}-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{k}D\varphi_{k}\sigma_{k}{\bar{\partial}}(D\varphi_{k})\big{)}$

in (7.12). We start by considering the contribution from the first term in
(7.15). By arguments as in the proof of Lemma 6.6, we get that

(7.16)
$-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{1}D\varphi_{1}{\bar{\partial}}\sigma_{1}D\varphi_{1}\big{)}+{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}D\varphi_{2}{\bar{\partial}}\sigma_{2}D\varphi_{2}\big{)}=\\\
{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}{\bar{\partial}}\sigma_{1}D\varphi_{1}D\varphi_{2}\big{)}-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}D{\bar{\partial}}\sigma_{1}D\varphi_{1}\big{)}+{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}D{\bar{\partial}}\sigma_{2}D\varphi_{2}\big{)}.$

Taking the limit of the first term in the right hand side of (7.16), we get

(7.17) $\begin{split}\lim_{\epsilon\to
0}{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}(\sigma_{2}{\bar{\partial}}\sigma_{1}D\varphi_{1}D\varphi_{2})&=\lim_{\epsilon\to
0}{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}(D\varphi_{1}D\varphi_{2}\sigma_{2}{\bar{\partial}}\sigma_{1})\\\
&=\operatorname{tr}(D\varphi_{1}D\varphi_{2}R_{2}^{0})\\\ \
&=\left(\frac{2\pi}{i}\right)^{2}\ell(2k-m)[0].\end{split}$

Here, the first equality follows from (2.7), the second equality from (2.18),
and the third equality is computed in [LW18, Example 5.2]. Next, by (2.7),
$\operatorname{tr}(D{\bar{\partial}}\sigma_{1}D\varphi_{1})=-\operatorname{tr}(D\varphi_{1}D{\bar{\partial}}\sigma_{1})$.
Using (2.10), (2.11) and the fact that
$\varphi_{1}^{\prime}\sigma_{1}^{\prime}=1$, so
$\varphi_{1}^{\prime}{\bar{\partial}}\sigma_{1}^{\prime}=0$, we have

${\bar{\partial}}\chi_{\epsilon}^{2}\wedge
D\varphi_{1}D{\bar{\partial}}\sigma_{1}={\bar{\partial}}\chi_{\epsilon}^{2}\wedge
D(D\varphi_{1}{\bar{\partial}}\sigma_{1})={\bar{\partial}}\chi_{\epsilon}^{2}\wedge
D(D\varphi^{\prime}_{1}{\bar{\partial}}\sigma^{\prime}_{1})={\bar{\partial}}\chi_{\epsilon}^{2}\wedge
D\varphi^{\prime}_{1}D{\bar{\partial}}\sigma^{\prime}_{1}.$

If we let $f$ be the section of ${\mathcal{O}}(k-m)\oplus{\mathcal{O}}(\ell)$
defined by $f=\left[\begin{array}[]{cc}y^{k-m}&x^{\ell}\end{array}\right]$,
then $\varphi_{2},\varphi_{1}^{\prime}$ are the morphisms in the Koszul
complex defined by (contraction with) $f$. If we let $\sigma$ be the minimal
inverse of $f$, when $f$ is viewed as a section of
$\operatorname{Hom}({\mathcal{O}}(-(k-m))\oplus{\mathcal{O}}(-\ell),{\mathcal{O}})$,
then $\sigma_{2}$ and $\sigma_{1}^{\prime}$ are given by multiplication with
$\sigma$. One may verify that $D\varphi_{2}$ and $D\varphi_{1}^{\prime}$ are
given by contraction with $Df$, and that $D{\bar{\partial}}\sigma_{2}$ and
$D{\bar{\partial}}\sigma_{1}^{\prime}$ are given by multiplication with
$D{\bar{\partial}}\sigma$. A calculation then yields that

$\operatorname{tr}\big{(}D\varphi^{\prime}_{1}D{\bar{\partial}}\sigma^{\prime}_{1}\big{)}=-\operatorname{tr}\big{(}D{\bar{\partial}}\sigma_{2}D\varphi_{2}\big{)},$

so by (7.16),

(7.18)
$-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{1}D\varphi_{1}{\bar{\partial}}\sigma_{1}D\varphi_{1}\big{)}+{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}D\varphi_{2}{\bar{\partial}}\sigma_{2}D\varphi_{2}\big{)}={\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}{\bar{\partial}}\sigma_{1}D\varphi_{1}D\varphi_{2}\big{)}.$

Thus, in view of (7.18) and (7.17),

(7.19)
$-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{1}D\varphi_{1}{\bar{\partial}}\sigma_{1}D\varphi_{1}\big{)}+{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}D\varphi_{2}{\bar{\partial}}\sigma_{2}D\varphi_{2}\big{)}\xrightarrow[\epsilon\to
0]{}\left(\frac{2\pi}{i}\right)^{2}\ell(2k-m)[0].$

Next, let us consider the contribution from the second term in (7.15). As
above, using (2.12), _cf._ (7.9),

${\bar{\partial}}\chi_{\epsilon}^{2}\wedge\big{(}\sigma_{1}D\varphi_{1}\sigma_{1}{\bar{\partial}}(D\varphi_{1})\big{)}=-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\big{(}\sigma_{1}D\varphi_{1}\sigma_{1}\varphi_{1}{}_{1}\big{)}={\bar{\partial}}\chi_{\epsilon}^{2}\wedge\big{(}\frac{Dy^{m}}{y^{m}}\sigma_{1}^{\prime}\varphi_{1}^{\prime}{}_{1}\big{)}+O_{\epsilon}.$

Note that
${\bar{\partial}}\chi_{\epsilon}^{2}\wedge\sigma_{2}D\varphi_{2}\sigma_{2}{\bar{\partial}}(D\varphi_{2})$
is in $O_{\epsilon}$. Thus, by (7.10) and (7.7),

(7.20)
${\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{1}D\varphi_{1}\sigma_{1}{\bar{\partial}}(D\varphi_{1})\big{)}-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}D\varphi_{2}\sigma_{2}{\bar{\partial}}(D\varphi_{2})\big{)}\xrightarrow[\epsilon\to
0]{}\frac{2\pi}{i}m(\ell+m)\hat{\omega}\wedge[y=0],$

_cf._ (7.14).

From (7.15), (7.19), and (7.20), we conclude that

(7.21)
$-{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{1}D\varphi_{1}{\bar{\partial}}(\sigma_{1}D\varphi_{1})\big{)}+{\bar{\partial}}\chi_{\epsilon}^{2}\wedge\operatorname{tr}\big{(}\sigma_{2}D\varphi_{2}{\bar{\partial}}(\sigma_{2}D\varphi_{2})\big{)}\xrightarrow[\epsilon\to
0]{}\\\
\left(\frac{2\pi}{i}\right)^{2}\ell(2k-m)[0]+\frac{2\pi}{i}m(\ell+m)\hat{\omega}\wedge[y=0].$

Next, from (7.12), (7.13), (7.14), and (7.21), we conclude that

(7.22)
$\hat{p}_{2}=-\operatorname{tr}(\widehat{\Theta}_{1})^{2}_{(1,1)}+\operatorname{tr}(\widehat{\Theta}_{2})_{(1,1)}^{2}\xrightarrow[\epsilon\to
0]{}-(2\pi/i)m(\ell+m)\hat{\omega}\wedge[y=0]+(2\pi/i)^{2}\ell(2k-m)[0].$

Finally from (7.5), (7.11), and (7.22) we conclude that

$c_{2}^{\mathrm{Res}}(E,D)=\left(\frac{i}{2\pi}\right)^{2}\frac{1}{2}\,\lim_{\epsilon\to
0}(\hat{p}_{1}^{2}-\hat{p}_{2})=\frac{1}{2}\big{(}m(2m+\ell)\omega\wedge[y=0]-\ell(2k-m)[0]\big{)}$

and that

$(c_{1}^{\mathrm{Res}})^{2}(E,D)=\left(\frac{i}{2\pi}\right)^{2}\lim_{\epsilon\to
0}\hat{p}_{1}^{2}=m^{2}\omega[y=0].$

Taking cohomology, since $[[0]]=[[y=0]\wedge\omega]=[\omega^{2}]$, we get

$\displaystyle\big{[}c_{2}^{\mathrm{Res}}(E,D)\big{]}=\big{(}m^{2}+\ell(m-k)\big{)}[\omega^{2}]=c_{2}(\mathcal{F})\quad\text{
and
}\quad\big{[}(c_{1}^{\mathrm{Res}})^{2}(E,D)\big{]}=m^{2}[\omega^{2}]=c_{1}(\mathcal{F})^{2},$

see (7.4).

## References

  * [AW07] M. Andersson and E. Wulcan, _Residue currents with prescribed annihilator ideals_ , Ann. Sci. École Norm. Sup. 40 (2007), 985–1007.
  * [AW10] by same author, _Decomposition of residue currents_ , J. Reine Angew. Math. 638 (2010), 103–118.
  * [AW18] by same author, _Direct images of semi-meromorphic currents_ , Ann. Inst. Fourier 68 (2018), 875–900.
  * [BB72] P. Baum and R. Bott, _Singularities of holomorphic foliations_ , J. Differential Geometry 7 (1972), 279–342.
  * [BGS90a] J.-M. Bismut, H. Gillet, and C. Soulé, _Bott-Chern currents and complex immersions_ , Duke Math. J. 60 (1990), 255–284.
  * [BGS90b] by same author, _Complex immersions and Arakelov geometry_. In: The Grothendieck Festschrift, Vol. I, pp. 249–331, Progr. Math., 86, Birkhäuser Boston, Boston, MA, 1990.
  * [BSW21] J.-M. Bismut, S. Shen, and Z. Wei, _Coherent sheaves, superconnections, and RRG_ , preprint arXiv:2102.08129 (2021).
  * [BS58] A. Borel and J.-P. Serre, _Le théorème de Riemann-Roch_ , Bull. Soc. Math. France 86 (1958), 97–136.
  * [CH78] N. R. Coleff and M. E. Herrera, _Les courants résiduels associés à une forme méromorphe_ , Lecture Notes in Mathematics 633, Springer, Berlin, 1978.
  * [EH16] D. Eisenbud and J. Harris, _3264 and all that—a second course in algebraic geometry_ , Cambridge University Press, Cambridge, 2016.
  * [Ful98] W. Fulton, _Intersection theory_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 2, Springer-Verlag, Berlin, 1998.
  * [Gre80] H. I. Green, _Chern classes for coherent sheaves_ , Ph.D. Thesis, University of Warwick, (1980). Available from http://wrap.warwick.ac.uk/40592/1/WRAP_THESIS_Green_1980.pdf.
  * [Gri10] J. Grivaux, _Chern classes in Deligne cohomology for coherent analytic sheaves_ , Math. Ann. 347 (2010), 249–284.
  * [Hos20a] T. Hosgood, _Simplicial Chern-Weil theory for coherent analytic sheaves, part I_ , preprint arXiv:2003.10023 (2020).
  * [Hos20b] by same author, _Simplicial Chern-Weil theory for coherent analytic sheaves, part II_ , preprint arXiv:2003.10591 (2020).
  * [HL93] F. R. Harvey and H. B. Lawson Jr., _A theory of characteristic currents associated with a singular connection_ , Astérisque 213, Société Mathématique de France, 1993.
  * [HL71] M. Herrera and D. Lieberman, _Residues and principal values on complex spaces_ , Math. Ann. 194 (1971), 259–294.
  * [JL21] J. Johansson and R. Lärkäng, _An explicit isomorphism of different representations of the Ext functor using residue currents_ , preprint arXiv:2109.00480 (2021).
  * [Kar08] M. Karoubi, _$K$ -theory_, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
  * [LW18] R. Lärkäng, and E. Wulcan, _Residue currents and fundamental cycles_ , Indiana Univ. Math. J. 67 (2018), 1085–1114.
  * [LW21] by same author, _Residue currents and cycles of complexes of vector bundles_ , Ann. Fac. Sci. Toulouse 30 (2021), no. 5, 961–984.
  * [Mac95] I. G. Macdonald, _Symmetric functions and Hall polynomials_ , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  * [Qia16] H. Qiang, _On the Bott-Chern characteristic classes for coherent sheaves_ , preprint arXiv:1611.04238 (2016).
  * [Suw98] T. Suwa, _Indices of vector fields and residues of singular holomorphic foliations_ , Actualités Mathématiques, Hermann, Paris, 1998.
  * [TT86] D. Toledo and Y. L. L. Tong, _Green’s theory of Chern classes and the Riemann-Roch formula_. In: _The Lefschetz centennial conference, Part I_ (Mexico City, 1984), pp. 261–275, Contemp. Math. 58, Amer. Math. Soc., Providence, RI, 1986.
  * [Tu17] L. W. Tu, _Differential geometry_ , Graduate Texts in Mathematics, 275, Springer, Cham, 2017.
  * [Wu20] X. Wu, _Intersection theory and Chern classes in Bott-Chern cohomology_ , preprint arXiv:2011.13759 (2020).