# Characteristic Currents on Cohesive Modules

Zhaobo Han

(April 2024)

###### Abstract

Let $\mathcal{F}$ be a coherent sheaf on a complex variety $X$ that has a
locally free resolution $E^{\bullet}$. In [19], the authors constructed a
pseudomeromorphic current whose support is contained in $supp(E^{\bullet})$
that represents products of Chern classes of $\mathcal{F}.$ In this paper, we
show that their construction works for general de-Rham characteristic classes
and then generalize it to represent products (in de-Rham cohomology) of
characteristic forms of cohesive modules defined by Block[8]. Finally, we
state a corollary to a transgression result in [16] that show that it is
sufficient to only use the degree-$0$ and degree-$1$ parts of the
superconnection to construct currents[6][5] that represent characteristic
forms of cohesive modules in the Bott-Chern cohomology.

##  1 Introduction

While coherent sheaves on any quasi-projective scheme over a Noetherian affine
scheme admits a locally free resolution [14, Example 6.5.1], this is not
generally true. An example is certain coherent sheaves on $Spec(K[x]/(x^{2})$
for any field $K.$[12, Example 4.18]. To circumvent such issue, in [8,
Definition 2.3.2][9], Block introduced the differential-graded category
$\mathcal{P}_{\mathcal{D}}$ of Cohesive Modules over the differential-graded
algebra (dga)
$\mathcal{D}=(\mathcal{A}^{\bullet}(X),d,0)=(\mathcal{A}^{\bullet,0}(X),\overline{\partial},0)$
the Dolbeault dga of a complex manifold $X$ (and also over general curved
dga’s) and studied their properties. It has the important property that

######  Theorem 1.1.

[8, Theorem 4.1.3] Let $X$ be a compact complex manifold, and $D_{coh}^{b}(X)$
be the bounded derived category of complexes $\mathcal{O}_{X}$-sheaves with
coherent cohomology. Then the homotopy category
$Ho(\mathcal{P}_{\mathcal{D}}),$ whose objects are exactly those of
$\mathcal{P}_{\mathcal{D}}$ and whose morphisms
$Ho(\mathcal{P}_{\mathcal{D}})(x,y)=H^{0}(\mathcal{P}_{\mathcal{D}}(x,y)),$ is
equivalent to $D_{coh}^{b}(X).$

Thus many statements about coherent sheaves admitting a locally free
resolution can be translated into more general statements about cohesive
modules. Our main result in this paper (Theorem 5.1) generalizes Theorem 5.1
in [19], which constructs, for a coherent sheaf $\mathcal{F}$ that admits a
locally free resolution on a complex manifold $X$, a current having the same
support as a resolution for $\mathcal{F}$ that represents products of Chern
classes/Chern forms of $\mathcal{F},$ to a more general class of
characteristic forms on cohesive modules which define classes in both the de-
Rham cohomology and Bott-Chern cohomology of the manifold $X.$ At the end, we
ask some questions about transgression formulae for superconnections and
superconnection currents that represent certain Bott-Chern characteristic
forms.

##  2 Acknowledgements

I would like to sincerely express my gratitude towards Prof.Jonathan Block,
whose patience, guidance, insight, and encouragement helped me overcome many
difficulties during both the learning process and the research process. I
would also like to thank him for teaching an awesome Algebraic Topology
sequence, where I learned many tools necessary for this project. Special
thanks to Prof.Tony Pantev for teaching an awesome Complex Algebraic Geometry
sequence, and especially for completing the course despite his health
conditions. This project would have been impossible without what I learned
from his lectures. I also sincerely thank Prof.Ron Donagi for serving on my
thesis committee and for holding the algebraic geometry seminar, which exposed
me to cool topics that furthered my interest in algebraic geometry. I would
like to also extend my gratitude to other faculty members and graduate
students who offered me advice throughout the project, and with whom I had
fruitful discussions with: Prof.Ted Chinburg, Prof.Ryan Hynd, Fangji Liu,
Tianyue Liu, Xingyu Meng, Zixuan Qu, Zhecheng Wu, Shengjing Xu and David Zhu.

##  3 Cohesive Modules and Unitary Connections

###  3.1 The $\overline{\partial}$-superconnection

Let $X$ be a complex manifold, and
$\mathcal{D}=(\mathcal{A}^{0,\bullet}(X),\overline{\partial})$ be its
Dolbeault differential graded algebra, we can define the dg-category
$\mathcal{P}_{\mathcal{D}}$ of $\mathcal{D}-$cohesive modules as follows[16]:
the objects are $E=(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}}).$ Here,
$E^{\bullet}=\bigoplus_{k=0}^{N}E_{k},$ with each $E^{k}$ a finite dimensional
complex vector bundle over $X.$

######  Proposition 3.1.

Here are some basic facts about the sheaves of $E^{\bullet}-$ and
$End(E^{\bullet})-$valued differential forms

  1. 1.

$\mathcal{A}^{0,\bullet}(X,E^{\bullet})\cong\mathcal{A}^{0,\bullet}(X)\otimes_{\mathcal{A}^{0}(X)}\mathcal{A}^{0}(X,E^{\bullet})$
and $\mathcal{A}^{\bullet,0}(X,E^{\bullet})\cong$
$\mathcal{A}^{\bullet,0}(X)\otimes_{\mathcal{A}^{0}(X)}\mathcal{A}^{0}(X,E^{\bullet}).$
Therefore
$\mathcal{A}^{\bullet}(X,E^{\bullet})\cong\mathcal{A}^{\bullet}(X)\otimes_{\mathcal{A}^{0}(X)}$
$\mathcal{A}^{0}(X,E^{\bullet}).$

  2. 2.

Same can be said if we replace $E^{\bullet}$ by
$End_{\mathbb{C}}(E^{\bullet})$ in (1).

  3. 3.

Therefore
$End_{\mathcal{O}_{X}}(\mathcal{A}^{\bullet}(X,E^{\bullet}))\cong\mathcal{A}^{\bullet}(X,End_{\mathbb{C}}(E^{\bullet}))$
as $\mathcal{O}_{X}-$modules.

Now let
$\mathbb{E}^{{}^{\prime\prime}}:\mathcal{A}^{0,\bullet}(X,E^{\bullet})\rightarrow\mathcal{A}^{0,\bullet}(X,E^{\bullet})$
be $\mathcal{O}_{X}$-linear of total degree$-1$ and satisfy the following:

  1. 1.

$\mathbb{E}^{{}^{\prime\prime}}\circ\mathbb{E}^{{}^{\prime\prime}}=0;$ i.e.
$\mathbb{E}^{{}^{\prime\prime}}$ is flat.

  2. 2.

The $\overline{\partial}-$Leibniz formula $\forall
s\in\mathcal{A}^{0}(X,E^{\bullet}),\forall\omega\in\mathcal{A}^{0,\bullet}(X),$

$\mathbb{E}^{{}^{\prime\prime}}(s\otimes\omega):=\mathbb{E}^{{}^{\prime\prime}}(s)\otimes\omega+(-1)^{\deg(\omega)}s\otimes\overline{\partial}(\omega)$
(1)

The meaning of total degree$-1$ is that $\forall p,q\in\mathbb{N}$

$\mathbb{E}^{{}^{\prime\prime}}(\mathcal{A}^{0,p}(X,E^{q}))\subseteq\bigoplus_{k\geq\max\\{-p,-q+1\\}}\mathcal{A}^{p+k}(X,E^{q-k+1})\Rightarrow\mathbb{E}^{{}^{\prime\prime}}=\bigoplus_{k\in\mathbb{Z}}\mathbb{E}^{{}^{\prime\prime}}_{k},$

with $\mathbb{E}_{k}^{{}^{\prime\prime}}=0,\forall k<\max\\{-p,-q+1\\}.$ Note
that by definition of $\mathbb{E}^{{}^{\prime\prime}}$ and degree, we know
that $\mathbb{E}^{{}^{\prime\prime}}_{k}$ is $\mathcal{A}^{\bullet}(X)-$linear
$\forall k\neq 1.$ Now consider $\forall 0\leq k\leq n,$ we have
$\mathbb{E}^{{}^{\prime\prime}}(\mathbb{E}^{{}^{\prime\prime}}|_{\mathcal{A}^{0}(X,E^{k})})=0,$
so its projection onto $\mathcal{A}^{0}(X,E^{k+2})$ is also $0.$ Therefore
$\mathbb{E}^{{}^{\prime\prime}}_{0}\circ\mathbb{E}^{{}^{\prime\prime}}_{0}(\mathcal{A}^{0}(X,E^{k}))=0$,
and we know that

${0}$${\mathcal{A}^{0}(X,E^{0})}$${\mathcal{A}^{0}(X,E^{1})}$${\cdots}$${\mathcal{A}^{0}(X,E^{N})}$${0}$$\scriptstyle{\mathbb{E}^{{}^{\prime\prime}}_{0}}$$\scriptstyle{\mathbb{E}^{{}^{\prime\prime}}_{0}}$$\scriptstyle{\mathbb{E}^{{}^{\prime\prime}}_{0}}$

is a complex of coherent sheaves.

###  3.2 Extended Hermitian Metric and the d-connection

Let $h$ be a Hermitian metric on $E^{\bullet}.$ Then, using Proposition
2.1.(1), we can extend $h$ to $\mathcal{A}^{\bullet}(X,E^{\bullet})$ via

$h(\alpha\otimes f,\beta\otimes g)=\overline{\alpha}\wedge
h(f,g)\wedge\beta,\hskip
14.22636pt\forall\alpha,\beta\in\mathcal{A}^{\bullet}(X),\hskip
14.22636pt\forall f,g\in\mathcal{A}^{0}(X,E^{\bullet})$

We will write a cohesive module as
$(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h)$ to emphasize the dependence
of various properties/constructions on the Hermitian metric. Now we state an
important structure theorem:

######  Theorem 3.1.

[16] For a Hermitian cohesive module
$(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h),$ there is a unique
$\mathbb{E}^{{}^{\prime}}:\mathcal{A}^{\bullet,0}(X,E^{\bullet})\rightarrow\bigcup_{q\in\mathbb{Z}}\mathcal{A}^{\bullet+q+1,0}(X,E^{\bullet-q})$
satisfying the following:

  1. 1.

$\mathbb{E}^{{}^{\prime}}$ is a $\partial-$superconnection, i.e.111Note that
$\mathcal{A}^{\bullet,0}(X,E^{\bullet})\cong\mathcal{A}^{\bullet,0}(X)\otimes_{\mathcal{A}^{0}(X)}\mathcal{A}^{0}(X,E^{\bullet})$
$\forall\alpha\otimes f$, with
$\alpha\in\mathcal{A}^{\bullet}(X),f\in\mathcal{A}^{0}(X,E^{\bullet}),$ we
have $\mathbb{E}^{{}^{\prime}}(\alpha\otimes
f)=\mathbb{E}^{{}^{\prime}}(\alpha)\otimes
f+(-1)^{deg(\alpha)}\alpha\otimes(\partial f).$

  2. 2.

Extending $\mathbb{E}^{{}^{\prime\prime}}$ and $\mathbb{E}^{{}^{\prime}}$ to
$\mathcal{A}^{\bullet,\bullet}(X,E^{\bullet})$ linearly (considering that
$\mathcal{A}^{\bullet,\bullet}(X,E^{\bullet})$
$\cong\mathcal{A}^{0,\bullet}(X)\otimes_{\mathcal{A}^{0}(X)}\mathcal{A}^{\bullet,0}(X)\otimes_{\mathcal{A}^{0}(X)}\mathcal{A}^{0}(X,E^{\bullet})$),
and we have
$\mathbb{E}=\mathbb{E}^{{}^{\prime\prime}}+\mathbb{E}^{{}^{\prime}}$ is a
$d-$superconnection.

  3. 3.

$\mathbb{E}$ is $h-$unitary, i.e. $\forall
s,t\in\mathcal{A}^{\bullet}(X,E^{\bullet}),$ we have
$(-1)^{deg(s)}d(h(s,t))=-h(\mathbb{E}(s),t)+h(s,\mathbb{E}(t)).$

  4. 4.

Writing
$\mathbb{E}^{{}^{\prime}}=\bigoplus_{q\in\mathbb{Z}}\mathbb{E}^{{}^{\prime}}_{q},\nabla=\mathbb{E}^{{}^{\prime\prime}}_{1}+\mathbb{E}^{{}^{\prime}}_{1}:\mathcal{A}^{\bullet,\bullet}(X,E^{\bullet})\rightarrow\mathcal{A}^{\bullet+1,\bullet+1}(X,E^{\bullet})$
is a unitary $d-$connection.

From the last statement we see that $\nabla$ restricts to connections on each
$E^{k},0\leq k\leq N.$ Therefore, writing $\nabla_{k}=\nabla|_{E_{k}}$ and
$\nabla=\bigoplus_{k=0}^{N}\nabla_{k}$, we know that $(E^{k},\nabla_{k})$ is a
vector bundle with a $d-$connection. Note that $\nabla$ induces a connection
$\nabla^{End}$ on $End(E^{\bullet})$ by

$\nabla^{End}:\mathcal{A}^{\bullet}(X,End(E^{\bullet}))\rightarrow\mathcal{A}^{\bullet+1}(X,End(E^{\bullet})),\hskip
14.22636pt\phi\mapsto\nabla\circ\phi-(-1)^{deg(\phi)}\phi\circ\nabla.$

###  3.3 Construction of a Compatible Unitary Connection

######  Definition 3.1.

[19, Section 4] Connections $\\{\Theta_{k}\\}_{0\leq k\leq N}$ on
$\\{E_{k}\\}$ are compatible with the complex
${0}$${E_{0}}$${E_{1}}$${\cdots}$${E_{N}}$${0}$$\scriptstyle{\phi_{0}}$$\scriptstyle{\phi_{1}}$$\scriptstyle{\phi_{N-1}}$
if
$\Theta_{k+1}\circ\phi_{k}=-\phi_{k}\circ\Theta_{k}\Leftrightarrow\Theta_{k+1}\circ\phi_{k}+\phi_{k}\circ\Theta_{k}=0.$

The reason why compatibility is important will be seen in the next section. In
the case of cohesive modules, the $\nabla_{k}$’s might not be compatible with
$\mathbb{E}^{{}^{\prime\prime}}_{0}.$ Let $Z_{i}$ be the set where
$\mathbb{E}^{{}^{\prime\prime}}_{0}$ is not exact, and let
$Z=\bigcup_{i=0}^{N}Z_{i}.$ We will call $Z$ the support of the cohesive
module. Let $\chi:\mathbb{R}\rightarrow[0,1]$ be a smooth characteristic
function such that $\chi\equiv 0$ on $(-\infty,1-\delta)$ and $\chi\equiv 1$
on $(1+\delta,\infty),$ for some arbitrarily small positive $\delta.$

Now for a positive $\epsilon>0,$ if $Z=\emptyset,$ define
$\chi_{\epsilon}\equiv 1$ on $X.$ Otherwise, define
$F=\boxtimes_{k=1}^{N}F_{k}=\boxtimes_{k=1}^{N}det(\mathbb{E}_{0}^{{}^{\prime\prime}}|_{E_{k-1}})^{\bigwedge
rank(E_{k-1})}$ on $X$[2, Section 2], which is a section to the coherent sheaf
$\mathcal{F}=\boxtimes_{k=1}^{N}\left(\bigwedge^{rank(E_{k-1})}E_{k-1}^{*}\otimes\bigwedge^{rank(E_{k-1})}E_{k}\right)$.
Then it is clear that $Z=\\{F=0\\}=\bigcup_{k=1}^{N}\\{F_{k}=0\\}$. If it is
impossible to find an $F$ that is generically nonvanishing such that
$Z\subseteq\\{F=0\\}$, define $\chi_{\epsilon}\equiv 1.$ Otherwise define
$\chi_{\epsilon}(x)=\chi\left(\frac{|F(x)|^{2}}{\epsilon}\right),\forall x\in
X.$ In this case $\chi_{\epsilon}\equiv 1$ except on a small neighborhood of
$Z$ when $\epsilon$ is small. (Since $F$ is generically nonvanishing, we can
modify $F$ such that $F\equiv 1$ except in a small neighborhood of $Z.$) Now
we need another concept before constructing the compatible connections: the
minimal inverse.

######  Definition 3.2.

[19, 2] For $0\leq k\leq N-1,$ write
$\mathcal{A}^{0}(X,E_{k+1})=\mathbb{E}^{{}^{\prime\prime}}_{0}(\mathcal{A}^{0}(X,E_{k}))$
$\oplus F_{k+1}.$ Define the minimal inverse $\sigma_{k}:E_{k+1}\rightarrow
E_{k}$, a morphism between vector bundles, by the following conditions: if
$e\in\mathbb{E}^{{}^{\prime\prime}}_{0}(E_{k})$, then $\sigma_{k}(e)\equiv n,$
where $\mathbb{E}^{{}^{\prime\prime}}_{0}(n)=e,$ and $n$ has pointwise the
minimal $h-$norm among all such vectors. If under
$h,e\perp\mathbb{E}^{{}^{\prime\prime}}_{0}(E_{k}),$ then $\sigma_{k}(e)\equiv
0.$ It then follows that
$\mathbb{E}^{{}^{\prime\prime}}_{0}\circ\sigma_{k}\circ\mathbb{E}^{{}^{\prime\prime}}_{0}=\mathbb{E}^{{}^{\prime\prime}}_{0}.$

######  Remark.

Here are some properties of $\sigma_{k}$

  1. 1.

The minimality of $\sigma_{k}(e)$ is equivalent to stating that $n\perp
Ker(\mathbb{E}^{{}^{\prime\prime}}_{0}|_{E_{k}}),$ since any complement of
$Ker(\mathbb{E}^{{}^{\prime\prime}}_{0}|_{E_{k}})$ injects onto
$\mathbb{E}^{{}^{\prime\prime}}_{0}(E_{k})$ under
$\mathbb{E}^{{}^{\prime\prime}}_{0}.$[1]

  2. 2.

From Remark (1), we know that $Im(\sigma_{k})\perp
Ker(\mathbb{E}^{{}^{\prime\prime}}_{0}|_{E_{k}})\Rightarrow
Im(\sigma_{k})\perp\mathbb{E}^{{}^{\prime\prime}}_{0}(E_{k-1}),$ since
$(\mathbb{E}^{{}^{\prime\prime}}_{0})^{2}=0.$ This means that
$\sigma_{k-1}\sigma_{k}=0.$[19]

  3. 3.

$\sigma_{k}$ is smooth on $X\backslash Z_{k}.$ This is because on $X\backslash
Z_{k},$ the rank of $\mathbb{E}^{{}^{\prime\prime}}_{0}|_{E_{k}}$ is constant.
It then suffices to show that
$\sigma_{k}|_{\mathbb{E}^{{}^{\prime\prime}}_{0}(E_{k})}$ is smooth, since
$\sigma_{k}$ on the orthogonal complement is constant. This follows from a
description of $\sigma_{k}$ in [1, Section 3].

Now we construct the connections
$\nabla_{k}^{\epsilon}=\nabla_{k}-\chi_{\epsilon}(\sigma_{k}\circ\nabla^{End}\circ\mathbb{E}^{{}^{\prime\prime}}_{0})$
on $E_{k}$’s, and we write
$\nabla^{\epsilon}=\bigoplus_{k=0}^{N}\nabla_{k}^{\epsilon}.$ Then we have

######  Theorem 3.2.

[19, Lemma 4.4] For any $\epsilon>0$, the connections
$\\{\nabla_{k}^{\epsilon}\\}_{0\leq k\leq N}$ are compatible with
$\mathbb{E}^{{}^{\prime\prime}}_{0}$ exactly where $\chi_{\epsilon}\equiv 1.$

##  4 Characteristic Class in de-Rham Cohomology

###  4.1 Characteristic Forms of
$(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h)$

Let $(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h)$ be a Hermitian cohesive
module. Define the curvature
$R_{h}=\mathbb{E}^{2}=\frac{1}{2}[\mathbb{E},\mathbb{E}]=[\mathbb{E}^{{}^{\prime}},\mathbb{E}^{{}^{\prime\prime}}].$

######  Remark.

Noting that $\mathbb{E}\in
End(\mathcal{A}^{\bullet}(X,E^{\bullet}))\cong\mathcal{A}^{\bullet}(X,End(E^{\bullet}))\cong$
$\mathcal{A}^{\bullet}(X)\otimes_{\mathcal{A}^{0}(X)}\mathcal{A}^{0}(X,End(E^{\bullet})),$
so if we write $\mathbb{E}=\alpha\otimes f,$ then we have
$R_{h}=(\alpha\wedge\alpha)\otimes(f\circ f).$

Then, following Quillen’s notion of the supertrace[20], for a fixed convergent
complex power series $f(T),$ we define its characteristic form to be
$Tr_{s}(f(R_{h}))$, where
$Tr_{s}:\mathcal{A}^{\bullet}(X,End(E^{\bullet}))\rightarrow\mathcal{A}^{\bullet}(X)$
is defined as follows [20, 2]: letting $E^{\bullet}=E^{+}\oplus E^{-},$ with
$E^{+}=\bigoplus_{2|k}E_{k},E^{-}=\bigoplus_{2\not|k}E_{k},$ we define
$Tr_{s}:End(E^{\bullet})\rightarrow\mathbb{C},X\mapsto tr(\epsilon X),$ where
for $e\in E^{+},\epsilon X(e)=X(e)$, and for $e\in E^{-},\epsilon X(e)=-X(e).$
Now extend $Tr_{s}$ linearly to $\mathcal{A}^{\bullet}(X,End(E^{\bullet})).$
Then we have the following facts

######  Theorem 4.1.

[16][Corollary 2.26] Characteristic forms are closed, so they define classes
in $H^{\bullet}_{dR}(X,\mathbb{C}).$ These classes are well-defined by Serre’s
Vanishing Theorem. We then have
$[Tr_{s}(f(R_{h}))]=[Tr_{s}(f(\Theta_{\nabla}))]$222This $\nabla$ was defined
in Theorem 2.1.(4). in $H^{\bullet}_{dR}(X,\mathbb{C}),$ where
$\Theta_{\nabla}=\nabla^{2}$ is the curvature form associated to $\nabla.$

###  4.2 Characteristic Forms of Exact Chain Complexes

We first show the following claim establishing a more explicit relation
between the characteristic form and the curvature form. Let $(E,\nabla)$ be a
vector bundle on $X.$ Denote
$tr:\mathcal{A}^{\bullet}(X,End(E))\rightarrow\mathcal{A}^{\bullet}(X)$ the
extension of the trace function on $\mathcal{A}^{0}(X,End(E)).$ Then we have

######  Proposition 4.1.

$[tr(f(\Theta_{\nabla}))]\in H^{*}_{dR}(X)$ is a polynomial in the Chern
classes of $E.$ Specifically, it is a symmetric polynomial in
$[\Theta_{\nabla}].$ It is also a sum of homogeneous polynomimals in
$[\Theta_{\nabla}].$

###### Proof.

Let $X,Y$ be two complex algebraic varieties, and $(E,\Delta)$ a vector bundle
of rank $k$ on $X.$ We show that $(E,\Delta)\mapsto[tr(f(\Theta_{\Delta}))]$
is a natural transformation from $\mathsf{Vect}_{k}(-;\mathbb{C})$ to
$H^{*}(-).$ For a morphism $\phi:Y\rightarrow X$, let
$(\phi^{*}E,\phi^{*}\Delta)$be the pullback vector bundle on $Y$ with the
pullback connection which is functorial (as defined in [21, Theorem 3.6(a)]),
we know that $\Theta_{\phi^{*}\Delta}=\phi^{*}\Theta_{\Delta}.$ Now it
suffices to show that $\forall
i\in\mathbb{N},\phi^{*}[tr(\Theta_{\Delta}^{i})]=[tr((\phi^{*}\Theta_{\Delta})^{i})].$
Recall the splitting principle

######  Lemma 4.2.

[10, Section 21] Let $E\rightarrow X$ a $C^{\infty}$ complex vector bundle and
$p:\mathbb{P}(E)\rightarrow M$ be the projection map. Then
$p^{*}(E)\rightarrow\mathbb{P}(E)$ splits into a direct sum of line bundles
and $p^{*}:H^{*}(X)\rightarrow H^{*}(\mathbb{P}(E))$ is an embedding.

Using the lemma and the fact that
$\phi^{*}\mathbb{P}(E)=\mathbb{P}(\phi^{*}(E))$, and considering the
commutative diagram

${{L_{1}\oplus\cdots\oplus
L_{n}\rightarrow\mathbb{P}(E)}}$${{\phi^{*}(L_{1})\oplus\cdots\oplus\phi^{*}(L_{n})\rightarrow\mathbb{P}(\phi^{*}(E))}}$${{E\rightarrow
X}}$${{\phi^{*}(E)\rightarrow
Y}}$$\scriptstyle{\phi^{*}}$$\scriptstyle{p_{E}^{*}}$$\scriptstyle{\phi^{*}}$$\scriptstyle{P^{*}_{\phi^{*}(E)}}$

and then noting that $\mathbb{P}(\phi^{*}(E))=\phi^{*}(\mathbb{P}(E)),$ we can
reduce to when $E\rightarrow X$ is a line bundle, in which case $\phi^{*}$
amounts to multiplication by an element in $\mathcal{O}_{X}$ on both sides.

Now recall that every natural transformation
$\mathsf{Vect}_{k}(-,\mathbb{C})\rightarrow H^{*}_{dR}(-)$ can be expressed as
a polynomial in the Chern classes [10, Proposition 23.11], it remains to show
that the Chern classes $c_{n}(E)$ is a polynomial of $[\Theta_{\Delta}].$ This
follows directly from [21, Definition 3.4]. ∎

######  Remark.

To show that $[tr(f(\Theta_{\nabla}))]$ is a polynomial, not a series, in the
Chern classes, we implicitly used the fact that degrees$\geq
2dim_{\mathbb{C}}(X)$ vanish in $H^{*}_{dR}(X).$

Now we recall a Whitney formula [4, Lemma 4.22]

######  Theorem 4.3.

Let $\phi$ be a symmetric homogeneous polynomial of degree less than or equal
to $dim_{\mathbb{C}}(X),$ then let $\\{D_{k}\\}_{0\leq k\leq N}$ be compatible
connections on $(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}}).$ If the complex
$(E^{\bullet},\mathbb{E}_{0}^{{}^{\prime\prime}})$ is exact, then in de-Rham
cohomology,
$[\phi(\Theta_{D_{0}})]=-\left(\frac{2\pi}{i}\right)^{deg(\phi)}\cdot[\phi(\sum_{i=1}^{N}(-1)^{i}E_{i})]$[4,
Equation 4.15], which is defined as follows: letting $\sigma_{k}(D_{j})=$
$\left(\frac{2\pi}{i}\right)^{k}\cdot c_{k}(E_{j},D_{j})$333Here $\sigma_{k}$
denotes the elementary symmetric polynomial of degree $k.$[4, Equation 3.34],
and
$\phi(D_{j})=\tilde{\phi}(\sigma_{1}(D_{j}),\cdots,\sigma_{deg(\phi)}(D_{j})),$
and writing
$\sum_{i=1}^{N}(-1)^{i}E_{i}$ as $\widehat{E^{\bullet}},$ we define
$\phi(\widehat{E^{\bullet}})\equiv\tilde{\phi}(c_{1}(\widehat{E^{\bullet}}),\cdots,c_{deg(\phi)}(\widehat{E^{\bullet}})).$

######  Remark.

Here, the Chern classes of $\widehat{E^{\bullet}}$ satisfy the following:

$1+\sum_{i=1}^{2dim_{\mathbb{C}}(X)}c_{i}(\widehat{E^{\bullet}})=c(\widehat{E^{\bullet}})\equiv\bigoplus_{k=0}^{N}c(E_{i})^{(-1)^{i}}$

Since $\phi$ is homogeneous, we know that
$[\phi(\Theta_{D_{0}})]=\left(\frac{2\pi}{i}\right)^{deg(\phi)}$
$[\tilde{\phi}(c_{1}(E_{0},D_{0}),\cdots,c_{deg(\phi)}(E_{0},D_{0})].$
Therefore From this and Proposition 3.1, we automatically have

######  Corollary 4.3.1.

On $X\backslash Z,$

$Tr_{s}(f(\Theta_{\nabla}))=\bigoplus_{k=0}^{N}Tr_{s}(f(\Theta_{\nabla_{k}}))=\sum_{k=0}^{N}(-1)^{k}tr(\Theta_{\nabla_{k}})=0.$

##  5 Characteristic Currents

Now we define the characteristic currents on a cohesive module
$(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h).$ For $a\in H^{*}_{dR}(X),$
denote $a_{k}$ to be the degree-$k$ component of $a.$

######  Definition 5.1.

For $(p_{1},\cdots,p_{k})\in[0,2dim_{\mathbb{C}}(X)]^{k}$ define the
characteristic current to be $Tr_{p_{1}}(E^{\bullet},\nabla)\wedge\cdots\wedge
Tr_{p_{k}}(E^{\bullet},\nabla)\equiv\lim_{\epsilon\to
0}[Tr_{s}(f(\nabla^{\epsilon}))]_{p_{1}}\wedge\cdots\wedge[Tr_{s}(f(\nabla^{\epsilon}))]_{p_{k}}.$

######  Remark.

We will explain the meaning of wedge product right after Definition 5.2
(assuming Theorem 5.1 a priori).

######  Definition 5.2.

[3] Let $f$ be a holomorphic function on $X.$ For $a\in\mathbb{N},$ the
current $[\frac{1}{f^{a}}]$ is defined as the functional on test forms
$\xi\mapsto\lim_{\epsilon\to 0}\int_{|f|>\epsilon}\frac{\xi}{f^{a}}$ and
$\overline{\partial}[\frac{1}{f^{a}}]$ sends test form $\xi$ to
$\lim_{\epsilon\to
0}\int_{|f|>\epsilon}\frac{\overline{\partial}(\xi)}{f^{a}}.$ These are well-
defined by[15, Theorem 7.1]. Let
$\Pi=\Pi_{1}\circ\Pi_{2}\circ\cdots\circ\Pi_{r}$ be a sequence of resolutions
of singularities, with $\Pi_{i}:Y_{i}\rightarrow Y_{i-1}$ with $Y_{0}=X.$ Then
a current on $X$ is pseudomeromorphic if it can be written as
$\sum_{\ell}\Pi_{*}\tau_{\ell},$ where $\tau_{\ell}$ is a current on some
$Y_{\ell}$ of the form
$\left(\prod_{i=1}^{k}[\frac{1}{f^{a_{i}}}]\right)\overline{\partial}[\frac{1}{f^{b_{1}}}]\wedge\cdots\wedge\overline{\partial}[\frac{1}{f^{b_{m}}}]$
for some holomorphic $f$ on $Y_{\ell}.$

Here, we need to define a notion of ”wedge product” of currents. This is given
by the Coleff-Herrera product[11, Theorem 1.7.2][18, Theorem 2]. Call
$(\epsilon_{1},\cdots,\epsilon_{p})\rightarrow(0,\cdots,0)$ along an
admissible path if $\forall k\in\mathbb{N}$ and $\forall j\geq
2,\frac{\epsilon_{j-1}}{\epsilon_{j}^{k}}\to 0.$ in this case we write
$\epsilon_{1}\ll\cdots\ll\epsilon_{p}\to 0.$ Then for $f_{1},\cdots,f_{p}$
holomorphic, we define

$\overline{\partial}[\frac{1}{f_{1}}]\wedge\cdots\wedge\overline{\partial}[\frac{1}{f_{p}}]=\lim_{\epsilon_{1}\ll\cdots\ll\epsilon_{p}\to
0}\frac{\overline{\partial}\chi(|f_{1}|^{2}/\epsilon_{1})}{f_{1}}\wedge\cdots\wedge\frac{\overline{\partial}\chi(|f_{p}|^{2}/\epsilon_{p})}{f_{p}},$

where for each $(\epsilon_{1},\cdots,\epsilon_{p})$ and for any test form
$\phi$ of bi-degree $(dim_{\mathbb{C}}(X),$ $dim_{\mathbb{C}}(X)-p),$ the
expression on the right hand side denotes
$\phi\mapsto\int_{X}\frac{\overline{\partial}\chi(|f_{1}|^{2}/\epsilon_{1})}{f_{1}}\wedge\cdots\wedge\frac{\overline{\partial}\chi(|f_{p}|^{2}/\epsilon_{p})}{f_{p}}\wedge\phi.$
Then for holomorphic $f_{1},\cdots,f_{k},g_{1},\cdots,g_{m}$, and for a
$(dim_{\mathbb{C}}(X),$ $dim_{\mathbb{C}}(X)-p)-$test form $\phi,$ define
$\left(\prod_{i=1}^{k}[\frac{1}{f_{i}}]\right)\overline{\partial}[\frac{1}{g_{1}}]\wedge\cdots\wedge\overline{\partial}[\frac{1}{g_{m}}](\phi)$
as

$\lim_{\begin{subarray}{c}\delta_{1},\cdots,\delta_{k}\to 0\\\
\epsilon_{1}\ll\cdots\ll\epsilon_{m}\to
0\end{subarray}}\int_{|f_{i}|>\delta_{i},\forall
i}\frac{\bigwedge_{j=1}^{m}\overline{\partial}\chi\left(\frac{|g_{j}|^{2}}{\epsilon_{j}}\right)\wedge\phi}{\prod_{i=1}^{k}f_{i}\prod_{j=1}^{m}g_{j}}$

######  Remark.

It follows from Definition 5.2 that pushforwards of pseudomeromorphic currents
under resolutions of singularities are still pseudomeromorphic.

######  Theorem 5.1.

The characteristic current $Tr_{p_{1}}(E^{\bullet},\nabla)\wedge\cdots\wedge
Tr_{p_{k}}(E^{\bullet},\nabla)$ is a well-defined closed pseudomeromorphic
current, with support contained in $Z,$ an analytic subvariety of positive
codimension, that represents
$Tr_{s}(f(\Theta_{\nabla}))_{p_{1}}\wedge\cdots\wedge
Tr_{s}(f(\Theta_{\nabla}))_{p_{k}}$ for any $k-$tuple $(p_{1},\cdots,p_{k}).$

######  Remark.

Denote $(\mathbb{D}^{*}(X),d)$ the complex of currents on $M.$ Here
$\mathbb{D}^{q}(X)$ is the dual space to
$\Omega^{2dim_{\mathbb{C}}(X)-q}_{c}(X),$ the vector space of compactly
supported smooth forms on $X,$ with the dual topology. Also, the chain map
$d:\mathbb{D}^{q}(X)\rightarrow\mathbb{D}^{q+1}(X)$ is given by
$(dT)(\phi)=(-1)^{q+1}T(d\phi).$ Then we have
$H^{*}_{dR}(X)\overset{\cong}{\longrightarrow}H^{*}(\mathbb{D}^{*}(X),d).$[13,
Chapter 3, Section 1]

###### Proof.

By Proposition 4.1 and linearity, it suffices to consider the case where
$Tr_{s}(f(\Theta_{\nabla}))$ is a monomial in the Chern classes. Then from
[19, Lemma 2.1] and [19, Theorem 5.1], it suffices to show that
$\forall\ell_{1},\ell_{2}$ we have

$\lim_{\epsilon\to
0}c_{\ell_{1}}(E^{\bullet},\nabla^{\epsilon})\wedge\lim_{\delta\to
0}c_{\ell_{2}}(E^{\bullet},\nabla^{\delta})=\lim_{\epsilon\to
0}c_{\ell_{1}}(E^{\bullet},\nabla^{\epsilon})\wedge
c_{\ell_{2}}(E^{\bullet},\nabla^{\epsilon})$

and then proceed inductively (for wedge products of more Chern classes). Since
$[tr(f(\Theta_{\nabla}))]$ is a characteristic class, it does not depend on
the choice of connection. As in the proof of [19, Theorem 5.1], for any
$\epsilon$ and $\delta,$ we can write $c_{\ell_{1}}(E^{\bullet},\epsilon)$ as
$A_{1}+\sum_{j\geq 1}\chi_{\epsilon}^{j}B_{j}+\sum_{j\geq
1}\chi_{\epsilon}^{j-1}\wedge d\chi_{\epsilon}\wedge B_{j}^{\prime}$ and
$c_{\ell_{2}}(E^{\bullet},\delta)$ as $A_{2}+\sum_{j\geq
1}\chi_{\delta}^{j}C_{j}+\sum_{j\geq 1}\chi_{\delta}^{j-1}\wedge
d\chi_{\delta}\wedge C_{j}^{\prime},$ where
$A_{1},A_{2},B_{j},C_{j},B_{j}^{\prime},C_{j}^{\prime}$ are independent of
$\epsilon$ and $\delta,$ with $A_{1},A_{2}$ smooth and
$B_{j},C_{j},B_{j}^{\prime},C_{j}^{\prime}$ polynomials in the entries of the
minimal inverses $\sigma_{k}$ (cf. Definition 2.2),
$D_{End(E^{\bullet})}\mathbb{E}_{0}^{{}^{\prime\prime}},$ and $\theta_{k}$,
which are the matrices representing $\nabla_{k}$ ($0\leq k\leq N$). Also,
$\chi_{\epsilon}=\chi(\frac{|F|^{2}}{\epsilon})$ as defined in Section 3.3.
Therefore it suffices to consider where $\chi_{\epsilon}\not\equiv 1$ for
$\epsilon$ small enough and show that for any $i,j$ and any $s,t$ products of
entries of $\sigma_{k},D_{End(E^{\bullet})}\mathbb{E}_{0}^{{}^{\prime\prime}}$
and $\theta_{k}$, we have $\chi_{\epsilon},\lim_{\epsilon\to
0}\chi_{\epsilon}^{i}s\wedge\lim_{\delta\to
0}\chi_{\delta}^{j}t=\lim_{\epsilon}\chi_{\epsilon}^{i+j}s\wedge t,$ and also
$\lim_{\epsilon\to 0}\chi_{\epsilon}^{i}d\chi_{\epsilon}\wedge
s\wedge\lim_{\delta\to 0}\chi_{\delta}^{j}d\chi_{\delta}\wedge
t=\lim_{\epsilon\to 0}\chi_{\epsilon}^{i+j}d\chi_{\epsilon}\wedge s\wedge
d\chi_{\epsilon}\wedge t.$

We use similar ideas to [19, Lemma 2.1]. By resolution of singularities [17,
Theorem 3.36], since in Section 2.3 we found a section
$F=\boxtimes_{j=0}^{N}F_{j}$ to a coherent sheaf (actually a vector bundle)
$\mathcal{F}$ such that $Z\subseteq Z(F),$ which implies that
$(F)\cdot\mathcal{O}_{X}$444Here $(F)$ is the ideal of $\mathcal{O}_{X}$
generated by $(F).$ is not invertible on $Z(F),$ we know that there exists a
(composition of) birational and projective modification(s) $\pi:Y\rightarrow
X$ such that $\pi|_{Y-\pi^{-1}(Z(F))}:Y-\pi^{-1}(Z(F))\rightarrow X-Z(F)$ is
an isomorphism, and the coherent sheaf of ideals on $Y$ generated by pullbacks
of local sections to $(F)\cdot\mathcal{O}_{X},$ which we write as
$\pi^{-1}\left((F)\cdot\mathcal{O}_{X}\right),$ is a monomonial sheaf of
ideals.555Also reference [17, Note 3.16] for the equivalent characterizations
of monomial ideal sheaves. Equivalently, this is the subsheaf of
$\mathcal{O}_{Y}$ generated by $\pi^{*}F=\boxtimes_{j=0}^{N}\pi^{*}F_{j}.$
This means that $\pi^{*}F_{j}=F_{j0}F_{j1},$ where
$F_{j0}=\prod_{i=1}^{n}z_{i}^{c_{i}}$ is a monomial in local coordinates
$\\{z_{1},\cdots,z_{n}\\}$ and $Z(F_{j0})\subseteq\pi^{-1}(Z(F))$ and $F_{j1}$
is holomorphic and nonvanishing. Then on $Y$, we have the local formula

$\pi^{*}\sigma_{k}=\frac{1}{F_{k0}}\phi_{k},\forall 0\leq k\leq N$ (2)

with $\phi_{k}$ smooth everywhere on $Y.$ (cf.the definition of $\sigma_{k}$
in Definition 2.2. This can also be found in [2, Section 2]) Note that
$D_{End(E^{\bullet})}\mathbb{E}_{0}^{{}^{\prime\prime}}$ and $\theta_{k}$ are
everywhere smooth. Now by [19, Equation 2.2], writing
$\pi^{*}s=\frac{1}{\psi_{1}}\tilde{s},\pi^{*}t=\frac{1}{\psi_{2}}\tilde{t}$,
with $\psi_{1},\psi_{2}$ products of monomials (thus also monomials) of local
coordinates on $Y$ and $\tilde{s},\tilde{t}$ smooth.666More accurately
$\pi^{*}s$ might be a sum of such terms, but we can certainly apply a
normalization argument to achieve a fraction whose denominator is a monomial
in local coordinates. In view of Equation (2), it suffices to show that for
any test $2dim_{\mathbb{C}}(X)$-form $\xi$ on $Y$ (here we note that
$\chi\sim\chi^{i},\forall i\in\mathbb{N},$ and also
$\\{\psi_{1}=0\\}\bigcup\\{\psi_{2}=0\\}\subseteq\pi^{-1}(Z)=\\{\pi^{*}F=0\\},$
using the fact that $\pi$ is an isomorphism on $X-Z(F)$ and $s,t$ are smooth
outside of $Z$)

$\bigg{[}\lim_{\epsilon\to
0}\frac{\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)^{i+j}}{\psi_{1}\psi_{2}}\bigg{]}(\xi)=\bigg{[}\lim_{\epsilon\to
0}\frac{\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)^{i}}{\psi_{1}}\bigg{]}\bigg{[}\lim_{\delta\to
0}\frac{\chi\left(\frac{|\pi^{*}F|^{2}}{\delta}\right)^{j}}{\psi_{2}}\bigg{]}(\xi)\Longleftrightarrow$
$\lim_{\epsilon\to
0}\int_{|\psi_{1}\psi_{2}|>\epsilon}\frac{\xi}{\psi_{1}\psi_{2}}=\lim_{\begin{subarray}{c}\epsilon\to
0\\\ \delta\to 0\end{subarray}}\int_{\begin{subarray}{c}|\psi_{1}|>\epsilon\\\
|\psi_{2}|>\delta\end{subarray}}\frac{\xi}{\psi_{1}\psi_{2}}$ (3)

since this current does not depend on the choice of characteristic
function.[19, Lemma 2.1] Then the difference between the two is (noting that
$\xi$ is a test form so $||\xi||_{L^{\infty}(X)}$ exists)

$\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \delta\to
0\end{subarray}}\int_{|\psi_{1}|>\epsilon}\frac{(\mathbbm{1}_{|\psi_{2}|>\epsilon/|\psi_{1}|}-\mathbbm{1}_{|\psi_{2}|>\delta})\xi}{\psi_{1}\psi_{2}}+\lim_{\epsilon\to
0}\int_{|\psi_{1}|<\epsilon}\frac{\mathbbm{1}_{|\psi_{2}|>\epsilon/|\psi_{1}|}\xi}{\psi_{1}\psi_{2}}$
$\leq\left(\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \delta\to
0\end{subarray}}\int_{\begin{subarray}{c}|\psi_{1}|>\epsilon\\\
\delta>\epsilon/|\psi_{1}|\\\
\epsilon/|\psi_{1}|<|\psi_{2}|<\delta\end{subarray}}\left|\frac{\xi}{\psi_{1}\psi_{2}}\right|+\int_{|\psi_{1}\psi_{2}|<\epsilon}\left|\frac{\xi}{\psi_{1}\psi_{2}}\right|\right)+\lim_{\epsilon\to
0}\int_{|\psi_{1}|<\epsilon}\left|\frac{\xi}{\psi_{1}}\right|\leq$
$||\xi||_{L^{\infty}(X)}\left(\lim_{\begin{subarray}{c}\epsilon\to 0\\\
\delta\to 0\end{subarray}}\int_{\begin{subarray}{c}|\psi_{1}|>\epsilon\\\
\delta>\epsilon/|\psi_{1}|\\\
\epsilon/|\psi_{1}|<|\psi_{2}|<\delta\end{subarray}}\left|\frac{\mathbbm{1}_{Supp(\xi)}}{\psi_{1}\psi_{2}}\right|+\lim_{\epsilon\to
0}\int_{|\psi_{1}\psi_{2}|<\epsilon}\left|\frac{\mathbbm{1}_{Supp(\xi)}}{\psi_{1}\psi_{2}}\right|+\int_{|\psi_{1}|<\epsilon}\left|\frac{\mathbbm{1}_{Supp(\xi)}}{\psi_{1}}\right|\right)$

We have

$\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \delta\to
0\end{subarray}}\int_{\begin{subarray}{c}|\psi_{1}|>\epsilon\\\
\delta>\epsilon/|\psi_{1}|\\\
\epsilon/|\psi_{1}|<|\psi_{2}|<\delta\end{subarray}}\left|\frac{\mathbbm{1}_{Supp(\xi)}}{\psi_{1}\psi_{2}}\right|\leq\lim_{\begin{subarray}{c}\epsilon\to
0\\\ \delta\to 0\end{subarray}}\int_{\begin{subarray}{c}|\psi_{1}|>\epsilon\\\
|\psi_{2}|<\delta\end{subarray}}\left|\frac{\mathbbm{1}_{Supp(\xi)}}{\psi_{1}\psi_{2}}\right|$
$\leq\lim_{\epsilon\to
0}\sqrt{\int_{|\psi_{1}|>\epsilon}\frac{\mathbbm{1}_{Supp(\xi)}}{|\psi_{1}|^{2}}}\cdot\lim_{\delta\to
0}\sqrt{\int_{|\psi_{2}|<\delta}\frac{\mathbbm{1}_{Supp(\xi)}}{|\psi_{2}|^{2}}}=O\left(\lim_{\delta\to
0}\int_{|\psi_{2}|<\delta}\frac{\mathbbm{1}_{Supp(\xi)}}{|\psi_{2}|}\right),$

which will follow from Lemma 5.3. To prove that this is $0,$ and also the
second and third terms vanish, it suffices to prove the following two lemmas:

######  Lemma 5.2.

Let $\phi$ be a holomorphic function on $M$ whose vanishing locus is a
measure-zero set (specifically, a subvariety of positive codimension), and
such that $\frac{1}{\phi}\in L^{1}(Supp(\xi)),$ then $\lim_{\epsilon\to
0}\int_{|\phi|<\epsilon}\frac{1}{|\phi|}=0.$ Here, $|\cdot|$ is the complex
norm.

###### Proof.

Note that $codim(Z(\phi))>0\Rightarrow\mu(Z(\phi))=0,$ where $\mu$ is the
pullback (under the coordinate maps) of the Lebesgue measure on
$\mathbb{C}^{dim_{\mathbb{C}}(X)}.$ Define

$f_{n}:X\rightarrow[0,+\infty],\quad
x\mapsto\begin{cases*}\frac{1}{|\phi(x)|}&$|\phi(x)|<\frac{1}{n}$\\\
0&otherwise\end{cases*}$ (4)

Then note that $f_{i}(x)\geq f_{j}(x),\forall i<j,$ and
$\lim_{n\to\infty}f_{n}(x)\neq 0\Leftrightarrow\phi(x)=0.$ Now since
$\int_{Supp(\xi)}|f_{1}|\leq\int_{Supp(\xi)}\frac{1}{|\phi|}<\infty$, then by
monotone convergence, we have

$\lim_{\epsilon\to
0}\int_{|\phi|<\epsilon}\frac{\mathbbm{1}_{Supp(\xi)}}{|\phi|}=\lim_{n\to\infty}\int_{|\phi|<\frac{1}{n}}\frac{\mathbbm{1}_{Supp(\xi)}}{|\phi|}=\lim_{n\to\infty}\int_{Supp(\xi)}f_{n}$
$=\int_{Supp(\xi)}\lim_{n\to\infty}f_{n}=\int_{Supp(\xi)\cap
Z(\phi)}\lim_{n\to\infty}f_{n},$

which is $0$ since $\mu(Supp(\xi)\cap Z(\phi))\leq\mu(Z(\phi))=0.$ ∎

######  Lemma 5.3.

We can apply a further sequence of blow-ups $\Pi:\tilde{Y}\rightarrow Y$ such
that $\frac{1}{\Pi^{*}\psi_{1}}$ and $\frac{1}{\Pi^{*}\psi_{2}}\in
L^{1}({Supp(\xi)}).$ Therefore we can assume WLOG that
$\frac{1}{\psi_{1}},\frac{1}{\psi_{2}}\in L^{1}({Supp(\xi)}).$

###### Proof.

We will show this for $\psi_{1}$ only. In local coordinates, write
$\psi_{1}=\prod_{i=1}^{n}z_{i}^{c_{i}}.$ Then by Fubini, and writing
$r=diam(Supp(\xi))<\infty$ we have
$||\psi_{1}||_{L^{1}({Supp(\xi)})}\leq\prod_{i=1}^{n}\int_{|z_{i}|\leq
r}\frac{1}{|z_{i}|^{c_{i}}}$, which is finite if $c_{i}$ is smaller than
$2dim_{\mathbb{C}}(Y),\forall i.$ Now since $Supp(\xi)\cap Z(\pi^{*}F)$ is
compact, we can cover it by finitely many coordinate neighborhoods, and
$|\psi_{1}\psi_{2}|$ will have a strictly positive lower bound outside these
neighborhoods. Thus the problem reduces to finding $\Pi:\tilde{Y}\rightarrow
Y$ such that $\Pi^{*}\psi_{1}$ and $\Pi^{*}\psi_{2}$ are monomials in local
coordinates covering $Supp(\xi)\cap Z(\pi^{*}F)$ in which the degree of every
coordinate does not exceed $2dim_{\mathbb{C}}(Y)-1.$ This will follow directly
from [17, Theorem 3.68], which states that if $I\subseteq\mathcal{O}_{X}$ is
an ideal sheaf with $\mathsf{Maxord}_{Supp(\xi)\cap Z(\pi^{*}F)}(I)\leq m$ for
some $m\in\mathbb{N}$, then there is a composition of blow-ups
$\Pi=\Pi_{r}\circ\cdots\circ\Pi_{1}$ such that
$\mathsf{Maxord}_{\Pi^{-1}(Supp(\xi)\cap
Z(\pi^{*}F))}\Pi^{-1}(I)=\mathsf{Maxord}_{\Pi^{-1}(Supp(\xi))\cap
Z(\Pi^{*}\pi^{*}F)}<m.$ Here, for a point $y\in
Y,\mathsf{ord}_{y}(I)\overset{def}{=}\max\\{r:\mathfrak{m}_{y}^{r}\mathcal{O}_{Y,y}\supset
I_{x}\\},$ where $\mathfrak{m}_{y}$ is the maximal ideal of
$\mathcal{O}_{Y,y};$ for a subset $Z\subseteq Y,$ define
$\mathsf{Maxord}_{Z}(I)=\sup_{y\in Z}\mathsf{ord}_{y}(I).$[17, Definition
3.47] In our case, we let $I=(\psi_{1},\psi_{2})\mathcal{O}_{Y}.$ Then
$\mathsf{ord}_{y}(I)$ is just the maximal order of the two monomials. ∎

(Proof of Theorem 5.1–Continued)
Now we show that $\lim_{\epsilon\to
0}\pi^{*}(\chi_{\epsilon})^{i}d\pi^{*}\chi_{\epsilon}\wedge\pi^{*}s\wedge\lim_{\delta\to
0}(\pi^{*}\chi_{\delta})^{j}d\pi^{*}\chi_{\delta}\wedge\pi^{*}t=\lim_{\epsilon\to
0}(\pi^{*}\chi_{\epsilon})^{i+j}d\pi^{*}\chi_{\epsilon}\wedge\pi^{*}s\wedge
d\pi^{*}\chi_{\epsilon}\wedge\pi^{*}t$. Since $\pi^{*}s,\pi^{*}t$ have nice
expressions, for a test form $\xi$ of matching degree, it suffices to consider

$\bigg{[}\lim_{\epsilon\to
0}\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)^{i}\frac{d\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)}{\psi_{1}}\bigwedge\lim_{\delta\to
0}\chi\left(\frac{|\pi^{*}F|^{2}}{\delta}\right)^{j}\frac{d\chi\left(\frac{|\pi^{*}F|^{2}}{\delta}\right)}{\psi_{2}}\bigg{]}(\xi)$
$=\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \delta\to
0\end{subarray}}\int_{X}\frac{d\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)\wedge
d\chi\left(\frac{|\pi^{*}F|^{2}}{\delta}\right)\wedge\xi}{\psi_{1}\psi_{2}}$

and

$\bigg{[}\lim_{\epsilon\to
0}\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)^{i+j}\frac{\bigwedge^{2}d\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)}{\psi_{1}\psi_{2}}\bigg{]}(\xi)=\lim_{\epsilon\to
0}\int_{X}\frac{\bigwedge^{2}d\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)\wedge\xi}{\psi_{1}\psi_{2}}.$

The difference is

$\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \delta\to 0\\\ \tau\to
0\end{subarray}}\int_{X}\frac{\xi}{\psi_{1}\psi_{2}}\wedge\left(d\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)\bigwedge
d\chi\left(\frac{|\pi^{*}F|^{2}}{\delta}\right)-\bigwedge^{2}d\chi\left(\frac{|\pi^{*}F|^{2}}{\tau}\right)\right)$

Note that the term in the parenthesis is nonzero only when
$|\pi^{*}F|^{2}<(1+\nu)\cdot\min\\{\epsilon,\delta,\tau\\},$ where
$supp(\chi)\subseteq[0,1+\nu).$ By definition of
$\chi,||\nabla\chi||_{L^{\infty}(\mathbb{R})}<\infty,$ and

$\left|\left|d\chi\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)\right|\right|\leq||\nabla\chi||_{L^{\infty}(\mathbb{R})}\cdot\left|\left|\nabla\left(\frac{|\pi^{*}F|^{2}}{\epsilon}\right)\right|\right|,$

so the difference does not exceed (writing
$k=||\xi||_{L^{\infty}(X)}\cdot||\nabla\chi||_{L^{\infty}(\mathbb{R})}$
$\cdot||\nabla(|\pi^{*}F|^{2})||_{L^{\infty}(Supp(\xi))},$ and denoting the
region $\\{|\pi^{*}F|^{2}<(1+\nu)\cdot\min\\{\epsilon,\delta,\tau\\}\\}$ by
$C_{\epsilon,\delta,\tau}$),

$k\cdot\lim_{\begin{subarray}{c}\epsilon\to 0\\\ \delta\to 0\\\ \tau\to
0\end{subarray}}\int_{C_{\epsilon,\delta,\tau}\cap
Supp(\xi)}\frac{1}{|\psi_{1}\psi_{2}|}\left(\frac{1}{\epsilon\delta}-\frac{1}{\tau^{2}}\right).$

Note that by Hölder’s Inequality, the integrand does not exceed

$\sqrt{\int_{Supp(\xi)\cap
C_{\epsilon,\delta,\tau}}\frac{1}{|\psi_{1}\psi_{2}|^{2}}}\cdot\sqrt{\int_{Supp(\xi)\cap
C_{\epsilon,\delta,\tau}}\left|\frac{1}{\epsilon\delta}-\frac{1}{\tau^{2}}\right|^{2}}$
$\leq\sqrt{\int_{Supp(\xi)}\frac{1}{|\psi_{1}\psi_{2}|^{2}}}\cdot\sqrt{\int_{Supp(\xi)\cap
C_{\epsilon,\delta.\tau}}\frac{2}{|\pi^{*}F|^{2}}}.$

Since $Supp(\xi)$ is compact, from what we have shown before the first term is
finite. Also, the second term goes to $0$ as $\epsilon,\delta$ and $\tau$ go
to $0$ simultaneously, since $\mu(Z(|\pi^{*}F|^{2}))=0$ and after applying a
further sequence of resolutions making the degrees of local coordinates low
enough in $|\pi^{*}F|^{2}$, just as in the proof of Lemma 5.3, we can follow
essentially the same proof as for Equation (3)). Then the statement will then
follow from Proposition 5.1. ∎

######  Proposition 5.1.

$\pi_{*}\left(\lim_{\epsilon\to
0}\pi^{*}(\chi_{\epsilon})^{i}d\pi^{*}\chi_{\epsilon}\wedge\pi^{*}s\wedge\lim_{\delta\to
0}(\pi^{*}\chi_{\delta})^{j}d\pi^{*}\chi_{\delta}\wedge\pi^{*}t\right)=\lim_{\epsilon\to
0}\chi_{\epsilon}^{i}d\chi_{\epsilon}\wedge s\wedge\lim_{\delta\to
0}\chi_{\delta}^{j}d\chi_{\delta}\wedge t$, and we can say the same about the
other three currents we are considering. 777i.e. Pushing forward by $\pi$ is
the same as taking away all the $\pi^{*}$ in the expressions.

###### Proof.

This is clear from the definition. ∎

######  Remark.

It is clear from our method that [19, Theorem 5.1] also applies to any other
characteristic class in de-Rham cohomology.

##  6 Chern Currents in Bott-Chern Cohomology

###  6.1 The Bott-Chern Character

We first define the double complex of Bott-Chern cohomology classes of the
cohesive module $(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h).$[7] Letting
$d=\partial+\overline{\partial}$ be the de-Rham differential, we have

######  Definition 6.1.

$H_{BC}^{p,q}(X)\equiv\left(\mathcal{A}^{p,q}(X)\cap
Ker(d)\right)/\overline{\partial}\partial\mathcal{A}^{p-1,q-1}(X).$

The Bott-Chern character of $(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h)$
is defined by
$ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}},h)=Trs(exp(-\mathcal{R}_{h})).$
By [16, Lemma 2.20], this defines a class in $H_{BC}(X)$. The Bott-Chern
character is also independent of the Hermitian metric $h$ by [16, Corollary
3.14][7, Theorem 8.2]. Therefore it makes sense to write it as
$ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}}).$ If the complex
$(\mathcal{A}^{\bullet}(X,E^{\bullet}),\mathbb{E}^{{}^{\prime\prime}}_{0})$ is
exact, then we have

######  Theorem 6.1.

[16, Theorem 4.21] Let
$\mathbb{E}_{t}^{{}^{\prime\prime}}=\sum_{k=0}^{N}t^{\frac{1-k}{2}}\mathbb{E}^{{}^{\prime\prime}}|_{E_{k}}$
for $t>0$ and $\mathcal{R}_{t}$ denote the corresponding curvatures. Let
$N_{H}$ be the number operator $h\mapsto kh$ if
$h\in\mathcal{A}^{\bullet}(X,E_{k})$[16, Definition 4.18][6, Definition 2.3],
and we have

$ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}})=\partial\overline{\partial}\int_{1}^{\infty}Trs\left(\frac{N_{H}\cdot
exp(-\mathcal{R}_{t})}{t}\right)dt\Rightarrow
ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}})=0.$

Now we consider the case where $\mathcal{A}^{\bullet}(X,E^{\bullet})$ is not
exact. Again, denote the support of this complex by $Z.$

###  6.2 Transgression Formulae and Superconnection Currents

There are Bott-Chern currents representing
$ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}}).$ Denote
$Z(F)=X^{\prime}$ and $\phi:\bigwedge T^{*}X\rightarrow\bigwedge
T^{*}X,a\mapsto(2\pi i)^{-|a|/2}a.$ Now write
$\delta_{X^{\prime}}\in\mathbb{D}^{*}(X)$ be the current of integration along
$X^{\prime}.$ Let $\mathscr{HE}$ be the sheaf of cohomology groups of
$E^{\bullet}.$ Note that $\forall x\in X$ there is a canonical isomorphism
$\mathscr{HE}_{x}\cong\\{y\in
E^{\bullet}:\mathbb{E}_{0}^{{}^{\prime\prime}}(y)=0,\mathbb{E}_{0}^{{}^{\prime}}(y)=0\\},$
so by [6, Theorem 1.2] it inherits a Hermitian metric from $h.$ Let
$\nabla^{\mathscr{HE}}$ be the connection compatible with the inherited
Hermitian metric. Now we define the following superconnections:

  1. 1.

Fix a $y\in(N_{X/X^{\prime}})_{\mathbb{R}}$ with
$\overline{y}\in\overline{(N_{X/X^{\prime}})_{\mathbb{R}}},$ define
$B=\nabla^{\mathscr{HE}}+\partial_{y}\mathbb{E}^{{}^{\prime\prime}}_{0}+\partial_{\overline{y}}\mathbb{E}^{{}^{\prime}}_{0}.$

  2. 2.

For $t>0,$ let $A_{t}=\nabla^{E^{\bullet}}+\sqrt{t}\mathbb{E}_{0}.$

Now for any $t>0,$ define currents
$\zeta_{E^{\bullet}}(t),\zeta^{{}^{\prime}}_{E^{\bullet}}(0)$ and $T(h)$ by

$\zeta_{E^{\bullet}}(t)=\frac{1}{\Gamma(t)}\int_{0}^{\infty}u^{t-1}\left(Trs(N_{H}\cdot
exp(-A_{u}^{2}))-\int_{X^{\prime}}Trs(N_{H}\cdot
exp(-B^{2}))\cdot\delta_{X^{\prime}}\right)du$
$\int_{X}\mu\zeta^{{}^{\prime}}_{E^{\bullet}}(0)=\frac{\partial}{\partial
t}|_{t=0}\int_{X}\mu\zeta_{E^{\bullet}}(t),\text{
$\forall\mu\in\mathcal{A}^{\bullet}(X).$}$
$T(h)=\phi(\zeta^{{}^{\prime}}_{E^{\bullet}}(0)).$

Then we have the following representation formula

######  Theorem 6.2.

[5, Theorem 2.5]

$ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}})=\left(\int_{N_{X/X^{\prime}}}\phi(Trs(exp(-B^{2})))\right)\delta_{X^{\prime}}-\frac{\overline{\partial}\partial}{2\pi
i}T(h).$

By [6, Theorem 3.2], We know that the wave-front set of
$\left(\int_{N_{X/X^{\prime}}}\phi(Trs(exp(-B^{2})))\right)\delta_{X^{\prime}}$
is contained in $(N_{X/X^{\prime}})_{\mathbb{R}^{*}},$ and there is the
following convergence resembling our previous construction:

######  Theorem 6.3.

As $t\to\infty,$ we have

$Trs(exp(-A_{t}^{2}))\to\left(\int_{N_{X/X^{\prime}}}Trs(exp(-B^{2}))\right)\delta_{X^{\prime}}$

and also (abusing the notation a bit and writing $A_{t}$ also as the current
$\xi\mapsto\int_{X}Trs(exp(-A_{t}^{2}))\wedge\xi$ and $B$ by the current
$\xi\mapsto\left(\int_{N_{X/X^{\prime}}}Trs(exp(-B^{2}))\right)\delta_{X^{\prime}}(\xi).$)

$\lim_{t\to\infty}\sup_{\xi\in\Gamma}|\xi|^{m}\cdot\left|\widehat{\phi\cdot(A_{t}-B)}(\xi)\right|=\lim_{t\to\infty}\sup_{\xi\in\Gamma}|\xi|^{m}\cdot\left|(A_{t}-B)(\phi\widehat{\xi})\right|=0,$

with the following nice convergence: $\exists C^{\prime}>0$ such that for
$t\ll 1,$ we have

$\lim_{t\to\infty}\sup_{\xi\in\Gamma}|\xi|^{m}\cdot\left|\widehat{\phi\cdot(A_{t}-B)}(\xi)\right|\leq\frac{C^{\prime}}{\sqrt{t}},$

for any fixed $m\geq 1,$ for any open $U\subseteq X$ biholomorphic to a ball
and contained in a trivializing neighborhood of $T_{\mathbb{R}}^{*}X$ and any
smooth function $\phi$ supported on $U,$ and any $\Gamma$ a closed cone such
that on $U\cap
X^{\prime},\Gamma\cap(N_{X/X^{\prime}})_{\mathbb{R}}^{*}=\\{0\\}.$

Observe that all the above constructions involve only the degree-$0$ and
degree-$1$ terms of $\mathbb{E}$. It is then natural to ask the following
question:

######  Question 6.1.

What effects do the $\mathbb{E}_{k}^{{}^{\prime\prime}}$ terms ($k\geq 2$)
have on $ch_{BC}(E^{\bullet},\mathbb{E}^{{}^{\prime\prime}})$?

The answer is that they have no effects. This follows from Qiang’s
transgression formula with respect to superconnections[16].

####  6.2.1 Known Transgression Formulae

There are two main types of transgression formulae. The first type is with
respect to the moduli space (with the topology of uniform $C^{\infty}$
convergence on compact sets) $\mathscr{M}$ of Hemitian metrics on
$E^{\bullet}.$[7, Theorem 8.1.2][6, Theorems 2.1, 2.2, 2.4][16, Proposition
3.10, Corollary 3.13, Theorem 3.19]. The second type is with respect to the
moduli space of superconnections. We will need the following (combining
Corollary 4.8 and Proposition 4.15 of [16]):

######  Theorem 6.4.

Let $f$ be a convergent power series. Let $\mathcal{E}$ be the space of all
$\overline{\partial}-$superconnections of degree-$1$ on $(E^{\bullet},h).$
Then $\exists\delta_{1},\delta_{2},$ which are $1-$forms on the subspace of
$\mathcal{A}^{\bullet,0}(X,End(E^{\bullet}))$ of exotic degree888From
Proposition 3.1, we can write an element
$A\in\mathcal{A}^{p,q}(X,End(E^{\bullet}))$ as $\phi\otimes\tau,$ with
$\phi\in\mathcal{A}^{p,q}(X)$ and $\tau\in End^{d}(E^{\bullet})$ for some $d.$
Then we define the exotic degre to be $d+q-p.$ $-1$ and the subspace of
$\mathcal{A}^{0,\bullet}(X,End(E^{\bullet}))$ of exotic degree $1$
respectively, and $\gamma_{1},\gamma_{2}$ which are sections to the subspaces
of exotic degree $0$ of $\mathcal{A}^{0,\bullet}(X,End(E^{\bullet}))$ and
$\mathcal{A}^{\bullet,0}(X,End(E^{\bullet}))$ respectively such that

$-d^{\mathcal{E}}Trs(f(\mathcal{R}_{h}))=\partial
Trs(f^{\prime}(\mathcal{R}_{h})\cdot\delta_{1})+\overline{\partial}Trs(f^{\prime}(\mathcal{R}_{h})\cdot\delta_{2})$

and

$\overline{\partial}Trs(f^{\prime}(\mathcal{R}_{h}\cdot\gamma_{1}))=Trs(f^{\prime}(\mathcal{R}_{h})\cdot\delta_{1})$
$\partial
Trs(f^{\prime}(\mathcal{R}_{h}\cdot\gamma_{2}))=Trs(f^{\prime}(\mathcal{R}_{h})\cdot\delta_{2}).$

For the construction of $\gamma_{1},\gamma_{2},\delta_{1},\delta_{2}$ refer to
Definition 4.11 and Definition 4.5 in [16]. It then directly follows that in
$H_{BC}^{\bullet}(X),$

######  Corollary 6.4.1.

Let $\psi_{t}(\mathbb{E}^{{}^{\prime\prime}})\in\mathcal{E}$ be the
superconnection
$\mathbb{E}^{{}^{\prime\prime}}_{0}+\mathbb{E}^{{}^{\prime\prime}}_{1}+t\sum_{t\geq
2}\mathbb{E}^{{}^{\prime\prime}}_{k},$ and let the associated curvature forms
be $\phi_{t}(\mathcal{R}_{h}).$ Then $Trs(f(\phi_{1}(\mathcal{R}_{h})))\simeq
Trs(f(\phi_{0}(\mathcal{R}_{h})))=Trs(f(\mathcal{R}_{h}))$ in
$H_{BC}^{\bullet}(X).$

## References

  * [1] M. Andersson.  Residue currents of holomorphic morphisms.  J. reine angew. Math., 596:215–234, 2006.
  * [2] M. Andersson and E. Wulcan.  Residue currents with prescribed annihilator ideals.  Ann. Scient. Éc. Norm. Sup., 40:985–1007, 2007.
  * [3] M. Andersson and E. Wulcan.  Decomposition of residue currents.  2009\.  https://arxiv.org/pdf/0710.2016v3.pdf.
  * [4] P. Baum and R. Bott.  Singularities of holomorphic foliations.  J. Differential Geometry, 7:279–342, 1972.
  * [5] H. Bismut, Jean-Michel; Gillet and C. Soulé.  Bott-chern currents and complex immersions.  Duke Mathematical Journal, 60(1):255–284, 1990.
  * [6] J.-M. Bismut.  Superconnection currents and complex immersions.  Inventiones Mathematicae, 99:59–113, 1990.
  * [7] S. Bismut, Jean-Michel; Shen and Z. Wei.  Coherent Sheaves, Superconnections, and RRG, volume Progress in Mathematics, Volume 347.  Birkhäuser, 2022.
  * [8] J. Block.  Duality and equivalence of module categories in noncommutative geometry.  2005\.  https://arxiv.org/abs/math/0509284v2.
  * [9] J. Block.  Duality and equivalence of module categories in noncommutative geometry ii: Mukai duality for holomorphic noncommutative tori.  2006\.  https://arxiv.org/abs/math/0604296v1.
  * [10] R. Bott and L. Tu.  Differential Forms in Algebraic Topology.  Springer-Verlag, New York, 1982.
  * [11] N. Coleff and M. Herrera.  Les Courants Reésiduels Associés à une Forme Méromorphe.  Springer Berlin, Heidelberg, 1978.
  * [12] E. Friedlander.  An introduction to k-theory, 2007.  https://archive.org/details/Eric_M_Friedlander__An_Introduction_to_Ktheory.
  * [13] P. Griffiths and J. Harris.  Principles of Algebraic Geometry.  John Wiley & Sons, Inc., 1978.
  * [14] R. Hartshorne.  Algebraic Geometry.  Springer Science+Business Media, Inc., 1977.
  * [15] M. Herrera and D. Lieberman.  Residues and principal values on complex spaces.  Mathematische Annalen, 194:259–294, 1971.
  * [16] Q. Hua.  On the bott-chern characteristic classes for coherent sheaves.  https://arxiv.org/abs/1611.04238.
  * [17] J. Kollár.  Lectures on Resolution of Singularities.  Number 166. Annals of mathematics studies, 2007.
  * [18] R. Lärkäng and H. S. Kalm.  Various approaches to products of residue currents.  Journal of Functional Analysis, 264:118–138, 2013.
  * [19] R. Lärkäng and E. Wulcan.  Chern currents of coherent sheaves.  Épijournal de Géométrie Algébrique, 6(14), 2022.  https://doi.org/10.46298/epiga.2022.8653.
  * [20] D. Quillen.  Superconnections and the chern character.  Topology, 24(1):89–95, 1985.  https://doi.org/10.1016/0040-9383(85)90047-3.
  * [21] R. O. Wells Jr.  Differential Analysis on Complex Manifolds.  Springer Science+Business Media, New York, 2008.


# An assessment of peer instruction in large first year mathematics courses

Raymond Vozzo  School of Computer and Mathematical Sciences
University of Adelaide
Adelaide, SA 5005
Australia<EMAIL_ADDRESS>,  Stuart Johnson  School of Computer
and Mathematical Sciences
University of Adelaide
Adelaide, SA 5005
Australia<EMAIL_ADDRESS>and  Jonathan Tuke  School of
Computer and Mathematical Sciences
University of Adelaide
Adelaide, SA 5005
Australia<EMAIL_ADDRESS>
###### Abstract.

Many recent studies have expounded the benefits of active learning in tertiary
education. It can be challenging to implement these techniques at large scale
(for example in first year mathematics courses). A common method for actively
engaging students in large classes is through online quizzes, which may
include peer instruction. In this paper, we investigate the effect of having
students answer quiz-style questions during class both with and without
discussion in a first year mathematics course. We also investigate the short-
and long-term effects of each protocol.

We find that peer instruction improves student learning in mathematics in the
following ways: First, when the responses to questions was measured before and
after peer instruction the proportion of questions answered correctly
increased by 0.2; second, when correct responses were compared to similar
questions the following week the proportion correct increased by 0.34
(compared to 0.07 for the control); finally, when measured at the end of the
semester the proportion of questions answered correctly increased by 0.42
(compared to 0.2 for the control).

###### Contents

  1. 1 Introduction
  2. 2 Methodology
    1. 2.1 Background
    2. 2.2 Statistical methodology
  3. 3 Results
  4. 4 Limitations and outlook
  5. 5 Conclusion
  6. A Examples of questions

##  1\. Introduction

Active learning has been proposed as a method for improving learning and
outcomes in STEM areas (see for example [5]). For large classes many of the
most popular methods for introducing active learning involve so-called
clickers [4], which involve students using technology to answer quiz
questions. This style of active learning may also include peer instruction, in
the sense originally due to Mazur [3], which requires students to answer a
question, discuss with their fellow students and then answer again. Methods
that can be employed effectively in large classes (for example, more than 400
students) are particularly important in mathematics as many science and
engineering faculties have large mathematics service courses in first year
undergraduate degrees.

Following the work of Mazur in employing peer instruction in physics, it has
since been adapted to many other areas [10]. In mathematics, studies related
to the use of peer instruction have considered various aspects including: the
effectiveness against traditional workshops in linear algebra [9] and in
calculus [6] and the problem of improving questions to achieve optimal
effectiveness [2, 8, 11].

In this article, we study the effectiveness of peer instruction in
undergraduate mathematics (both in linear algebra and in calculus) by
comparing student performance in answering questions during class with or
without structured discussion. This allows us to measure the value of
augmenting quiz-based active learning in large mathematics with peer
interactions.

##  2\. Methodology

###  2.1. Background

This study was conducted in a first year undergraduate mathematics course
containing parallel streams of calculus and linear algebra, with a diverse
cohort including mathematics, computer science, engineering and science
students. Enrolments for the semester were approximately 550 students. For
most students this is their second semester of university mathematics. The
course utilises a flipped classroom model, with videos and notes on the course
material available online in the learning management system; a one hour
workshop each week for the entire class (in a very large workshop theatre)
where the students participate in quizzes, with follow up discussion from the
workshopr where required; and weekly tutorials, where students work in groups
of 4 or 5 at whiteboards solving problems. There are several components of
assessment throughout the semester (including weekly written and online
assignments, and an invigilated test) and a final written exam.

In the workshop, peer instruction is often used, where a question is asked and
students provide answers (using their phones or other devices to access
Mentimeter111www.mentimeter.com), and are then invited to discuss the question
with their peers and potentially change answers.

For the study, each question asked was randomly allocated one of two
treatments:

  * •

control, in which the question was posed and students have only one
opportunity to answer, after which the solution would be revealed and
explained if necessary. No particular attempt was made either to force
students to discuss with their peers or to not discuss at all, they were
allowed to answer in a natural way;

  * •

discussion, in which students were instructed to give an initial answer
without any interactions with their peers, with the collective answers shown
to the class before giving the students a chance to discuss their answers, and
change them if they wish. Then the solutions are shown.

The questions asked were mostly conceptual in nature, highlighting fundamental
aspects of the material or particular topics that students typically have
difficulty with. Generally it was expected that students could answer
questions without the need for a great deal of calculation. Some examples of
questions used can be found in Appendix A.

The effect of peer instruction was measured in a number of different ways. In
cases where students are asked to discuss questions and then given the
opportunity to change their answer we can compare the initial responses with
those obtained after the discussion. To obtain a measure of the effect of peer
instruction against a control group we measured responses over consecutive
weeks. In each weekly class, students are first asked new questions about the
same concepts as the questions from the previous week. These are designed to
test conceptual understanding rather than recall of answers from the previous
week (see Appendix A for examples). This allows us to measure the effect of
peer instruction by comparing performance on these questions for each of the
“control” and “discussion” treatments.

In addition, some questions were repeated at the end of the semester in a
revision class, allowing longer term retention of knowledge to be measured.

###  2.2. Statistical methodology

Each week, four new questions were asked in the quiz. These consisted of two
algebra questions, and two calculus questions. Within each area (calculus or
algebra), one question was a control question, and one was a discussion
question. The allocation to either control or discussion was done by
randomization.

In weeks 2-11 four additional questions were asked, these were related to the
questions from the previous week. In week 12, there were 15 questions that
were all related to questions from previous weeks.

The form of the experiment is based on a matched pairs design, in that we have
a pair of questions that were discussed in the previous week, and also a pair
of question that were not discussed and acted as the control. All of the
analysis was performed in R [7] using the program
RStudio222https://posit.co/download/rstudio-desktop/. We fitted four models to
address the effect of repetition on the retention rates of students. In each
case, we fitted a generalised linear mixed-effects model (GLMM) with the
outcome variable being the number of correct answers, and the predictors the
offering and whether that the question had been reviewed—where appropriate. As
well, to account for the repeated measurements, we included a random intercept
for each question. The GLMM were fitted using the lme4 package [1]. For each
model, we then predicted the probability of getting a question correct as
given in Tables 1, 2, 3 and 4.

##  3\. Results

Figure 1 gives an overview of the experimental design, and indicates when
questions were repeated. Some questions are the same question (brown
triangle), while some are different questions, but the same concept (blue
circle). The offering on the $x$-axis gives the week and the order, so for
example 02-3 is the third question in the second week. The ID identifies each
question. The first part gives the week, the letter indicates the subject (A
is Algebra, while C is Calculus) and the number indicates whether there is
more than one question in a subject, so W02A3 is the third question in Algebra
for Week 2. We see that some questions were repeated once, for example W01A1,
while some may be repeated up to four times (e.g. W04C2).

Figure 1. Figure of experimental design showing when questions were repeated.
Some questions are the same question (brown triangle), while some are
different questions, but the same concept (blue circle). The offering on the
$x$-axis gives the week and the order, the ID identifies each question.

Figure 2 shows the proportion correct for each attempt of a question. The
thick lines show the average proportion.

In more detail, we fitted four seperate models:

  * •

Repeat Model, which compares the first and second attempt at discussion
questions in the same workshop.

  * •

Short-term Model A, which compares the first attempt at control and discussion
questions with the attempt at the related questions in the following week.
Note that it is the first attempt at the discussion question being used.

  * •

Short-term Model B, which compares the control and the second attempt at the
discussion questions with the attempt at the related questions in the
following week. Note that it is the second attempt at the discussion question
being used.

  * •

Long-term Model, which compares the first attempt at control and discussion
questions with the attempt in Week 12.

The results of each of these models is shown in Tables 1, 2, 3 and 4,
respectively.

First we see that over time there is an improvement in understanding as seen
by an increased proportion of correct answers for second attempts and beyond.
This is also seen in Tables 1, 2, 3 and 4, where in all cases there is a
positive improvement in proportion correct. Also we see that the increase in
understanding from the first attempt is much larger for discussion questions
compared to the control questions (Table 4). This may be that we can see that
the proportion correct for discussion questions on the first attempt is much
lower than the control questions. As the questions were randomly allocated to
control or discussion, this is surprising—see Section 4 for a discussion on
the authors’ thought on why this occurred.

Figure 2. Proportion answered correctly for each attempt of the questions. The
colour indicates if the question is a discussion or a control question. The
lines correspond to a single question.
Treatment | 1st | 2nd | Improvement
---|---|---|---
Discussed | 0.42 | 0.63 | 0.2
Table 1. Repeat Model. Proportion correct for questions at the first and then
second attempt in the same workshop.
Treatment | First week | Second week | Improvement
---|---|---|---
Control | $0.67$ | $0.74$ | $0.07$
Discussion | $0.43$ | $0.77$ | $0.34$
Table 2. Short-term Model A. Proportion correct for questions at the first and
then second attempt in the next workshop. Note that we are using the first
attempt for the discussed questions in the first week.
Treatment | First week | Second week | Improvement
---|---|---|---
Control | $0.67$ | $0.74$ | $0.07$
Discussion | $0.64$ | $0.79$ | $0.15$
Table 3. Short-term Model B. Proportion correct for questions at the first and
then second attempt in the next workshop. Note that we are using the second
attempt for the discussed questions in the first week.
Treatment | First offering | Week 12 | Improvement
---|---|---|---
Control | $0.55$ | $0.75$ | $0.2$
Discussion | $0.28$ | $0.70$ | $0.42$
Table 4. Long-term Model. Proportion correct for questions at the first and
then second attempt in the Week 12 workshop.

Table 5 contains the estimates and P-value for each of the models that we
considered. We see a statistically significant effect (5% level) of offering
and treatment (control versus discussion). Also we see that there is a
statistically significant interaction of offering and treatment for the Short-
term and Long-term models. As the interpretation of two-way interactions are
difficult to interpret, we advise looking at the predicted probabilities of
getting the answers correct as given in Tables 1, 2, 3 and 4.

Model | Predictor | Estimate | P-value
---|---|---|---
Repeat | Repeat | $0.83$ | $1.8\times 10^{-33}$
Short-term Model A | Discussion | $-0.99$ | $3.01\times 10^{-3}$
| Second week | $0.34$ | $1.05\times 10^{-6}$
| Interaction | $1.16$ | $1.11\times 10^{-29}$
Short-term Model B | Discussion | $-0.13$ | $7.25\times 10^{-1}$
| Second week | $0.35$ | $1.04\times 10^{-6}$
| Interaction | $0.42$ | $4.57\times 10^{-5}$
Long-term | Control | $-1.13$ | $2.34\times 10^{-2}$
| Week 12 | $0.91$ | $5.97\times 10^{-11}$
| Interaction | $0.88$ | $7.84\times 10^{-6}$
Table 5. Coefficients and P-values for each of the three models fitted to look
at the effect of offering and discussion on the probability of getting a
question correct.

##  4\. Limitations and outlook

Here we outline some of the limitations of this project and possible future
directions for study. We also offer an explanation as to the interesting
difference noted from Figure 2 between the proportion of correct answers on
the initial attempt for the control questions and the discussion questions.

The main limitation was the fact that the discussion environment was largely
determined by how students chose to engage. The class was in a large workshop
theatre and it was up to students to choose to sit with others with whom they
could discuss the questions, so some students may not have engaged in
discussion. It is possible that if they changed their answer after the
discussion period it may have simply been the result of seeing which answer
the majority chose, rather than the outcome of a reasoned discussion.

Furthermore, this also meant that the control and repeated questions were not
strictly discussion free, which could explain the difference in success of the
first attempts at questions. It is possible that during the discussion
questions, students were explicitly instructed (and monitored) to _not_
discuss the questions on their first attempt; whereas during the control
questions this was not done, so some discussion naturally occurred during this
time. This actually represents a more natural classroom situation than the
somewhat artificial enforcement of silence during a question. Thus the
“control” questions can be viewed as “non-peer-instruction” questions, while
the discussion questions are, as previously described, structured peer
instruction questions.

This observation prompted the comparison of the (first and only) attempt at
the control questions with the _second_ , post-discussion attempt at the
discussion questions, which is the model labelled Short-term Model B, shown in
table 3. This model shows that there is still an improvement in answering the
related questions the following week, suggesting that having a genuine solo
attempt at a question before discussion resulted in improved understanding.

Further study is warranted here, with a potential improvement being to ensure
that the review questions that, which are asked the following week, have a “no
discussion” policy while being answered. This would ensure that the short term
improvements described in Table 2 are accurate. It would also be interesting
to repeat this with different classroom setups; for example, in smaller rooms,
or flat-floor rooms that are more amenable to collaborative discussions than a
large workshop theatre.

##  5\. Conclusion

The effect of peer instruction was measured in three ways. First, by comparing
the responses of students to the same question immediately after discussion
with peers (Repeat model). Second, by asking related questions the following
week and measuring performance in comparison with control questions for which
no peer instruction was used (Short-term models). Finally by repeating some
questions in a revision session at the end of the semester (Long-term model).

In response to the significant difference in first attempts at control and
discussion questions, the Short-term models were split into two versions,
comparing either the first or second attempt at a discussion question with the
attempt at a related question the following week, to account for the
possibility that the control might involve student discussion and hence be
more comparable with the post discussion question.

The results show a positive effect of peer instructions in all four measures.

Specifically, using the first measure we find that peer discussion improves
the proportion of correct responses by 0.2. Using the second measure, we find
that while the proportion correct for the control questions increased by 0.07,
for the peer instruction questions this improvement was 0.34 in model A or
0.15 in model B. Finally, at the end of the semester, while there was
improvement for both sets of questions, for the control questions this was 0.2
while for the peer instruction questions, this was 0.42.

## Acknowledgments

The authors would like to thank Nickolas Falkner for helpful feedback on the
interpretation of the results and Tanya Evans for helpful comments on the
first draft.

##  Appendix A Examples of questions

We provide some examples of questions used during the study that demonstrate
the sort of thing that students were typically asked. The examples below show
both conceptual questions and computational questions. Note that if any
computations were required they were both fairly simple and designed to test
understanding of a particular technique or result. In each case two versions
of a question are presented, the initial one, and the related question which
was asked the following week. The related questions are designed to test the
same concept, and to require genuine understanding of the concept, rather than
being able to be correctly answered purely by knowing the answer to the
original question.

######  Example.

Which of the following are linear transformations?

  1. A.

$F(x,y)=(x,y)$

  2. B.

$F(x,y)=(0,0)$

  3. C.

$F(x,y)=(1,1)$

  4. D.

$F(x,y)=(xy,0)$

  5. E.

$F(x,y)=(y,y)$.

Related question. Which of the following are linear transformations?

  1. A.

$F(x,y)=(y,x)$

  2. B.

$F(x,y)=(0,1)$

  3. C.

$F(x,y)=(x+y,0)$

  4. D.

$F(x,y)=(xy,xy)$

  5. E.

$F(x,y)=(y,y+1)$.

######  Example.

Suppose $A=\begin{bmatrix}2&*\\\ *&3\end{bmatrix}$ is not invertible. What are
the eigenvalues of $A$?

  1. A.

$0$

  2. B.

$2$

  3. C.

$3$

  4. D.

$5$

  5. E.

$6$

  6. F.

There is not enough information to determine this

Related question. Suppose $A=\begin{bmatrix}-1&*\\\ *&2\end{bmatrix}$ is not
invertible. What are the eigenvalues of $A$?

  1. A.

$-2$

  2. B.

$-1$

  3. C.

$0$

  4. D.

$1$

  5. E.

$3$

  6. F.

There is not enough information to determine this

######  Example.

Fill in the blank.

$\displaystyle\sum_{n=1}^{\infty}a_{n}$ converges
$\underline{\phantom{BLANK}}\ \displaystyle\lim_{n\to\infty}a_{n}=0$.

  1. A.

$\Rightarrow$

  2. B.

$\Leftarrow$

  3. C.

$\Leftrightarrow$

Related question. Fill in the blank.

Suppose that $a_{n}>0$ is a decreasing sequence.
Then $\displaystyle\sum_{n=1}^{\infty}(-1)^{n}a_{n}$ converges
$\underline{\phantom{BLANK}}\ \displaystyle\lim_{n\to\infty}a_{n}=0$.

  1. A.

$\Rightarrow$

  2. B.

$\Leftarrow$

  3. C.

$\Leftrightarrow$

######  Example.

Consider the graph of the function $f(x,y)$ below.

Which of the following is $f_{x}$ and which is $f_{y}$?

A.  B.

Related question. Consider the graph of the function $f(x,y)$ below.

Which of the following is $f_{x}$ and which is $f_{y}$?

A.  B.

## References

  * [1] Douglas Bates, Martin Mächler, Ben Bolker, and Steve Walker.  Fitting linear mixed-effects models using lme4.  Journal of Statistical Software, 67(1):1–48, 2015.
  * [2] Kelly Cline, Holly Zullo, Jonathan Duncan, Ann Stewart, and Marie Snipes.  Creating discussions with classroom voting in linear algebra.  International Journal of Mathematical Education in Science and Technology, 44(8):1131–1142, 2013.
  * [3] Catherine H. Crouch and Eric Mazur.  Peer Instruction: Ten years of experience and results.  American Journal of Physics, 69(9):970–977, 09 2001.
  * [4] Douglas Duncan and Eric Mazur.  Clickers in the Classroom: How to Enhance Science Teaching Using Classroom Response Systems.  Pearson Series in Educational Innovation: Instructor Resources for Physics Series. Pearson Education, 2005.
  * [5] Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth.  Active learning increases student performance in science, engineering, and mathematics.  Proc. Natl. Acad. Sci. USA, 111(23):8410–8415, 2014.
  * [6] Scott Pilzer.  Peer instruction in physics and mathematics.  PRIMUS, 11(2):185–192, 2001.
  * [7] R Core Team.  R: A Language and Environment for Statistical Computing.  R Foundation for Statistical Computing, Vienna, Austria, 2023.
  * [8] Everilis Santana-Vega Robyn L. Miller and Maria S. Terrell.  Can good questions and peer discussion improve calculus instruction?  PRIMUS, 16(3):193–203, 2006.
  * [9] Katiuscia Costa Barros Teixeira.  Pedagogical strategies to enhance learning in a linear algebra course.  PRIMUS, 33(2):152–174, 2023.
  * [10] Jonathan G. Tullis and Robert L. Goldstone.  Why does peer instruction benefit student learning?  Cognitive Research: Principles and Implications, 5(1):15, 2020.
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# A Dual Convolutional Neural Network Pipeline for Melanoma Diagnostics and
Prognostics

Marie Bø-Sande1*, Edvin Benjaminsen1*, Neel Kanwal1, Saul Fuster1, Helga
Hardardottir2,3, Ingrid Lundal2,3, Emiel A.M. Janssen2,3, Kjersti Engan1,†
1 Department of Electrical Engineering and Computer Science, University of
Stavanger, Stavanger, Norway
2 Department of Chemistry, Bioscience and Environmental Engineering,
University of Stavanger, Norway
3 Department of Pathology, Stavanger University Hospital, Stavanger, Norway
* These authors contributed equally
$\dagger$ Corresponding author<EMAIL_ADDRESS>

###### Abstract

Melanoma is a type of cancer that begins in the cells controlling the pigment
of the skin, and it is often referred to as the most dangerous skin cancer.
Diagnosing melanoma can be time-consuming, and a recent increase in melanoma
incidents indicates a growing demand for a more efficient diagnostic process.
This paper presents a pipeline for melanoma diagnostics, leveraging two
convolutional neural networks, a diagnosis, and a prognosis model. The
diagnostic model is responsible for localizing malignant patches across whole
slide images and delivering a patient-level diagnosis as malignant or benign.
Further, the prognosis model utilizes the diagnostic model’s output to provide
a patient-level prognosis as good or bad. The full pipeline has an F1 score of
0.79 when tested on data from the same distribution as it was trained on.

###### Index Terms:

Cancer Diagnosis, Deep Learning, Melanoma, Image Classification, Segmentation,
Prognosis.

##  I Introduction

Melanoma cancer is the leading cause of death from skin disease. It begins in
the skin cells called melanocytes, and around 30% starts in existing moles.
According to a recent worldwide study, the number of newly diagnosed melanoma
cases will rise by more than 50%, up to 510,000 by 2040, while the number of
melanoma deaths will rise by almost 68%, from 57,000 in 2020 to 96,000 in 2040
[1]. In regards to Norway, the annual cancer report shows a 20% increase in
melanoma cases from 2021 to 2022 [2]. Coupled with the increased incidence
rate, the estimated survival rate for five years following diagnosis varies
depending on the stage of melanoma [3]. Consequently, early detection of
melanoma plays a crucial role in the prognostic outcome.

Figure 1: An overview of our proposed deep learning pipeline. _Preprocessing:_
Background/foreground segmentation is applied on the whole slide image (WSI)
to find tissue regions. Later, morphological operations remove small holes and
separate small areas, and then patches are extracted at 10x magnification for
the diagnosis model. _Diagnosis:_ Prediction for every patch is used to
calculate patient-level prediction as benign vs. malignant. _Prognosis:_
Detected lesion regions on WSIs predicted as malignant are used to evaluate
prognosis at the patient level as good or bad.

In recent years, digital pathology is becoming mainstream, producing whole
slide images (WSIs) as digital microscopy gigapixel images of tissue slides.
The process of producing WSIs from biopsies, with its potential artifacts, is
described in [4]. Computational pathology (CPATH) is a growing field dealing
with automated solutions for visualization, diagnostics, and prognostics from
WSI, using image processing and deep learning (DL).

Melanoma detection using DL techniques has shown promising results, which can
help with early diagnosis and treatment decisions [5, 6] [7]. DL algorithms
can detect potential regions with melanoma by identifying various cellular and
tissue-level features and enhancing diagnostic accuracy [8]. Some existing
methods focus on detecting melanoma by classifying tissue samples and moles
either as melanoma or benign nevi [6, 9, 5]. Others address the question of
prognostic prediction from lesion-tissue of verified melanoma cases, typically
with the lesion manually delineated [10, 11, 12]. Clinical labels are usually
patient-based, and providing manually annotated regions for the melanoma
tissue is time-consuming. The challenge of having a complete CPATH pipeline
that can differentiate between WSIs with benign nevi and melanoma (malignant),
segment the melanoma region from a WSI, and make prognostic predictions in the
case of melanoma is currently unexplored.

To address this challenge in this work, we are developing a pipeline by
integrating two convolutional neural networks (CNN), as illustrated in Figure
1. By leveraging DL methodologies, the first model of the pipeline identifies
melanoma in WSIs and produces a patient-level diagnosis, whereas the second
model provides a prognosis for identified melanoma patients. The performance
of each CNN model is evaluated individually before integrating them into a
comprehensive pipeline.

##  II Data Materials

The dataset is collected at Stavanger University Hospital (SUH), Stavanger,
Norway. A Hamamatsu Nanozoomer s60 scanner was used to scan a cohort of
Hematoxylin and Eosin (H&E) stained glass slides at 40x magnification,
thereafter saved in NDPI format. Clinical labels give the patient diagnosis as
$y_{i}^{d}\in\\{0,1\\}$ where 1 indicates melanoma and 0 benign nevi, and $i$
is a patient (or WSI) index, and $d$ indicates diagnostic label. For a patient
diagnosed with melanoma, the prognostic label was established using follow-up
data by considering the occurrence of either local or distant metastasis
(bad), or the absence of metastasis (good), within a five-year time frame;
$y_{i}^{p}\in\\{0,1\\}$ where 1 indicates bad prognosis and 0 good prognosis.
The dataset is divided into annotated WSIs, $D^{a}$, and not-annotated WSIs,
$D^{n/a}$, explained more in the following subsections. Later, patches were
extracted from the WSIs for analysis. A breakdown of the number of patches
extracted from the different sets is shown in Tab. reftab:patches.

Table I: The amount of patches from each subset of $D^{a}_{d}$ after patch extraction for the diagnostic model. Label | $D_{train}$ | $D_{val}$ | $D_{test}$ | Total
---|---|---|---|---
B | 42 420 | 11 660 | 7 190 | 61 270
M | 215 320 | 25 169 | 37 310 | 27 799
NE | 2 801 | 657 | 604 | 4 062
Total | 260 541 | 37 486 | 45 104 | 343 131
Table II: The amount of patches of $D^{a}_{p}$ after patch extraction for the prognostic model. Label | Total
---|---
Good | 5 542
Bad | 6 358
Total | 11 900

###  II-A Annotated WSIs

$D^{a}$ is a set of 125 WSIs, 47 benign nevi, and 78 with melanoma. The
lesion, or region of interest (ROI), in all WSI, is roughly annotated by a
pathologist. The annotated regions of the lesion have two different classes,
corresponding to melanoma M and benign nevi B. In addition, some areas of
normal epidermal tissue NE are annotated, but not all such areas. Tissue
outside these regions is _not_ annotated. Labels associated with these regions
are defined as $y_{ji}^{d}$, where $j$ is a patch index, $i$ is a patient or
WSI index. In addition, there are large tissue regions that are _not_
annotated in all WSI. The diagnosis model used a sub-dataset $D^{a}_{d}$ of 90
WSIs, where 73 of them were used for training, 8 for validating, and 9 for
testing, with approximately 50% benign nevi and melanoma. The prognostic model
used a sub-dataset $D^{a}_{p}$ of 52 WSIs, all with melanoma, 50% with bad and
50% with good prognosis. Some patients were excluded as they were present in
both $D^{a}_{d}$ and $D^{a}_{p}$. A total of 9 with bad prognosis and 5 with
good prognosis. A sub dataset $\hat{D}^{a}_{p}$ was defined comprising all the
images from $D^{a}_{p}$ except for the aforementioned excluded patients, as
they were employed for the development of the diagnostic model. There is no
overlap from training to validation or test in the pipeline experiments we
show.

###  II-B Non-annotated WSIs

A dataset $D^{n/a}$ containing 243 WSIs from SUH cohort is provided with
patient-level clinical labels without any manual annotation around lesion
areas. Of all 243 WSIs, 110 of them were diagnosed with a benign nevis, and
133 with melanoma, 10 of these with bad prognosis (metastasis within 5 years).
The dataset is divided into a train/validation set of 203 WSIs and a test set
of 40 WSIs. In the test set, 18 WSIs are labeled as benign and 22 as melanoma,
2 of them having a bad prognosis (metastasis within 5 years).

##  III Method

###  III-A Preprocessing

To enable the analysis of WSIs using CNN, the tissue regions within the WSIs
were divided into smaller patches of size 256$\times$256 pixels at different
magnification levels (2.5x, 10x, and 40x). Let ${\bf x}_{ji}^{10x}$ denote
patch $j$ from WSI $i$ at magnification level 10x. The index $i$ denotes the
WSI and is sometimes omitted. To separate the tissue from the background,
background-foreground segmentation was performed by transforming the RGB
images to the HSV color space, and the Hue channel was thresholded within the
range of [100-180] to identify purple and pink tones. Morphological opening
and closing operations were applied to close holes in the foreground and
remove small areas. Grid extraction was applied to extract valid patches as
described in [13].

###  III-B Diagnosis

Valid patches, ${\bf x}_{ji}^{10x}$, from the preprocessing of WSI $i$ are fed
into the diagnosis model, providing a patch-level prediction: $f^{d}({\bf
x}_{ji}^{10x})=\hat{y}_{ji}^{d}$. The feature extractor of the model is based
on the VGG16 [14] architecture with pre-trained weights from ImageNet [15]. A
three-layer classifier of fully connected layers is added. The diagnostic
model is fine-tuned using the annotated training data from $D_{d}^{a}$, as in
[6]. The models output layer consists of a softmax giving an array ${\bf
v}_{ji}$ of three probability values for each patch, benign nevi (B), melanoma
(M), and normal epidermal tissue (NE). The patch-level diagnosis predictions
are denoted as $\hat{y}_{ji}^{d}$, for patch $j$, and WSI $i$. If $max({\bf
v}_{ji})>t_{p}$, $\hat{y}_{ji}^{d}$ is set to the most probable class label
(M, B, NE). Else, $\hat{y}_{i}^{d}$ is set to _T_ for tissue (i.e., none of
the other classes). Thus, we train the model with three labels, but we
classify the patches into four classes, $\hat{y}_{ji}^{d}\in\\{$M,B,NE,T}. The
patient-level prediction $\hat{y}_{i}$ is determined by calculating the ratio
${\psi}$ of number of patches predicted as malignant (i.e. melanoma) over
other patches. In this work, two different methods are used to calculate the
ratio as shown in Eq. (1) and (2). The first ratio, ${\psi}^{MB}$, calculates
the ratio between predicted malignant and benign patches, while the second
ratio, ${\psi}^{MT}$ calculates the ratio between malignant patches and
patches in the entire tissue mask. Let the indicator function,
$I(\hat{y}_{ji},\\{\textit{M,B}\\})=1$ if $\hat{y}_{ji}=\textit{M}$ or B and 0
otherwise:

$\psi^{MB}_{i}=\frac{\sum_{j}I(\hat{y}_{ji}^{D},M)}{\sum_{j}I(\hat{y}_{ji}^{D},\\{\textit{M,B}\\})}$
(1)

$\psi^{MT}_{i}=\frac{\sum_{j}I(\hat{y}_{ji}^{D},M)}{\sum_{j}I(\hat{y}_{ji}^{D},\\{\textit{M,B,NE,T}\\})}$
(2)

Thereafter, the ratio is compared with a threshold $t_{r}$. If
${\psi}_{i}<t_{r}$ the WSI $i$ is predicted as benign (0); conversely, if
${\psi}_{i}>t_{r}$ the WSI $i$ is predicted as melanoma (1), i.e. finding
patient-level diagnosis label $\hat{y}_{i}^{d}$ as shown in Eq. (3).

$\hat{y}_{i}^{d}=\begin{cases}1,(melanoma)&\text{if }\psi_{i}\geq t_{r}\\\
0,(benign)&\text{else}\end{cases}$ (3)

Let $\\{{\bf x}_{j}\\}_{M}$ denote the set of patches where
$I(\hat{y}_{j}^{d},M)\cap\hat{y}^{d}$, i.e. when the patch is predicted as
malignant and the patient is predicted as melanoma, defines the ROId for
further prognostic analysis.

###  III-C Prognosis

The prognostic model utilizes a VGG16 backbone, and transfer learning is used
with pretrained weights. The classifier of the VGG16 is replaced with three
fully-connected layers, the last having softmax activation function, giving a
binary output for good or bad prognosis, trained on $D_{p}^{a}$ as in [10].
The prognosis model uses malignant patches of the predicted melanoma WSIs for
further analysis, i.e. the ROId defined by $\\{{\bf x}_{j}\\}_{M}$ as
described in the previous section. However, the prognostic model operates on a
different magnification scale than the diagnostic model. Thus, the ROI from
the malignant patches is used to extract new patches at 20x magnification with
the method of [13], requiring a minimum 70% overlap between a valid patch and
the ROI.

The prognosis model provides a patch-level prediction: $f^{p}({\bf
x}_{ki}^{20x})=\hat{y}_{ki}^{p}$ for image $i$ and patch $k$
$\in\text{ROI}_{d}$ defined by $\\{{\bf x}_{j}\\}_{Mi}$.
$\hat{y}_{ki}^{p}\in\\{0,1\\}$ where "1" indicates bad prognosis at patch
level. A threshold $t_{m}$ is used to calculate patient-level $\hat{y}^{p}$
with "1" indicating a bad prognosis and "0" indicating a good prognosis at the
WSI level, as shown in Eq. (4).and Eq. (5).

$\phi_{i}=\frac{\sum_{j}\hat{y}_{ji}^{p}}{\sum_{j}(I(\hat{y}_{ji}^{p},0)+I(\hat{y}_{ji}^{p},1))}$
(4)

$\hat{y}^{p}_{i}=\begin{cases}1,&\text{if }\phi_{i}\geq t_{m}\\\
0,&\text{else}\end{cases}$ (5)

Figure 2: Examples of the diagnosis model’s patch predictions at different
$t_{p}$ thresholds for $D^{n/a}_{val}$. Each column displays a WSI with its
ground truth label (patient level) at the top. Each row presents the
prediction masks for each WSI with the given $t_{p}$ on the left side.

###  III-D Evaluation Metrics

Let TP, TN, FP, and FN stand for true positive, true negative, false positive,
and false negative, respectively. The accuracy is calculated by
$(TP+TN)/(TP+FP+FN+TN)$. Recall/ sensitivity is calculated using $TP/(TP+FN)$.
Precision = $TP/(TP+DP)$ and specificity = $TN/TN+FP$. F1 is a weighted
harmonic mean of precision and recall.

##  IV Experiments and Results

Experiments are done by validating the models on the annotated datasets
$D^{a}$. The pipeline is tested by comparing prognosis prediction using
annotated melanoma ROI as inputs with prognosis prediction using automated
ROI, i.e., outputs from the diagnostic model, on the exact same dataset. The
performance of both models is tested with dataset $D^{n/a}$. We employed
5-fold cross-validation for training our models. We selected the best-
performing model and proceeded with it. The results we have shown correspond
to the values of the best-performing model trained using cross-validation. By
evaluating the performance of each model separately, it is possible to assess
their individual accuracy in addition to testing the combined pipeline.

###  IV-A Annotated data

The diagnostic model achieved a performance of 100% accuracy at the WSI level
on the 9 WSIs in the test set of dataset $D^{a}_{d}$.

####  IV-A1 Full pipeline test

The prognosis model’s validation process involves comparing the performance of
$f^{p}$ when running on a dataset $\hat{D}^{a}_{p}$ with ROI inputs from the
manual annotations, ROIa, and with ROI from the masks generated by the
predictions from the diagnosis model $f^{d}$, ROId. By utilizing the diagnosis
model’s outputs as inputs for the prognostic model’s evaluation, we can
effectively assess the performance and accuracy of the prognostic model in a
realistic setting. This approach allows us to better understand how the two
models work together and how well the prognostic model performs when applied
to new, unseen data.

Table III displays the evaluation metrics after running the prognosis on the
dataset ($\hat{D}^{a}_{p}$).

Table III: Evaluation Metrics after running the prognosis on dataset $\hat{D}^{a}_{p}$ with annotation masks and generated masks from the diagnosis. $\hat{D}^{a}_{p}$ | Sens. | Spec. | F1 | Accuracy
---|---|---|---|---
ROIa | 0.941 | 0.714 | 0.821 | 0.816
ROId | 1.000 | 0.571 | 0.791 | 0.763

The F1 score and Accuracy are slightly better when using ROIa from manual
annotations, but the results look promising for using the automatically found
ROId.

###  IV-B Non-annotated data

In this experiment, the diagnostic model’s performance on the non-annotated
dataset $D^{n/a}_{val}$ will be evaluated. The initial thresholds for patch-
level classification ($t_{p}$) and patient-level classification ($t_{r}$) are
set at 0.999 and 0.04, respectively. These thresholds were found in the
previous experiment, where the model predicted all images correctly on
$D^{a}_{d}$. The results of the experiment can be found in Table IV.

Table IV: Results from running inference with diagnosis model on $D^{n/a}_{val}$ with thresholds $t_{p}=0.999$ and $t_{r}=0.04$. Eval. Metric | Sens. | Spec. | F1 | Accuracy
---|---|---|---|---
Score | 0.977 | 0.146 | 0.728 | 0.601

The diagnostic model’s accuracy is measured to be 0.601, indicating that it
correctly classifies only 60% of the 243 WSIs in dataset $D^{n/a}$.
Furthermore, the recall value is calculated to be 0.977, indicating that the
model excels at correctly predicting almost all melanoma cases but faces
difficulties in accurately predicting benign cases.

Table V: Non-annotated data, pipeline test on the diagnosis $(f^{d})$ and prognostic model $(f^{p})$ on the $D^{n/a}_{test}$. For each of the three tests, performance metrics for the diagnostic model only (D), as well as the complete pipeline (P) is reported as D/P in the metrics. Test | Mod | $t_{p}$ | $t_{r}$ | $\mathbf{\psi}$ | Spec. | Sens. | $F_{1}$ | Acc
---|---|---|---|---|---|---|---|---
1 | D/P | 0.9990 | 0.04 | MB | 0.11/0.14 |  1.00/1.00 |  0.73/0.11 | 0.60/0.18
2 | D/P | 0.9990 | 0.04 | MT |  0.39/0.07 | 0.82/1.00 | 0.71/0.14 | 0.63/0.14
3 | D/P | 0.9999 | 0.01 | MT |  0.39/0.11 | 0.86/1.00 | 0.73/0.14 |  0.65/0.17

The results from this evaluation demonstrate that the diagnosis model shows
some level of generalizability for new data with the current settings.
However, it is evident that there is room for improvement to enhance its
performance further. This highlights the need to focus on parameter tuning and
optimization for the model.

Figure 3: ROC plot for diagnosis model with $t_{p}=0.9999$ and $MT_{rate}$
method on dataset $D^{n/a}$.

Dataset $D^{n/a}_{val}$ is used to find optimal threshold $t_{p}$ and $t_{r}$.
Figure 2 shows predictions based on different $t_{p}$. The first column shows
an example of a benign slide that is mistaken for melanoma. There are very few
predictions of the NE class, probably because the class is underrepresented
during training. The distinction between NE and T has no diagnostic or
prognostic relevance, but we keep it as a separate class in case it helps
separate benign from malignant cases. In future work, we will investigate this
further.

When a lesion is annotated, it is obvious to calculate the ratio of MB patches
within the lesion. However, in situations where the lesion size, as well as
the tissue size, varies and is unknown, focusing on MT appears to be a more
logical approach. For each $t_{p}$, different $t_{r}$ was tested with a ROC
plot to find the optimal threshold, Figure 3 shows the MT ROC plot for
$t_{p}=0.9999$ and $t_{r}=0.01$ marked as updated thresholds for the new,
larger dataset. For MB, we observed that the optimal threshold choice yields
relatively poorer performance across all $t_{r}$ values. All in all, the
increase in AUC score suggests that MT is more effective.

####  IV-B1 Non-annotated data pipeline test

A final test is done using the updated thresholds from the previous
experiment, and testing the entire pipeline on the test set from $D^{n/a}$.
Results are presented in Table V. The diagnose model performs reasonably well,
with F1 scores around 0.73. The model recall is higher than the specificity,
which is also what is desired since it is better to be sure that malignant
melanoma is discovered. The difference between the new and old thresholds is
not very large. The prognostic model, however, has a recall of 1 in all
experiments and a bad specificity, even if the results in Table III are
promising. Prognosis prediction is far more difficult than diagnostic
prediction in general, and this shows us that the model has not generalized
well enough, and a larger training set with both good and bad prognosis is
needed to get general models. In future work, we will investigate using
multiple instance learning with a larger dataset for the prognostic part.

##  V Conclusion

This paper presents a pipeline putting together two CNN models, one for
melanoma detection and localization and one for prognosis prediction on
melanoma cases. The pipeline test demonstrates that the prognostic model works
similarly well, with F1 scores of 0.82 and 0.79 if the input to the model
comes from manual annotation or the output of the diagnostic model when tested
on the data set from the same distribution as used to train the models, which
is very encouraging. Further updating of the parameters in the diagnostic
model showed reasonably good performance for the diagnosis part on a new data
set; however, the prognostic model overestimates bad prognosis and should be
trained on larger data sets. Prognosis is generally harder to predict.

## Compliance with ethical standards

This study was performed in line with the principles of the Declaration of
Helsinki. Approval was granted by the Regional Ethics Committee (No:
2019/747/RekVest). The authors have no relevant financial or non-financial
interests to disclose.

## Acknowledgements

Thanks to Andres David Mosquera Zamudio for verifying some of the slides.
This research has received funding from the European Union’s Horizon 2020
research and innovation program under grant agreements 860627 (CLARIFY) and
“Pathology Services in the Western Norway Health Region – a centre for applied
digitization (PiV)” from a Strategic investment from the Western Norway Health
Authority.

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