# The cohomology of the elliptic tangent bundle

Aldo Witte  Department of Mathematics, Utrecht University, 3508 TA Utrecht,
The Netherlands<EMAIL_ADDRESS>
###### Abstract.

In this note we compute the cohomology of the elliptic tangent bundle, a Lie
algebroid introduced in [1, 2] used to describe singular symplectic forms
arising from generalized complex geometry.

###### Contents

  1. 1 Elliptic divisors
  2. 2 Geometric structure on the normal bundle
  3. 3 Residue maps
  4. 4 Cohomology of the elliptic tangent bundle
  5. 5 Examples and outlook
  6. A Atiyah algebroids

## Introduction

Generalized complex structures [8, 6] are a simultaneous generalisation of
symplectic and complex structures. Infinitesimally these structures decompose
each tangent space as the direct sum of a complex and a symplectic vector
space. However, the number of complex directions, called the _type_ , might
vary throughout the manifold. A well-behaved and interesting class of
structures for which the type changes, called _self-crossing stable
generalized complex structures_ , was defined in [1, 2]. These are generalized
complex structures which have type zero on an open and dense set and allow the
type to change along an immersed codimension-two submanifold.

In [2] it is shown that these self-crossing stable generalized complex
structures are in one-to-one correspondence with (a subset of) _self-crossing
elliptic symplectic structures_. These are symplectic structures with well-
behaved singularities, and can be described as Lie algebroid symplectic forms
for a certain Lie algebroid: the _self-crossing elliptic tangent bundle_.

Much research on Lie algebroid symplectic structures has been carried out over
the past few years. Examples include b-symplectic [7], scattering-symplectic
[10], c-symplectic [11] (or self-crossing b-symplectic). In the
algebraic/complex setting holomorphic log symplectic structures, which are
holomorphic forms with logarithmic singularities in the sense of [4], are
well-studied.

In [1] deformations of stable generalized complex structures with embedded
type change locus and the corresponding elliptic symplectic structures are
described. It is shown there that deformations are completely controlled by
the cohomology of the relevant Lie algebroid.111The description of
deformations of stable generalized complex structures was obtained
independently in [5].

If we allow the type change locus to be immersed this will no longer be the
case, still the computation of the Lie algebroid cohomology is the first step
in understanding the deformation theory.

In this note we compute the cohomology of the self-crossing elliptic tangent
bundles. This cohomology is determined by the topological data of the
manifold, of the type change locus and of the normal bundle.
Organisation of the paper: This paper is organized as follows: In Section 1 we
recall the notion of self-crossing elliptic divisors and their associated Lie
algebroids. In Section 2 we describe the geometric structure a self-crossing
elliptic divisor induces on the directions normal to its degeneracy locus. In
Section 3 we will use this geometric structure to define residue maps, which
we will use in Section 4 to compute the Lie algebroid cohomology of the
elliptic tangent bundle. In Section 5 we compute this cohomology in some
explicit examples. We end this section by giving an outlook on the deformation
theory of self-crossing stable generalized complex structures.

Acknowledgements. The author would like to thank his supervisor Gil Cavalcanti
and Ralph Klaasse for useful discussions. The author was supported by the NWO
through the Utrecht Geometry Centre Graduate Programme.

##  1\. Elliptic divisors

The singularities we encounter are governed by the notion of a real divisor, a
notion which is inspired by the notion of divisor in algebraic geometry. In
this section we will recall the definition of elliptic divisors, and their
associated Lie algebroids, the elliptic tangent bundles. We will keep our
discussion brief and refer to [1, 2] for more information.

######  Definition 1.1.

A real divisor $(R,q)$ consists of a real line bundle $R$ together with a
section $q$ with nowhere dense zero-set. Its associated ideal $I_{q}$ is the
image of the map $q:\Gamma(R^{*})\rightarrow\mathbb{R}$.

An isomorphism of divisors $(L_{i},q_{i})$ is a vector bundle isomorphism
$\Phi:R_{1}\rightarrow R_{2}$ covering the identity and intertwining the
sections, $\Phi^{*}(q_{2})=q_{1}$. A real divisor is, up to isomorphism,
determined by its associated ideal and therefore we may use ideals and
divisors interchangeably.

######  Definition 1.2 ([1]).

A smooth elliptic divisor is a real divisor $(R,q)$ such that the critical set
of $q$ is a codimension two submanifold along which the normal Hessian is
positive definite. We call $D=q^{-1}(0)$ the vanishing locus.

The normal Hessian is the leading part of the Taylor expansion of $q$ around
$D$, and defines a section
$\operatorname{Hess}^{\nu}q\in\Gamma(\text{Sym}^{2}N^{*}D\otimes R)$. Phrased
in other words, Definition 1.2 states that $q$ locally defines a particular
type of Morse-Bott function.

######  Definition 1.3 ([2]).

A  self-crossing elliptic divisor is a real divisor $\left|D\right|=(R,q)$ on
$M$ such that for every point $p\in M$ there exists a neighbourhood $U$ of $p$
such that

$I_{q}(U)=I_{1}\cdot\ldots\cdot I_{j},$

where the $I_{1},\ldots,I_{j}$ are smooth elliptic divisors with transversely
intersecting vanishing loci.

We call $I_{1},\ldots,I_{j}$ as above a choice of local normal crossing
elliptic divisors near a point $p$. If $I_{q}$ is globally of the form
$I_{1}\cdot\ldots\cdot I_{j}$, then say that $I_{q}$ is a global normal
crossing elliptic divisor.

In what follows if we write “elliptic divisor” it is understood to have self-
crossings, if it is a smooth elliptic divisor we will explicitly state this.

The vanishing locus of an elliptic divisor is an immersed submanifold,
stratified by the amount of submanifolds intersecting:

######  Definition 1.4.

Let $I_{\left|D\right|}$ be an elliptic divisor on $M$. The intersection
number of a point $p\in M$ is the minimum of the integers $j$ from Definition
1.3 over all neighbourhoods $U$ of $p$. The intersection number of the divisor
is the maximum of the intersection numbers of all points $p\in M$. If
$I_{\left|D\right|}$ has intersection number equal to $n$, the sets $D(j)$ of
points of intersection number at least $j$ induce a filtration of $M$:

$M=D(0)\supset D(1)=D\supset\cdots\supset D(n)\supset D(n+1)=\emptyset,$

with induced stratification $D[j]:=D(j)/D(j+1)$. These strata are embedded
submanifolds and consist of the points with intersection number precisely $j$.

The fact that the $D(i)$ indeed induce a stratification follows readily from
the normal forms of the elliptic divisor in Lemma 1.6 below.

Restricting the divisor to the set of points with at most a given intersection
number produces another divisor:

######  Lemma 1.5.

If $I_{\left|D\right|}$ is an elliptic divisor with intersection number $n$
and $i\leq n$, then the restriction ${I_{\left|D\right|}}|_{M\backslash
D(i+1)}$ defines a divisor with intersection number $i$.

Note that on a manifold $M^{m}$ the maximum intersection number of an elliptic
divisor is $\lfloor m/2\rfloor$. Using the Morse–Bott lemma inductively we can
establish the following local normal form for elliptic divisors:

######  Lemma 1.6 ([2]).

Let $\left|D\right|=(R,q)$ be an elliptic divisor with intersection number
$n\leq\lfloor m/2\rfloor$ on $M^{m}$, and let $x\in D[i]$ with $i\leq n$. Then
there exists coordinates
$(x_{1},y_{1},\ldots,x_{i},y_{i},w_{2i+1},\ldots,w_{2i+l})$ around $x$ such
that $(R,q)$ is isomorphic to the elliptic divisor defined by the ideal
$I_{\left|D\right|}=\left\langle(x_{1}^{2}+y_{1}^{2})\cdot\ldots\cdot(x_{i}^{2}+y_{i}^{2})\right\rangle$.

We call $I_{\left|D\right|}$ as above, the standard elliptic divisor with
intersection number $i$ on $\mathbb{R}^{2i}\times\mathbb{R}^{l}$.

If $(R,q)$ is an elliptic divisor, then $R$ is trivialisable away from the
codimension-two submanifold $D[1]$ and thus globally trivialisable:

######  Lemma 1.7.

Let $(R,q)$ be an elliptic divisor on $M$. Then $R$ is trivialiasible, and
consequently $I_{q}=\left\langle f\right\rangle$ for some $f\in
C^{\infty}(M)$.

Although $R$ is trivialisable, it is not canonically so and thus we prefer to
work with the ideal $I_{q}$ rather then one particular function generating it.

######  Remark 1.0 (Metrics).

Let $(R,q)$ be a smooth elliptic divisor with vanishing locus $D$, and let $f$
be any global function generating $I_{\left|D\right|}$. Because $f$ is nowhere
vanishing on $M\backslash D$, and $D$ has codimenion-two in $M$ the sign of
$f$ is constant on the entirety of $M$. Consequently, we can always choose a
non-negative representative of the elliptic ideal. Therefore the normal
Hessian of $f$, $\operatorname{Hess}^{\nu}f\in\Gamma(\text{Sym}^{2}N^{*}D)$ is
positive definite, and thus defines a metric.

######  Definition 1.8.

Let $\left|D\right|=(R,q)$ be an elliptic divisor. The vector fields
preserving $I_{\left|D\right|}$ define a Lie algebroid
$\mathcal{A}_{\left|D\right|}\to TM$, called the elliptic tangent bundle.

The existence of this Lie algebroid is established through the Serre–Swan
theorem: if $x\in D[i]$, then in the coordinates of Lemma 1.6 vector fields
preserving $I_{\left|D\right|}$ are given by

$\Gamma(\mathcal{A}_{\left|D\right|})=\left\langle
r_{1}\partial_{r_{1}},\partial_{\theta_{1}},\ldots,r_{i}\partial_{r_{i}},\partial_{\theta_{i}},\partial_{w_{2i+1}},\ldots,\partial_{w_{2i+l}}\right\rangle,$

where $r_{j}\partial_{r_{j}}:=x_{j}\partial_{x_{j}}+y_{j}\partial_{y_{j}}$ and
$\partial_{\theta_{j}}:=x_{j}\partial_{y_{j}}-y_{j}\partial_{x_{j}}$.
Consequently, vector fields preserving $I_{\left|D\right|}$ define a locally
free sheaf and thus induce a Lie algebroid.

Lie algebroid forms for the elliptic tangent bundle are locally given by

$\Gamma(\mathcal{A}_{\left|D\right|}^{*})=\left\langle d\log
r_{1},d\theta_{1},\ldots,d\log
r_{i},d\theta_{i},dw_{2i+1},\ldots,dw_{2i+l}\right\rangle,$

with $d\log r_{j}=(x_{j}^{2}+y_{j}^{2})^{-1}(x_{j}dx_{j}+y_{j}dy_{j})$ and
$d\theta_{j}=(x_{j}^{2}+y_{j}^{2})^{-1}(x_{j}dy_{j}-y_{j}dx_{j})$.

######  Definition 1.9.

Let $\left|D\right|=(R,q)$ be an elliptic divisor. A (self-crossing) elliptic
symplectic structure is a Lie algebroid symplectic form for the elliptic
tangent bundle: That is, a two-form
$\omega\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ which is non-degenerate
($\omega^{\flat}:(\mathcal{A}_{\left|D\right|})_{x}\rightarrow(\mathcal{A}_{\left|D\right|})_{x}^{*}$
is an isomorphism) and closed ($d_{\mathcal{A}_{\left|D\right|}}\omega=0$).

In [2] we study these structures and show that a subset corresponds to a class
of generalized complex structures, called self-crossing stable.

##  2\. Geometric structure on the normal bundle

In this section we study the geometric structure present on the normal bundles
of the strata of the vanishing locus of an elliptic divisor. In the next
section, this geometric structure will be used to describe the Lie algebroid
cohomology of the elliptic tangent bundle.

Remark 1 explains how a smooth elliptic divisor $(R,q)$ induces a metric on
the normal bundle to $D$. The metric depends on the choice of particular
trivialisation of $R$, but the conformal class does not. Phrased in another
way, $ND$ inherits a canonical $O(2)\times\mathbb{R}_{+}$ structure group
reduction. For self-crossing elliptic divisors we have the following result:

######  Proposition 2.1.

Let $I_{\left|D\right|}$ be an elliptic divisor with intersection number $n$.
Then $ND[n]$ admits a canonical structure group reduction to
$(O(2)\times\mathbb{R}_{+})^{n}\rtimes S_{n}$.

###### Proof.

For a point $x\in D[n]$, let
$I_{\left|D_{1}\right|},\ldots,I_{\left|D_{n}\right|}$ be any choice of local
normal crossing divisors and let the $f_{i}$ be any representatives of the
$I_{\left|D_{i}\right|}$, which we may choose to be all non-negative. Because
the $D_{i}$ all intersect transversely, we have

$N_{x}(D[n])\simeq\bigoplus_{i=1}^{n}{N_{x}(D_{i})}|_{D[n]}.$

If $u_{i}=(u_{i_{1}},u_{i_{2}})$ forms a local frame for ${ND_{i}}|_{D[n]}$,
then $(u_{1_{1}},u_{1_{2}},\ldots,u_{n_{1}},u_{n_{2}})$ forms a local frame
for $N(D[n])$. We consider all such local frames for which $u_{i}$ is
orthonormal with respect to $\operatorname{Hess}^{\nu}(f_{i})$. There are two
important points to remark. First, the choice of representatives $f_{i}$ is
not canonical. Secondly, the ordering of the local divisors is not well-
defined, and instead there is an $S_{n}$-symmetry present which permutes the
divisors. Therefore we consider the frames with respect to all these choices
of representatives $f_{i}$ and all orderings for the local divisors. This
provides a reduction of the structure group to
$(O(2)\times\mathbb{R}_{+})^{n}\rtimes S_{n}$. ∎

######  Corollary 2.2.

Let $I_{\left|D\right|}$ be an elliptic divisor with intersection number $n$,
then

  * •

the structure group of $ND[n]$ reduces to $O(2)^{n}\ltimes S_{n}$.

  * •

If $I_{\left|D\right|}$ is a global normal crossing divisor, then a choice of
non-negative representatives $f_{1},\ldots,f_{n}$ of the smooth elliptic
divisor ideals induces a further structure group reduction to $O(2)^{n}$.

  * •

If $D[1]$ is furthermore co-orientable, then a further choice of co-
orientation of $D[1]$ induces a structure group reduction to $T^{n}$.

###### Proof.

  * •

The quotient of $(O(2)\times\mathbb{R}_{+})^{n}\rtimes S_{n}$ by
$O(2)^{n}\ltimes S_{n}$ is $(\mathbb{R}_{+})^{n}$, and thus contractible.
Therefore the required structure group reduction exists.

  * •

Let $I_{\left|D_{1}\right|},\ldots,I_{\left|D_{n}\right|}$ be global smooth
elliptic divisors for $I_{\left|D\right|}$. We consider frames
$(u_{1},\ldots,u_{n})$ of $ND[n]$, where $u_{i}$ is a local frame of
${ND_{i}}|_{D[n]}$ orthonormal with respect to
$\operatorname{Hess}^{\nu}(f_{i})$. This will give the required
$O(2)^{n}$-reduction.

  * •

The co-orientation on $D[1]$ induces a co-orientation on each $D_{i}$ away
from a codimension-two submanifold, and therefore a co-orientation on the
entirety of $D_{i}$. Proceeding as in the previous point with oriented frames
provides a structure group reduction to $SO(2)^{n}$.

∎

######  Remark 2.0.

By Lemma 1.5 the restriction of an elliptic divisor $I_{\left|D\right|}$ to
$M\backslash D(i+1)$ is an elliptic divisor of intersection number $i$. The
highest stratum of this divisor is precisely $D[i]$, and consequently we can
apply the results in this section to obtain structure group reduction of
$ND[i]$.

##  3\. Residue maps

In this section we describe residue maps for elliptic divisors. These are maps
which pick out the coefficient of a singular generator of a Lie algebroid
form, much akin to the residue of a meromorphic differential form. In [9] a
general theory of residue maps of Lie algebroids is described. We will first
recall the general definition and then specialise to the case of elliptic
divisors.

Let $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be a short exact
sequence of vector spaces with $l=\dim A$. A splitting of this sequence
induces an isomorphism $B\simeq A\oplus C$. Consequently $\wedge^{k}B^{*}$
decomposes as a direct sum
$\oplus_{i}(\wedge^{k-i}C^{*}\otimes\wedge^{k}A^{*})$. The projections from
$\wedge^{k}W^{*}$ to each of these factors depend on the particular splitting,
however, the projection to $\wedge^{k-l}C^{*}\otimes\wedge^{l}A^{*}$ does not.
We define the residue map to be this projection:

$\operatorname{Res}:\wedge^{k}B^{*}\rightarrow\wedge^{k-l}C^{*}\otimes\wedge^{l}A^{*}.$

Let
$0\rightarrow\mathcal{A}\rightarrow\mathcal{B}\rightarrow\mathcal{C}\rightarrow
0$ be a short exact sequence of Lie algebroids with
$\text{rk}(\mathcal{A})=l$. Applying the residue map fibre-wise gives a map
$\operatorname{Res}:\Omega^{k}(\mathcal{B})\rightarrow\Omega^{k-l}(\mathcal{C};\det(\mathcal{A}^{*}))$.
To endow $\Omega^{\bullet}(\mathcal{C};\det(\mathcal{A}^{*}))$ with a
differential one needs a flat $\mathcal{C}$-connection on
$\det(\mathcal{A}^{*})$. We now describe a situation in which such a
connection exists and the residue map is a cochain morphism with respect to
this differential.

######  Proposition 3.1.

Let $\mathcal{B}\rightarrow M$ be a Lie algebroid and let
$i:\mathcal{A}\hookrightarrow\mathcal{B}$ be an abelian ideal subalgebroid of
rank $l$. Then there is a canonical flat
$(\mathcal{B}/\mathcal{A})$-connection on $\det(\mathcal{A}^{*})$.

Furthermore assume that for every point $x\in M$, there exists a neighbourhood
$U$ of $p$ and closed sections
$\beta_{1},\ldots,\beta_{l}\in\Omega^{1}(\mathcal{B}|_{U})$ such that
$i^{*}\beta_{1},\ldots,i^{*}\beta_{l}$ generate
$\Omega^{1}(\mathcal{A}|_{U})$. Then
$\operatorname{Res}:\Omega^{k}(\mathcal{B})\rightarrow\Omega^{k-l}(\mathcal{B}/\mathcal{A};\det(\mathcal{A}^{*}))$
is a cochain morphism.

###### Proof.

Let $\sigma:\mathcal{B}/\mathcal{A}\rightarrow\mathcal{B}$ be any splitting
and define:

$\nabla_{c}(a)=[\sigma(c),a],\quad
c\in\Gamma(\mathcal{B}/\mathcal{A}),a\in\Gamma(\mathcal{A}).$

It is straightforward to verify that this is connection is flat and does not
depend on the choice of splitting. If we let
$\beta_{1},\ldots,\beta_{l}\in\Omega^{1}(\mathcal{B})$ be local closed forms
with the property that their restriction to $\mathcal{A}$ defines a local
frame for $\Omega^{1}(\mathcal{A})$, then

$\operatorname{Res}(\beta_{1}\wedge\ldots\wedge\beta_{l}\wedge\gamma)=\gamma\otimes(\beta_{1}\wedge\ldots\wedge\beta_{l}),$

and one easily verifies that $\operatorname{Res}$ is a cochain morphism. ∎

If $I_{\left|D\right|}$ is an elliptic ideal with intersection number $n$ then
the restriction of the anchor of the elliptic tangent bundle to $D[n]$ has
image inside $TD[n]$, and hence ${\mathcal{A}_{\left|D\right|}}|_{D[n]}$ is
again a Lie algebroid. We will describe this Lie algebroid using the language
of Atiyah algebroids, which we recall in Appendix A.

######  Lemma 3.2.

Let $I_{\left|D\right|}$ be an elliptic divisor with intersection number $n$
and let $P$ denote the $(O(2)\times\mathbb{R}_{+})^{n}\rtimes S_{n}$-structure
group reduction of $ND[n]$ from Proposition 2.1. Then
${\mathcal{A}_{\left|D\right|}}|_{D[n]}\simeq\operatorname{At}(P)$.

###### Proof.

Let $I_{1},\ldots,I_{n}$ be local normal crossing elliptic divisors and let
$f_{1},\ldots,f_{n}$ by representatives of these ideals. Let $Q_{f_{i}}\in
C^{\infty}(ND_{i})$ denote the quadratic approximations of these functions,
and define the ideal $I_{\left|D\right|}^{ND[n]}=\left\langle Q_{f_{1}}\cdots
Q_{f_{n}}\right\rangle$. Although the representatives $f_{i}$ are not unique
nor global this ideal is and thus defines an elliptic divisor on $ND[n]$. Let
$\mathcal{A}_{\left|D\right|}^{ND[n]}$ denote the corresponding elliptic
tangent bundle. And let
$\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})_{\text{lin}}$ denote the
sections which are send to linear vector fields by the anchor. Remark that
$\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})_{\text{lin}}$ is only a
$C^{\infty}(D[n])$-module, not a $C^{\infty}(ND[n])$-module.

Given $X\in\Gamma({\mathcal{A}_{\left|D\right|}}|_{D[n]})$ one can show that
there exists a unique linear vector field
$\widetilde{X}\in\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})_{\text{lin}}$
with ${\widetilde{X}}|_{D[n]}=X$. Existence of such a vector field is obtained
by linearising any extension
$X^{\prime}\in\Gamma({\mathcal{A}_{\left|D\right|}}|_{\mathcal{U}})$ of $X$
and observing that because $X^{\prime}$ preserves the ideal
$I_{\left|D\right|}$ its linearisation $\widetilde{X}$ preserves the ideal
$I_{\left|D\right|}^{ND[n]}$. We thus obtain a map

$\varphi\colon\Gamma({\mathcal{A}_{\left|D\right|}}|_{D[n]})\rightarrow\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})_{\text{lin}},\qquad
X\mapsto\widetilde{X}.$

This map is a bijection and bracket preserving. To finish the argument we are
left to show that sections of the right-hand side coincide with sections of
$\operatorname{At}(P)$. Let $F$ be a representative of
$I_{\left|D\right|}^{ND[n]}$, then

$\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})_{\text{lin}}=\\{X\in\mathfrak{X}^{1}(ND[n])_{\text{lin}}:\mathcal{L}_{X}(F)=\lambda
F\text{ for some }\lambda\in C^{\infty}(D[n])\\}.$

Note that a priori for an element of
$\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})$ we only have
$\mathcal{L}_{X}(F)=\lambda F$ with $\lambda\in C^{\infty}(ND[n])$, however
because $X$ is linear we must have $\lambda\in C^{\infty}(D[n])$.

As $F$ is locally of the form $Q_{f_{1}}\cdots Q_{f_{n}}$ and all these
functions are functionally indivisible, we must have that for all $i$ there
exists precisely one $j$ such that
$\mathcal{L}_{X}(Q_{f_{i}})=\lambda^{i}_{j}Q_{f_{j}}$ for some function
$\lambda^{i}_{j}\in C^{\infty}(D[n])$.

On the other side, using Lemma A.4 one can show that

$\operatorname{At}(P)=\\{X\in\mathfrak{X}(ND[n])_{\text{lin}}:\forall~{}i~{}\exists!~{}j~{}\text{s.t.
}\mathcal{L}_{X}(\operatorname{Hess}^{\nu}(f_{i}))=\lambda^{i}_{j}\operatorname{Hess}^{\nu}(f_{j}),\text{
for some }\lambda^{i}_{j}\in C^{\infty}(D[n])\\}.$

Using the fact that for a linear vector field $X$ and definite Morse–Bott
functions $f,g$ we have $\mathcal{L}_{X}(\operatorname{Hess}^{\nu}f)=g$ if and
only if $\mathcal{L}_{X}(Q_{f})=Q_{g}$ we see that indeed
$\Gamma(\mathcal{A}_{\left|D\right|}^{ND[n]})_{\text{lin}}=\Gamma(\operatorname{At}(P))$,
which finishes the proof. ∎

###  3.1. Residues for smooth elliptic divisors

In this section we describe the radial residue map for a smooth elliptic
divisor. The radial residue map was already used in [1] to compute the Lie
algebroid cohomology of the elliptic tangent bundle of a smooth elliptic
divisor. We recall their construction and add some details which are
especially important in the general case.

######  Lemma 3.3.

Let $I_{\left|D\right|}$ be a smooth elliptic divisor on a manifold $M$, and
let $\rho:\mathcal{A}_{\left|D\right|}\rightarrow TM$ be the corresponding
elliptic tangent bundle. There exists a unique nowhere vanishing section
$\mathcal{E}_{D}$ in ${\ker\rho}|_{D}$ with the property that every extension
is an Euler like vector field222That is, a vector field which vanishes along
$D$ and has linearisation the Euler vector field..

###### Proof.

Let $\mathcal{U}$ be any tubular neighbourhood of $D$, and let $\mathcal{E}$
be the corresponding Euler vector field. Let $f\in I_{\left|D\right|}$ be a
local representative of the ideal, and pick local Morse–Bott coordinates in
which $f=r^{2}$. In these coordinates $\mathcal{E}$ is given by
$r\partial_{r}$. Therefore, $\mathcal{E}$ defines a section of the elliptic
tangent bundle and its restriction to $D$ defines a section $\mathcal{E}_{D}$
of ${\ker\rho}|_{D}$. It is readily verified that $\mathcal{E}_{D}$ is the
unique vector field with the given property. ∎

Because $L_{r}=\mathbb{R}\cdot\mathcal{E}_{D}$ is an ideal of
${\mathcal{A}_{\left|D\right|}}|_{D}$ we can consider the quotient Lie
algebroid, which can be described as follows:

######  Lemma 3.4.

Let $I_{\left|D\right|}$ be a smooth elliptic divisor with non-negative
generator $f\in I_{\left|D\right|}$. Let $Q=O(2)ND$ denote the induced
structure group reduction from Corollary 2.2. Then

$\displaystyle{\mathcal{A}_{\left|D\right|}}|_{D}/L_{r}$
$\displaystyle\rightarrow\operatorname{At}(Q)$ $\displaystyle[X]$
$\displaystyle\mapsto X-\frac{\mathcal{L}_{X}(f)}{f}\mathcal{E}_{D}$

is an isomorphism of Lie algebroids.

Because $L_{r}$ is an abelian ideal of ${\mathcal{A}_{\left|D\right|}}|_{D}$
we may apply Proposition 3.1 to obtain a residue map
$\Omega^{\bullet}({\mathcal{A}_{\left|D\right|}}|_{D})\rightarrow\Omega^{\bullet-1}(\operatorname{At}(O(2)ND);L_{r}^{*})$.
By Lemma 1.6 $\Omega^{1}(\mathcal{A}_{\left|D\right|})$ is locally generated
by closed forms and therefore Proposition 3.1 implies that the residue map is
a cochain morphism. After pre-composing with the restriction to $D$ to obtain
the the radial residue map:

$\operatorname{Res}_{r}:\Omega^{\bullet}(\mathcal{A}_{\left|D\right|})\rightarrow\Omega^{\bullet-1}(\operatorname{At}(O(2)ND);L_{r}^{*})=\Omega^{\bullet-1}(\operatorname{At}(O(2)ND)),$

where we used that $L_{r}^{*}$ is trivial as an
$\operatorname{At}(O(2)ND)$-representation. Note that this residue map depends
on the choice of representative $f\in I_{\left|D\right|}$, but any two
different choices result into isomorphic Atiyah algebroids. In the local
coordinates $(x_{1},y_{1},w_{3},\ldots,w_{n})$ of Lemma 1.6 the radial residue
map is given by:

$\operatorname{Res}_{r}(\alpha)=(\iota_{r_{1}\partial_{r_{1}}}\alpha)|_{D},\qquad\alpha\in\Omega^{\bullet}(\mathcal{A}_{|D|}).$

###  3.2. Residues for self-crossing elliptic divisors

We will now describe residue maps for the self-crossing elliptic tangent
bundle. As before, the Euler vector fields generate an abelian ideal:

######  Lemma 3.5.

Let $I_{\left|D\right|}$ be an elliptic divisor with intersection number $n$.
Then there exists a canonical abelian ideal subalgebroid $L_{r}$ of
${\mathcal{A}_{\left|D\right|}}|_{D[n]}$. In local coordinates of Lemma 1.6 we
have

$\Gamma(L_{r})\simeq\left\langle{r_{1}\partial_{r_{1}}}|_{D[n]},\ldots,{r_{n}\partial_{r_{n}}}|_{D[n]}\right\rangle.$

###### Proof.

Let $x\in D[n]$, and let
$I_{\left|D_{1}\right|},\ldots,I_{\left|D_{n}\right|}$ be local normal
crossing divisor on a neighbourhood $U$ of $x$ and let
$\rho_{i}:\mathcal{A}_{\left|D_{i}\right|}\rightarrow TU$ be the corresponding
elliptic tangent bundles. By transverality of the loci $D_{i}$ we obtain the
following splitting:

${\ker\rho}|_{D[n]\cap U}\simeq{\ker\rho_{1}}|_{D[n]\cap
U}\oplus\cdots\oplus{\ker\rho_{n}}|_{D[n]\cap U}.$

By Lemma 3.3 there exist canonical vector subbundles $L_{r_{i}}$ of
${\ker\rho_{i}}|_{D(n)}$ with generators $\mathcal{E}_{D_{i}}$. Define
${L_{r}}|_{U}$ as the direct sum of these vector subbundles. Although each of
the summands $L_{r_{i}}$ is not globally defined and neither is their order,
$L_{r}$ does define a global vector bundle. Finally, in the local coordinates
of Lemma 1.6 we have $\mathcal{E}_{D_{i}}=r_{i}\partial_{r_{i}}$ as desired. ∎

Because $L_{r}$ is an ideal we can consider the quotient Lie algebroid which
can be described as follows:

######  Lemma 3.6.

There is an $O(2)^{n}\rtimes S_{n}$-principal bundle $Q$ such that
$\operatorname{At}(Q)={\mathcal{A}_{\left|D\right|}}|_{D[n]}/L_{r}$.

###### Proof.

Let $P=(O(2)\times\mathbb{R}_{+})^{n}\rtimes S_{n}$ be structure group
reduction from Proposition 2.1. As locally the sections of $L_{r}$ are
generated by the Euler vector fields of the normal bundles to the local loci
$D_{i}$ we have that $L_{r}$ generates an $(\mathbb{R}_{+})^{n}$-action on
$P$. If $x\in M$ and $(u_{1},\ldots,u_{n})\in P_{x}$ is a frame with $u_{i}$ a
frame of $N_{x}D_{i}$, then this action is given by
$(\lambda_{1},\ldots,\lambda_{n})\cdot(u_{1},\ldots,u_{n})=(\lambda_{1}u_{1},\ldots,\lambda_{n}u_{n})$.
We can consider the quotient bundle $Q:=P/(\mathbb{R}_{+})^{n}$, which is an
$O(2)^{n}\rtimes S_{n}$-principal bundle. Consequently
${\mathcal{A}_{\left|D\right|}}|_{D[n]}/L_{r}=\operatorname{At}(Q)$ which
finishes the proof. ∎

As in the smooth case we apply Proposition 3.1 to obtain a residue map, which
we compose with the restriction to $D[n]$ to obtain the total radial residue
map:

$\operatorname{Res}_{r}^{n}:\Omega^{\bullet}(\mathcal{A}_{\left|D\right|})\rightarrow\Omega^{\bullet-n}(\operatorname{At}(Q);\det(L_{r}^{*})),$

In the local coordinates of Lemma 1.6 we have that the total radial residue
map is given by

$\operatorname{Res}_{r}^{n}(\alpha)=(\iota_{r_{1}\partial_{r_{1}}\wedge\cdots\wedge
r_{n}\partial_{r_{n}}}\alpha)|_{D[n]}.$

When the elliptic divisor is a global normal crossing divisor the situation
simplifies significantly:

######  Lemma 3.7.

Let $I_{\left|D\right|}=I_{\left|D_{1}\right|}\cdot\ldots\cdot
I_{\left|D_{n}\right|}$ be a global normal crossing elliptic divisor. Let
$Q=O(2)^{n}ND[n]$ denote the structure group reduction from Corollary 2.2.
Then ${\mathcal{A}_{\left|D\right|}}|_{D[n]}/L_{r}\simeq\operatorname{At}(Q)$.

###### Proof.

Let $f_{1},\ldots,f_{n}$ be non-negative representatives of the smooth
elliptic ideals. Let $L_{r}=L_{r_{1}}\oplus\cdots\oplus L_{r_{n}}$ be the
ideal subalgebroid from Lemma 3.5 and let
$\mathcal{E}_{D_{1}},\ldots,\mathcal{E}_{D_{n}}$ denote the preferred
generators. Let $f_{1},\ldots,f_{n}$ be non-negative representatives of
$I_{\left|D_{1}\right|},\ldots,I_{\left|D_{n}\right|}$. Then

$\displaystyle{\mathcal{A}_{\left|D\right|}}|_{D[n]}/L_{r}$
$\displaystyle\rightarrow\operatorname{At}(Q),$ $\displaystyle[X]$
$\displaystyle\mapsto
X-\frac{\mathcal{L}_{X}(f_{1})}{f_{1}}\mathcal{E}_{D_{1}}-\ldots-\frac{\mathcal{L}_{X}(f_{n})}{f_{n}}\mathcal{E}_{D_{n}},$

is the desired isomorphism. ∎

In this case $\det(L_{r}^{*})$ is trivial as an
$\operatorname{At}(O(2)^{n})$-representation and we conclude that the total
radial residue map becomes a map:

$\operatorname{Res}_{r}^{n}:\Omega^{\bullet}(\mathcal{A}_{\left|D\right|})\rightarrow\Omega^{\bullet-n}(\operatorname{At}(O(2)^{n}ND[n])).$

##  4\. Cohomology of the elliptic tangent bundle

Using the total radial residue map introduced in the previous section, we will
compute the Lie algebroid cohomology of the elliptic tangent bundle.

Let $I_{\left|D\right|}$ be an elliptic divisor of intersection number $n$, in
this section we will write $\mathcal{A}_{\left|D\right|}^{n}$ to keep track of
this number and to simpify notion we write:

$\Omega^{\bullet}_{\text{res}}(\mathcal{A}_{\left|D\right|}^{n}):=\Omega^{\bullet}(\operatorname{At}(Q));\det(L_{r}^{*})),$

and denote its cohomology by
$H^{\bullet}_{\text{res}}(\mathcal{A}_{\left|D\right|}^{n})$.

Recall that by Lemma 1.5 if $i\leq n$, then the restriction of
$I_{\left|D\right|}$ to $M\backslash D(i+1)$ defines an elliptic divisor of
intersection number $i$, which we denote by $I_{\left|D\backslash
D(i+1)\right|}$. Therefore the inclusion $\iota_{i}:M\backslash
D(i+1)\hookrightarrow M$ induces a cochain map:

$\iota^{*}_{i}:\Omega^{\bullet}(\mathcal{A}_{\left|D\right|}^{n})\rightarrow\Omega^{\bullet}(\mathcal{A}^{i}_{\left|D\backslash
D(i)\right|}).$

Consequently, we can consider the radial residue map for the divisor
$I_{\left|D\backslash D(i+1)\right|}$ and the composition:

$\text{Res}_{r}^{i}\hskip 0.28453pt\circ\hskip
0.28453pt\iota^{*}_{i}:\Omega^{\bullet}(\mathcal{A}_{\left|D\right|}^{n})\rightarrow\Omega^{\bullet-i}_{\text{res}}(\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|}),$

for each $1\leq i\leq n-1$.

######  Theorem 4.1.

Let $I_{\left|D\right|}$ be an elliptic divisor with intersection number $n$.
Then

(4.1) $\displaystyle H^{k}(\mathcal{A}_{\left|D\right|}^{n})$
$\displaystyle\rightarrow H^{k}(M\backslash
D)\oplus\bigoplus_{i=1}^{n}H^{k-i}_{\text{\emph{res}}}(\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|})$ $\displaystyle[\alpha]$
$\displaystyle\mapsto(\iota_{0}^{*}[\alpha],\operatorname{Res}_{r}^{1}\hskip
0.28453pt\circ\hskip
0.28453pt\iota_{1}^{*}[\alpha],\ldots,\operatorname{Res}_{r}^{n-1}\hskip
0.28453pt\circ\hskip
0.28453pt\iota_{n-1}^{*}[\alpha],\operatorname{Res}_{r}^{n}[\alpha])$

is an isomorphism.

###### Proof.

The argument uses the observation from [4], that it suffices to proof that the
above map induces an isomorphism in sheaf cohomology. Therefore, let $x\in
D[l]$ and let $U$ be a neighbourhood of $x$ as in Lemma 1.6, that is $U\cap
D[l+1]=\emptyset$, $U=\mathbb{R}^{2l}\times\mathbb{R}^{m}$ and
$I_{\left|D\right|}$ is the standard elliptic divisor of intersection number
$l$; $I_{\left|D_{1}\right|}\cdot\ldots\cdot I_{\left|D_{l}\right|}$.

Below we will implicitly push-forward sheaves using the inclusion maps
$\iota_{j}$ whenever required.

The left hand side of equation (4.1) can be easily described as the following
free algebra:

$H^{\bullet}(U,\mathcal{A}_{\left|D\right|}^{n})=\left\langle 1,d\log
r_{1},\ldots,d\log r_{l},d\theta_{1},\ldots,d\theta_{l}\right\rangle.$

Now let $i\leq l$. By Lemma 3.7
$\Omega_{\operatorname{res}}^{\bullet}(\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|})=\Omega^{\bullet}(\operatorname{At}(O(2)^{i}ND[i]))$. Because
$D[i]$ is orientable, Lemma A.5 furthermore implies
$\Omega_{\operatorname{res}}^{\bullet}(\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|})=\Omega^{\bullet}(\operatorname{At}(T^{i}ND[i]))$. And using
invariant invariant cohomology
$H^{\bullet}(\operatorname{At}(T^{i}ND[i]))=H^{\bullet}_{\text{dR}}(T^{i}ND[i])$.
We will thus view $\operatorname{Res}_{r}^{i}\hskip 0.28453pt\circ\hskip
0.28453pt\iota_{i}^{*}$ as a map into the latter cohomology.

First, let us consider the map $\iota_{0}^{*}$. By an elementary argument
$U\backslash D$ is homotopic to $T^{l}$, and $\iota_{0}^{*}$ sends the forms
$\alpha$, with $\alpha\in\left\langle
1,d\theta_{1},\ldots,d\theta_{l}\right\rangle$ precisely to the generators of
$H^{\bullet}(T^{l})$, proving that $\iota_{0}^{*}$ is surjective. For $i>l$,
$\iota^{*}_{i}:H^{\bullet}(U,\mathcal{A}_{\left|D\right|}^{n})\rightarrow
H^{\bullet}_{\text{res}}(U,\mathcal{A}^{i}_{\left|D\backslash D(i+1)\right|})$
is the zero map as $\mathcal{A}^{i}_{\left|D\backslash D(i+1)\right|}=TU$ on
$U$ and $U$ is contractible.

We will prove that all maps $\iota_{i}^{*}\hskip 0.28453pt\circ\hskip
0.28453pt\operatorname{Res}_{r}^{i}$ with $1\leq i\leq l$ are surjective and
have disjoint kernels.

First we need to establish the homotopy type of $D[i]$; remark that

$D[i]=\overset{\cdot}{\bigcup_{(j_{1},\ldots,j_{i})}}(D_{j_{1}}\cap\cdots\cap
D_{j_{i}})\backslash D(i+1),$

where the sum runs over all multi-indices of length $i$. By an elementary
argument, $(D_{j_{1}}\cap\cdots\cap D_{j_{i}})\backslash D(i+1)$ is homotopic
to $T^{l-i}$. Moreover, $ND[i]$ decomposes as a direct sum, and thus so does
the associated $T^{i}$-bundle:

$T^{i}ND[i]=\bigoplus_{(j_{1},\ldots,j_{i})}S^{1}ND_{j_{1}}|_{D[i]}\oplus\cdots\oplus
S^{1}ND_{j_{i}}|_{D[i]}$

Because each $D_{j_{k}}$ is contractible, the above is a sum of trivial circle
bundles, and thus $T^{i}ND[i]=T^{i}\times D[i]$. Combining this with the
description of the homotopy type of $D[i]$ we conclude that:

(4.2) $H^{\bullet}_{\text{res}}(U,\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|})\simeq\bigoplus_{(j_{1},\ldots,j_{i})}H^{\bullet}(T^{l}).$

Given $\alpha\in\Omega^{k}(\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|})$, then $\operatorname{Res}^{i}_{r}(\alpha)$ is only non-zero
if $\alpha$ contains a term of the form $d\log r_{j_{1}}\wedge\cdots\wedge
d\log r_{j_{i}}\wedge\beta$, with $\beta\in\langle 1,d\theta_{1},\ldots
d\theta_{l}\rangle$. Moreover, $\operatorname{Res}_{r}^{i}$ takes forms of
this form precisely to the generators of the $H^{\bullet}(T^{n})$ in (4.2)
corresponding to the multi-index $(j_{1},\ldots,j_{i})$. This shows that
$\iota_{i}^{*}\hskip 0.28453pt\circ\hskip 0.28453pt\operatorname{Res}_{r}^{i}$
is surjective, and that the kernels of these maps for different values of $i$
are all disjoint. We conclude that $(\iota_{0}^{*},\iota_{1}^{*}\hskip
0.28453pt\circ\hskip
0.28453pt\operatorname{Res}_{r}^{1},\ldots,\iota_{n-1}^{*}\hskip
0.28453pt\circ\hskip
0.28453pt\operatorname{Res}_{r}^{n-1},\operatorname{Res}_{r}^{n})$ is an
isomorphism. ∎

When the elliptic divisor is a co-orientable normal crossing divisor the
cohomology becomes easier to describe:

######  Corollary 4.2.

If $I_{\left|D\right|}$ is a global normal crossing elliptic divisor for which
$D[1]$ is co-orientable. Then

$H^{k}(\mathcal{A}_{\left|D\right|})\simeq H^{k}(M\backslash
D)\oplus\bigoplus_{i=1}^{n}H^{k-i}(T^{i}N(D[i])).$

###### Proof.

By Corollary 2.2 $ND[i]$ admits a $T^{i}$-structure group reduction, and using
Lemma A.5 we see that
$\Omega_{\operatorname{Res}}^{\bullet}(\mathcal{A}^{i}_{\left|D\backslash
D(i+1)\right|})=\Omega^{\bullet}(\operatorname{At}(T^{i}ND[i]))$. Therefore by
Theorem 4.1 we obtain that the cohomology is isomorphic to

$\displaystyle H^{k}(\mathcal{A}_{\left|D\right|})\simeq H^{k}(M\backslash
D)\oplus\bigoplus_{i=1}^{n}H^{k-i}(\operatorname{At}(T^{i}N(D[i])))$

Using invariant cohomology we have
$H^{\bullet}(\operatorname{At}(T^{i}ND[i]))=H^{\bullet}_{\text{dR}}(T^{i}ND[i])$,
and therefore we arrive at the required result. ∎

In applications we are often interested in the case when the manifold is four-
dimensional, and hence $D[2]$ consists of a collection of points. In this case
the cohomology also simplifies for general divisors:

######  Corollary 4.3.

Let $I_{\left|D\right|}$ be an elliptic divisor on $M^{4}$ for which $D[1]$ is
co-orientable. Then

$H^{k}(\mathcal{A}_{\left|D\right|})\simeq H^{k}(M\backslash D)\oplus
H^{k-1}(S^{1}ND[1])\oplus H^{k-2}(T^{2})^{(\\#D[2])}.$

##  5\. Examples and outlook

In this section we give some explicit examples of elliptic divisors and
compute their associated cohomology. One class of examples arises through
complex divisors:

######  Definition 5.1.

A smooth complex log divisor $(L,\sigma)$ consists of a complex line bundle
$L$ together with a transversely vanishing section $\sigma$.

######  Definition 5.2.

A (self-crossing) complex log divisor $(L,\sigma)$ is a complex divisor on $M$
with the property that for every point $p\in M$ there exists a neighbourhood
$U$ of $p$ such that

$\displaystyle I_{\sigma}(U)=I_{1}\cdots\ldots\cdots I_{j},$

where the $I_{1},\ldots,I_{j}$ correspond to complex log divisors with
transversely intersecting vanishing loci.

######  Remark 5.0.

Let $(L,\sigma)$ be a complex log divisor on $M$. Then then
$(L\otimes\overline{L},\sigma\otimes\overline{\sigma})$ is the
complexification of a real line bundle $(R,q)$. For every point $p\in D$ there
exists local coordinates $z_{1},\ldots,z_{n}$ such that $\sigma$ corresponds
to the function $z_{1}\cdot\ldots\cdot z_{n}$. Consequently, $q$ corresponds
to the function $\left|z_{1}\right|^{2}\cdot\ldots\cdot\left|z_{n}\right|^{2}$
and thus $(R,q)$ defines an elliptic divisor. We call this the associated
elliptic divisor.

Just as for the elliptic tangent bundle, we can define a Lie algebroid
associated to a complex divisor:

######  Definition 5.3.

Let $(L,\sigma)$ be a complex log divisor. The complex vector fields presering
$I_{\sigma}$ define a complex Lie algebroid $\mathcal{A}_{D}\rightarrow
T_{\mathbb{C}}M$, called the complex log tangent bundle.

In [2] we use this Lie algebroid to study certain generalized complex
structures. There, we also compute the Lie algebroid cohomology:

######  Theorem 5.4.

Let $(L,\sigma)$ be a complex log divisor on a manifold $M$. Then the
inclusion $\iota:M\backslash D\hookrightarrow M$ induces an isomorphism:

$\displaystyle H^{\bullet}(\mathcal{A}_{D})\simeq H^{\bullet}(M\backslash
D;\mathbb{C}).$

The following example plays an important role in constructing examples of
stable generalized complex structures in [2]:

######  Example 5.5.

Let $M=\mathbb{C}P^{2}$ and let $D$ be a normal crossings divisor consisting
of three lines intersecting transversely, for instance
$D=\\{z_{0}z_{1}z_{2}=0\\}$. The associated holomorphic line bundle with
section $(L,\sigma)$ is a complex log divisor, and let $(R,q)$ be the
associated elliptic divisor. We will now compute the cohomology of the
associated elliptic tangent bundle:

  * •

The complement $M\backslash D=(\mathbb{C}^{*})^{2}$ and thus homotopic to
$T^{2}$.

  * •

The stratum $D[1]$ consists of three disjoint cylinders, and therefore every
complex line bundle on it is trivial. Consequently $S^{1}ND[1]$ is homotopic
to three disjoint tori.

  * •

The stratum $D[2]$ consists of three points.

Corollary 4.3 now gives us the following description of the elliptic
cohomology:

$k$ | 0 | 1 | 2 | 3 | 4 | otherwise
---|---|---|---|---|---|---
$H^{k}(\mathcal{A}_{\left|D\right|})$ | $\mathbb{R}$ | $\mathbb{R}^{5}$ | $\mathbb{R}^{10}$ | $\mathbb{R}^{9}$ | $\mathbb{R}^{3}$ | 0

The comohology of the complex log tangent bundle is much simpler and given by:

$k$ | 0 | 1 | 2 | otherwise
---|---|---|---|---
$H^{k}(\mathcal{A}_{D})$ | $\mathbb{C}$ | $\mathbb{C}^{2}$ | $\mathbb{C}$ | 0

The following example plays an important role in the construction of
codimension-one symplectic foliations in work in progress of Cavalcanti and
Crainic:

######  Example 5.6.

Let $M=\mathbb{C}P^{2}$ and let $D$ be the smooth complex divisor given by the
zero-set of a smooth cubic, for instance
$D=\\{z_{0}^{3}+z_{1}^{3}+z_{2}^{3}=0\\}$. By the genus-degree formula we know
that $D$ is isomorphic to a torus. The cohomology of the complement can be
computed using the long exact sequence associated to the inclusion
$D\hookrightarrow M$:

$\displaystyle
0\rightarrow\mathbb{R}\overset{\text{id}}{\rightarrow}\mathbb{R}\rightarrow
H^{0}(\mathbb{C}P^{2},D)\rightarrow 0\rightarrow\mathbb{R}^{2}\rightarrow
H^{1}(\mathbb{C}P^{2},D)\rightarrow\mathbb{R}\overset{f}{\rightarrow}\mathbb{R}\rightarrow
H^{2}(\mathbb{C}P^{2},D)\rightarrow 0.$

Because $D$ is a complex submanifold of a Kahler manifold it is also a
symplectic submanifold with respect to $\omega_{FS}$. Therefore the map
$H^{2}(\mathbb{C}P^{2})\rightarrow H^{2}(D)$ is an isomorphism, and the map
$f$ becomes the identity, and we can easily read of the cohomology of
$M\backslash D$ form the above sequence. To compute the cohomology of
$P=S^{1}ND$ we employ the Thom-Gysin sequence, which takes the following form:

$\displaystyle 0\rightarrow\mathbb{R}\rightarrow H^{0}(P)\rightarrow
0\rightarrow\mathbb{R}^{2}\rightarrow
H^{1}(P)\rightarrow\mathbb{R}\overset{\cup~{}e}{\rightarrow}\mathbb{R}\rightarrow
H^{2}(P)\rightarrow\mathbb{R}^{2}\rightarrow 0.$

We have $ND=\mathcal{O}(3)|_{D}$, because $\mathcal{O}(3)$ is non-trivial and
$H^{2}(\mathbb{C}P^{2})\rightarrow H^{2}(D)$ is an isomorphism, we have that
the Euler-class of $P$ is non-trivial. Therefore the map $\cup~{}e$ is an
isomorphism, and we can easily read of the cohomology of $P$ from the
sequence. Corollary 4.3 now gives us the following description of the elliptic
cohomology:

$k$ | 0 | 1 | 2 | 3 | otherwise
---|---|---|---|---|---
$H^{k}(\mathcal{A}_{\left|D\right|})$ | $\mathbb{R}$ | $\mathbb{R}^{3}$ | $\mathbb{R}^{2}$ | $\mathbb{R}^{2}$ | 0

Again, this is quite different from the complex log tangent bundle:

$k$ | 0 | 1 | otherwise
---|---|---|---
$H^{k}(\mathcal{A}_{D})$ | $\mathbb{C}$ | $\mathbb{C}^{2}$ | 0

######  Example 5.7.

Let $f:M^{4}\rightarrow\Sigma^{2}$ be a Lefschetz fibration with singular
value $p$ and corresponding singular fibre $F=f^{-1}(p)$. Let $(E,s)$ be the
complex divisor on $\Sigma$ corresponding to the divisor
$\\{p\\}\subset\Sigma$, and define a self-crossing complex log divisor on $M$
by $(f^{*}E,f^{*}s)$. This divisor has as vanishing locus precisely the
singular fibre $F$, and $D[2]$ consist precisely of the critical points on
$F$.

Let $n$ be the number of critical points of $F$. Then $D[1]$ consist of a
disjoint union of $n$ cylinders, and consequently $S^{1}ND[1]$ is homotopic to
$n$ tori.

Corollary 4.3 now gives us the following description of the elliptic
cohomology:

$\displaystyle H^{k}(\mathcal{A}_{\left|D\right|})=H^{k}(M\backslash
F)\oplus\begin{tabular}[]{|l|l|l|l|l|l|}\hline\cr$k$&1&2&3&4&otherwise\\\
\hline\cr&$\mathbb{R}^{n}$&$\mathbb{R}^{3n}$&$\mathbb{R}^{3n}$&$\mathbb{R}^{n}$&0\\\
\hline\cr\end{tabular}$

###  5.1. Outlook: Deformation theory

We end by given an outlook on studying the deformation theory of self-crossing
stable generalized complex structures and self-crossing elliptic symplectic
structures.

Already the deformation theory of all smooth elliptic symplectic structures is
more difficult than that of smooth stable generalized complex structures. The
reason is that smooth complex log divisors are very rigid: if one deforms a
complex function which vanishes transversely a bit, than one again obtains a
function which vanishes transversely. This is used in [1] to show that any two
smooth complex log divisors which are deformations of each other are in fact
isomorphic. Therefore, deformations of smooth stable generalized complex
structures can be reduced to deformations on a fixed Lie algebroid,
simplifying the discussion significantly. In the end, this ensures that
deformations of smooth stable generalized complex structures are completely
controlled by the cohomology of the Lie algebroid.

Smooth elliptic divisors are much less rigid than smooth complex log divisors.
If $I_{\left|D\right|}$ is a smooth elliptic ideal with non-negative generator
$f$ then for all $\epsilon>0$, the function $f+\varepsilon$ will be nowhere
vanishing. Therefore small deformations of elliptic divisors will not
necessarily be isomorphic to the original divisor, which significantly
complicates the deformation theory of general elliptic symplectic structures.

Allowing for self-crossings allows for even more flexibility, so that in this
case also complex log divisors are no longer rigid. The reason is that the
self-crossings can be smoothend. Take for instance the complex divisor
$I_{D}=\left\langle z_{1}z_{2}\right\rangle$ on $\mathbb{C}^{2}$, which has as
vanishing locus the coordinate hyperplanes. The divisor
$I_{D_{t}}=\left\langle z_{1}z_{2}+t\right\rangle$ is a smooth complex log
divisor for all $t\in[0,1]$ and provides a deformation between $I_{D}$ and a
complex log divisor with embedded vanishing locus. This shows that for self-
crossing complex log divisors, deformation equivalent divisors need not be
isomorphic.

One might restrict the discussion to deformations for which the divisors are
isomorphic. We have good reasons to believe that in that case the results of
[1] can be generalized and the deformations will be controlled by the
cohomology. However, deformations as the one above are very important to the
theory. The particular deformation $I_{D_{t}}$ is used in [2] to show that in
four dimensions any self-crossing stable generalized complex structure can be
deformed into one with embedded degeneracy locus.

Therefore it is most natural to study general deformations, which would thus
combine the deformation theory of the Lie algebroid with that of the elliptic
symplectic structures. To study this one will have to combine the cohomology
governing the deformations of the Lie algebroid, which is the deformation
cohomology of Crainic-Moerdijk ([12]) together with the Lie algebroid
cohomology of the elliptic tangent bundle.

##  Appendix A Atiyah algebroids

This section recalls some results on Atiyah algebroids needed in this paper.
These results are all classical and can be found in the literature, see for
instance [3].

######  Definition A.1.

Let $P$ be a principal $G$-bundle over $M$. The Atiyah algebroid of $P$, is
defined as $\operatorname{At}(P)=TP/G$. The anchor is induced by the
differential of the projection $P\rightarrow M$. The bracket is obtained by
viewing sections of $\operatorname{At}(P)$ as $G$-invariant vector fields on
$P$.

Given a vector bundle $E$, we can consider its frame bundle $\text{Fr}(E)$ and
its associated Atiyah algebroid, which we will denote by
$\operatorname{At}(E)$. We can describe its sections as fibre-wise linear
vector fields:

######  Lemma A.2.

Let $E\rightarrow M$ be a vector bundle. Then
$\Gamma(\operatorname{At}(E))\simeq\mathfrak{X}^{1}(E)_{\operatorname{lin}}=\\{X\in\mathfrak{X}^{1}(E):[X,\mathcal{E}]=0\\}$.

We are mainly interested in the case of metrics, for which we have the
following descriptions:

######  Lemma A.3.

Given a metric $g$ on a vector bundle $E$ let $Q$ denote the corresponding
$O(n)$ structure group reduction of $\operatorname{Fr}(E)$. We have

$\displaystyle\Gamma(\operatorname{At}(Q))\simeq\\{\widetilde{X}\in\mathfrak{X}^{1}(E)_{\operatorname{lin}}:\mathcal{L}_{\widetilde{X}}g=0\\}.$

######  Lemma A.4.

Given a conformal class of metrics $[g]$ let $Q$ denote the corresponding
$O(n)\times\mathbb{R}_{+}$ structure group reduction of
$\operatorname{Fr}(E)$. We have

$\displaystyle\Gamma(\operatorname{At}(Q))\simeq\\{\widetilde{X}\in\mathfrak{X}^{1}(E)_{\operatorname{lin}}:\mathcal{L}_{\widetilde{X}}g=\lambda
g,~{}\text{for some }\lambda\in C^{\infty}(M)\\}.$

The Atiyah algebroid cannot detect discrete parts of the structure group:

######  Lemma A.5.

If $P$ is a principal $G$-bundle and $Q$ is a structure group reduction to
$H$, and $\mathfrak{g}\simeq\mathfrak{h}$ then
$\operatorname{At}(P)\simeq\operatorname{At}(Q)$.

## References

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  * [2] G. R. Cavalcanti, R. L. Klaasse, and A. Witte, _Self-crossing stable generalized complex structures_ , arXiv:2004.07559 [math.DG].
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Rohit Paturi∗, Sundararajan Srinivasan∗, Xiang Li

# Lexical Speaker Error Correction: Leveraging Language Models for Speaker
Diarization Error Correction

###### Abstract

Speaker diarization (SD) is typically used with an automatic speech
recognition (ASR) system to ascribe speaker labels to recognized words. The
conventional approach reconciles outputs from independently optimized ASR and
SD systems, where the SD system typically uses only acoustic information to
identify the speakers in the audio stream. This approach can lead to speaker
errors especially around speaker turns and regions of speaker overlap. In this
paper, we propose a novel second-pass speaker error correction system using
lexical information, leveraging the power of modern language models (LMs). Our
experiments across multiple telephony datasets show that our approach is both
effective and robust. Training and tuning only on the Fisher dataset, this
error correction approach leads to relative word-level diarization error rate
(WDER) reductions of 15-30% on three telephony datasets: RT03-CTS, Callhome
American English and held-out portions of Fisher.

**footnotetext: These authors contributed equally to this work

Index Terms: Speaker Diarization, Large Language Models, Automatic Speech
Recognition, Error Correction

##  1 Introduction

Figure 1: (a) Speaker Error Correction as a 2nd-pass post-processing step to
the traditional SD-ASR system, (b) Lexical SEC: Word embeddings from the LM,
Speaker IDs from SD are passed to the Transformer Encoder to get the corrected
Speaker IDs, (c) SEC inference performed on sliding windows with 2 hypothesis
speakers.

Speech transcription systems have advanced significantly in the past decade
but even with these remarkable advances, machines have difficulties
understanding natural conversations with multiple speakers such as in
broadcast interviews, meetings, telephone calls, videos or medical recordings.
One of the first steps in understanding natural conversations is to recognize
the words spoken and their corresponding speakers. Speaker Diarization (SD) is
the process of determining "who spoke when" in a multi-speaker audio signal
and is a key component in any speech transcription system. SD is used in
conjunction with Automatic Speech Recognition (ASR) to assign a speaker label
to each transcribed speaker turn and has widespread applications in generating
meeting/interview transcripts, medical notes, automated subtitling and
dubbing, downstream speaker analytics, among others (we refer to this combined
system as SD-ASR in this paper). This is typically performed in multiple steps
that include (1) transcribing the words using an ASR system, (2) predicting
“who spoke when” using a speaker diarization (SD) system, and, finally, (3)
reconciling the output of those two systems.

Recent advances in SD systems are outlined in [1] and the independent module
optimized SD systems typically consists of the following main sub-tasks: (a)
segment the input audio into speech segments using a Voice activity detector
(VAD), (b) generate speaker segments from the speech segments by either using
a uniform window size [2, 3, 4] or by detecting speaker turns [5, 6, 7], (c)
extract speaker embeddings [8, 9, 2, 10] for each of the speaker segments and
(d) cluster the resulting speaker embeddings using clustering algorithms like
Spectral Clustering [2], Agglomerative Hierarchical Clustering [4] among
others. These sub-tasks of most of the diarization systems in literature rely
only acoustic information and can thus lead to speaker errors, mainly around
the speaker turns. This can happen in uniform speaker segmentation as long
segments very likely contain speaker turn boundaries, while short segments
carry insufficient speaker information. It is also shown that detecting
speaker turns using only acoustic information is also error-prone [7]. In
addition to the SD errors, speakers can be attributed to the wrong words in
the SD-ASR reconciliation phase due to errors in ASR word timings.
Reconciliation errors can also occur in regions of speech overlap as SD can
identify one of the speakers while ASR can identify words corresponding to a
different speaker.

Lexical information can contain complementary information which can be very
useful in accurately predicting speaker turns [6, 7]. For instance, analyzing
only the written transcript of a conversation such as "how are you i am good",
enables us to infer that there is likely a speaker change between the
utterances "how are you" and "i am good". There have been a handful of works
[6, 7, 11, 12, 13, 14] which leverage the ASR transcripts to infuse lexical
information in the SD module. In [7], lexical cues are used to estimate the
speaker turns for diarization. [11] made use of turn probabilities from
lexical cues in the clustering stage by enhancing the adjacency matrix. Though
these approaches showed good SD improvements, these systems can still produce
errors around speaker turns due to ASR and Diarization errors in overlapped
speech as well are sensitive to ASR word timings as they rely on ASR timings
in the diarization sub-tasks as well as in the Reconciliation phase. [12]
modeled SD and ASR jointly but is confined to 2 speakers with specific
distinct roles.

In this paper, we propose a Speaker Error Correction (SEC) module which can
correct speaker errors at the word level without modifying the underlying ASR
or the acoustic SD system. This SEC module makes use of the any of the readily
available pre-trained LMs [15, 16, 17, 18] to infuse the lexical knowledge to
correct speaker errors while also leveraging speaker scores from the SD system
to prevent over-corrections. The reliance on LMs also significantly reduces
the amount of speaker labelled text data needed to train the system. Our
approach has components which are modular and don’t need paired audio, text
data to train while only needing a small amount of paired data for fine-
tuning. This approach is also easier to integrate with existing systems than
other lexical-based diarization approaches, since the first-pass acoustic SD
system can be run independently of the ASR system. Using experiments across
three telephony datasets, we demonstrate that the proposed system is both
effective as well as capable of generalization.

##  2 Speaker Error Corrector

The overall pipeline of the proposed two-pass Speaker Error Corrector (SEC)
framework is shown in Fig 1a. The conventional Speaker Transcription system
consists of an ASR module, a SD module and a reconciliation stage. The SEC
follows the reconciliation stage and takes in two streams of inputs: acoustic
features from the SD module and lexical features from the ASR module. The ASR
and acoustic SD models can continue to run in parallel, making it easier to
integrate with existing systems. The core component of the SEC is the Lexical
Correction module which takes in the transcribed words from ASR along with the
speaker labels from the SD module. These are explained in more detail in the
following sub-sections.

###  2.1 Lexical Diarization Corrector

While lexical features have complementary information to the acoustic features
and can be leveraged to correct some of the errors from a naïve reconciliation
of ASR and SD, lexical features alone can’t accurately predict the speaker
labels especially in realistic conversations. So, we propose a simple yet
efficient way to correct the speakers based on both the decisions from the 1st
pass diarizer and the ASR transcriptions. Our proposed Lexical Speaker Error
Corrector consists of two main components: a backbone language model (LM) and
a Transformer Encoder Front-end to predict the speaker labels. After
reconciling ASR and diarization outputs, we have speaker labels
$\\{\bm{S_{i}}\\}_{i=1}^{N},\bm{S_{i}}\in\mathbb{R}^{1\times K}$ for every
word $\\{\bm{W_{i}}\\}_{i=1}^{N}$, where $N$ is the number of words in the
sequence and K is the number of speakers the SEC is trained to handle. The
words $W_{i}$ are tokenized and passed to the backbone LM to obtain contextual
word embeddings $\\{\bm{E_{j}}\\}_{j=1}^{M},\bm{E_{j}}\in\mathbb{R}^{1\times
W}$ where $M$ is the number of tokens in the word sequence and W is the word
embedding dimension. The word level speaker labels $S_{i}$ are mapped to token
level by mapping the speaker ID corresponding to the word to its first token
if the word has more than 2 tokens and assigning a special “don’t care” token
to any of the subsequent tokens of the word. These token level embeddings
$E_{j}$ are concatenated with the speaker IDs $S_{j}$ to form the fused
features for the Front-end Transformer Encoder as shown in Figure 1b. The
posteriors from the Front-end Encoder
$\\{\bm{L_{ij}}\\}_{j=1}^{K},\bm{L_{i}}\in\mathbb{R}$ are used to optimize the
classification loss on the ground-truth speaker labels.

###  2.2 Training Methodology

The SEC model can be trained only using speaker turn transcripts and doesn't
require paired audio data and we show that training the lexical corrector on
just the transcripts also improves the performance of the baseline. Since the
relatively smaller number of speaker errors produced by 1st pass diarizer
system limits the training of the error corrector, we train the corrector by
simulating speaker errors based on the ground truth as well by simulating ASR
substitution errors.

We define the probability of ASR errors as $P_{ASR}$ and the probability of
speaker errors as $P_{Spk}$. Setting $P_{ASR}=1$ implies that all the words in
the training transcripts are substituted with random words and $P_{ASR}=0$
implies the original ground-truth transcripts. Similarly, $P_{Spk}=1$ implies
all the speaker labels are randomly substituted whereas $P_{Spk}=0$ implies
the ground-truth speaker labels. We simulate ASR, Speaker errors using a
curriculum learning paradigm [19] to make sure that we don’t under or over
correct the speakers and balance the information flow from the SD labels and
ASR word lexical information. We start the curriculum for $P_{Spk}$ at a low
value and increase $P_{Spk}$ as the training progresses. Conversely, $P_{ASR}$
starts at a high value at the first epoch and decreases as the training
progress. The intuition for this curriculum with $P_{ASR}$ being higher and
$P_{Spk}$ being lower in the initial epochs is to train the model without any
meaningful lexical information and to train the model to at least copy the 1st
pass speaker labels in the initial epochs. More meaningful lexical information
with a smaller $P_{ASR}$ is used in the later epochs along with a higher
$P_{Spk}$ to train the model on more complex speaker errors as the training
progresses.

In addition to the errors simulated text data, we also use paired audio data
to train, fine-tune the model on real data. For this, we generate speaker
labels using the baseline 1st SD and use the ground-truth speaker labels as
the targets. In this work, we train the SEC on two speaker cases, i.e., $K=2$

###  2.3 Inference Setup

During inference, we perform error correction on sliding windows with a fixed
number of ASR transcribed words as shown in Fig 1c. Though the lexical
corrector is trained to only correct two speakers locally, we can still handle
use-cases where more than two speakers are detected globally in the audio. We
achieve this by only correcting sliding windows comprising of two speakers and
by bypassing the remaining windows as shown in Figure 1c. The size of the
sliding window is a parameter we tune on a validation set.

##  3 Experiments

###  3.1 Data and Metrics

In this work, we use the full Fisher dataset [20, 21] to train the Speaker
Corrector system. We split the Fisher data into train, validation and test
splits as defined in [22]. We also the Fisher train set to fine-tune the
backbone LM model as well as to train, fine-tune the Corrector model. We only
use the Fisher validation split for tuning our model. For evaluation, in
addition to Fisher test split, we use the standard dev, test splits of
CALLHOME American English (CHAE) [23] and RT03-CTS [24] which are majorly two
speaker calls. We also evaluate on the two-speaker only set of CHAE, the
CH-109 dataset [25] by fixing the number of clusters to 2 as well as
automatically determining the number of speakers in the 1st pass SD system.

In order to evaluate the full ASR, SD system, we use the Word Diarization
Error Rate (WDER) proposed in [12] as it aptly captures both ASR and SD errors
at the word level. We also account for words transcribed in regions of speech
overlap in the WDER metric. This is achieved by using asclite [26] as it can
align multiple speaker hypotheses against multiple reference speaker
transcriptions and can also efficiently handle words in regions of speaker
overlaps.

Figure 2: Window Size tuning on the validation set

###  3.2 Baseline System

Our baseline SD system follows the pipeline in [2] and consists of a speaker
embedding model followed by Spectral Clustering and the number of speakers is
identified using the maximum eigengap of the Spectral Clustering. The speaker
embedding model is based on a ResNet-34 architecture trained with a
combination of classification, metric loss [27] and channel loss [28] on about
12k speakers and 4k hours of CTS data . We use a uniform speaker segmentation
[3, 4] with a duration of 500ms to extract the speaker embeddings followed by
the Clustering phase for the SD system. Our baseline SD system is comparable
to state-of-the-art diarization systems across several datasets and achieves a
a DER of 3.72 and SER of 1.1 on CHAE test set which is a stronger baseline
than the one reported in [11]. We use a hybrid ASR system [29, 30, 31] with a
Conformer Acoustic model [32] and a n-gram Language model trained on several
tens of thousands of audio, text data. For the reconciliation phase, the SD
system provides speaker turns with time boundaries and these labels are mapped
to recognized words using the associated word boundaries from the ASR system.
When the speaker turn boundary falls in the middle of a word, we assign the
word to the speaker with the largest overlap with the word similar to the
baseline system in [12]. We use a neural-network based Speech Activity
detector (SAD) similar to [33] as a front end for both SD-ASR systems above.

###  3.3 SEC System

For the SEC model, we use a pre-trained Roberta-base model [16] as the
backbone LM and a Transformer Encoder of size 128 hidden states for the Front-
end model. The curriculum for $P_{ASR}$ starts at 1 at the 1st epoch and
decreases to 0.08 in the 10th and subsequent epochs in uniform steps. The
curriculum for $P_{Spk}$ starts at 0 in the 1st epoch and increase to 0.14 in
the 10th and subsequent epochs in uniform steps. The model is trained with
Adam Optimizer with a batch size of 32 and an average sequence length of 30
words per batch. We use a learning rate of 1e-4 and train the model for 30
epochs on a machine with 8 GPUs.

Figure 3: Correction Examples: a) Errors due to overlapping speech, b) Errors
around speaker turns.

We use the SEC as a 2nd pass post-processing step to the baseline SD-ASR
system in Section 3.2. In order to determine the number of simulated errors
needed to effectively train the lexical SEC to correct the speaker errors, we
follow the error curricula mentioned in Section 2.4 and pick the checkpoint
with $P_{ASR}$, $P_{Spk}$ that achieves the lowest WDER on the Fisher
validation set. The values that achieve the best validation WDER are 0.1 for
both $P_{ASR}$ and $P_{Spk}$. In addition to the training parameters, we also
tune an inference parameter, the sliding window size as mentioned in Section
2.1 also on the Fisher validation set.

Table 1: WDER of different models on Fisher test, RT03-CTS and CHAE dev, test sets. CHAE-109 is evaluated with and without fixing the number of speakers in 1st pass to 2. SimSEC: SEC model trained using simulated transcript errors, RealSEC: SEC trained/tuned on real paired data Model Type |  Fisher Test | RT03-CTS | CHAE | CH-109
---|---|---|---|---
|  |  Validation |  Test |  Validation |  Test |  Known Spkrs |  Unknown Spkrs
Baseline (No Correction) |  2.26 |  2.30 |  2.18 |  4.23 |  2.82 |  3.69 |  4.28
SimSEC_v1 (Base Roberta) |  1.72 |  2.18 |  1.98 |  4.16 |  2.68 |  3.41 |  4.29
SimSEC_v2 (Tuned Roberta) |  1.63 |  1.90 |  1.67 |  3.52 |  2.49 |  3.16 |  3.74
RealSEC (flat-start Training) |  1.53 |  1.73 |  1.58 |  3.31 |  2.30 |  2.98 |  3.57
SimSEC_v2 init + RealSEC Tuning |  1.53 |  1.73 |  1.59 |  3.28 |  2.26 |  2.97 |  3.56
Table 2: WDER of models with different amounts of Synthetic/Real Data on Fisher test set. Model Type |  Fraction of Train Data |  WDER
---|---|---
Baseline (No Correction) |  2.26
SimSEC_v2 | 0.2 |  1.73
| 0.4 |  1.68
| 0.8 |  1.65
| 1.0 |  1.63
SimSEC_v2 init + RealSEC Tuned | 0.2 |  1.58
| 0.4 |  1.57
| 0.8 |  1.54
| 1.0 |  1.54

###  3.4 Results

We tune for the sliding window size on our Fisher validation subset and plot
the WDER with the corresponding values as shown in Figure 2. From the plot, we
see that WDER decreases as the window size increases up to 30 due to increased
lexical context for the backbone LM as well as the corrector model. The WDER
further increases beyond the window size of 30 likely due to the corrector
model being trained with an average seq length of 30 and more sliding windows
with greater than 2 speakers being bypassed with a larger window size. We have
also tried training with larger average sequence lengths but that did not show
any additional gains compared to the sequence length of 30 words. So, we use
the sliding window size as 30 words for the remainder of the experiments. We
also show some qualitative examples of the correction performance on the
Fisher test set using the SEC model with the best sliding window size in
Figure 3. Figure 3a shows that the correction model is able to effectively
correct errors due to overlapping speech when the SD hypothesizes one of the
overlapping speakers and ASR hypothesizes the words of the other speaker. The
model is also effective in correcting the lexically implausible errors around
speaker turns which is one of the major error-prone scenario [34] for SD
systems as seen in Figure 3b.

The quantitative WDER improvements of the correction models on the held out
validation and test sets are outlined in Table 1. We call the model trained on
ground truth transcripts with simulated speaker, ASR errors as the "SimSEC"
model. SimSEC_v2 is the "SimSEC" model with a Fisher tuned backbone Roberta
and trained with a custom curriculum as mentioned in Section 3.3 SimSEC_v1 is
similar to SimSEC_v2 but with the Roberta-base as a backbone without any
further fine-tuning. We evaluate SimSEC_v1 to quantify the gains attributed to
fine-tuning of the backbone LM on conversational datasets. "SimSEC init +
RealSEC Tuning" model is the paired data tuned model initialized with
SimSEC_v2 and tuned using the 1st pass acoustic SD labels instead of the
simulated speaker errors. RealSEC model is similar to "SimSEC_v2 init +
RealSEC Tuning" but is only trained by flat-starting the model on real paired
data.

From Table 1, we can see that almost all of the corrector models produce
considerable WDER gains over the Baseline SD-ASR reconciled system across all
the datasets, except from SimSEC_v1 on CH-109 with unknown speakers. It can be
observed that tuning the backbone Roberta LM in SimSEC_v2 can produce moderate
WDER gains over the pretrained Roberta-base LM, especially on CHAE validation
set and CH-109 with unknown speakers. The model trained on Paired data, either
by tuning the SimSEC_v2 model or by flat-start training (RealSEC) produces
further gains over the models train with errors simulated (SimSEC_v1 and
SimSEC_v2). The performance improvement of the models on CH-109 without fixing
the speakers to 2 in the Clustering phase is comparatively limited due to
hypothesising more than 2 speakers on few of the audio files leading to
smaller average WDER gains. With the best model "SimSEC_v2 init + RealSEC
Tuning", we observe relative WDER gains in the range 15-30% across all the
datasets.

To further analyze the importance of dataset sizes needed to train or tune the
models, we perform an ablation by only using a fraction of the Fisher train
data as shown in Table2. We evaluate The models SimSEC_v2 and "SimSEC init +
RealSEC Tuning" with different fractions of ground truth text and paired data
respectively. We see that the WDER of the SimSEC_v2 model and "SimSEC init +
RealSEC Tuning" model only improves moderately and saturates at a point as the
amount of text data and paired data increases respectively. This shows that
the corrector model can be trained purely on small amounts of ground truth
transcripts by simulating speaker, ASR errors and can also be fine-tuned on a
small amount of paired data to achieve significant WDER gains.

##  4 Conclusion and Future Work

In this work, we propose a novel Speaker Error Corrector (SEC) to correct
word-level speaker label errors from a conventional audio-only speaker
diarization system. We achieve this using a language model over the ASR
transcriptions to correct the speaker labels. The proposed lexical SEC can be
trained effectively using only text data by simulating speaker errors without
the need for any paired audio-text data. A small amount of paired data can
further improve model performance, leading to overall relative reduction of
WDER by over 15% across three telephony datasets. The proposed SEC framework
is also lightweight and is easy to integrate as a post-processing module over
existing systems.

One limitation of our current work is that it has been applied only to
conversations in English. One future work can include training a multi-lingual
SEC to make the system language-agnostic. To increase the robustness of this
approach, in addition to the first-pass SD labels, we can leverage additional
complementary acoustic cues to further improve the performance. Also, the
current SEC model can only handle 2 speakers in a sliding window, which we
plan to generalize to handle more number of speakers. We will also explore
leveraging large generative models to synthesize conversational transcripts
across multiple domains using curated prompts [35].

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