# Non-principal $T$-duality, generalized complex geometry and blow-ups

Gil R. Cavalcanti  Department of Mathematics, Utrecht University, 3508 TA
Utrecht, The Netherlands<EMAIL_ADDRESS>and  Aldo Witte  Departement of
Mathematics, Celestijnenlaan 200B, B-3001 Leuven (Heverlee), Belgium
<EMAIL_ADDRESS>
###### Abstract.

We extend the notion of $T$-duality to manifolds endowed with non-principal
torus actions. The singularities of the torus action are controlled by a
certain Lie algebroid, called the elliptic tangent bundle. Using this Lie
algebroid, we explain how certain invariant generalized complex structures can
be transported via $T$-duality. Along the way, we use the elliptic tangent
bundle to define connections for these torus action, and give new insight to
the classification of such actions by Haefliger-Salem.

_Dedicated to the memory of Gilles Castel_

###### Contents

  1. 1 Introduction
    1. 1.1 $T$-duality
    2. 1.2 Blow-ups
    3. 1.3 Torus actions and connections
    4. 1.4 Organisation of the paper
    5. 1.5 Acknowledgements
  2. 2 Divisors and Lie algebroids; generalized complex structures and their lifts
    1. 2.1 Divisors
    2. 2.2 Lie algebroids and residue maps
    3. 2.3 Generalized complex structures
    4. 2.4 Lifting generalized complex structures
  3. 3 Classification of torus actions and connections
    1. 3.1 Haefliger–Salem classification
    2. 3.2 Torus actions, divisors and connections
    3. 3.3 Smoothness of the curvature
    4. 3.4 Cech–de Rham isomorphism
    5. 3.5 Connection approach to Haefliger-Salem class
  4. 4 The singular differential geometry of the correspondence space
  5. 5 Non prinicipal $T$-duality
    1. 5.1 $T$-duality
    2. 5.2 Courant algebroids
  6. 6 Blow-ups
    1. 6.1 Blow-ups in generalized complex geometry
    2. 6.2 Blowing up for the elliptic tangent bundle
    3. 6.3 $T$-duality and blow-up
  7. 7 Examples
    1. 7.1 Basic examples
    2. 7.2 Non-zero residue elliptic symplectic structures

##  1\. Introduction

###  1.1. $T$-duality

_$T$ -duality_ is an equivalence between quantum field theories with very
different properties. The precise relationship between $T$-dual Riemannian
structures was first understood by Buscher in [Bus87] and was developed
further by Roček and Verlinde in [RV92]. Topologically, $T$-duality can be
described as an equivalence between principal torus bundles, and this approach
has been studied in [BEM04].

In [CG10] it was observed that $T$-duality can be understood in the language
of _generalized geometry_ [Hit03]. Generalized geometry concerns itself with
geometric structures on the double tangent bundle: the direct sum of the
tangent and cotangent bundle. The double tangent bundle comes with the Courant
bracket, extending the Lie bracket of vector fields, giving it the structure
of a _Courant algebroid_. Almost complex structures on the double tangent
bundle which are integrable with respect to the Courant bracket are known as
_generalized complex structures_ ; these are geometric structures which
interpolate between complex and symplectic structures.

The main contribution of [CG10] is that $T$-duality may be phrased as an
isomorphism of Courant algebroids between the double tangent bundles.
Consequently, it can be used to transport generalized complex structures which
are invariant under the torus action.

Every generalized complex structure decomposes each tangent space into the
direct sum of a complex and symplectic vector space. The number of symplectic
directions is called the _type_ of the generalized complex structure, and may
vary across the manifold. The simplest type-change of generalized complex
structures happens in _stable generalized complex structures_ ([CG18, GH16,
CKW20b]). These have the property that they are symplectic almost everywhere,
and degenerate in a controlled fashion along an immersed codimension two
submanifold. Many examples of these structures have been constructed in the
past decade ([CG09, Tor12, TY14]), especially in four dimensions. The geometry
of these four-dimensional generalized complex structures is very reminiscent
of symplectic Landau–Ginzburg models from physics [HIV00].

However very few examples of stable generalized complex structures are
invariant under a principal torus action. Nevertheless, in [CKW20a] we saw
that many are invariant under well-behaved singular torus actions.

The goal of the current article is to develop a framework for $T$-duality,
which allows for these types of singular torus actions. More precisely, we
will consider torus actions which we will call _standard_ : $T^{k}$-actions on
manifolds, which locally decompose as standard rotations of the plane. The
main tool for obtaining this framework are _elliptic divisors_
$I_{\left|D\right|}$ and _elliptic tangent bundles_
$\mathcal{A}_{\left|D\right|}$. An elliptic divisor is an ideal of smooth
functions, which comes naturally associated to a standard torus actions. The
elliptic tangent bundle is a Lie algebroid defined as the vector fields which
preserve the elliptic divisor.

As with principal $T$-duality, $T$-duality for these actions will provide an
isomorphism of Courant algebroids, but now using the elliptic tangent bundle:

###### Theorem 5.11.

Let $M,\tilde{M}$ be manifolds with standard $T^{k}$ actions and let
$\mathcal{A}_{\left|D\right|}\rightarrow
M,\mathcal{A}_{\left|\hat{D}\right|}\rightarrow\hat{M}$ denote the induced
elliptic tangent bundles. Let
$H\in\Omega^{3}_{T^{k}}(\mathcal{A}_{\left|D\right|}),\hat{H}\in\Omega^{3}(\mathcal{A}_{\left|\hat{D}\right|})$
be closed invariant forms. If $(M,H)$ and $(\hat{M},\hat{H})$ are $T$-dual
then there exists an isomorphism of Courant algebroids:

(1.1)
$(\mathcal{A}_{\left|D\right|}\oplus\mathcal{A}_{\left|D\right|}^{*})/T^{k}\overset{\simeq}{\longrightarrow}(\mathcal{A}_{\left|\hat{D}\right|}\oplus\mathcal{A}_{\left|\hat{D}\right|}^{*})/T^{k}.$

This isomorphism of Courant algebroids allows us to transport geometric
structures on $\mathcal{A}_{\left|D\right|}$ which are invariant under the
torus action between $M$ and $\hat{M}$. Stable generalized complex structures
can be viewed as symplectic structures on $\mathcal{A}_{\left|D\right|}$, and
can hence be transported in this manner. There is however an intriguing
difference with $T$-duality for principal torus actions: A generalized complex
structure on $M$ which lifts to $\mathcal{A}_{\left|D\right|}$ will always
induce a generalized complex structure on the $T$-dual Lie algebroid
$\mathcal{A}_{\left|\hat{D}\right|}\rightarrow\hat{M}$, however it might not
induce a generalized complex structure on $M$. We will, for instance,
encounter complex structures on the elliptic tangent bundle on manifolds which
are not almost complex.

###  1.2. Blow-ups

The generalized complex structures we will consider turn out to be especially
well-behaved under blow-ups. The theory of blow-ups within generalized complex
geometry was developed in [BCD16]. We will show that the elliptic tangent
bundle behaves very well with respect to blowing up:

###### Theorem 6.9.

Let $I_{\left|D\right|}$ be an elliptic divisor, and let
$\mathcal{A}_{\left|D\right|}$ be the associated elliptic tangent bundle. Let
$Y\subset D$ be a submanifold compatible with $D$ in a suitable sense. Then
there exists a natural elliptic divisor $I_{\left|\tilde{D}\right|}$ on
$\tilde{M}$ such that the blow-down map $p:\tilde{M}\rightarrow M$ provides a
Lie algebroid submersion

$p:\mathcal{A}_{\left|\tilde{D}\right|}\rightarrow\mathcal{A}_{\left|D\right|}.$

When $M$ comes equipped with a torus action and $Y=\\{*\\}$ is a fixed point
of this action, the blow-up $\tilde{M}$ naturally inherits a torus action.
Consequently we will show that:

###### Proposition 6.15.

$T$-duality and blowing up commute.

This procedure will allow us to construct many examples of $T$-dual
generalized complex structures.

###  1.3. Torus actions and connections

An integral part of this paper is the classification of torus actions on
manifolds. For principal torus actions, the relevant datum is its
characteristic class. For general torus actions, this question is studied by
Haefliger and Salem in [HS91]. This classification is fundamentally different
from the principal case, because there is no trivial $T^{k}$-bundle over a
given base $B$. Therefore, the classification in [HS91] is affine in nature:
given any manifold $M$ with $T^{k}$-action, any other manifold $M^{\prime}$
endowed with a similar $T^{k}$-action is classified by a class in
$\check{H}^{2}(M;\mathbb{Z}^{k})$.

Although the class of [HS91] resembles a curvature class, no connection
approach exists. In this article we will show that one may use the elliptic
tangent bundle to give a definition of connections for standard
$T^{k}$-actions.The curvature of these connections then induce the class of
Haefliger-Salem in the following way:

###### Proposition 3.19.

Let $\pi:M\rightarrow B$ be a manifold endowed with a standard $T^{k}$-action
and let $\pi^{\prime}:M^{\prime}\rightarrow B$ be a manifold endowed with a
locally equivalent $T^{k}$-action. Let $\Theta,\Theta^{\prime}$ be a
connection one-form on $M$ respectively $M^{\prime}$. Then
$[d\Theta-d\Theta^{\prime}]\in
H^{2}(\mathcal{A}_{\left|D\right|};\mathfrak{t}^{k})$ is representable by a
smooth form, and descends to the Haefliger-Salem class in
$\check{H}^{2}(B;\mathfrak{t}^{k})$.

###  1.4. Organisation of the paper

This paper is organized as follows. In Section 2 we will recall the notions of
elliptic divisors, and their associated Lie algebroids. Furthermore, we recall
how stable generalized complex structures can be viewed as symplectic
structures on Lie algebroids. More generally, we discuss lifting generalized
complex structures to Lie algebroids. In Section 3 we recall the
classification of torus actions by Heafliger-Salem. Afterwards we introduce
the notion of a connection for a standard torus action, and show how these can
be used to understand this classification. In Section 4 we study singular
fibre products, which we will encounter in the description of $T$-duality. In
Section 5 we introduce the the notion of non-principal $T$-duality, using the
Lie algebroids from Section 2. In Section 6 we show that the elliptic tangent
bundle interacts very well with (generalized) complex blow-ups, and show that
$T$-dualizing commutes with blowing-up. Finally, in Section 7 we discuss
concrete examples.

###  1.5. Acknowledgements

The second author was supported by the NWO through the UGC Graduate Programme
(Netherlands), and through the FWO and FNRS under EOS project G0H4518N, and
FWO Project G083118N (Belgium). The authors would like to thank Maarten Mol
and Álvaro del Pino for useful discussions. The authors are grateful to the
organizers of the workshop “Generalized Geometry in Interaction”, Madrid 2022,
were a preliminary version of this article was presented.

##  2\. Divisors and Lie algebroids; generalized complex structures and their
lifts

As we mentioned in the introduction, the main objective of this paper is to
extend $T$-duality to torus actions with fixed points without giving up on
compactness of the spaces involved. The object that will allow us to
accomplish this task are Lie algebroids, specifically Lie algebroids
associated to divisors as introduced in [CG18, CKW20b]. Not only will these
allow us to lift torus actions, but also we will arrive at the very natural
problem of lifting geometric structures from the underlying manifold to the
Lie algebroid.

In this section give the first step towards our goal and recall the theory of
divisors and Lie algebroids (for more details, see [CG18, CKW20b]). We also
introduce several relevant geometric structures, including generalized complex
structures, and their lifts to these Lie algebroids

###  2.1. Divisors

Inspired by divisors in algebraic geometry, we will make use of the following
notion to deal with singularities of geometric structures:

######  Definition 2.1.

A real (resp. complex) divisor on $M$ is a locally principal ideal $I\subset
C^{\infty}(M;\mathbb{R})$ (respectively $C^{\infty}(M;\mathbb{C}))$ which is
locally generated by functions with a nowhere dense zero-set.

######  Definition 2.2.

Let $(M,I_{M})$ and $(N,I_{N})$ be manifolds together with divisors. A smooth
map $\varphi:M\rightarrow N$ is a morphism of divisors if
$\varphi^{*}I_{N}=I_{M}$. It is said to be a diffeomorphism of divisors if
$\varphi$ is moreover a diffeomorphism.

We will encounter three types of divisors in this article:

######  Definition 2.3.

Let $M^{n}$ be a smooth manifold.

  * •

A real log divisor on $M$ is a real divisor, $I_{Z}$, such that for each $p\in
M$ there are coordinates $(x_{1},\ldots,x_{n})\colon U\to\mathbb{R}^{n}$
centred at $p$ and an integer $k$ such that

$I_{Z}|_{U}=\left\langle x_{1}\cdot\ldots\cdot x_{k}\right\rangle.$

  * •

A complex log divisor on $M$ is a complex divisor $I_{D}$, such that for each
$p\in M$ there are coordinates
$(z_{1},\ldots,z_{k},x_{2k+1},\ldots,x_{n})\colon
U\to\mathbb{C}^{k}\times\mathbb{R}^{n-2k}$ centred at $p$ such that

$I_{D}|_{U}=\left\langle z_{1}\cdot\ldots\cdot z_{k}\right\rangle.$

  * •

An elliptic divisor on $M$ is a real divisor $I_{\left|D\right|}$, such that
for each $p\in M$ there are coordinates $(x_{1},\ldots,x_{n})\colon
U\to\mathbb{R}^{n}$ centred at $p$ and an integer $k$ such that

$I_{\left|D\right|}|_{U}=\left\langle r_{1}^{2}\cdot\ldots\cdot
r_{k}^{2}\right\rangle,$

with $r_{i}^{2}=x_{2i-i}^{2}+x_{2i}^{2}$.

In these three cases, the number $k$ is the multiplicity of $p$. The vanishing
set of the ideal is called the vanishing locus. We call each of these divisors
smooth if their vanishing locus is an embedded smooth submanifold, that is, if
the multiplicity of each point is either $0$ or $1$.

In general, the vanishing locus is only (the image of) an immersed submanifold
with transverse intersections, and is naturally stratified by the multiplicity
of its points. It is convenient to introduce some notation to refer to the
different strata of the vanishing locus, so if we let, say, $D$, denote the
vanishing locus of any of the divisors above, the set of points with
multiplicity at least $k$ will be denoted by $D(k)$ and the set of points with
multiplicity at precisely $k$ will be denoted by $D[k]$. Each of the $D(k)$ is
itself an immersed submanifold, and hence we may speak of:

######  Definition 2.4.

Let $I_{\left|D\right|}$ be an elliptic divisor, and let $\iota:N\rightarrow
D(k)$ be an immersion. The image of a connected component of $N$ is called an
irreducible component of $D(k)$.

Given a complex log divisor $I_{D}$, one may consider its associated elliptic
divisor defined by the condition
$I_{\left|D\right|}\otimes\mathbb{C}=I_{D}\otimes\overline{I}_{D}$. In this
case, $D[1]$ is automatically co-oriented by the differential of any local
generator of $I_{D}$. Conversely, given an elliptic divisor
$I_{\left|D\right|}$ for which $D[1]$ is co-oriented, there exists a unique
complex log divisor $I_{D}$ inducing $I_{\left|D\right|}$ and the given co-
orientation.

###  2.2. Lie algebroids and residue maps

Using the Serre-Swann theorem, one may show that vector fields preserving the
divisors from Definition 2.3 define vector bundles and the natural
identification of sections of these vector bundles with vector fields makes
these vector bundles into Lie algebroids:

######  Definition 2.5.

Let $M$ be a manifold, and $I_{Z},I_{D},I_{\left|D\right|}$ a real log,
complex log, respectively elliptic divisor on $M$. Vector fields preserving
these ideals define Lie algebroids:

  * •

$\mathcal{A}_{Z}\rightarrow TM$, the real log-tangent bundle.

  * •

$\mathcal{A}_{D}\rightarrow T_{\mathbb{C}}M$, the complex log-tangent bundle.

  * •

$\mathcal{A}_{\left|D\right|}\rightarrow TM$, the elliptic tangent bundle.

As the divisors under consideration have local normal forms, so do have the
associated Lie algebroids:

######  Remark 2.0.

In the coordinates of Definition 2.3, we have:

  * •

$\Gamma(\mathcal{A}_{Z})=\left\langle
x_{1}\partial_{x_{1}},\ldots,x_{k}\partial_{x_{k}},\partial_{x_{k+1}},\ldots,\partial_{x_{n}}\right\rangle$.

  * •

$\Gamma(\mathcal{A}_{D})=\left\langle
z_{1}\partial_{z_{1}},\partial_{\overline{z}_{1}},\ldots,z_{k}\partial_{z_{k}},\partial_{\overline{z}_{k}},\partial_{x_{2k+1}},\ldots,\partial_{x_{n}}\right\rangle$.

  * •

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

with
$r_{i}\partial_{r_{i}}=x_{2i-1}\partial_{x_{2i-1}}+x_{2i}\partial_{x_{2i}}$
and
$\partial_{\theta_{i}}=x_{2i-1}\partial_{x_{2i}}-x_{2i}\partial_{x_{2i-1}}$.

For each of the Lie algebroids above, we obtain a corresponding de Rham
complex on the exterior power of the dual vector bundle. The relation between
complex and elliptic tangent bundles translates into the existence of maps
between their corresponding de Rhams complexes. Indeed, given a complex log
divisor, $I_{D}$, and associated elliptic divisor, $I_{\left|D\right|}$,
taking the imaginary part of a complex log form is a map
$\Im^{*}\colon(\Omega^{\bullet}(\mathcal{A}_{D}),d_{\mathcal{A}_{D}})\rightarrow(\Omega^{\bullet}(\mathcal{A}_{\left|D\right|}),d_{\mathcal{A}_{\left|D\right|}})$.

There are several residue maps associated to these Lie algebroids, obtained by
taking the coefficients of certain singular generators. We quickly recall the
ones we need in this article. Given a smooth elliptic divisor, with $D$ co-
orientable, it is explained in [CG18] that there are residue maps:

$\displaystyle\operatorname{Res}_{q}:(\Omega^{\bullet}(\mathcal{A}_{\left|D\right|}),d_{\mathcal{A}_{\left|D\right|}})\rightarrow(\Omega^{\bullet-2}(D),d),\qquad\alpha$
$\displaystyle\mapsto\iota_{D}^{*}(\iota_{r\partial_{r}\wedge\partial_{\theta}}\alpha)$
$\displaystyle\operatorname{Res}_{r}:(\Omega^{\bullet}(\mathcal{A}_{\left|D\right|}),d_{\mathcal{A}_{\left|D\right|}})\rightarrow(\Omega^{\bullet-1}(S^{1}ND),d),\qquad\alpha$
$\displaystyle\mapsto\iota_{r\partial_{r}}\alpha,$

where $S^{1}ND$ is the sphere bundle of the normal bundle of $D$.

Let $I_{\left|D\right|}$ be a general elliptic divisor with $D[1]$ co-
orientable. The inclusion $\iota_{M\backslash D(2)}:M\backslash
D(2)\rightarrow M$, pulls the elliptic divisor $I_{\left|D\right|}$ back to a
smooth elliptic divisor with vanishing locus $D[1]$. Hence we may define
residue maps by composition:

$\displaystyle\operatorname{Res}_{q}:\Omega^{\bullet}(\mathcal{A}_{\left|D\right|})\rightarrow\Omega^{\bullet-2}(D[1]),\qquad\alpha$
$\displaystyle\mapsto\iota_{D[1]}^{*}(\iota_{r\partial_{r}\wedge\partial_{\theta}}(\iota_{M\backslash
D(2)}^{*}\alpha))$
$\displaystyle\operatorname{Res}_{r}\Omega^{\bullet}(\mathcal{A}_{\left|D\right|})\rightarrow\Omega^{\bullet-1}(S^{1}ND[1]),\qquad\alpha$
$\displaystyle\mapsto\iota_{r\partial_{r}}(\iota^{*}_{M\backslash
D(2)}\alpha)$

There exists several higher order residue maps, however defining these very
quickly becomes a combinatorial task. As we are mainly interested in two-
forms, we will define:

######  Definition 2.6.

Let $I_{\left|D\right|}$ be an elliptic divisor,
$\omega\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ and let $p\in D[2]$. Using
the coordinates from Remark 2.2 we define

$\operatorname{Res}_{r_{i}r_{j}}\omega(p)=\omega_{p}(r_{i}\partial_{r_{i}},r_{j}\partial_{r_{j}}),\quad\operatorname{Res}_{r_{i}\theta_{j}}\omega(p)=\omega_{p}(r_{i}\partial_{r_{i}},\partial_{\theta_{j}}),\quad\operatorname{Res}_{\theta_{i}\theta_{j}}(\omega)(p)=\omega_{p}(\partial_{\theta_{i}},\partial_{\theta_{j}}).\hfill$

Note that these residues are only well-defined up to sign. Similarly, one may
define residue maps for the real log tangent bundle. If $I_{Z}$ is a smooth
real log divisor, we may define

$\operatorname{Res}:\Omega^{\bullet}(\mathcal{A}_{Z})\rightarrow\Omega^{\bullet-1}(Z),\quad\alpha\mapsto\iota^{*}_{Z}(\iota_{x\partial_{x}}\alpha).$

Again, for a general real log divisor we may define
$\operatorname{Res}^{1}:\Omega^{\bullet}(\mathcal{A}_{Z})\rightarrow\Omega^{\bullet-1}(Z[1])$
by precomposing with the pullback $M\backslash D(2)\hookrightarrow M$. The
higher residues are again somewhat involved to define. For two-forms we
define:

(2.1)
$\operatorname{Res}^{2}_{x_{i}x_{j}}(\omega)(p)=\omega_{p}(x_{i}\partial_{x_{i}},x_{j}\partial_{x_{j}}),\quad
p\in Z[2].$

###  2.3. Generalized complex structures

In this section we recall the definition of generalized complex structures and
focus our attention on the class of stable generalized complex structures
called stable. We will recall that these structures are in correspondence with
symplectic structures on the elliptic tangent bundle.

Given a manifold $M$, we form $\mathbb{T}M=TM\oplus T^{*}M$. This bundle is
endowed with the natural symmetric pairing given by evaluation of forms on
vectors.

######  Definition 2.7.

A generalized complex structure on a manifold $M$, is a pair $(\mathbb{J},H)$,
where $H\in\Omega^{3}(M)$ is a closed real 3-form and $\mathbb{J}$ is a
complex structure on $\mathbb{T}M$ orthogonal with respect to the natural
paring and whose $+i$-eigenbundle, $L\subset\mathbb{T}_{\mathbb{C}}M$ is
involutive with respect to the Courant–Dorfmann bracket:

$[\\![X+\xi,Y+\eta]\\!]_{H}=[X,Y]+\mathcal{L}_{X}\eta-\iota_{Y}d\xi+\iota_{X}\iota_{Y}H,\quad
X+\xi,Y+\eta\in\mathbb{T}M.\hfill$

Given a generalized complex structure one may decompose it as a block matrix
with respect to the splitting of $\mathbb{T}M$. The skew-symmetry of
$\mathbb{J}$ ensures this decomposition is of the form

$\mathbb{J}=\begin{pmatrix}J&\pi_{\mathbb{J}}^{\sharp}\\\
\sigma^{\flat}&-J^{*},\end{pmatrix},\qquad
J\in\text{End}(TM),\sigma\in\Omega^{2}(M),\pi_{\mathbb{J}}\in\mathfrak{X}^{2}(M).$

The bivector $\pi_{\mathbb{J}}$ is Poisson. Two generalized complex structures
$(\mathbb{J}_{1},H_{1}),(\mathbb{J}_{2},H_{2})$ are said to be gauge
equivalent, if there exists a two-form $B\in\Omega^{2}(M)$ such that
$dB=H_{1}-H_{2}$ and

$\mathbb{J}_{2}=\begin{pmatrix}1&0\\\
B^{\flat}&1\end{pmatrix}\mathbb{J}_{1}\begin{pmatrix}1&0\\\
-B^{\flat}&1\end{pmatrix}.$

If two generalized complex structures $\mathbb{J}_{1},\mathbb{J}_{2}$ are
gauge equivalent, then $\pi_{\mathbb{J}_{1}}=\pi_{\mathbb{J}_{2}}$.

The Clifford action of $X+\xi\in\mathbb{T}_{\mathbb{C}}M$ on
$\rho\in\wedge^{n}T_{\mathbb{C}}M$ is defined via
$(X+\xi)\cdot\rho=\iota_{X}\rho+\xi\wedge\rho$. Given a generalized complex
structure, one may define its canonical bundle
$K\subset\wedge^{\text{top}}T_{\mathbb{C}}M$ via

$L=\\{u\in T_{\mathbb{C}}M:u\cdot K=0\\}.$

The dual bundle $K^{*}$ has a natural section $s$, called the anticanonical
section which is defined via $s:\Gamma(K)\rightarrow
C^{\infty}(M;\mathbb{C}),\rho\mapsto\rho_{0}$.

######  Definition 2.8.

A generalized complex structure $(\mathbb{J},H)$ on $M$ is called stable if
$s(\Gamma(K))\subset C^{\infty}(M;\mathbb{C})$ defines a complex log divisor.

######  Example 2.9.

Let $(M,J,\pi)$ be a holomorphic Poisson manifold with $\pi=\pi_{R}+i\pi_{I}$
and denote by $\pi_{I}^{\sharp}\colon T^{*}M\to TM$ the map associated to
$\pi_{I}$. Then

$\mathbb{J}_{J,\pi}:=\begin{pmatrix}-J&4\pi_{I}^{\sharp}\\\
0&J^{*}\end{pmatrix}$

defines a generalized complex structure. If furthermore
$\wedge^{n}\pi(\Gamma(\det(T^{*}M))\subset C^{\infty}(M;\mathbb{C})$ defines a
complex log divisor, $\mathbb{J}_{J,\pi}$ defines a stable generalized complex
structure. Holomorphic Poisson structures with these properties are called
_log symplectic_.

These generalized complex structures can be described as symplectic structures
on Lie algebroids:

######  Theorem 2.10 ([CKW20b]).

There is a one-to-one correspondence between gauge equivalence classes
$[(\mathbb{J},H)]$ of stable generalized complex structures on $M$ and
symplectic forms $\omega\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ on the
elliptic tangent bundle with $D[1]$ co-orientable and satisfying the following
equations:

  * •

$\operatorname{Res}^{q}(\omega)=0$

  * •

$\operatorname{Res}_{\theta_{i}r_{j}}\omega=\operatorname{Res}_{r_{i}\theta_{j}}\omega$

  * •

$\operatorname{Res}_{r_{i}r_{j}}\omega=-\operatorname{Res}_{\theta_{i}\theta_{j}}\omega$

Given $(\mathbb{J},H)$, the associated
$\omega\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ is given by
$\pi_{\mathbb{J}}^{-1}$.

###  2.4. Lifting generalized complex structures

Lifting Poisson structures to Lie algebroids is a well-established field of
study. However, lifts of generalized complex and Dirac structures to Lie
algebroids have not yet received much attention. As we will see, the questions
of which structures lift to Lie algebroids and which structures descend from
Lie algebroids arise naturally.

####  2.4.1. Severa class

Let us first shortly study Courant algebroids where we replace the role of the
tangent bundle by a general Lie algebroid. Given any Lie algebroid
$\mathcal{A}$ and $H\in\Omega^{3}(\mathcal{A})$ closed one may define the
standard $\mathcal{A}$-Courant algebroid, in the usual manner. We set
$\mathbb{A}=\mathcal{A}\oplus\mathcal{A}^{*}$ and for
$X+\xi,Y+\eta\in\Gamma(\mathcal{A}\oplus\mathcal{A}^{*})$.

$\displaystyle\left\langle X+\xi,Y+\eta\right\rangle$
$\displaystyle=\frac{1}{2}(\eta(X)+\xi(Y)),$
$\displaystyle[\\![X+\xi,Y+\eta]\\!]_{H}$
$\displaystyle=[X,Y]+\mathcal{L}_{X}\eta-\iota_{Y}d\xi+\iota_{X}\iota_{Y}H.$

This notion, which is a particular case of a Lie bialgebroid, can be
axiomatised as follows:

######  Definition 2.11.

An exact $\mathcal{A}$-Courant algebroid is a vector bundle extension

(2.2)
$0\rightarrow\mathcal{A}^{*}\overset{\pi^{*}}{\rightarrow}\mathbb{A}\overset{\pi}{\rightarrow}\mathcal{A}\rightarrow
0,$

together with a non-degenerate pairing $\left\langle\cdot,\cdot\right\rangle$
on $\mathbb{A}$, and a bracket $[\\![\cdot,\cdot]\\!]$ on $\Gamma(\mathbb{A})$
satisfying:

  * •

$[\\![e_{1},[\\![e_{2},e_{3}]\\!]]\\!]=[\\![[\\![e_{1},e_{2}]\\!],e_{3}]\\!]+[\\![e_{2},[\\![e_{1},e_{3}]\\!]]\\!]$

  * •

$\pi([\\![e_{1},e_{2}]\\!])=[\pi(e_{1}),\pi(e_{2})]_{\mathcal{A}}$

  * •

$[\\![e_{1},fe_{2}]\\!]=f[\\![e_{1},e_{2}]\\!]+\mathcal{L}_{\rho(\pi(e_{1}))}(f)e_{2}$

  * •

$\mathcal{L}_{\rho(\pi(e_{1}))}(\left\langle
e_{2},e_{3}\right\rangle)=\left\langle[\\![e_{1},e_{2}]\\!],e_{3}\right\rangle+\left\langle
e_{2},[\\![e_{1},e_{3}]\\!]\right\rangle$

  * •

$[\\![e_{1},e_{1}]\\!]=\pi^{*}d_{\mathcal{A}}\left\langle
e_{1},e_{1}\right\rangle$.

Given an exact $\mathcal{A}$-Courant algebroid, the composition $\rho\hskip
0.28453pt\circ\hskip 0.28453pt\pi:\mathbb{A}\rightarrow TM$ together with
$\left\langle\cdot,\cdot\right\rangle$ and $[\\![\cdot,\cdot]\\!]$ will give
$\mathbb{A}$ the structure of a non-exact Courant algebroid.

Just as for ordinary exact courant algebroids, isotropic splittings
$s:\mathcal{A}\rightarrow\mathbb{A}$ of (2.2) are classified by closed three-
forms $H\in\Omega^{3}(\mathcal{A})$ appearing as the curvature

$\iota_{X}\iota_{Y}H=s^{*}[s(X),s(Y)].$

The difference between two splittings $s-s^{\prime}$, defines a two-form
$B\in\Omega^{2}(\mathcal{A})$ via $(s-s^{\prime})(X)=\iota_{X}B$. Consequently
$[H]\in H^{3}(\mathcal{A})$ is independent of the isotropic splitting and
determines the $\mathcal{A}$-Courant algebroid structure completely. Hence,
this cohomology class is the natural extension of the Severa-class of exact
Courant algebroids.

####  2.4.2. Lifting Dirac structures

The answer to the question on how to lift structures takes different forms
depending on what one is trying to lift. Here we will focus on Dirac and
generalized complex structures, whose definitions on $\mathbb{A}$ is
completely analogous to the definition on $\mathbb{T}M$:

######  Definition 2.12.

Let $\mathcal{A}\rightarrow M$ be a Lie algebroid. An $\mathcal{A}$-Dirac
structure is a subbundle $E_{\mathcal{A}}\subset\mathbb{A}$ together with a
three-form $H\in\Omega^{3}_{\rm cl}(\mathcal{A})$, such that $E_{\mathcal{A}}$
is isotropic with respect to the pairing and involutive with respect to the
bracket on $\mathbb{A}$.

An $\mathcal{A}$-generalized complex structure is a pair
$(\mathcal{J}_{\mathcal{A}},H_{\mathcal{A}})$ where
$H_{\mathcal{A}}\in\Omega^{3}_{\rm cl}(\mathcal{A})$ and
$\mathcal{J}_{\mathcal{A}}$ is an orthogonal complex structure
$\mathcal{J}_{\mathcal{A}}$ on $\mathbb{A}$ for which $\mathbb{A}^{1,0}$, the
$+i$-eigenbundle, defines an $\mathbb{A}_{\mathbb{C}}$-Dirac structure.

######  Definition 2.13.

Let $\mathcal{A}\rightarrow M$ be a Lie algebroid. An $\mathcal{A}$-Dirac
structure $(E_{\mathcal{A}},H_{\mathcal{A}})$ is a lift of a Dirac structure
$(E,H)$ on $M$ if $H_{\mathcal{A}}=\rho_{\mathcal{A}}^{*}H$ and
$\mathfrak{F}_{\rho}(E_{\mathcal{A}})=E$.

Here $\mathfrak{F}_{\rho}$ denotes the Dirac pushforward via the anchor map.

This notion matches the notion of a lift of a Poisson structure, when that is
considered as a Dirac structure. Indeed, given a Lie algebroid
$\mathcal{A}\rightarrow M$, a Poisson structure
$\pi_{\mathcal{A}}\in\Gamma(\wedge^{2}\mathcal{A})$ and a Poisson structure
$\pi\in\mathfrak{X}^{2}(M)$ one can readily see that
$\rho_{*}\pi_{\mathcal{A}}=\pi$ if and only if $E_{\pi_{\mathcal{A}}}={\rm
graph}(\pi_{\mathcal{A}})\subset\mathbb{A}$ is a lift of $E_{\pi}={\rm
graph}(\pi)$.

For generalized complex structures, however, working only with the push-
forward map is too restrictive, so the notion of lift requires some tweaking.

######  Definition 2.14.

Let $\rho\colon\mathcal{A}\to TM$ be a Lie algebroid over $M$

  * •

$\tilde{X}+\tilde{\xi}\in\mathbb{A}$ is related to $X+\xi\in\mathbb{T}M$ if
$\rho(\tilde{X})=X$ and $\rho^{*}\xi=\tilde{\xi}$.

  * •

A generalized complex structure $(\mathcal{J}_{\mathcal{A}},H_{\mathcal{A}})$
on $\mathbb{A}$ is a lift of a generalized complex structure $(\mathcal{J},H)$
on $M$ if whenever $\tilde{X}+\tilde{\xi}\in\mathbb{A}$ is related to
$X+\xi\in\mathbb{T}M$, $\mathcal{J}_{\mathcal{A}}(\tilde{X}+\tilde{\xi})$ is
related to $\mathcal{J}(X+\xi)$.

######  Example 2.15.

In the situation above, if $J\colon TM\rightarrow TM$ and
$J_{\mathcal{A}}\colon\mathcal{A}\to\mathcal{A}$ are complex structures we can
promote them to generalized complex structures and ask when $J_{\mathcal{A}}$
is a lift of $J$. It turns out this happens if and only if the following
diagram commutes

${\mathcal{A}}$${\mathcal{A}}$${TM}$${TM}$$\scriptstyle{J_{\mathcal{A}}}$$\scriptstyle{\rho}$$\scriptstyle{\rho}$$\scriptstyle{J}$

which agrees with what one would like to call the lift of a complex structure.

######  Remark 2.0.

In this paper we will work with Lie algebroids $\mathcal{A}$ with dense
isomorphism locus, $U$. In this case, we see that $\mathcal{J}_{\mathcal{A}}$
is a lift of $\mathcal{J}$ on $U$ if and only if
$\mathcal{J}_{\mathcal{A}}|_{U}=(\rho|_{U})\hskip 0.28453pt\circ\hskip
0.28453pt\mathcal{J}\hskip 0.28453pt\circ\hskip 0.28453pt(\rho|_{U})^{-1}$. If
that holds, by continuity $\mathcal{J}_{\mathcal{A}}$ is related to
$\mathcal{J}$ in all of $M$. Therefore we see that the question of whether
$\mathcal{J}$ admits a lift to a Lie algebroid with dense isomorphism locus
becomes the question of existence of a smooth extension of $(\rho|_{U})\hskip
0.28453pt\circ\hskip 0.28453pt\mathcal{J}\hskip 0.28453pt\circ\hskip
0.28453pt(\rho|_{U})^{-1}$ beyond the isomorphism locus.

######  Example 2.16.

It is also good to keep in mind that there are (generalized) complex
structures on Lie algebroids, which do arise as lifts of structures on $M$.
Consider, for example, the elliptic tangent bundle
$\mathcal{A}_{\left|D\right|}$ on $\mathbb{C}^{2}$, associated to the ideal
$\left\langle r_{1}^{2}r_{2}^{2}\right\rangle$. The following defines a
complex structure on $\mathcal{A}_{\left|D\right|}$:

$\displaystyle z_{1}\partial_{z_{1}}\mapsto z_{2}\partial_{z_{2}},$
$\displaystyle\quad\overline{z_{1}}\partial_{\overline{z_{1}}}\mapsto\overline{z_{2}}\partial_{\overline{z_{2}}},$
$\displaystyle z_{2}\partial_{z_{2}}\mapsto-z_{1}\partial_{z_{1}},$
$\displaystyle\quad\overline{z_{2}}\partial_{\overline{z_{2}}}\mapsto-\overline{z_{1}}\partial_{\overline{z_{1}}}.$

However, the induced complex structure on $\mathbb{C}^{2}\setminus D$ does not
extend over $D$, as we have, for example,

$\partial_{z_{1}}\mapsto\frac{z_{2}}{z_{1}}\partial_{z_{2}}.\hfill$

##  3\. Classification of torus actions and connections

Having introduced the geometric structures of interest last section, now we
move on to study manifolds with the relevant torus actions. We begin this
section by recalling the classification of certain torus actions by Haefliger
and Salem in [HS91]. The relevant cohomology class for this classification is
$\check{H}^{2}(B;\mathbb{Z}^{n})$, where $B$ is the quotient space of the
torus action. Although the Haefliger–Salem class greatly resembles the Chern
class of a principal $S^{1}$-bundle, no such description exists at the moment
in terms of connections and curvature. We will use the elliptic tangent bundle
to provide a connection point of view on this class.

###  3.1. Haefliger–Salem classification

The first ingredient of the classification is the following result:

######  Theorem 3.1 (Theorem 3.1 in [HS91]).

Let $M$ be a manifold with a $T^{k}$-action, and let $\pi\colon M\rightarrow
B$ denote the quotient map. Any diffeomorphism $h\colon M\to M$ commuting with
the action of $T^{k}$ and preserving all orbits is of the form
$h(x)=f(\pi(x))\cdot x$, where $f\in C^{\infty}(B;T^{k})$.

Here we say that $f\in C^{\infty}(B)$ if $\pi^{*}f$ is smooth on $M$.
Similarly, we define $\Omega^{\bullet}(B)$ to correspond to
$\Omega^{\bullet}_{\rm bas}(M)$.

######  Remark 3.0.

The discussion in [HS91] concerns itself with the more general setting in
which $M$ is also allowed to be an orbifold. As our case of interest is when
the total space is a smooth manifold we will immediately restrict to that
case.

######  Definition 3.2.

Let $M,M^{\prime}$ be two manifolds endowed with $T^{k}$-actions and let
$\pi:M\rightarrow B,\pi^{\prime}:M^{\prime}\rightarrow B$ denote the quotient
maps. Then $M$ and $M^{\prime}$ are said to be locally equivalent if each
point in $B$ has a neighbourhood $U$ such that there exists a
$T^{k}$-equivariant diffeomorphism from $\pi^{-1}(U)$ to $\pi^{\prime-1}(U)$,
covering the identity on $B$.

When fixing an effective $T^{k}$-action on a manifold $M$ it is possible to
classify all other such actions which are locally equivalent to $M$:

######  Proposition 3.3 (Proposition 4.2 in [HS91]).

Let $\pi:M\rightarrow B$ be a manifold endowed with an effective torus action
with connected isotropy groups. The equivalence classes of manifolds endowed
with locally equivalent torus actions form an affine space modelled on
$\check{H}^{2}(B;\mathbb{Z}^{k})$.

Since we will need to get our hands on the cohomology classes prescribed by
the proposition, we give a sketch of the proof. Notice that there is no
preferred manifold endowed with a torus action locally equivalent to that of
$M$ and the class in question measures how different from $M$ another such
manifold is.

###### Proof.

Let $\pi^{\prime}:M^{\prime}\rightarrow B$ be a manifold endowed with a
locally equivalent torus action. Therefore, there exists a cover
$\mathcal{U}=\\{U_{i}\\}_{i\in I}$ of $B$ together with equivariant
diffeomorphisms $h_{i}:\pi^{-1}(U_{i})\rightarrow\pi^{\prime-1}(U_{i})$. The
transition maps $h_{j}^{-1}\hskip 0.28453pt\circ\hskip
0.28453pth_{i}:\pi^{-1}(U_{i}\cap U_{j})\rightarrow\pi^{-1}(U_{i}\cap U_{j})$
satisfy the assumptions of Theorem 3.1 and therefore there exists $f_{ji}\in
C^{\infty}(U_{i}\cap U_{j};T^{k})$ such that $h_{j}^{-1}\hskip
0.28453pt\circ\hskip 0.28453pth_{i}(x)=f_{ji}(\pi(x))\cdot x$. These functions
combine to give a cocycle
$\\{f_{ji}\\}\in\check{C}^{1}(\mathcal{U};\mathcal{T}^{k})$, where
$\mathcal{T}^{k}$ is the sheaf of smooth functions with values in $T^{k}$.
Using the exponential sequence one obtains the desired class in
$\check{H}^{2}(\mathcal{U};\mathbb{Z}^{k})$.

Conversely, given a class in $\check{H}^{2}(\mathcal{U};\mathbb{Z}^{k})$, one
uses again the exponential sequence to obtain a cocyle
$\\{f_{ji}\\}\in\check{C}^{1}(\mathcal{U};\mathcal{T}^{k})$. Using this
cocycle one may glue the disjoint unions of the $\pi^{-1}(U_{i})$ to obtain a
manifold $M^{\prime}$ endowed with a locally equivalent torus action. One
readily checks that cohomologous cocycles result in equivalent $T^{k}$
manifolds. ∎

######  Definition 3.4.

Let $\pi:M\rightarrow B$ be a manifold endowed with an effective torus action
with connected isotropy groups, and $\pi^{\prime}:M^{\prime}\rightarrow B$ be
a locally equivalent action. We will refer to the corresponding class in
$\check{H}^{2}(B,\mathbb{Z}^{k})$ as the Haefliger-Salem class of $M^{\prime}$
with respect to $M$.

######  Remark 3.0.

A key difference with the classification of principal torus bundles in terms
of their Chern class, is that the classification in Proposition 3.3 is affine
in nature. Given a base $B$, there is no canonical torus actions playing the
role $B\times T^{k}$ has in the principal case.

######  Remark 3.0.

Note that the results in [HS91] are about orbifolds, but as we assume that $M$
is a smooth manifold, any locally equivalent spaces will be smooth manifolds
as well.

###  3.2. Torus actions, divisors and connections

We will lift torus actions to the elliptic tangent bundle in such a way that
the associated infinitesimal action becomes free. The required notion we need
is the following:

######  Definition 3.5.

An effective $T^{k}$ action on a manifold with an elliptic divisor
$(M,I_{|D|})$ is compatible with the divisor if

  * •

the isotropy group of each point is connected and

  * •

there exists an injective Lie algebroid homomorphism
$\tilde{a}:\mathfrak{t}^{k}\rightarrow\mathcal{A}_{\left|D\right|}$ such that
the following diagram commutes:

${\mathcal{A}_{\left|D\right|}}$${\mathfrak{t}^{k}}$${TM,}$$\scriptstyle{\tilde{a}}$$\scriptstyle{a}$

where $a$ is the infinitesimal action $a\colon\mathfrak{t}^{k}\to TM$.

Studying general torus actions goes beyond the scope of this article. For
instance, one problem which occurs is that there is too much freedom in
choosing a compatible elliptic divisor:

######  Example 3.6.

Consider $\mathbb{C}^{2}$, endowed with the $S^{1}$-action
$\theta_{1}\cdot(z_{1},z_{2})=(e^{i\theta_{1}}z_{1},z_{2})$ and the compatible
elliptic divisor generated by $\left|z_{1}\right|^{2}\left|z_{2}\right|^{2}$.
The quotient space $B=\mathbb{R}_{\geq 0}\times\mathbb{C}$, which obtains a
quotient divisor $I_{\operatorname{red}}=\left\langle
t\left|z_{2}\right|^{2}\right\rangle$. Which is a mixture between an elliptic
and real log divisor. More generally, for a $T^{k}$-action on $\mathbb{C}^{n}$
the induced divisor on the base can have any kind of mixture between elliptic
and real log divisors.

We restrict ourselves to a very well-behaved class of singular torus actions,
but still interesting enough to allow for many examples:

######  Definition 3.7.

We say that a $T^{k}$-action on a manifold $M^{2k+l}$ is standard if each
orbit has a neighbourhood $U$ such that

$U\simeq T^{k-l}\times\mathbb{C}^{l}\times\mathbb{R}^{k},$

where $T^{k}$ acts on $T^{k-l}$ by multiplication, acts trivially on
$\mathbb{R}^{k}$ and on $\mathbb{C}^{l}$ by

(3.1)
$(e^{i\theta_{1}},\ldots,e^{i\theta_{l}})\cdot(z_{1},\ldots,z_{l})=(e^{i\theta_{1}}z_{1},\ldots,e^{i\theta_{l}}z_{l}),$

possibly after changing the generators of $T^{k}$.

For general $T^{k}$-actions, deriving conditions when they are standard is not
simple. However, for full torus actions one can show the following:

######  Lemma 3.8.

A $T^{n}$-action on a manifold $M^{2n}$ which is effective and has connected
isotropy groups is standard.

The local normal form of a standard $T^{k}$-action immediately implies the
following:

######  Proposition 3.9.

Let $\pi:M\rightarrow B:=M/T^{k}$ be a manifold endowed with a standard
$T^{k}$-action. Then

  * •

$B$ is a manifold with corners.

  * •

The action is compatible with the elliptic divisor
$I_{\left|D\right|}:=\pi^{*}I_{\partial B}$.

  * •

$\pi$ induces a Lie algebroid submersion
$\mathcal{A}_{\left|D\right|}\rightarrow\mathcal{A}_{\partial B}$.

In particular there is a canonical compatible elliptic divisor associated to a
standard $T^{k}$-action. Therefore, in what follows when given a standard
$T^{k}$-action on $M$, $I_{\left|D\right|}$ will always denote the canonical
associated elliptic divisor.

######  Example 3.10.

Consider $S^{n}\subset\mathbb{R}^{n+1}$, and the circle action given by
rotating the first two coordinates. This action is standard with quotient
space diffeomorphic to $D^{n}$ and elliptic divisor generated by
$x_{1}^{2}+x_{2}^{2}$.

Because a standard $T^{k}$-action lifts to the Lie algebroids so does the
projection map, and we obtain a diagram as follows:

${\mathcal{A}_{\left|D\right|}}$${\mathcal{A}_{\partial_{B}}}$${M}$${B}$$\scriptstyle{\tilde{\pi}}$$\scriptstyle{\pi}$

Hence, we obtain a short exact sequence of vector bundles over $M$:

$0\rightarrow\ker
d\tilde{\pi}\rightarrow\mathcal{A}_{\left|D\right|}\rightarrow\pi^{*}\mathcal{A}_{\partial_{B}}\rightarrow
0$

We may quotient by $T^{k}$, to obtain the sequence of vector bundles over $B$:

$0\rightarrow
B\times\mathfrak{t}^{k}\rightarrow\mathcal{A}_{\left|D\right|}/T^{k}\rightarrow\mathcal{A}_{\partial_{B}}\rightarrow
0$

One may think of this sequence as an $\mathcal{A}_{\left|D\right|}$ version of
the Atiyah sequence for principal bundles. With this in mind, vector bundle
splittings of this sequence give rise to the notion of connections for the
$T^{k}$-action on $M$. Concretely, in terms of differential forms we have:

######  Definition 3.11.

Let $M$ be a manifold endowed with a $T^{k}$ action, and let
$I_{\left|D\right|}$ be a compatible elliptic divisor. A connection-one form
on $M$ is a Lie algebroid one-form
$\Theta\in\Omega^{1}(\mathcal{A}_{\left|D\right|};\mathfrak{t}^{k})$ such that

  * •

$m_{\lambda}^{*}\Theta=\Theta$ for all $\lambda\in T^{k}$

  * •

$\iota_{a(v)}\Theta=v$, for all $v\in\mathfrak{t}^{k}$.

####  3.2.1. Complex line bundles

Let $L\rightarrow M$ be a complex line bundle. Then $\Gamma(L^{1,0*})\subset
C^{\infty}(M;\mathbb{C})$ naturally defines a smooth complex log divisor. The
associated elliptic divisor $I_{\left|D\right|}$ is naturally compatible with
the standard $S^{1}$-action on $L$.

Connections for the $S^{1}$-action arise from linear connections on $L$:

######  Lemma 3.12.

Let $\nabla$ be a complex linear connection on a complex line bundle
$L\rightarrow M$, and let $\zeta\in\Omega^{1}(L\backslash M;\mathbb{C})$
denote the corresponding connection one-form on the associated
$\mathbb{C}^{*}$-bundle. Then $\zeta$ extends over the zero-section as a
complex logarithmic form, and
$\Theta:=\Im(\zeta)\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ defines a
connection one-form for the associated $S^{1}$-action

###### Proof.

Let $z_{\alpha}:L|_{U_{\alpha}}\rightarrow\mathbb{C}$ be local fibre
coordinates of $L$. Write $z_{\alpha}=r_{\alpha}e^{i\theta_{\alpha}}$, and
consider the complex log forms $d\log z_{\alpha}=d\log
r_{\alpha}+id\theta_{\alpha}$. If $z_{\alpha}=g^{\beta}_{\alpha}z_{\beta}$,
with $g^{\alpha}_{\beta}$ transition functions, we find that $d\log
z_{\alpha}-d\log z_{\beta}=d\log g^{\alpha}_{\beta}$. Given a complex linear
connection $\nabla$ on $L$, we may consider the corresponding connection one-
forms $\xi_{\alpha}\in\Omega^{1}(U_{\alpha};\mathbb{C})$, which satisfy
$\xi_{\alpha}-\xi_{\beta}=d\log g^{\alpha}_{\beta}$. We conclude that
$\zeta:=d\log z_{\alpha}+\xi_{\alpha}$ defines a global complex logarithmic
form ∎

If we let $K=K_{1}+iK_{2}$ denote the curvature of the connection $\nabla$ on
$L$, we find that $d\Theta=K_{2}$ is actually a smooth form.

######  Remark 3.0.

The form $\Theta$, restricted to the complement of the zero-section, also goes
by the name of the global angular form. The above discussion shows that the
global angular form can be extended over the zero-section as an elliptic
differential form.

###  3.3. Smoothness of the curvature

Given a standard $T^{k}$-action and a corresponding connection one-form
$\Theta\in\Omega^{1}(\mathcal{A}_{\left|D\right|};\mathfrak{t}^{k})$, the
curvature, $d\Theta$, is basic and descends to a form in
$\Omega^{2}(\mathcal{A}_{\partial B};\mathfrak{t}^{k})$. We saw for complex
line bundles that there are connections for which the curvature was in fact a
smooth form. This is a general phenomenon for standard torus actions. To proof
this, we need to understand the cohomology of the log-tangent bundle on the
base:

######  Proposition 3.13 ([MS18],[GLPR17]).

Let $(M,I_{Z})$ be a manifold endowed with real log divisor. Then there exists
an isomorphism

(3.2) $H^{\bullet}(\mathcal{A}_{Z})\simeq H^{\bullet}(M)\oplus
H^{\bullet-1}(Z[1])\oplus H^{\bullet-2}(Z[2])\oplus\cdots,$

where $Z[n]$ denotes the intersection stratification of $Z$. The maps
$H^{\bullet}(\mathcal{A}_{Z})\rightarrow H^{\bullet-i}(Z[i])$ are induced by
the residue maps $\operatorname{Res}^{i}$, and the map
$H^{\bullet}(M)\rightarrow H^{\bullet}(\mathcal{A}_{Z})$ is the pullback via
the anchor.

######  Remark 3.0.

The map $\operatorname{Res}^{2}$ is defined combining the local residues
$\operatorname{Res}_{x_{i}x_{j}}$, as in (2.1) We refer to ([MS18],[GLPR17])
for the details.

Using (3.2) we see that we can associate a well-defined smooth cohomology
class to a class in $H^{\bullet}(\mathcal{A}_{Z})$. Explicitly we may also
phrase this as follows:

######  Lemma 3.14.

Let $(M,Z)$ be a manifold endowed with a immersed hypersurface, with self-
transverse intersections. Let $\alpha\in\Omega^{\bullet}(M)$ be such that
$[\rho^{*}\alpha]=0\in H^{\bullet}(\mathcal{A}_{Z})$. Then $[\alpha]=0\in
H^{\bullet}(M)$.

Recall from Lemma 3.9 that the quotient map of a standard $T^{k}$-action
induces a Lie algebroid submersion
$\pi:\mathcal{A}_{\left|D\right|}\rightarrow\mathcal{A}_{\partial B}$. This
map is compatible with the residues in the following manner.

######  Lemma 3.15.

Let $\pi:M\rightarrow B$ be endowed with a standard $T^{k}$-action. Then for
$i=1,2$ the following diagram commutes:

${\Omega^{2}(\mathcal{A}_{\left|D\right|})}$${\Omega^{2-i}(T^{i}ND[i])}$${\Omega^{2}(\mathcal{A}_{\partial
B})}$${\Omega^{2-i}(\partial
B[i])}$$\scriptstyle{\operatorname{Res}_{r}^{i}}$$\scriptstyle{\operatorname{Res}^{i}}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi^{*}}$

As $D[2]$ is a discrete set of points, $\operatorname{Res}^{2}_{r}\omega$ is
defined to be $\operatorname{Res}_{r_{1}r_{2}}(\omega)(p)$. A similar result
holds for higher order residues, but is more involved to state.

Using the description of the cohomlogy of $\mathcal{A}_{\partial B}$, we can
show that the curvature of a connection, $d\Theta$, is cohomologically smooth:

######  Lemma 3.16.

Let $M$ be endowed with a standard $T^{k}$-action and let
$\Theta\in\Omega^{1}(\mathcal{A}_{\left|D\right|};\mathfrak{t}^{k})$ be a
connection-one form with curvature $K\in\Omega^{2}(\mathcal{A}_{\partial
B},\mathfrak{t}^{k})$. Then there exists a smooth form
$\tilde{K}\in\Omega^{2}(B,\mathfrak{t}^{k})$ with $[\rho^{*}\tilde{K}]=[K]\in
H^{2}(\mathcal{A}_{\partial_{B}};\mathfrak{t}^{k})$.

###### Proof.

Using (3.2) we have to show that the residues $\operatorname{Res}^{i}$ of $K$
vanish. By rank reasons $\operatorname{Res}^{i}$ vanishes for $i\geq 3$. We
have that
$p^{*}\operatorname{Res}^{1}(K)=\operatorname{Res}^{1}_{r}(d\Theta)=d\operatorname{Res}^{1}_{r}(\Theta)$.
By $T^{n}$-invariance of $\Theta$ we must have that
$\operatorname{Res}^{1}_{r}(\Theta)$ is $T^{n}$-invariant as well, and thus of
the form $p^{*}f$, with $f\in C^{\infty}(B)$. Therefore
$p^{*}\operatorname{Res}^{1}(K)=p^{*}df$, and we conclude that
$\operatorname{Res}^{1}(K)$ is exact. We also have that
$p^{*}\operatorname{Res}^{2}(K)=\operatorname{Res}^{2}_{r}(d\Theta)$, but the
latter is necessarily zero by smoothness of the coefficient functions of
elliptic one-forms. We conclude that all the residues of $[K]$ vanish, as
these are precisely the maps $H^{\bullet}(\mathcal{A}_{\partial B})\rightarrow
H^{\bullet-i}(\partial B[i])$ in Equation (3.2), we conclude that
$[K]=[\rho^{*}\tilde{K}]$, for some smooth
$\tilde{K}\in\Omega^{2}(B;\mathfrak{t}^{k})$. ∎

The following example shows that Lemma 3.14 is not a given when the elliptic
divisor and action are independent:

######  Example 3.17.

Let $M$ be a manifold endowed with a smooth elliptic divisor
$I_{\left|D\right|}$, with $D$ co-orientable, and consider the trivial
$S^{1}$-bundle, $\pi:M\times S^{1}\rightarrow M$. The elliptic divisor is
trivially compatible with the $S^{1}$-action, and hence we can choose a
connection one-form for this action. Any such form will be given by
$\Theta=\alpha+d\theta$, with
$\alpha\in\Omega^{1}(\mathcal{A}_{\left|D\right|})$ and
$d\theta\in\Omega^{1}(S^{1})$.

We may pick $\alpha=0$, and then the curvature of $\Theta$ simply vanishes.
However, we may change $\alpha$ in such a manner that $d\Theta$ remains smooth
but is no longer exact as an element in $H^{2}(M)$, contradicting the
conclusion of Lemma 3.14.

The construction goes as follows: Consider a metric on a tubular neighbourhood
$\mathcal{U}$ of $D$ in $M$, and let $r^{2}:\mathcal{U}\rightarrow\mathbb{R}$
denote the associated squared distance function. Let
$\Theta_{M}\in\Omega^{1}(\mathcal{A}_{\left|D\right|})$ be a connection one-
form on $\mathcal{U}$, obtained as in Lemma 3.12 by viewing $\mathcal{U}$ as
an (open) in a complex line bundle.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a smooth function with ${\rm
supp}f\subset[-1,1]$, and $f|_{[-\epsilon,\epsilon]}\equiv 1$ for some
$\varepsilon<1$. Define an elliptic form $\alpha=f(r^{2})\Theta_{M}$. The
cohomology class $d\alpha$ is concentrated in $\mathcal{U}$, where it is
precisely the curvature of the normal bundle of $D$. Consequently, we see that
$\Theta=\alpha+d\theta$ has smooth curvature, which is not cohomologous to
zero.

####  3.3.1. Local description of connection one-forms

For future reference we recall the local description for connection one-forms
on principal bundles:

######  Proposition 3.18.

Let $\pi:P\rightarrow B$ be a principal $G$-bundle, and let
$h_{i}:\pi^{-1}(U_{i})\rightarrow U_{i}\times G$ be local trivialisations for
$i=1,2$. Let $\Theta\in\Omega^{1}(P;\mathfrak{g})$ be a connection one-form on
$P$. Then

$h_{i}^{*}\Theta=\omega_{\rm MC}+\pi^{*}\alpha_{i},$

with $\omega_{\rm MC}\in\Omega^{1}(G;\mathfrak{g})$ the Maurer-Cartan form,
and $\alpha_{i}\in\Omega^{1}(U_{i};\mathfrak{g})$. Moreover, if we let
$f_{ji}\in C^{\infty}(U_{i}\cap U_{j};G)$ be the transition functions given by
$h_{j}^{-1}\hskip 0.28453pt\circ\hskip 0.28453pth_{i}(x,g)=(x,f_{ji}(x)\cdot
g)$, we have

$\alpha_{i}-{\rm Ad}^{*}_{f_{ji}}\alpha_{j}=\pi^{*}d\log f_{ji}.$

###  3.4. Cech–de Rham isomorphism

To continue, we need to recall a basic version of the Čech-de Rham
isomorphism.

Let $M$ be endowed with a proper group action by some group $G$. Let
$\Omega^{n}_{\operatorname{bas}}$ denote the sheaf of basic differential forms
and let $Z^{n}_{\rm bas}$ denote the sheaf of closed basic differential forms.
Both are sheaves of $C^{\infty}(M)_{\rm bas}$-modules and fit into the short
exact sequence

(3.3) $0\rightarrow Z^{n}_{\rm
bas}\rightarrow\Omega^{n}_{\operatorname{bas}}\rightarrow Z^{n+1}_{\rm
bas}\rightarrow 0.$

Consider the basic Čech cocycle groups
$C_{\operatorname{bas}}^{k}(\mathcal{U},\mathcal{F})$, for which the maps
$U_{i_{1}\ldots i_{k}}\rightarrow\mathcal{F}$ are invariant with respect to
the $G$-action on $U_{i_{1}\ldots i_{k}}$.

By existence of a $G$-invariant partition of unity on $M$, the sheaves
$\Omega^{n}_{\rm bas}$ are fine, and hence
$H^{i}_{\operatorname{bas}}(M,\Omega^{n}_{\rm bas})=\\{0\\}$ for all $i\geq
1$. Therefore the connecting morphism of the above exact sequence induces an
isomorphism $H^{i}_{\operatorname{bas}}(M,Z^{n+1}_{\rm bas})\rightarrow
H^{i}_{\operatorname{bas}}(M,Z^{n}_{\rm bas})$. Applying this consecutively,
one obtains the isomorphism between
$H^{i+1}_{\operatorname{bas}}(M,\mathbb{R})\rightarrow
H^{1}_{\operatorname{bas}}(M,Z^{i}_{\operatorname{bas}})$. As the latter
corresponds to $H^{i+1}_{dR,\operatorname{bas}}(M)$ one obtains the desired
isomorphism.

We also consider a basic version of the exponential sequence:

(3.4) $0\rightarrow\mathbb{Z}^{n}\rightarrow C^{\infty}_{\rm
bas}(M;\mathbb{R}^{n})\rightarrow
C^{\infty}_{\operatorname{bas}}(M;T^{n})\rightarrow 0.$

The long exact sequence will induce an isomorphism
$\check{H}^{1}_{\operatorname{bas}}(M,C^{\infty}_{\operatorname{bas}}(M;T^{n}))\rightarrow\check{H}^{2}_{\rm
bas}(M;\mathbb{Z}^{n})$. Combining this with the Čech-de Rham isomorphism we
obtain a map to $H^{2}_{\rm bas}(M;\mathbb{Z}^{n})$.

###  3.5. Connection approach to Haefliger-Salem class

We now have all the theory in place to compare the curvature of a connection
for a standard $T^{k}$-action with the Haefliger-Salem class:

######  Proposition 3.19.

Let $\pi:M\rightarrow B$ be a manifold endowed with a standard $T^{k}$-action
and let $\pi^{\prime}:M^{\prime}\rightarrow B$ be a manifold endowed with a
locally equivalent action. Let $\Theta,\Theta^{\prime}$ be connection one-
forms on $M$ respectively $M^{\prime}$. Then $[d\Theta-d\Theta^{\prime}]\in
H^{2}(A_{\partial B};\mathfrak{t}^{k})$ is representable by a smooth form, and
descends to the Haefliger-Salem class of $M^{\prime}$ in
$\check{H}^{2}(B;\mathfrak{t}^{k})$.

###### Proof.

Let $I_{\left|D\right|},I_{\left|D^{\prime}\right|}$ denote the induced
elliptic divisors on $M$ respectively $M^{\prime}$. By Lemma 3.16, we have
that both $[d\Theta]$ and $[d\Theta^{\prime}]$ are representable by smooth
forms, therefore we can modify $\Theta$ and $\Theta^{\prime}$ such that
$d\Theta$ and $d\Theta^{\prime}$ are smooth forms. We will still denote the
modified connections by $\Theta$ and $\Theta^{\prime}$.

Let $\mathcal{U}=\\{U_{i}\\}$ be an open cover of $B$, together with
equivariant diffeomorphism
$h_{i}:\pi^{-1}(U_{i})\rightarrow\pi^{\prime-1}(U_{i})$ provided by the fact
that the actions on $M$ and $M^{\prime}$ are locally equivalent. Furthermore,
assume that over $U_{i}\backslash\partial B$ both $\pi$ and $\pi^{\prime}$ are
trivalisable as principal $T^{k}$-bundles.

Because the $h_{i}$ are equivariant, they are morphisms of divisors, and hence
we may consider
$h_{i}^{*}\Theta^{\prime}\in\Omega^{1}(\mathcal{A}_{\left|D\right|}|_{\pi^{-1}(U_{i})};\mathfrak{t}^{k})$.
One readily shows that $h_{i}^{*}\Theta^{\prime}$ is a connection on
$\pi^{-1}(U_{i})$ and thus $\eta_{i}:=\Theta-
h_{i}^{*}\Theta^{\prime}\in\Omega^{1}_{\operatorname{bas}}(\mathcal{A}_{\left|D\right|}|_{\pi^{-1}(U_{i})};\mathfrak{t}^{k})$
is a basic form.

Let $f_{ji}\in C^{\infty}(U_{i}\cap U_{j};T^{k})$ be the transition functions
given by $h_{j}\hskip 0.28453pt\circ\hskip 0.28453pth_{i}(x)=(x,f_{ji}(x))$.
By Lemma 3.18,
$\eta_{i}-\eta_{j}=h_{i}^{*}\Theta^{\prime}-h_{j}^{*}\Theta^{\prime}$
coincides with $\pi^{*}d\log f_{ji}$ on $\pi^{-1}((U_{i}\cap
U_{j})\backslash\partial B)$. By density, we have that
$\eta_{i}-\eta_{j}=\pi^{*}d\log f_{ji}$ on the entirety of $\pi^{-1}(U_{i}\cap
U_{j})$. As the $\eta_{i}$ are closed, we conclude that
$\\{\eta_{i}-\eta_{j}\\}_{ij}$ defines a cocycle in $\check{C}^{1}_{\rm
bas}(M,Z^{1}_{\operatorname{bas}})$.

The Haefliger-Salem class is defined by applying the exponential sequence to
the class $\\{f_{ji}\\}$ in $\check{H}^{1}(\mathcal{U};\mathcal{T}^{k})$.
Inspecting the Čech-de Rham isomorphism, we conclude that this class
corresponds to the class defined by $\\{\eta_{i}-\eta_{j}\\}_{ij}$. We are
left to show that
$\\{\eta_{i}-\eta_{j}\\}_{ij}\in\check{H}^{1}(\mathcal{U},Z^{1}_{\operatorname{bas}})$
corresponds to $[d\Theta-d\Theta^{\prime}]$. By exactness of (3.3), we know
that there exist a refinement $\tilde{\mathcal{U}}=\\{\tilde{U}_{i}\\}$ of
$\mathcal{U}$ and
$\tilde{\eta_{i}}\in\Omega^{1}_{\operatorname{bas}}(\pi^{-1}(\tilde{U}_{i}));\mathfrak{g})$
such that $\tilde{\eta}_{i}-\tilde{\eta}_{j}=\eta_{i}-\eta_{j}$. The class in
$H^{2}_{dR,\operatorname{bas}}(M,\mathfrak{t}^{k})$ corresponding to
$\\{\eta_{i}-\eta_{j}\\}_{ij}$ is given by $d\tilde{\eta}_{i}$.

Note that $\tilde{\eta}_{i}-\eta_{i}=\tilde{\eta}_{j}-\eta_{j}$, and hence
these forms glue to give a globally defined form
$\eta\in\Omega^{1}_{\operatorname{bas}}(M;\mathfrak{g})$. Because
$d\eta=d\tilde{\eta}_{i}-d\eta_{i}$, and $d\eta_{i}=d\Theta-
h_{i}^{*}d\Theta^{\prime}$, we have that $d\eta$ is smooth. Hence, by Lemma
3.14 we have that $d\eta$ is exact in $H^{2}(B;\mathfrak{t}^{k})$.
Consequently $[d\tilde{\eta}_{i}]=[d\eta_{i}]\in H^{2}(B;\mathfrak{t}^{k})$.
But as $d\eta_{i}=d\Theta-h_{i}^{*}d\Theta^{\prime}$, we conclude that the
class in $H^{2}_{dR,\operatorname{bas}}(M,\mathfrak{t}^{k})$ corresponding to
$\\{\eta_{i}-\eta_{j}\\}_{ij}$ is given by $d\Theta-d\Theta^{\prime}$, which
finishes the proof. ∎

The description of the Haefliger-Salem class in terms of connections has the
following consequence:

######  Lemma 3.20.

Let $\pi:M\rightarrow B$ be a standard $T^{k}$-action. Then there exists a
locally equivalent action $\pi^{\prime}:M^{\prime}\rightarrow B$ which admits
a flat connection.

###### Proof.

Let $\Theta$ be a connection on $M$, with integral curvature. The class
$[d\Theta]\in H^{2}(B;\mathbb{Z}^{k})$ corresponds to a locally equivalent
torus action $M^{\prime}\rightarrow B$. Given a connection $\Theta^{\prime}$
on $M^{\prime}$, we have by Proposition 3.19 that $[d\Theta-d\Theta^{\prime}]$
defines the Haefliger-Salem class of $M^{\prime}$. But as this class is given
by $[d\Theta]$, we conclude that $[d\Theta^{\prime}]=0$ which finishes the
proof. ∎

The importance of this lemma is that in the classification of standard
$T^{k}$-actions, it is now possible to pick as a reference an action
$M_{0}\rightarrow B$ with a flat connection. Using this reference, and a
locally equivalent action $M^{\prime}\rightarrow B$, we find that the
Haefliger-Salem class of $M_{0}$ with respect to $M^{\prime}$ is given exactly
by $[d\Theta^{\prime}]$ for some connection $\Theta^{\prime}$ on $M^{\prime}$.

######  Definition 3.21.

Let $M\rightarrow B$ be a standard $T^{k}$-action. The locally equivalent
action $M_{0}\rightarrow B$ which admits a flat connection is called the
reference action.

When $M\rightarrow B$ is a principal $T^{k}$-action, the reference action is
simply the trivial bundle $B\times T^{k}$. Contrary to in the principal case,
there might be several non locally equivalent standard $T^{k}$-actions over
the same base with flat connections. We will see examples of this in Section
5.2.1.

##  4\. The singular differential geometry of the correspondence space

Let $M\rightarrow B,\hat{M}\rightarrow B$ be manifolds endowed with standard
torus actions. The fibre-product $M\times_{B}\hat{M}\subset M\times\hat{M}$
fails to be a smooth submanifold. This space plays an important role in
$T$-duality, and therefore we will shortly study notions for smooth objects on
this space.

Let us start with the simplest example:

######  Example 4.1.

Let $M,\hat{M}$ both denote $\mathbb{C}$ endowed with the standard
$S^{1}$-action, and $B=\mathbb{R}_{\geq 0}$ the quotient. Then

$M\times_{B}\hat{M}=\\{(z,\tilde{z}):\left|z\right|=\left|\tilde{z}\right|\\},$

and hence we may consider singular coordinates
$(r,\theta,\hat{\theta})\in\mathbb{R}_{\geq 0}\times T^{2}$. These coordinates
are well-defined except at $r=0$. We see that $M\times_{B}\hat{M}$ is a cone
over a torus.

The space $\mathbb{C}\times_{\mathbb{R}_{\geq 0}}\mathbb{C}$ will serve as
local model for the spaces we want to consider, so we will describe smoothness
in terms of it:

######  Definition 4.2.

A function $f:\mathbb{C}\times_{\mathbb{R}_{\geq
0}}\mathbb{C}\rightarrow\mathbb{R}$ is smooth, if it is the restriction of a
smooth function $\tilde{f}\in C^{\infty}(\mathbb{C}\times\mathbb{C})$.

######  Definition 4.3.

A map $\Phi:\mathbb{C}\times_{\mathbb{R}_{\geq
0}}\mathbb{C}\rightarrow\mathbb{C}\times_{\mathbb{R}_{\geq 0}}\mathbb{C}$ is
said to be smooth, if for every $f\in
C^{\infty}(\mathbb{C}\times_{\mathbb{R}_{\geq 0}}\mathbb{C})$, $f\hskip
0.28453pt\circ\hskip 0.28453pt\Phi$ is smooth.

Taking products, we can talk about smooth maps and diffeomorphisms of
$(\mathbb{C}\times_{\mathbb{R}_{\geq
0}}\times\mathbb{C})^{l}\times\mathbb{R}^{k}$, and charts and atlases of this
form. From the local form of standard $T^{k}$-actions it immediately follows
that:

######  Lemma 4.4.

Let $M,\hat{M}$ be standard $T^{k}$-actions over the same base $B$. Then
$M\times_{B}\hat{M}$ admits an atlas of coordinate charts of the form
$(\mathbb{C}\times_{\mathbb{R}_{\geq
0}}\times\mathbb{C})^{l}\times\mathbb{R}^{k}$.

A point in $M\times_{B}\hat{M}$ has singular coordinates

(4.1)
$(r_{1},\ldots,r_{l},\theta_{1},\ldots,\theta_{l},\hat{\theta}_{1},\ldots,\hat{\theta}_{l},\psi_{1},\ldots,\psi_{n-l},x_{1},\ldots,x_{n-l},\hat{x}_{1},\ldots,\hat{x}_{n-l}).$

Here the $\\{r_{i}\\}$ are one set of radial coordinates on $\mathbb{C}^{l}$,
where $\\{\theta_{i}\\},\\{\hat{\theta}_{i}\\}$ are two sets of polar
coordinates on $\mathbb{C}^{l}$. The $\\{\psi_{i}\\}$ are honest coordinates
on $T^{n-l}$ and the $\\{x_{i}\\}$ and $\\{\hat{x}_{i}\\}$ are coordinates on
the two copies of $\mathbb{R}^{n-l}$.

######  Remark 4.0 (Induced divisor).

Let $M,\hat{M}$ be manifolds endowed with standard $T^{k}$-actions and same
base $B$ and let $I_{\left|D\right|}$ and $I_{\left|\hat{D}\right|}$ denote
the corresponding elliptic divisors. Then the restriction of
$I_{\left|D\right|}$ and $I_{\left|\hat{D}\right|}$ to $M\times_{B}\hat{M}$
coincides. In the coordinates of (4.1) the induced elliptic divisor on
$M\times_{B}\hat{M}$ is given by $\left\langle
r_{1},\ldots,r_{l}\right\rangle$.

Rather than trying to find a notion of tangent bundle for
$M\times_{B}\hat{M}$, we use the ambient ellipitc tangent bundle on
$M\times\hat{M}$ to define a Lie algebroid over $M\times_{B}\hat{M}$.

The first observation is that the way $M\times_{B}\hat{M}$ sits inside the
smooth manifold $M\times\hat{M}$ is compatible with the Lie algebroid
$\mathcal{A}_{\left|D\right|}\boxtimes\mathcal{A}_{\left|\hat{D}\right|}$ in
the following manner:

######  Definition 4.5.

Let $\mathcal{A}\rightarrow M$ be a Lie algebroid and $N\subset M$ a subset,
and let $\Gamma_{N}(\mathcal{A})$ denote the sections of $\mathcal{A}$ tangent
to $N$. Then $N$ is adapted to $\mathcal{A}$ if
$\operatorname{Sing}(N)\subseteq\\{x\in M:\rho_{x}\text{ is not of full
rank}\\}$ and the pull-back sheaf $\iota^{*}_{N}\Gamma_{N}(\mathcal{A})$ is
locally free over $\iota^{*}_{N}C^{\infty}(M)$.

As $\iota^{*}_{N}\Gamma(\mathcal{A})$ is locally free, we may view it as a
topological vector bundle over $N$. By declaring sections to be smooth if they
are restrictions of smooth sections of $\mathcal{A}$, we can view
$\mathcal{A}$ as a smooth Lie algebroid over $N$.

######  Example 4.6.

A subspace $N\subset M$ is adapted to $TM$ if and only if it is an embedded
submanifold. In this case the pull-back sheaf is simply given by vector fields
on $N$.

The rationale behind this definition is as follows. We allow subspaces to have
singularities, but they need to occur precisely when the anchor of
$\mathcal{A}$ is not an isomorphism.

######  Example 4.7.

Let $M=\mathbb{R}^{2}$, and let $\mathcal{A}$ be the Lie algebroid generated
by $x\partial_{x}$ and $y\partial_{y}$. Let $N$ denote the union of the
diagonal and anti-diagonal. If $X=fx\partial_{x}+gy\partial_{y}$ is tangent to
the diagonal, it implies it must preserve the ideal generated by $x-y$.
Therefore, $X=f(x\partial_{x}+y\partial_{y})$. As these vector fields are also
tangent to the anti-diagonal we conclude that $N$ is an $\mathcal{A}$-adepted
subspace, and the restricted Lie algebroid is generated by
$x\partial_{x}+y\partial_{y}$

From the local form of standard $T^{k}$-actions, we immediately obtain the
following:

######  Lemma 4.8.

Let $M,\hat{M}$ be standard $T^{k}$-actions over the same base $B$. Then
$M\times_{B}\hat{M}$ is an
$\mathcal{A}_{\left|D\right|}\boxtimes\mathcal{A}_{\left|\hat{D}\right|}$-adapted
subspace.

We denote the induced Lie algebroid on $M\times_{B}\hat{M}$ by
$\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|}$.

##  5\. Non prinicipal $T$-duality

Using the Lie algebroids from Section 2 we will extend the notions of
$T$-duality to standard torus actions. To proof the existence of $T$-duals we
will use the classification of torus actions, and the connection point of view
thereof, as described in Section 3. Moreover, we will stress the differences
with $T$-duality for principal torus bundles; in particular the new source of
topology change arising from the singularities of the torus action.

###  5.1. $T$-duality

In this subsection we will recall the notions of $T$-duality for principal
torus actions. We will be brief and refer to [BEM04] and [CG10] for more
details.

######  Definition 5.1.

Let $M$ and $\hat{M}$ be principal $T^{k}$-bundles with base $B$ and let
$H\in\Omega^{3}_{T^{k}}(M),\hat{H}\in\Omega^{3}_{T^{k}}(\hat{M})$ be
$T^{k}$-invariant closed three-forms. Consider the fibre-product,
$M\times_{B}\hat{M}$, and the following diagram:

${(M\times_{B}\hat{M},p^{*}H-\hat{p}^{*}\hat{H}),}$${(M,H)}$${(\hat{M},\hat{H})}$${B}$$\scriptstyle{p}$$\scriptstyle{\hat{p}}$$\scriptstyle{\pi}$$\scriptstyle{\hat{\pi}}$

We say that $(M,H)$ and $(\hat{M},\hat{H})$ are $T$-dual if there exists a
$T^{2k}$-invariant two-form $F\in\Omega^{2}_{T^{2k}}(M\times_{B}\hat{M})$ such
that

$dF=p^{*}H-\hat{p}^{*}\hat{H},$

and the restriction

$F:\mathfrak{t}_{M}\times\mathfrak{t}_{\hat{M}}\rightarrow\mathbb{R},$

where $\mathfrak{t}_{M},\mathfrak{t}_{\hat{M}}$ denotes the tangent space to
the fibres of $p$, $\hat{p}$ respectively, is non-degenerate. We call
$M\times_{B}\hat{M}$ the correspondence space.

$T$-duals always exist, provided the three-form is integral:

######  Theorem 5.2 ([BEM04]).

Let $M\rightarrow B$ be a principal $T^{k}$-bundle, and
$H\in\Omega^{3}_{T^{k}}(M)$ with integral cohomology class. Then there exists
a principal $T^{k}$-bundle, $\hat{M}$ $T$-dual to $M$.

$T$-dual spaces have isomorphic complexes of invariant differential forms:

######  Theorem 5.3 ([BEM04]).

Let $(M,H)$ and $(\hat{M},\hat{H})$ be $T$-dual with
$dF=p^{*}H-\tilde{p}^{*}\hat{H}$. Then

$\tau:(\Omega^{\bullet}_{T^{k}}(M),d_{H})\rightarrow(\Omega^{\bullet}_{T^{k}}(\hat{M}),d_{\hat{H}}),\quad\rho\mapsto\int_{T^{k}}e^{F}\wedge\rho,$

is an isomorphism of cochain complexes. Here the integral is over the fibres
of $\tilde{p}$.

###  5.2. Courant algebroids

$T$-duality induces an isomorphism of invariant sections of the standard
Courant algebroid:

######  Theorem 5.4 ([CG10]).

If $(M,H)$ and $(\hat{M},\hat{H})$ are $T$-dual, then there is an isomorphism
of Courant algebroids

$\varphi:(TM\oplus T^{*}M)/T^{k}\rightarrow(T\hat{M}\oplus
T^{*}\hat{M})/T^{k},$

such that $\tau(v\cdot\rho)=\varphi(v)\cdot\tau(\rho)$, for all $v\in(TM\oplus
T^{*}M)/T^{k}$ and $\rho\in\Omega^{\bullet}(M)$.

The strength of this description is that it immediately allows one to
transport structures:

######  Theorem 5.5 ([CG10]).

Let $(M,H)$ and $(\hat{M},\hat{H})$ be $T$-dual spaces. Then any Dirac,
generalized complex, generalized Kähler or SKT structure on $M$ which is
invariant under the torus action is transformed via $\varphi$ into a structure
of the same kind on $\hat{M}$.

We are now ready to introduce the notion of $T$-duality for standard torus
actions:

######  Definition 5.6.

Let $M,\hat{M}$ be two manifolds endowed with standard $T^{k}$-actions over
the same base $B$ and let $I_{\left|D\right|},I_{\left|\hat{D}\right|}$ be the
induced elliptic divisors. Furthermore, suppose we are given
$H\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|D\right|}),\hat{H}\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|\hat{D}\right|})$ closed and $T^{k}$-invariant. We
consider the following diagram:

${(\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|}\rightarrow
M\times_{B}\hat{M},p^{*}H-\hat{p}^{*}\hat{H})}$${(\mathcal{A}_{\left|D\right|}\rightarrow
M,H)}$${(\mathcal{A}_{\left|\hat{D}\right|}\rightarrow\hat{M},\hat{H})}$${\mathcal{A}_{\partial
B}\rightarrow
B}$$\scriptstyle{p}$$\scriptstyle{\hat{p}}$$\scriptstyle{\pi}$$\scriptstyle{\hat{\pi}}$

We say that $(M,H)$ and $(\hat{M},\hat{H})$ are $T$-dual if there exists a
$T^{2k}$-invariant two-form
$F\in\Omega^{2}(\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|})$
such that $p^{*}H-\hat{p}^{*}\hat{H}=dF$ and

(5.1) $F:\mathfrak{t}_{M}\otimes\mathfrak{t}_{\hat{M}}\rightarrow\mathbb{R},$

is non-degenerate.

Let us expand on this definition. First
$\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|}$ is
the Lie algebroid induced by viewing $M\times_{B}\hat{M}$ as an
$(\mathcal{A}_{\left|D\right|}\boxtimes\mathcal{A}_{\left|\hat{D}\right|})$-adapted
subspace of $M\times\hat{M}$. The two-form $F$ is a form for this Lie
algebroid, and can in practise be thought of as the restriction of a two-form
$\tilde{F}\in\Omega^{2}(\mathcal{A}_{\left|D\right|}\boxtimes\mathcal{A}_{\left|\hat{D}\right|})$.
Here $\mathfrak{t}_{M_{i}}$ denotes the kernel of the Lie algebroid submersion
$\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|}\rightarrow\mathcal{A}_{\left|D_{i}\right|}$
induced by $p_{i}$.

######  Example 5.7.

Consdier $\mathbb{C}$ with the standard $S^{1}$-action, and $H=0$. Then
$F=d\theta_{1}\wedge d\theta_{2}$ provides a $T$-dual from $(\mathbb{C},0)$ to
itself.

A connection one-form
$\Theta\in\Omega^{1}(\mathcal{A}_{\left|D\right|};\mathfrak{t})$ induces an
isomorphism
$\mathcal{A}_{\left|D\right|}/T^{k}\simeq\mathfrak{t}^{k}\oplus\mathcal{A}_{\partial
B}$, therefore we obtain a decomposition

$\Omega^{3}_{T^{k}}(\mathcal{A}_{\left|D\right|})=\sum_{i=0}^{3}\Omega^{i}(\mathcal{A}_{\partial
B},\wedge^{3-i}\mathfrak{t}^{*}).$

The $T$-duality conditions forces the component of $H$ in
$\Omega^{0}(\mathcal{A}_{\partial B};\wedge^{3}t^{*})$ to vanish, and the
component in $\Omega^{1}(\mathcal{A}_{\partial B};\wedge^{2}t^{*})$ to be
exact. When considering geometric structures (up to gauge equivalence), it is
the cohomology class $[H]$, rather than the form itself, which is of
importance. Hence we can impose that we consider $H$ which only have two terms
in this decomposition:

(5.2) $H=\left\langle\hat{c},\Theta\right\rangle+h,$

for some $\hat{c}\in\Omega^{2}(\mathcal{A}_{\partial B};\mathfrak{t}^{*})$ and
$h\in\Omega^{3}(\mathcal{A}_{\partial B})$. Using this decomposition, and the
relation of the curvature with the Haefliger-Salem class, we can now prove:

######  Proposition 5.8.

Given a manifold $M\rightarrow B$ endowed with a standard $T^{k}$ and an
invariant closed three-form $H\in\Omega^{3}(\mathcal{A}_{\left|D\right|})$. If
$[\hat{c}]$ as in Equation (5.2) defines a class in $H^{2}(B;\mathbb{Z}^{k})$,
then there a $T$-dual $\hat{M}\rightarrow B$.

###### Proof.

Let $M_{0}\rightarrow B$, be the reference torus action of $M$. Using
Proposition 3.3, with respect to $M_{0}$ and $\hat{M}$, the class
$[\hat{c}]\in H^{2}(B;\mathbb{Z}^{k})$ corresponds to a manifold
$\hat{M}\rightarrow B$ endowed with a locally equivalent action. Moreover,
there exists a connection
$\hat{\Theta}\in\Omega^{1}(\mathcal{A}_{\left|\hat{D}\right|};\mathfrak{t}^{k})$
for which $d\hat{\Theta}=\hat{c}$. Now define

$\hat{H}=\left\langle
c,\hat{\Theta}\right\rangle+h\in\Omega^{3}(\mathcal{A}_{\left|\hat{D}\right|}),$

with $c=d\Theta$, for some connection
$\Theta\in\Omega^{1}(M;\mathfrak{t}^{k})$. Now $(M,H)$ and $(\hat{M},\hat{H})$
are $T$-dual because

$p^{*}H-\tilde{p}^{*}\hat{H}=\left\langle\hat{c},\Theta\right\rangle-\left\langle
c,\hat{\Theta}\right\rangle=\left\langle
d\hat{\Theta},\Theta\right\rangle-\left\langle
d\Theta,\hat{\Theta}\right\rangle=-d\left\langle\Theta,\hat{\Theta}\right\rangle,$

and as $\Theta$ and $\hat{\Theta}$ are connection-one-forms, we obtain that
$F=-\left\langle\Theta,\hat{\Theta}\right\rangle$ satisfies the requirements
of $T$-duality. ∎

######  Remark 5.0.

Note that it might be preferable to put a direct condition on $H$, rather than
the $[\hat{c}]$ in the decomposition. However, this will also impose further
conditions on the $h$ term which are not strictly necessary. Still, in
practice we will assume that $H$ is smooth and defines an integral cohomology
class, which implies the assumptions in Proposition 5.8

######  Remark 5.0 (Non-uniqueness).

This proposition provides the existence of $T$-duals, but these are by no
means unique. Already in the case without fixed points, the construction does
not take any torsion of the torus bundle into account. But also changing the
specific $F$ used may change the topology of the resulting $T$-dual, see also
Bunke-Schick [BS05]. Now the non uniqueness goes even further: because we made
use of the classification of Haefliger-Salem, any $T$-dual produced using
Proposition 5.8 will necessarily be locally equivalent to $M$. However, that
needn’t be the case as we will see momentarily.

####  5.2.1. Topology change from singularities

The singularities of the torus actions provide a new source of topology change
for $T$-duals:

######  Proposition 5.9.

Let $M,\hat{M}$ be manifolds with standard $T^{k}$-actions over the same
contractible base $B$. Then $(M,H=0)$ and $(\hat{M},\hat{H}=0)$ are $T$-dual.

###### Proof.

Because the base of $M$ and $\hat{M}$ is contractible there exists flat
connections $\Theta,\hat{\Theta}$ on $M$ respectively $\hat{M}$. Then, as in
the proof of Proposition 5.8 (the restriction of)
$F=-\left\langle\Theta,\hat{\Theta}\right\rangle$ provides a $T$-dual between
$(M,0)$ and $(\hat{M},0)$. ∎

In [OR70] a classification of four-manifolds with effective $T^{2}$-actions
and connected isotropy groups appears. All such manifolds are of the following
form:

  * •

$\\#n(S^{2}\times S^{2})$, $n\in\mathbb{N}$

  * •

$\\#n(\mathbb{C}P^{2})\\#m(\overline{\mathbb{C}P^{2}})$, $n,m\in\mathbb{N}$

  * •

$S^{4}$

All of these manifolds have a contractible base, so we conclude by the above
Theorem that if two have the same base they are $T$-dual. The only invariant
of the base is the number of corners, which can be read of from the Euler
characteristic of the total space. For instance
$2\mathbb{C}P^{2},\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$ and
$S^{2}\times S^{2}$ are all $T$-dual. The common base of these manifolds is
contractible, and hence the actions are necessarily not locally equivalent.

Just as in principal $T$-duality, we obtain an isomorphism of differential
complexes. The proof is omitted as it is identical:

######  Proposition 5.10.

Let $M,\hat{M}$ be manifolds endowed with standard $T^{k}$-actions over the
same base $B$. Let $H\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|D\right|}),\hat{H}\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|\hat{D}\right|})$, and let
$F\in\Omega^{2}(\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|})$
provide a $T$-dual. Then

(5.3)
$\tau:\Omega^{\bullet}_{T^{k}}(\mathcal{A}_{\left|D\right|})\rightarrow\Omega^{\bullet}_{T^{k}}(\mathcal{A}_{\left|\hat{D}\right|}),\quad\tau(p)=\iota_{X_{1}\wedge\cdots\wedge
X_{n}}(e^{F}\wedge\rho),$

is an isomorphism of differential complexes
$(\Omega^{\bullet}_{T^{k}}(M;\mathcal{A}_{\left|D\right|}),d_{H})$ and
$(\Omega^{\bullet}_{T^{k}}(\hat{M};\mathcal{A}_{\left|\hat{D}\right|}),d_{\hat{H}})$.
Here $X_{1},\ldots,X_{n}\in\Gamma(\mathcal{A}_{\left|D\right|})$ are the
infinitesimal generators of the torus action on $M$.

Moreover, we may consider the standard $\mathcal{A}_{\left|D\right|}$-Courant
algebroid,
$\mathcal{A}_{\left|D\right|}\oplus\mathcal{A}_{\left|D\right|}^{*}$, and
obtain Theorem 5.4 in our setting. Again, the proof is identical to the
principal case and thus omitted.

######  Theorem 5.11.

Let $M,\hat{M}$ be manifolds endowed with standard $T^{k}$-actions over the
same base $B$. Let $H\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|D\right|}),\hat{H}\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|\hat{D}\right|})$, and assume that $(M,H)$ and
$(\hat{M},\hat{H})$ are $T$-dual. Then there is an isomorphism between Courant
algebroids
$(\mathcal{A}_{\left|D\right|}\oplus\mathcal{A}_{\left|D\right|}^{*})/T^{k}\rightarrow
M$ and
$(\mathcal{A}_{\left|D\right|}\oplus\mathcal{A}_{\left|D\right|}^{*})/T^{k}\rightarrow\hat{M}$.

Using this isomorphism we are able to transport geometric structures on
$\mathcal{A}_{\left|D\right|}\rightarrow M$ to geometric structures on
$\mathcal{A}_{\left|D\right|}\rightarrow\hat{M}$:

######  Theorem 5.12.

Let $(M,H)$ and $(\hat{M},\hat{H})$ be $T$-dual. Then any Dirac, generalized
complex or SKT structure on $\mathcal{A}_{\left|D\right|}\rightarrow M$
invariant under the torus action will be transported to a structure of the
same kind on $\mathcal{A}_{\left|D\right|}\rightarrow\hat{M}$.

However, there is one key difference compared to the case of principal
$T$-duality. Given a $T^{k}$-invariant generalized complex structure on $M$
which lifts to a generalized complex structure on
$\mathcal{A}_{\left|D\right|}\rightarrow M$, we can transport it to a
generalized complex structure on
$\mathcal{A}_{\left|\hat{D}\right|}\rightarrow\hat{M}$. However, a prior there
is no guarantee that it again descends to a generalized complex structure on
$M$. We illustrate this phenomenon by the following examples:

######  Example 5.13 ($\mathbb{\mathbb{C}}$).

Consider $M=\mathbb{C}$ with the standard $S^{1}$-action, and endow it with
the elliptic symplectic form $d\log r\wedge d\theta=\frac{dx\wedge
dy}{x^{2}+y^{2}}$. If we let $\hat{M}=\mathbb{C}$ denote another another copy
of $\mathbb{C}$, we have that $F=d\theta\wedge d\hat{\theta}$ provides a
$T$-dual. Under this $T$-duality, the elliptic symplectic structures
$e^{id\log r\wedge d\theta}$ is send to spinor $d\log\hat{z}$. This spinor
corresponds to the complex structure on
$\mathcal{A}_{\left|D\right|}=\\{r\partial_{r},\partial_{\theta}\\}$ defined
by $r\partial_{r}\mapsto\partial_{\theta}$. It is immediate to see that this
complex structure is a lift of the standard complex structure on $\mathbb{C}$.
Hence we conclude that the elliptic symplectic form, which is not a
generalized complex structure, is $T$-dual to the standard complex structure.

The next example will illustrate that $T$-duality for standard torus actions
is very sensitive to the specific
$F\in\Omega^{2}(\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|})$
chosen:

######  Example 5.14 ($\mathbb{\mathbb{C}}^{2}$).

Consider $M=\mathbb{C}^{2}$ with coordinates $(z_{1},z_{2})$ and
$\hat{M}=\mathbb{C}^{2}$ with coordinates $(\hat{z}_{1},\hat{z}_{2})$. Endow
both with the standard $T^{2}$-action. There are ample choices for a
$T$-duality between $(M,0)$ and $(\hat{M},0)$. Indeed, any of the following
two-forms will suffice:

  * •

$F_{1,\pm}=d\theta_{1}\wedge d\hat{\theta}_{1}\pm d\theta_{2}\wedge
d\hat{\theta}_{2}$.

  * •

$F_{2,\pm}=-d\theta_{1}\wedge d\hat{\theta}_{2}\pm d\theta_{2}\wedge
d\hat{\theta}_{1}$.

Although each of these forms satisfy the conditions necessary of $T$-duality,
the behaviour with respect to specific generalized complex structures is quite
different.

We consider the stable generalized complex structure given by
$\rho=z_{1}z_{2}+dz_{1}\wedge dz_{2}$. The corresponding elliptic symplectic
form is $\omega=d\log r_{1}\wedge d\theta_{2}+d\theta_{1}\wedge d\log r_{2}$
with spinor $\rho=1+id\log r_{1}\wedge d\theta_{2}+id\theta_{1}\wedge d\log
r_{2}-d\log r_{1}\wedge d\theta_{1}\wedge d\log r_{2}\wedge d\theta_{2}$.

By Theorem 5.12 we know that $T$-duality with any of these forms will provide
generalized complex structures on the elliptic tangent bundle. However, a
direct computation will show that only $F_{2,+}$ provides a generalized
complex structure on $M$. In that case we have that the $T$-dual generalized
complex structure is given by:

$\rho_{2,+}=d\log z_{1}\wedge d\log z_{2}.$

One immediately checks that $\rho_{2,+}$ is a lift of the standard complex
structure on $\mathbb{C}$. In conclusion, this elliptic symplectic structure
on $\mathbb{C}^{2}$ is $T$-dual to the standard complex structure on
$\mathbb{C}^{2}$.

The story for the real part of $\rho$ is quite different. Consider the
elliptic symplectic form $\omega=d\log r_{1}\wedge d\log
r_{2}-d\theta_{1}\wedge d\theta_{2}$, with spinor

$\rho=1+id\log r_{1}\wedge d\log r_{2}-id\theta_{1}\wedge
d\theta_{2}+d\theta_{1}\wedge d\theta_{2}\wedge d\log r_{1}\wedge d\log r_{2}$

. Again, one can perform a direct computation showing that only $F_{1,-}$ will
provide a generalized complex structure on $M$. In that case the $T$-dual is
given by:

$-ie^{-i(d\log r_{1}\wedge d\log r_{2}+d\hat{\theta}_{1}\wedge
d\hat{\theta_{2}})}$

In conclusion we see that $\rho$ is $T$-dual to itself.

##  6\. Blow-ups

In this section we will study the interaction between elliptic geometry and
blow-ups. We will work with the formalism of generalized complex blow-ups
introduced in [BCD16]. We will show that blow-ups of submanifolds of the
degeneracy locus of an elliptic divisor induce Lie algebroid submersions
between the respective elliptic tangent bundles. Torus actions are preserved
by blow-ups at fixed points, and hence we may consider the relation with
$T$-duality. We will show that $T$-duality and blowing up commutes.

###  6.1. Blow-ups in generalized complex geometry

The contents of this subsection appear in [BCD16], we recall them for
completeness.

As we do not assume any holomorphicity of our manifolds, more data is
necessary in constructing a blow-up:

######  Definition 6.1 ([BCD16]).

Let $M^{n}$ be a smooth manifold, and let $Y$ be a closed embedded submanifold
of codimension $2l$. A holomorphic ideal for $Y$ is a sheaf of ideals
$I_{Y}\subset C^{\infty}(M;\mathbb{C})$ such that:

  * •

$I_{Y}|_{M\backslash Y}=C^{\infty}(M;\mathbb{C})|_{M\backslash Y}$.

  * •

For every $y\in Y$, there is a neighbourhood $U$ with coordinates
$z_{1},\ldots,z_{l},x_{2l+1},\ldots,x_{n}:U\rightarrow\mathbb{C}^{l}\times\mathbb{R}^{n-2l}$
such that $Y\cap U=\\{z_{1}=\ldots=z_{l}=0\\}$ and $I_{Y}|U=\left\langle
z_{1},\ldots,z_{l}\right\rangle$.

As shown in [BCD16], a submanifold $Y$ admits a holomorphic ideal if and only
if the normal bundle $NY$ admits a complex structure.

######  Theorem 6.2 ([BCD16]).

Given a submanifold $Y$ and a holomorphic ideal $I_{Y}$, there exists a,
unique up to isomorphism, manifold $\tilde{M}$ together with a map
$p:\tilde{M}\rightarrow M$ such that $p^{*}I_{Y}$ is a divisor satisfying the
following universal property: For any smooth map $f:X\rightarrow M$, such that
$f^{*}I_{Y}$ is a divisor, there exists a unique
$\tilde{f}:X\rightarrow\tilde{M}$ such that the following diagram commutes:

${X}$${\tilde{M}}$${M}$$\scriptstyle{\tilde{f}}$$\scriptstyle{f}$$\scriptstyle{p}$

We call $\tilde{M}$ the blow-up of $I_{Y}$ in $M$, and $p:\tilde{M}\rightarrow
M$ the blow-down map. The natural manifolds one can blow-up in generalized
complex geometry are the following:

######  Definition 6.3 ([BCD16]).

Let $(M,\mathbb{J},H)$ be a generalized complex manifold. A submanifold $Y$ is
called generalized Poisson if

$\mathbb{J}N^{*}Y=N^{*}Y.\hfill$

An immediate consequence of this definition is that $Y$ is a Poisson
submanifold of the underlying Poisson structure $\pi_{\mathbb{J}}$. Moreover,
if $Q$ is a holomorphic Poisson submanifold and $\mathbb{J}_{Q}$ is the
associated generalized complex structure, a submanifold $Y$ is a generalized
Poisson submanifold if and only if it is a holomoprhic Poisson submanifold.

By definition, the generalized complex structure induces a complex structure
on the normal bundle of a generalized Poisson submanifold. Hence:

######  Lemma 6.4 ([BCD16]).

Let $(M,\mathbb{J},H)$ be a generalized complex manifold and $Y$ a generalized
Poisson submanifold. Then there exist a canonical holomorphic ideal $I_{Y}$
for $Y$ for which the induced complex structure on $NY$ coincides with the one
induced by $\mathbb{J}$.

Given that a generalized Poisson submanifold gives rise to a holomorphic
ideal, and hence a canonical blow-up, the next step is to determine when the
generalized complex structure lifts to the blow-up. For this we need the
following notion:

######  Definition 6.5.

A Lie algebra $\mathfrak{g}$ is said to be degenerate if the map
$\Lambda^{3}(\mathfrak{g})\rightarrow\rm{Sym}^{2}(\mathfrak{g})$

$x\wedge y\wedge z\mapsto[x,y]\wedge z,$

vanishes.

Given a generalized Poisson submanifold $Y\subset(M,\mathbb{J})$, it is in
particular a Poisson submanifold and hence one obtains a fibre-wise Lie
algebra structure on $N^{*}Y$. This Lie algebra structure captures whether the
generalized complex structure lifts to the blow-up:

######  Theorem 6.6 ([BCD16]).

Let $(M,\mathbb{J},H)$ be a generalized complex manifold and $Y$ a generalized
Poisson submanifold. Then there exists a generalized complex structure
$\tilde{\mathbb{J}}$ on the blow-up $p:\tilde{M}\rightarrow M$, with respect
to the canonical holomorphic ideal $I_{Y}$, for which $p$ is a generalized
holomorphic map, if and only if the Lie algebra structure on $N^{*}Y$ is
degenerate.

We may blow-up components of the degeneracy locus of a stable generalized
complex structure:

######  Lemma 6.7.

Let $(M,\mathbb{J})$ be a stable generalized complex manifold, and let
$Y\subset D(l)$ be an irreducible component. Then $Y$ is a generalized Poisson
submanifold for which the fibre-wise Lie algebra structure on $N^{*}Y$ is
trivial.

###### Proof.

First we note that $Y$ is a union of symplectic leaves of the underlying
Poisson structure $\pi_{\mathbb{J}}$, and hence a Poisson submanifold. For
every $p\in Y$, there exists a neighbourhood $U$, with a complex structure and
a holomorphic Poisson structure $Q$ inducing the generalized complex structure
(up to gauge equivalence). We have that $Y\cap U$, can be written as the zero-
set of some of the complex coordinates, and hence it is a complex submanifold
of $U$. Therefore, $Y\cap U$ is a holomorphic Poisson submanifold of $Q$, and
consequently a generalized Poisson submanifold of $\mathbb{J}$ on $U$. As
being a generalized Poisson submanifold is a point-wise condition, this shows
that $Y$ is a generalized Poisson submanifold of $M$.

Because $\pi_{\mathbb{J}}$ is an elliptic symplectic structure it lifts to an
element of $\Gamma(\wedge^{2}\mathcal{A}_{\left|D\right|})$, in particular the
bivector is tangent to $D$. Because of the geometry of $D$, it follows that
$\pi_{\mathbb{J}}$ is tangent to each stratum $D[l]$, and therefore in
particular to $Y$. Consequently, the Lie algebra structure induced on $N^{*}Y$
is trivial. ∎

We conclude that irreducible components of the degeneracy locus of a stable
generalized complex structure may be blown-up in a generalized complex
fashion.

###  6.2. Blowing up for the elliptic tangent bundle

There is yet another explanation why one can blow up irreducible components of
the degeneracy locus of a stable generalized complex structure. Namely, the
elliptic tangent bundle is very well-behaved under blow-ups. To state a result
independent of a generalized complex structure, we need that the holomorphic
ideal is adapted to the elliptic ideal:

######  Definition 6.8.

Let $M^{n}$ be endowed with an elliptic divisor $I_{\left|D\right|}$, and
$Y^{n-2l}$ a closed embedded submanifold. We say that a holomorphic ideal
$I_{Y}$ for $Y$ is adapted to $I_{\left|D\right|}$ if there exists local
coordinates $z_{1},\ldots,z_{k},x_{2k+1},\ldots,x_{n-2k}$ for which
$z_{1},\ldots,z_{l}$ are as in Definition 6.1 and
$I_{\left|D\right|}|_{U}=\left\langle\left|z_{1}\right|^{2}\cdot\ldots\cdot\left|z_{k}\right|^{2}\right\rangle$.

Note that this definition in particular implies that $Y\subset D$. Given such
a holomorphic ideal, we may blow-up the elliptic divisor to obtain:

######  Theorem 6.9.

Let $(M,I_{\left|D\right|})$ be a manifold endowed with an elliptic divisor,
and let $Y\subset D(k)$ be a irreducible component. Assume that we are given a
holomorphic ideal $I_{Y}$ for $Y$ adapted to $I_{\left|D\right|}$. Then there
exists an elliptic ideal $I_{\left|\tilde{D}\right|}$ on the blow-up
$\tilde{M}$ such that the blow-down map induces a fibre-wise isomorphism:

$\displaystyle
p_{*}:(\mathcal{A}_{\left|\tilde{D}\right|})_{x}\overset{\simeq}{\longrightarrow}(\mathcal{A}_{\left|D\right|})_{p(x)},$

for all $x\in\tilde{M}$.

###### Proof.

To study the blow-down map we need to use local coordinates describing it.
Given $l\geq k$ let $x\in Y\cap D[l]$, let
$U=\mathbb{C}^{l}\times\mathbb{R}^{m}$ be a local chart with coordinates
$(z_{1},\ldots,z_{l},x_{1},\ldots,x_{m})$, with the $z_{i}$ as in Definition
6.1. Let $\tilde{U}=\tilde{\mathbb{C}}^{l}\times\mathbb{R}^{m}$, and
$p=(p^{\prime},\text{Id}):\tilde{U}\rightarrow U$ be the blow-down map. The
manifold $\tilde{\mathbb{C}}^{l}$ has a cover of $l$-coordinate charts given
by

$\displaystyle(v_{1},\ldots,v_{i-1},z_{i},v_{i+1},\ldots,v_{l})$
$\displaystyle\leftrightarrow(z_{i}\cdot(v_{1},\ldots,v_{i-1},1,v_{i+1},\ldots,v_{l}),$
$\displaystyle[v_{1}:\ldots:v_{i-1}:1:v_{i+1}:\ldots:v_{l}]).$

In these coordinate charts the blow-down map is given by

$\displaystyle p^{\prime}:(v_{1},\ldots,v_{i-1},z_{i},v_{i+1},\ldots,v_{l})$
$\displaystyle\mapsto(z_{i}v_{1},\ldots,z_{i}v_{i-1},z_{i},z_{i}v_{i+1},\ldots,z_{i}v_{l}).$

With these coordinates at hand, we can consider the local form of
$p^{*}I_{\left|D\right|}$. Note that this does not provide an elliptic divisor
yet, however $I_{\left|D\right|}:=\sqrt{p^{*}I_{\left|D\right|}}$ does.
Because a vector field preserves $p^{*}I_{\left|D\right|}$ if and only if it
preserves $\sqrt{p^{*}I_{\left|D\right|}}$ we find that the blow-down map
induces a Lie algebroid map
$p_{*}:\mathcal{A}_{\left|\tilde{D}\right|}\rightarrow\mathcal{A}_{\left|D\right|}$.

To check that the blow-down map $p$ induces a fibre-wise isomorphism, it
suffices to show that it pulls-back a volume element
$(\mathcal{A}_{\left|D\right|})_{p(x)}$ to a volume element of
$(\mathcal{A}_{\left|\tilde{D}\right|})_{x}$. In the above coordinates, the
following would define a local volume form on
$(\mathcal{A}_{\left|D\right|})_{x}$:

$\displaystyle\Omega=d\log r_{1}\wedge d\theta_{1}\wedge\cdots\wedge d\log
r_{l}\wedge d\theta_{l}\wedge dx_{1}\wedge\cdots\wedge dx_{m}.$

Therefore

$\displaystyle p^{*}\Omega=d\log\tilde{r}_{1}\wedge
d\hat{\theta}_{1}\cdots\wedge d\log\tilde{r}_{l}\wedge d\hat{\theta}_{l}\wedge
d\log r_{l+1}\wedge d\theta_{l+1}\wedge\cdots d\log r_{k}\wedge
d\theta_{k}\wedge dx_{1}\wedge\cdots dx_{m},$

where the $(\tilde{r}_{j},\hat{\theta}_{j})$ are the polar coordinates
corresponding to $(v_{1},\ldots,v_{i-1},z_{i},v_{i+1},\ldots,v_{l})$. We
conclude that $p$ indeed induces the fibrewise isomorphism as required. ∎

######  Example 6.10 (Blowing up points).

Let $M$ be an orientable manifold and let $Y=\\{p\\}$ be a point. Then we can
always find complex coordinates around $p$, $z_{1},\ldots,z_{l}$, such that
$I_{Y}=\left\langle z_{1},\ldots,z_{l}\right\rangle$ is a holomorphic ideal.
The topology of the blow-up will depend on the parity of $l$, and whether the
coordinates $z_{1},\ldots,z_{l}$ are orientation preserving. If $l$ is odd,
the blow-up will be $M\\#\mathbb{C}P^{l}$, if $l$ is even it will be
$M\\#\overline{\mathbb{C}P^{l}}$ if the coordinates are orientated and
$M\\#\mathbb{C}P^{l}$ if they are not.

######  Example 6.11 (Global normal crossing).

Let $I_{\left|D\right|}$ be a global normal crossing divisor, and let
$Y\subset D(l)$ be an irreducible component. We have that $Y$ must then
necessarily be of the form $D_{1}\cap\cdots\cap D_{l}$, for some of the
components of $D$. If $D[1]$ is co-orientable, it follows that each $D_{i}$ is
co-orientable as well, and one can show that there exists a complex structure
on $ND_{i}$, inducing the elliptic divisor (see [CKW20b]). Combining these
complex structures to a complex structure on $NY$, gives a holomorphic ideal
for $Y$ which is compatible with $I_{\left|D\right|}$.

Given a generalized complex structure on any of the Lie algebroids
$\mathcal{A}_{\left|D\right|}$ may lift it to a structure of the same kind on
the blown-up Lie algebroid. But similarly as for $T$-duality, it might happen
that said structure does not descend to give a generalized complex structure
on $\tilde{M}$. However, when the holomorphic ideal is induced by the
generalized complex structure, this will always be the case:

######  Lemma 6.12.

Let $(M,\mathbb{J})$ be a stable generalized complex manifold, and let
$Y\subset D(k)$ be an irreducible component. Let
$\omega\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ denote the elliptic
symplectic structure corresponding to $\mathbb{J}$. Let $I_{Y}$ be the
canonical holomorphic ideal for $Y$. Then this ideal is adapted to
$I_{\left|D\right|}$, and the blow-up $p^{*}\omega$ will correspond to the
generalized complex blow-up of $\mathbb{J}$ as in Lemma 6.7.

We conclude that the point of view of blowing-up using the elliptic tangent
bundle results in the same blow-up as in [BCD16]. However, Theorem 6.9 can be
used in a larger context: for instance, elliptic symplectic forms not
corresponding to generalized complex structures may be blown-up using this
result.

####  6.2.1. Complex structures on the blow-up

Let $(M,I_{\left|D\right|})$ is a manifold endowed with an elliptic divisor,
and $\mathbb{J}$ is a generalized complex structure on $M$, which lifts to
$\mathcal{A}_{\left|D\right|}$. As pointed out before, if we are given
$Y\subset D(k)$ an irreducible component, and $I_{Y}$ a holomorphic ideal,
then the blow-up of $\mathbb{J}$ induces a generalized complex structure on
$\mathcal{A}_{\left|\tilde{D}\right|}$, however this needn’t descend to a
generalized complex structure on $\tilde{M}$. The following example shows this
can already happen with complex structures:

######  Example 6.13 (Complex structures on
$n\mathbb{C}P^{2}\\#m\overline{\mathbb{C}P^{2}}$).

Consider the standard complex structure $J$ on $\mathbb{C}P^{2}$, and
$\mathcal{O}(3)$ and the section $s$ corresponding to the polynomial
$z_{0}z_{1}z_{2}$. Then $I_{D}:=s^{*}(\Gamma(\mathcal{O}(3)))\subset
C^{\infty}(\mathbb{C}P^{2},\mathbb{C})$ defines a complex log divisor with
degeneracy locus
$D=\\{[0:z_{1}:z_{2}]\\}\cup\\{[z_{0}:0:z_{2}]\\}\cup\\{[z_{0}:z_{1}:0]\\}$,
and let $I_{\left|D\right|}$ be the associated elliptic divisor. As $D$ is a
complex submanifold of $\mathbb{C}P^{2}$, $J$ will preserve the log tangent
bundle $A_{D}$, and consequently induce a complex structure
$J_{\mathcal{A}_{\left|D\right|}}$ on
$\mathcal{A}_{\left|D\right|}\rightarrow\mathbb{C}P^{2}$.

We may also consider $\overline{\mathbb{C}P^{2}}$, with the same elliptic
divisor $I_{\left|D\right|}$. Although this is not an almost complex manifold,
it does admit a complex structure on $\mathcal{A}_{\left|D\right|}$. Indeed,
one may show that in affine coordinates $(w_{1},w_{2})$, the spinor
$\rho=d\log w_{1}\wedge d\log\overline{w}_{2}$ gives rise to a global complex
structure on
$\mathcal{A}_{\left|D\right|}\rightarrow\overline{\mathbb{C}P^{2}}$.

Now, let $Y=\\{{\rm pt}\\}\subset\mathbb{C}P^{2}$ be any of the vertices of
$D$, and pick a holomorphic ideal for $Y$. As described in Example 6.10, the
corresponding total space will be either $\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$
or $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$. In either case, we obtain a
complex structure $J_{\mathcal{A}_{\left|\tilde{D}\right|}}$ on the blown-up
Lie algebroid $\mathcal{A}_{\left|\tilde{D}\right|}$.

On $\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$, the blow-up coincides with
the usual complex blow-up and $J_{\mathcal{A}_{\left|\tilde{D}\right|}}$ is
simply a lift of the complex structure on
$\mathbb{C}P^{2}\\#\overline{\mathbb{C}P^{2}}$. However,
$\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$ is not even an almost complex manifold,
and hence the induced complex structure on
$J_{\mathcal{A}_{\left|\tilde{D}\right|}}$ does not descend to a (generalized)
complex structure on $\mathbb{C}P^{2}\\#\mathbb{C}P^{2}$. Continuing
inductively, we obtain complex structures on elliptic tangent bundles over
$n\mathbb{C}P^{2}m\\#\mathbb{C}P^{2}$, for all $n,m\in\mathbb{N}$.

###  6.3. $T$-duality and blow-up

We now study how $T$-duality interacts which blow-ups. First we recall that
toric actions are preserved under blow-ups:

######  Lemma 6.14.

Let $M^{2n}$ admit a standard $T^{n}$ action, and let $p\in M$ be a fixed
point. Then, both $M\\#\mathbb{C}P^{n}$ and $M\\#\overline{\mathbb{C}P^{n}}$
admits a standard $T^{n}$-action for which the blow-down map is equivariant.

The fact that the blow-down map induces a fibre-wise isomorphism between
elliptic tangent bundles immediately implies the following:

######  Proposition 6.15 ($T$-duality commutes with blow-up).

Let $M^{2n}$ and $\hat{M}^{2n}$ be endowed with standard $T^{n}$-actions over
the same base $B$, and let $I_{\left|D\right|},I_{\left|\hat{D}\right|}$
denote the induced elliptic divisors. Let $H\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|D\right|}),\hat{H}\in\Omega^{3}_{\rm
cl}(\mathcal{A}_{\left|\hat{D}\right|})$, and
$F\in\Omega^{2}(\mathcal{A}_{\left|D\right|}\times_{B}\mathcal{A}_{\left|\hat{D}\right|})$
provide a $T$-dual between $(M,H)$ and $(\hat{M},\hat{H})$.

Let $x\in M,\hat{x}\in\hat{M}$ be fixed points for the action, and let $X,Y$
denote either $\mathbb{C}P^{n}$ or $\overline{\mathbb{C}P^{n}}$. Let
$q:M\\#_{x}X\rightarrow M$, $\hat{q}:\hat{M}\\#_{\hat{x}}Y\rightarrow\hat{M}$
denote the blow-down maps. Then we have a canonical $T$-duality between the
blow-ups:

${((M\\#X)\times_{\tilde{B}}(\hat{M}\\#Y),p^{*}q^{*}H-\hat{p}^{*}\hat{q}^{*}\hat{H}),}$${(M\\#X,q^{*}H)}$${(\hat{M}\\#Y,\hat{q}^{*}\hat{H})}$${\tilde{B}}$$\scriptstyle{p}$$\scriptstyle{\hat{p}}$$\scriptstyle{\pi}$$\scriptstyle{\hat{\pi}}$

Consequently, the following diagram commutes:

${\Omega^{\bullet}(\mathcal{A}_{\left|\tilde{D}\right|})}$${\Omega^{\bullet}(\mathcal{A}_{\left|\tilde{\hat{D}}\right|})}$${\Omega^{\bullet}(\mathcal{A}_{\left|D\right|})}$${\Omega^{\bullet}(\mathcal{A}_{\left|\hat{D}\right|})}$$\scriptstyle{\tilde{\tau}}$$\scriptstyle{\tau}$$\scriptstyle{q^{*}}$$\scriptstyle{\hat{q}^{*}}$

###### Proof.

By Theorem 6.9 $q$ and $\hat{q}$ induce fibre-wise isomorphisms between
elliptic tangent bundles. Let
$Q:(M\\#X)\times_{\tilde{B}}(\hat{M}\\#Y)\rightarrow M\times_{B}\hat{M}$
denote the induced map between the correspondence spaces. Consequently $Q$
induces a fibre-wise isomorphism between the elliptic tangent bundles on the
correspondence spaces. Therefore $Q^{*}F$ will satisfy the equation for
$T$-duality. ∎

######  Corollary 6.16.

Let $(M,H)$, $(\hat{M},\hat{H})$ be as in the setting of Proposition 6.15, and
assume there is a generalized complex structure $\mathbb{J}$ on $(M,H)$
$T$-dual to a generalized complex structure $\hat{\mathbb{J}}$ on
$(\hat{M},\hat{H})$. Then the $T$-duality from the previous Proposition 6.15
sends $\tilde{\mathbb{J}}$, to $\hat{\tilde{\mathbb{J}}}$.

###### Proof.

As $\tilde{\hat{M}}\backslash E\simeq\hat{M}\simeq\\{\hat{x}\\}$, we have that
the $T$-dual of $\tilde{\mathbb{J}}$ coincides with $\hat{\tilde{\mathbb{J}}}$
outside of $E$. By continuity they must coincide over the entirety of
$\tilde{M}_{2}$. ∎

##  7\. Examples

In this final section we will provide explicit examples of $T$-dual geometric
structures on manifolds endowed with standard torus actions. We will define
some basic examples, and use these to produce more examples by blowing up.

###  7.1. Basic examples

We start by consider simple examples of generalized complex structures. The
analysis heavily depends on the local description in Example 5.14. So recall
from there that any of the following two-forms:

  * •

$F_{1,\pm}=d\theta_{1}\wedge d\hat{\theta}_{1}\pm d\theta_{2}\wedge
d\hat{\theta}_{2}$.

  * •

$F_{2,\pm}=-d\theta_{1}\wedge d\hat{\theta}_{2}\pm d\theta_{2}\wedge
d\hat{\theta}_{1}$.

provides a $T$-duality from $(\mathbb{C},0)$ to itself.

######  Example 7.1 ($S^{1}\times S^{3}$).

Let $T^{2}$ act on $S^{3}\subset\mathbb{C}^{2}$, and extend the action
trivially to an action on $S^{1}\times S^{3}$. The quotient space $B$ is an
annulus. Viewing $S^{1}\times S^{3}$ as the Hopf surface
$(\mathbb{C}^{2}\backslash\\{0\\})/\mathbb{Z}$, the stable generalized complex
structure $\tilde{\rho}=z_{1}z_{2}+dz_{1}\wedge dz_{2}$ on
$\mathbb{C}^{2}\backslash\\{0\\}$ descends to a stable generalized complex
structure $\rho$ on $S^{1}\times S^{3}$. All the two forms $F_{i,\pm}$ are
$\mathbb{Z}$-invariant and thus descend to two-forms on $S^{1}\times S^{3}$.
Therefore, as in Example 5.14, we obtain that $T$-duality induced by the two-
form $F_{2,+}$ sends $\rho$ to the standard complex structure on $S^{1}\times
S^{3}$.

If we instead consider the generalized complex structure induced by
$\tilde{\rho}_{2}=z_{1}z_{2}+idz_{1}\wedge dz_{2}$, and use the two-form
induced by $F_{1,-}$, we find that $\rho_{2}$ is $T$-dual to itself.

######  Example 7.2 ($\mathbb{C}P^{2}$).

We consider $\mathbb{C}P^{2}$ in the usual fashion as
$(\mathbb{C}^{3}\backslash\\{0\\})/\mathbb{C}$ with torus action
$(\lambda_{1},\lambda_{2})\cdot[z_{0}:z_{1}:z_{2}]=[e^{i\lambda_{1}}z_{0}:e^{i\lambda_{2}}z_{1}:z_{2}]$.
Consider the complex log forms $\tilde{\zeta_{1}}=d\log z_{0}-d\log z_{2}$ and
$\tilde{\zeta_{2}}=d\log z_{1}-d\log z_{2}$. As these are basic with respect
to the $\mathbb{C}^{*}$-action, they descend to complex log forms
$\zeta_{1},\zeta_{2}$. We let
$\Theta_{1}=\Im^{*}\zeta_{1},\Theta_{2}=\Im^{*}\zeta_{2}$ denote the imaginary
parts. Then $(\Theta_{1},\Theta_{2})$ provides a connection-one form for the
$T^{2}$-action on $\mathbb{C}P^{2}$.

Consider another copy of $\mathbb{C}P^{2}$, and denote the same connection
one-form by $(\hat{\Theta}_{1},\hat{\Theta}_{2})$. Similarly to Example 5.14
we now have that any of the following two-forms provides a self $T$-dual for
$(\mathbb{C}P^{2},0)$.

$\displaystyle F_{1,\pm}$
$\displaystyle=\Theta_{1}\wedge\hat{\theta}_{1}\pm\Theta_{2}\wedge\hat{\theta_{2}},$
$\displaystyle F_{2,\pm}$
$\displaystyle=-\Theta_{1}\wedge\tilde{\theta}_{2}\pm\Theta_{2}\wedge\hat{\theta}_{1}.$

We will now the action of this $T$-duality to the generalized complex
structures on $\mathbb{C}P^{2}$.

Consider the holomorphic Poisson structure
$\tilde{\pi}=z_{0}z_{1}\partial_{z_{1}}\wedge\partial_{z_{2}}$. Because this
is $\mathbb{C}^{*}$-invariant it induces a holomorphic Poisson structure $\pi$
on $\mathbb{C}P^{2}$, which is moreover log symplectic. One can show, as in
Example 5.14, that $F_{2,+}$ provides a $T$-dual between the corresponding
stable generalized complex structure and the standard complex structure on
$\mathbb{C}P^{2}$. Similarly, consider the holomorphic Poisson structure
$\tilde{\pi_{2}}=iz_{0}z_{1}\partial_{z_{1}}\wedge\partial_{z_{2}}$, and the
induced generalized complex structure $\rho_{2}$. Then $F_{1,-}$, as in
Example 5.14, will provide a $T$-dual between $\rho_{2}$ and itself.

######  Example 7.3 ($S^{2}\times S^{2}$).

Identify $M=S^{2}\times S^{2}$ with $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$,
and consider the torus action given by
$(\lambda_{1},\lambda_{2})\cdot([z_{0}:z_{1}],[w_{0}:w_{1}])=([e^{i\lambda_{1}}z_{0}:z_{1}],[e^{i\lambda_{2}}w_{0}:w_{1}])$.
Let $\tilde{\zeta}_{1}=d\log z_{0}-d\log z_{1}$ and $\tilde{\zeta}_{2}=d\log
w_{0}-d\log w_{1}$ be complex log-forms on
$(\mathbb{C}^{2}\backslash\\{0\\})^{2}$. As these are basic they induce
complex log-forms $\zeta_{1},\zeta_{2}$ on $M$, and we may consider their
imaginary parts $\Theta_{1}=\Im^{*}\zeta_{1},\Theta_{2}=\Im^{*}\zeta_{2}$. The
form
$(\Theta_{1},\Theta_{2})\in\Omega^{2}(\mathcal{A}_{\left|D\right|};\mathfrak{t}^{2})$
provides a connection one-form for the torus action on $M$. Let $\hat{M}$
denote another copy of $S^{2}\times S^{2}$, and consider the same connection
one-form $(\hat{\Theta}_{1},\hat{\Theta}_{2})$. As in the previous example,
the form
$F=-\Theta_{1}\wedge\hat{\Theta}_{2}+\Theta_{2}\wedge\hat{\Theta_{1}}$
provides a self $T$-dual on $(M,0)$.

In affine coordinates $t=z_{0}/z_{1},s=w_{0}/w_{1}$ we consider the
holomorphic Poisson structure $\pi=tw\partial_{t}\partial_{w}$. In any other
set of affine coordinates, this Poisson structure has the same form and hence
defines a global holomorphic Poisson structure on $M$, which is moreover log
symplectic. As in the previous example, the corresponding stable generalized
complex structure is $T$-dual to the standard complex structure on
$S^{2}\times S^{2}$. And the stable generalized complex structure associated
to $\pi=itw\partial_{t}\partial_{w}$ is $T$-dual to itself via
$F=\Theta_{1}\wedge\hat{\Theta}_{1}-\Theta_{2}\wedge\hat{\Theta}_{2}$.

We may now blow-up $\mathbb{C}P^{2}$ to obtain more examples of stable
generalized complex structures, which are $T$-dual:

######  Example 7.4 ($n\mathbb{C}P^{2}\\#m\overline{\mathbb{C}P}^{2}$).

Let $\tilde{\pi}_{1}=z_{0}z_{1}\partial_{z_{1}}\wedge\partial_{z_{2}}$, and
$\tilde{\pi}_{2}=iz_{0}z_{1}\partial_{z_{1}}\wedge\partial_{z_{2}}$ be
holomorphic Poisson structures on $\mathbb{C}^{3}\backslash\\{0\\}$, and
$\rho_{1},\rho_{2}$ the induced stable generalized complex structures on
$\mathbb{C}P^{2}$, as in Example 7.2. We may perform consequtive blow-ups of
either of these structures, to obtain families of generalized complex
structures $\rho_{1,n,m},\rho_{2,n,m}$ on
$\mathcal{A}_{\left|D\right|}\rightarrow
n\\#\mathbb{C}P^{2}m\\#\overline{\mathbb{C}P^{2}}$. One readily shows that
these generalized complex structures on $\mathcal{A}_{\left|D\right|}$,
induced generalized complex structures on
$n\\#\mathbb{C}P^{2}m\\#\overline{\mathbb{C}P^{2}}$ if and only if $n$ is odd.

We have seen in Example 7.2 that $\rho_{1}$ can be made $T$-dual to the
standard complex structure, and $\rho_{2}$ dual to itself. By Proposition 6.15
it then follows that each $\rho_{2,n,m}$ is $T$-dual to itself, and
$\rho_{1,n,m}$ is $T$-dual to the complex structures on
$\mathcal{A}_{\left|D\right|}$ described in Example 6.13.

######  Remark 7.0.

We obtained the families $\rho_{2,n,m}$ also in Theorem 7.5 in [CKW20b] via a
connected sum procedure. But not the families $\rho_{1,n,m}$. This is of
relevance, as we see that their behaviour under $T$-duality is very different.

###  7.2. Non-zero residue elliptic symplectic structures

Another class of structures which behave well with respect to $T$-duality are
elliptic symplectic forms with non-zero elliptic residue. These structures
admit a $\mathbb{C}^{*}$-invariant normal form:

######  Lemma 7.5 ([Wit21]).

Let $(M,I_{\left|D\right|})$ be a manifold with a smooth elliptic divisor, and
let $\omega\in\Omega^{2}(\mathcal{A}_{\left|D\right|})$ be an elliptic
symplectic form with $\lambda:=\operatorname{Res}_{q}(\omega)\neq 0$. Then
there exists a canonical flat connection on $\nu_{D}$ and a tubular
neighbourhood of $D$ such that

(7.1) $\omega=\lambda\rho\wedge\Theta+p^{*}\omega_{D}.$

Here $\rho,\Theta$ are the connection-one forms corresponding to the flat
connection, as defined in 3.12 and
$\omega_{D}=\iota^{*}_{D}\omega\in\Omega^{2}(D)$.

Conversely, given any complex line bundle $L\rightarrow D$, $\nabla$ a flat
connection on $L$, $\lambda\neq 0$ and $\omega_{D}\in\Omega^{2}(D)$
symplectic, (7.1) defines an elliptic symplectic form on $L$.

The form $\omega$ is $S^{1}$-invariant, and hence we can consider its
$T$-dual. It will appear via the following lemma:

######  Lemma 7.6.

Let $(M,J)$ be a generalized complex manifold, and $L\rightarrow M$ a complex
vector bundle, endowed with a flat connection $\nabla$. Then ${\rm tot}(L)$
inherits a natural generalized complex structure.

###### Proof.

We define $\tilde{J}$ on $\text{tot}(L)$ by specifying it on linear and
fibrewise constant vector fields. Given $X\in T_{p}M$, we define
$\tilde{J}(\operatorname{hor}^{\nabla}(X))=\operatorname{hor}^{\nabla}(J(X))$.
And given $v\in T^{\rm vert}_{p}L$, we define $\tilde{J}(v)=iv$, under the
identification of $T^{\rm vert}_{p}L\simeq L_{p}$. Because the connection is
flat, $\tilde{J}$ will be integrable. ∎

######  Lemma 7.7.

Given $(D,\omega_{D})$ a symplectic manifold, $p:L\rightarrow M$ a complex
line bundle with flat connection $\nabla$, and
$\lambda\in\mathbb{R}\setminus\\{0\\}$ then $\omega$ as in (7.1) is $T$-dual
to the type 1 generalized complex structure induced by Lemma 7.6

###### Proof.

Let $\hat{\Theta}$ denote the same connection one-form, on another copy of
$L\rightarrow M$. Then $F=\Theta\wedge\hat{\Theta}$ provides the desired
$T$-dual. We have that, using $\tau$ from Proposition 5.10 that

$\tau(e^{i\omega})=i\lambda(\rho+i\Theta)\wedge e^{i\omega_{D}}.$

Locally, this corresponds to the spinor given by $i\lambda dz\wedge
e^{i\omega_{D}}$, which is precisely the generalized complex structure form
Lemma 7.6. ∎

######  Remark 7.0.

Given a smooth elliptic divisor $I_{\left|D\right|}$, elliptic symplectic
forms with zero elliptic residue have a similar $S^{1}$-invariant normal form
as in (7.1). Unfortunately, the corresponding $T$-dual generalized complex
structure on $\mathcal{A}_{\left|D\right|}$ does not descend to a generalized
complex structure on $M$.

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