#  GENERALIZED LUTTINGER SURGERY AND OTHER CUT-AND-PASTE CONSTRUCTIONS IN
GENERALIZED COMPLEX GEOMETRY

Lorenzo Sillari  LS: Scuola Internazionale Superiore di Studi Avanzati
(SISSA), Via Bonomea 265, 34136 Trieste (TS), Italy<EMAIL_ADDRESS>
###### .

Abstract. Exploiting the affinity between stable generalized complex
structures and symplectic structures, we explain how certain constructions
coming from symplectic geometry can be performed in the generalized complex
setting. We introduce generalized Luttinger surgery and generalized Gluck
twist along $\mathcal{J}$-symplectic submanifolds. We also export branched
coverings to the generalized complex setting. As an application, stable
generalized complex structures are produced on a variety of high-dimensional
manifolds. Remarkably, some of them have non-homotopy-equivalent path-
connected components of their type change locus.

00footnotetext:  2020 Mathematics Subject Classification. Primary: 53C15,
53D18; Secondary: 53D05, 57R17.
Keywords: Branched coverings, generalized complex geometry, Gluck twist,
Luttinger surgery, stable generalized complex structures, torus surgery, type
change locus.

## 1\. Introduction

A _generalized complex structure_ ([28, 26, 27]) $\mathcal{J}$ on a
$2m$-manifold $M$ induces a symplectic structure on a subspace of $T_{p}M$ and
a transverse complex structure. The dimension of the latter is an integer
$t\in\\{0,\dots,m\\}$, called the _type_ of $\mathcal{J}$ at $p$. Symplectic
structures have type $0$ at every point and they are the most generic
situation, while complex structures have type $m$ at every point and they are
the less generic one.
In general, the type might change on $M$, and the set of points where it is
not locally constant is called the _type change locus_ of $\mathcal{J}$. The
most generic case with non-empty type change locus is that of _stable_
generalized complex structures and was extensively studied in [15]. Around
points where $\mathcal{J}$ has type $0$ (i.e. outside of the type change
locus), $M$ behaves like a symplectic manifold. This allows to consider ideas
and techniques from symplectic geometry and import them in generalized complex
geometry, leading to interesting consequences and fruitful applications ([13,
14, 33, 37, 24, 25, 6, 12, 15, 17, 7, 16, 19, 18]).

A fundamental notion in doing so is that of _$\mathcal{J}$ -symplectic
submanifold_ ([25, 31]), the generalized complex analogous of a symplectic
submanifold. The key observation is that as soon as we have a
$\mathcal{J}$-symplectic submanifold, several techniques involving symplectic
submanifolds extend with little or no effort to the generalized complex
setting. In this paper, we address three of them, namely _generalized
Luttinger surgery_ , a generalization of _torus surgery_ ([13]), _generalized
Gluck twist_ , a combination of generalized Luttinger surgery and _Gluck
twist_ ([20]), and _branched coverings_. However, this paper is not meant to
cover every tool that can be obtained in this way, nor to exhaust all the
possible examples that can be produced using them. Following our guidelines,
the list of available constructions can be easily enlarged and many more
examples will arise combining such techniques.

In more detail, torus surgery has been fruitfully employed in generalized
complex geometry in dimension $4$ ([13, 14, 33, 37, 25]) and $6$ ([31]) to
obtain many examples of stable generalized complex structures. The surgery is
performed along a $\mathcal{J}$-symplectic torus of codimension $2$ embedded
in a stable generalized complex manifold.
Generalized Luttinger surgery extends torus surgery in two ways: first, it can
be applied to manifolds of any even dimension $2m\geq 6$. This is the first
occurrence of a similar construction in dimension $2m\geq 8$. Second, it
modifies the type change locus by adding path-connected components that are
homotopy-equivalent to a product $T^{2}\times\Sigma$, where $\Sigma$ is a
symplectic $(2m-4)$-manifold. This is new even in dimension $6$, where the
only known possibilities were $T^{4}$ and $T^{2}\times\mathbb{S}^{2}$ ([31]).
More precisely, we prove the following

######  Theorem A.

Let $(M^{2m},\mathcal{J},H)$ be a stable generalized complex structure and let
$S$ be its type change locus. Let $(\Sigma,\omega_{\Sigma})$ be a symplectic
$(2m-4)$-manifold. Assume that there exists an embedded
$T^{2}\times\Sigma\hookrightarrow M$ such that

  * $(i)$

there exists a neighborhood $N_{T}$ of $T^{2}\times\Sigma$ that is
diffeomorphic to $D^{2}\times T^{2}\times\Sigma$;

  * $(ii)$

$T^{2}\times\Sigma$ is $\mathcal{J}$-symplectic and
$\omega_{|T^{2}\times\Sigma}=d\theta^{12}+\omega_{\Sigma}$, where $\omega$ is
the symplectic form induced by $\mathcal{J}$ on $N_{T}$, up to $B$-field
equivalence.

Then performing generalized Luttinger surgery along $T^{2}\times\Sigma$ yields
a stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type changes along
$S\sqcup\left(T^{2}\times\Sigma\right)$.

The possibility of performing surgeries along $T^{2}\times\Sigma$ is used to
prove the following

######  Theorem B.

There exist stable generalized complex structures with non-homotopy-equivalent
path-connected components of their type change locus.

We perform surgeries on stable generalized complex manifolds of the form
$M\times T^{2}$ to obtain stable generalized complex manifolds of the form
$\tilde{M}\times\mathbb{S}^{1}$. A wise choice of the surgery parameters
allows to produce manifolds $\tilde{M}$ with a wide choice of fundamental
groups. Under additional hypotheses, we pin down the diffeomorphism type of
our $6$-dimensional examples.

Generalized Luttinger surgery can be easily modified allowing more freedom for
the gluing diffeomorphism. We describe how to do that in general. This paves
the way to a possibly huge palette of constructions and enlarges the class of
diffeomorphism types of manifolds admitting a stable generalized complex
structure. Making an explicit choice of diffeomorphism, we introduce
generalized Gluck twist. It extends from dimension $6$ to any even dimension
$2m\geq 8$ the surgery described in [31], Theorem 3.3.2. Results similar to
those proved for generalized Luttinger surgery are true also for generalized
Gluck twist.

######  Theorem C.

Let $(M^{2m},\mathcal{J},H)$ be a stable generalized complex structure and let
$S$ be its type change locus. Let $(R,\omega_{R})$ be a symplectic
$(2m-6)$-manifold. Assume that there exists an embedded $T^{2}\times
R\times\mathbb{S}^{2}\hookrightarrow M$ such that

  * $(i)$

there exists a neighborhood $N_{T}$ of $T^{2}\times R\times\mathbb{S}^{2}$
that is diffeomorphic to $D^{2}\times T^{2}\times R\times\mathbb{S}^{2}$;

  * $(ii)$

$T^{2}\times R\times\mathbb{S}^{2}$ is $\mathcal{J}$-symplectic and
$\omega_{|T^{2}\times
R\times\mathbb{S}^{2}}=d\theta^{12}+dh\,d\theta^{S}+\omega_{R}$, where
$\omega$ is the symplectic form induced by $\mathcal{J}$ on $N_{T}$, up to
$B$-field equivalence.

Then performing generalized Gluck twist along $T^{2}\times
R\times\mathbb{S}^{2}$ yields a stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type changes along
$S\sqcup(T^{2}\times R\times\mathbb{S}^{2})$.

The third construction we study is that of branched coverings. The symplectic
side of the problem was already settled in Lemma 1 of [22]. Rephrasing the
result in the generalized complex language, we obtain that branched coverings
of stable generalized complex $2m$-manifolds, with $2m\geq 4$, are still
stable generalized complex.

######  Theorem D.

Let $(M^{2m},\mathcal{J},H)$ be a stable generalized complex structure whose
type change locus has $k\in\mathbb{N}$ path-connected components. Let $B$ be a
$\mathcal{J}$-symplectic submanifold of $M$, and let $\pi:\tilde{M}\rightarrow
M$ be a $d$-fold branched covering with branch locus $B$. Then there exists a
stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type change locus has
$d\cdot k$ path-connected components.

Combining generalized Luttinger surgery and coverings, we further exemplify
the versatility of our ideas and their potential in producing stable
generalized complex structures. Finally, using branched coverings we show a
new way to put stable generalized complex structures on
$T^{2}\times\Sigma_{g}$, where $\Sigma_{g}$ is the surface of genus $g\geq 1$,
and on the elliptic surfaces $E(2n)$.

Even though we stated our results for stable generalized complex structures,
they can be extended with no effort also to unstable structures (cf. Remark
2.7).

Acknowledgements. This paper is the first part of the ongoing PhD project of
the author. He is extremely grateful and deeply indebted to Rafael Torres, who
provided the main results describing the techniques and most of the
consequences we presented here. He also pointed out to the author many of the
examples. This paper would not have been written without his expertise and
patience. The author also thanks Roberto Rubio for comments that improved the
exposition of the paper.

## 2\. Preliminaries

In this section we recall basic definitions and results in generalized complex
geometry. Many of them can be found in [26], while the others are referenced
throughout the text. We also introduce our notation. Unless otherwise
specified, using the word _manifold_ we mean a compact orientable smooth
manifold with no boundary, and we shall often write $M^{k}$ to denote a
$k$-manifold. If $\\{x_{j}\\}$ are local (real or complex) coordinates, we
will write $dx^{j_{1}\dots j_{k}}$ referring to the differential form
$dx^{j_{1}}\wedge\dots\wedge dx^{j_{k}}$.

Let $M$ be a $2m$-manifold. Denote by $TM$, respectively $T^{*}M$, its tangent
bundle, respectively cotangent bundle, and set $E:=TM\oplus T^{*}M$.
Fixed any closed $3$-form $H$, the _$H$ -twisted Courant bracket_ can be
defined on sections of $E$ as follows:

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

Sections of $E$ also admit a natural symmetric pairing of signature $(n,n)$
defined by

$\langle X+\xi,Y+\eta\rangle:=\frac{1}{2}\left(\eta(X)+\xi(Y)\right).$

######  Definition 2.1.

A _generalized complex structure twisted by $H$_, $(M,\mathcal{J},H)$, is a
maximal isotropic sub-bundle $L$ of $E^{\mathbb{C}}$ satisfying the following
conditions:

  * •

_Non-degeneracy:_ $L\cap\bar{L}=\\{0\\}$;

  * •

_Integrability:_ $L$ is closed under the $H$-twisted Courant bracket.

Equivalently, $\mathcal{J}$ can be described in terms of a complex line bundle
$K\subset\Omega^{\bullet}$, generated by a _pure spinor_ , i.e., an element
$\rho\in\Omega^{\bullet}$ that has the local form

$\rho=e^{B+i\omega}\wedge\Omega,$

where $B$, and $\omega$ are real $2$-forms and $\Omega$ is a decomposable
$k$-form. Moreover the pure spinor must satisfy the conditions:

  * •

_Non-degeneracy:_ the forms $\Omega$ and $\omega$ must provide a non-
degenerate top-form

$\Omega\wedge\bar{\Omega}\wedge\omega^{m-k}\neq 0;$

  * •

_Integrability:_ there exists a section $X+\xi$ of $E$ such that

$d_{H}\rho:=d\rho+H\wedge\rho=(X+\xi)\cdot\rho,$

where $(X+\xi)\cdot\rho:=\iota_{X}\rho+\xi\wedge\rho$ is the _Clifford action_
of sections of $E$ on spinors.

The integer $\deg\Omega=k\in\\{0,1,\dots m\\}$, defined at a point $p$, is the
_type_ of the generalized complex structure at $p$, and it is not necessarily
constant on $M$. However, it is an upper semi-continuous function, thus
locally it can only decrease. Such a jump preserves the parity of $k$, that we
call the _parity_ of $\mathcal{J}$.

######  Definition 2.2.

The type change locus of a generalized complex structure $(M,\mathcal{J},H)$
is the set of points of $M$ where the type of $\mathcal{J}$ jumps, i.e., it is
not locally constant.

Given any $2$-form $B$, the _$B$ -field transform_ provided by $B$ is the
action on pure spinors

$\rho\longmapsto e^{B}\wedge\rho.$

If $\rho$ is integrable with respect to $H$, then any pure spinor obtained
from $\rho$ via $B$-field transform is integrable with respect to $H-dB$. Two
generalized complex structures are said to be _equivalent_ if there exists a
$B$-field transform sending one to the other.

######  Example 2.3.

Complex structures are generalized complex. If $(M,J)$ is a complex manifold,
then its canonical bundle $K:=\bigwedge^{n}T^{*}M^{1,0}$ defines a generalized
complex structure locally described by $\rho=\Omega\in K$, that has constant
type $m$. The converse also holds: if a generalized complex structure has
constant type $m$ then it is $B$-field equivalent to one induced by a complex
structure.

######  Example 2.4.

Symplectic forms induce a generalized complex structure. If $(M,\omega)$ is a
symplectic manifold, then the pure spinor $e^{i\omega}$ generates a canonical
bundle $K$. The associated generalized complex structure
$\mathcal{J}_{\omega}$ has constant type $0$. If a generalized complex
structure has constant type $0$ then it is $B$-field equivalent to one induced
by a symplectic form.

Among generalized complex structures, a special role is played by _stable
structures_ ([15]). Let $K$ be the canonical bundle associated to
$\mathcal{J}$ and let $K^{*}$ be the corresponding anticanonical bundle. Let
$s$ be a local section generating $K^{*}$, given by $s=\rho_{0}$, the zero-
degree part of $\rho$. The type of $\mathcal{J}$ jumps precisely along the
zero section, and it is constantly equal to $0$ outside of it.

######  Definition 2.5 ([15], Definition 2.10).

A generalized complex structure $(M,\mathcal{J},H)$ is called _stable_ if $s$
is transverse to the zero section $s^{-1}(0)$ of $K^{*}$.

Outside of the type change locus, $\mathcal{J}$ is $B$-field equivalent to the
structure induced by a symplectic form $\omega$. The type change locus is
locally the non-degenerate zero set of a function, thus it has codimension
$2$.

The main purpose of this paper is to carry techniques from symplectic geometry
into the generalized setting. To do so, we need an extension of the notion of
symplectic submanifold.

######  Definition 2.6 (cf. [25] and [31], Definition 2.3.6).

Let $(M,\mathcal{J},H)$ be a stable generalized complex structure. A
submanifold $N\subset M$ is said to be _$\mathcal{J}$ -symplectic_ if

  * •

there is a neighborhood $U$ of $N$ equipped with a symplectic form $\omega$
such that $N$ is a symplectic submanifold with respect to $\omega$;

  * •

when restricted to $U$, the generalized complex structure $\mathcal{J}$ is
$B$-field equivalent to the generalized complex structure
$\mathcal{J}_{\omega}$ induced by $\omega$.

######  Remark 2.7.

All the constructions we describe in the rest of the paper are presented for
stable generalized complex structures. However, they can be performed
independently of the stability of the considered structures: we just need the
notion of $\mathcal{J}$-symplectic submanifold, that can be easily extended to
unstable structures since it depends on the behaviour of $\mathcal{J}$ on a
neighborhood of a submanifold, and not on the global one.

We give a fundamental local example. Due to the local classification of
generalized complex structures ([1, 5, 4]), every generalized complex
structure is locally the product of a symplectic structure with a holomorphic
Poisson structure. If we restrict to stable structures, we have the following

######  Corollary 2.8 ([5], Proposition 4.2.7).

Any stable generalized complex structure is locally equivalent to the
structure on $\mathbb{C}^{2}\times\mathbb{R}^{2m-4}$ generated by

$\rho_{0}=(z_{1}+dz^{12})\wedge e^{i\omega_{0}},$

where $\mathbb{C}^{2}=\\{(z_{1},z_{2})\\}$ and $\omega_{0}$ is the standard
symplectic form on $\mathbb{R}^{2m-4}$.

The type change locus of $\rho_{0}$ consists in a single path-connected
component $\\{z_{1}=0\\}=\\{0\\}\times\mathbb{C}\times\mathbb{R}^{2m-4}$, on
which $\rho_{0}$ has type $2$. Everywhere else, $\rho_{0}$ has type $0$.
Indeed, taking coordinates $z_{1}=re^{i\theta}$, $z_{2}=x_{2}+iy_{2}$, such a
structure is equivalent to the one induced by the symplectic form

$\tilde{\omega}=d\log r\wedge dy^{2}+d\theta\wedge dx^{2}+\omega_{0}$ (2.1)

on the set $\\{z_{1}\neq 0\\}$.

######  Example 2.9.

Let $(\Sigma,\omega)$ be a symplectic $(2m-4)$-manifold. It is immediate to
put a stable generalized complex structure on $D^{2}\times T^{2}\times\Sigma$.
In fact, the product $\mathbb{C}^{2}\times\Sigma$ admits a stable generalized
complex structure generated by

$\rho=(z_{1}+dz^{12})\wedge e^{i\omega}.$

Taking the quotient of the second $\mathbb{C}$ factor by a lattice
$\mathbb{Z}[i]$, $\rho$ induces a stable structure $\mathcal{J}$ on
$D^{2}\times T^{2}\times\Sigma$, whose type changes from $0$ to $2$ along
$\\{0\\}\times T^{2}\times\Sigma$. If $p\in D^{2}\setminus\\{0\\}$, all the
submanifolds of the form $\\{p\\}\times T^{2}\times\Sigma$ are
$\mathcal{J}$-symplectic, and the restriction of $\mathcal{J}$ to them is
equivalent to the structure induced by a symplectic form $\tilde{\omega}$
defined as in (2.1), replacing $\omega_{0}$ by $\omega$.

## 3\. Generalized Luttinger surgery on $2m$-manifolds

In this section we introduce generalized Luttinger surgery on generalized
complex $2m$-manifolds, with $2m\geq 6$. We begin by describing the smooth
manifold that is obtained via the surgery and compute its basic topological
invariants, with special attention to the fundamental group. Next, we explain
how to put a stable generalized complex structure on it. We give examples and
applications of the surgery, enlarging the possible diffeomorphism types of
manifolds admitting stable generalized complex structures. Finally, we explain
how generalized Luttinger surgery can be further extended changing the gluing
diffeomorphism. Making a specific choice of diffeomorphism, we obtain
generalized Gluck twist.

### 3.1 The underlying smooth manifold.

Let $D^{2}_{\epsilon}$ be the closed disk of radius $\epsilon$ with
coordinates $re^{i\theta^{0}}$, let $T^{2}$ be the $2$-torus with coordinates
$(\theta^{1},\theta^{2})$ and let $(\Sigma,\omega_{\Sigma})$ be a symplectic
$(2m-4)$-manifold. Denote by $A(u,v)$ the annulus of radii $u,v$. Suppose that
$M$ is a $2m$-manifold admitting an embedded $D^{2}_{\epsilon}\times
T^{2}\times\Sigma$. Consider the manifold

$\tilde{M}:=M\setminus\left(D^{2}_{\epsilon_{0}}\times
T^{2}\times\Sigma\right)\cup_{\phi}\left(D^{2}_{1}\times
T^{2}\times\Sigma\right),$ (3.1)

obtained removing the embedded submanifold and gluing it back identifying
$A(r_{0},1)\times T^{2}\times\Sigma$ with $A(\epsilon_{0},\epsilon)\times
T^{2}\times\Sigma$ via a diffeomorphism

$\phi\colon A(r_{0},1)\times T^{2}\times\Sigma\longrightarrow
A(\epsilon_{0},\epsilon)\times T^{2}\times\Sigma.$

The gluing diffeomorphism $\phi$ is defined by

$\phi\left(re^{i\theta^{0}},\,\theta^{1},\,\theta^{2},\sigma\right)=\left(\sqrt{2\log(r\epsilon)}\,e^{i\left(p\theta^{0}+a\theta^{2}\right)},\,\theta^{1},\,q\theta^{0}+b\theta^{2},\sigma\right),$
(3.2)

where $\sigma\in\Sigma$ and $p,q,a,b$ are integers such that $pb-aq=1$.
Furthermore we can choose (cf. [25]) $p,q,a,b$ in such a way that

$pbt-aq\neq 0\qquad\text{for all $t\in[0,1]$.}$

Note that $\phi$ acts on $D^{2}\times T^{2}$ as a $4$-dimensional torus
surgery of multiplicity $p$ (see [23] for the construction in the smooth
setting and [25] for the generalized complex setting). This allows to give a
precise meaning to the parameters $p$, $q$, $a$, $b$. They are representing a
diffeomorphism of the boundary $\partial D^{2}\times T^{2}$ by a matrix

$\begin{bmatrix}p&0&q\\\ 0&1&0\\\ a&0&b\end{bmatrix}\in SL(3,\mathbb{Z}).$

Using handle decomposition of $D^{2}\times T^{2}$ it can be checked ([23], Ch.
4 and 8) that the resulting manifold is determined by the isotopy class of
$\phi(\partial D^{2}_{1}\times\\{pt\\})$. With this in mind, we proceed to
compute the basic topological invariants of $\tilde{M}$.

######  Proposition 3.1.

The manifolds $M$ and $\tilde{M}$ have the same Euler characteristic. If
$2m=4l$ for some $l\in\mathbb{N}$, they also have the same signature. The
fundamental group of $\tilde{M}$ is given by the quotient

$\left(\pi_{1}\left(M\setminus\left(D^{2}_{\epsilon_{0}}\times
T^{2}\times\Sigma\right)\right)\ast\pi_{1}(T^{2})\right)/N,$

where $N$ is the normal subgroup generated by $\phi_{*}(\pi_{1}(\partial
D^{2}_{1}\times T^{2}))$.

###### Proof.

see [9], proof of Proposition 2.2. ∎

The described surgery allows to obtain manifolds with a wide choice of
fundamental groups. We give two examples of that, but many more can be
produced changing the parameters $p,a,b,q$. In the first example we reduce the
fundamental group of the starting manifold. In the second one we show how to
add torsion to it. In Sections 3.3 and 5.3 we will endow our examples with a
stable generalized complex structure (Proposition 3.12, Example 5.6).

######  Example 3.2.

Suppose that $X$ is a $(2m-2)$-manifold admitting an embedded
$D^{2}\times\Sigma$, with $\Sigma$ a $(2m-4)$-manifold. Assume that the
inclusion

$\pi_{1}(X\setminus D^{2}\times\Sigma)\hookrightarrow\pi_{1}X$

is an isomorphism. Set $M:=X\times T^{2}$ and consider the manifold obtained
performing the surgery (3.1) with parameters $p=0$, $a=1$, $b=0$, $q=1$, along

$D^{2}\times\Sigma\times T^{2}\hookrightarrow X\times T^{2}.$

Since $\phi$ is the identity on $\theta^{1}$, the resulting manifold is a
product $\tilde{M}\cong\tilde{X}\times\mathbb{S}^{1}$. Moreover, $\phi$ is
gluing the loop generated by $\theta^{2}$ in $\pi_{1}(X\times T^{2})$ to the
loop $\partial D^{2}\times\\{pt\\}$. Thus, using Seifert-Van Kampen, we
conclude that

$\pi_{1}\tilde{X}\cong\pi_{1}(X\setminus D^{2}\times\Sigma)\cong\pi_{1}X.$

This reduces the fundamental group from
$\pi_{1}M\cong\pi_{1}X\times\mathbb{Z}^{2}$ to
$\pi_{1}\tilde{M}\cong\pi_{1}X\times\mathbb{Z}$.

######  Example 3.3.

Suppose that $X$ is a manifold as in Example 3.2. Assume that $X$ is also
simply connected. Set $M:=X\times T^{2}$ and consider the manifold obtained
performing the surgery (3.1) with parameters $p=0$, $a=1$, $b=0$, $q\geq 2$
along

$D^{2}\times\Sigma\times T^{2}\hookrightarrow X\times T^{2}.$

The resulting manifold $\tilde{M}$ will be diffeomorphic to
$\tilde{X}\times\mathbb{S}^{1}$ and

$\pi_{1}\tilde{M}\cong\pi_{1}\tilde{X}\times\mathbb{Z}\cong\mathbb{Z}_{q}\times\mathbb{Z}.$

### 3.2 The geometric structure.

Let $(M,\mathcal{J},H)$ be a stable generalized complex structure. Let
$(\Sigma,\omega_{\Sigma})$ be a $(2m-4)$-symplectic manifold. Suppose that
there exists an embedded, $\mathcal{J}$-symplectic, $D^{2}_{\epsilon}\times
T^{2}\times\Sigma\hookrightarrow M$ such that on it $\mathcal{J}$ is $B$-field
equivalent to the generalized complex structure induced by the product
symplectic form

$\tilde{\omega}:=r\,dr\,d\theta^{0}+d\theta^{12}+\omega_{\Sigma}.$

The choice of a diffeomorphism $\phi$ as in (3.2) induces a symplectic form
$\phi^{*}(\tilde{\omega})$ on the annulus $A(r_{0},1)\times
T^{2}\times\Sigma$. Let $\xi$ be a radial smooth increasing function vanishing
in a neighborhood of the origin and identically $1$ in a neighborhood of
$[r_{0},1]$. In the following lemma we use $\xi$ to extend
$\phi^{*}(\tilde{\omega})$ to a stable generalized complex structure defined
on $D^{2}_{\epsilon}\times T^{2}\times\Sigma$. Its proof is implicitly
contained in the proof of Theorem 3.1 in [25].

######  Lemma 3.4.

Let $\mathcal{J}_{0}$ be the stable generalized complex structure defined on
$D^{2}_{1}\times T^{2}\times\Sigma$ by the pure spinor

$\rho_{0}=z_{1}\,\exp\left(-\frac{p}{4}\xi(\lvert
z_{1}\rvert)\frac{dz^{1\bar{1}}}{\lvert
z_{1}\rvert^{2}}-\frac{b}{2}\,dz^{2\bar{2}}+\frac{dz^{1}}{2z_{1}}\left[(\frac{a}{2}-q)dz^{2}-(\frac{a}{2}+q)dz^{\bar{2}}\right]\right)\wedge
e^{i\phi^{*}(\omega_{\Sigma})}.$

The pullback form $\phi^{*}(\tilde{\omega})$ can be extended to a stable
generalized complex structure on $D^{2}_{1}\times T^{2}\times\Sigma$ that is
$B$-field equivalent to $\mathcal{J}_{0}$.

The geometric construction consists in removing from $M$ the embedded
$D^{2}_{\epsilon}\times T^{2}\times\Sigma$ and in gluing back a copy of it as
in (3.1) carrying a generalized complex structure that is $B$-field equivalent
to $\mathcal{J}_{0}$. More precisely, let $\rho_{\mathcal{J}}$ be the pure
spinor associated to $\mathcal{J}$, let $B_{\mathcal{J}}$ be the $2$-form
defined by $\rho_{\mathcal{J}}$ and $B_{0}$ the one defined by $\rho_{0}$. Let
$\rho$ be the pure spinor obtained from $\rho_{0}$ via the $B$-field transform

$\rho:=e^{\tilde{B}_{0}-B_{0}+\phi^{*}(B_{\mathcal{J}})}\rho_{0},$

where $\tilde{B}_{0}$ is a real $2$-form on $D^{2}_{1}\times
T^{2}\times\Sigma$ that vanishes on a neighborhood of its boundary and agrees
with $B_{0}$ on a neighborhood of $D^{2}_{r_{0}}\times T^{2}\times\Sigma$. We
set

$\tilde{\rho}:=\begin{cases}\rho_{\mathcal{J}}&\text{on
}M\setminus\left(D^{2}_{\epsilon_{0}}\times T^{2}\times\Sigma\right),\\\ &\\\
\rho&\text{on }D^{2}_{1}\times T^{2}\times\Sigma.\end{cases}$ (3.3)

Since $\rho_{\mathcal{J}}$ and $\rho$ agree on the overlap, $\tilde{\rho}$
defines a stable generalized complex structure $\tilde{\mathcal{J}}$ on
$\tilde{M}$ that is integrable with respect to the $3$-form

$\tilde{H}:=\begin{cases}H&\text{on
}M\setminus\left(D^{2}_{\epsilon_{0}}\times T^{2}\times\Sigma\right),\\\ &\\\
\phi^{*}(H)-d\tilde{B}_{0}&\text{on }D^{2}_{1}\times
T^{2}\times\Sigma.\end{cases}$

######  Definition 3.5.

We say that the generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ is obtained from
$(M,\mathcal{J},H)$ performing _generalized Luttinger surgery_ along
$T^{2}\times\Sigma$.

######  Remark 3.6.

We could have chosen to glue back a copy of $D^{2}_{1}\times
T^{2}\times\Sigma$ carrying any stable generalized complex structure extending
the pullback $\phi^{*}(\tilde{\omega})$. For a fixed $\phi$, the choice of an
extension not equivalent to $\rho_{0}$ will provide a different
$\tilde{\mathcal{J}}$ on the same underlying smooth manifold $\tilde{M}$.
Fixing the structure $\rho_{0}$ gives a definition of surgery that depends
only on the choice of $\phi$.

######  Remark 3.7.

Generalized Luttinger surgery presents similarities with another
generalization of Luttinger surgery _([29]):_ _coisotropic Luttinger surgery_
_([9, 3, 35, 2])_. They are both performed along codimension $2$ submanifolds
of the form $T^{2}\times\Sigma$. However, they differ in the choice of the
gluing diffeomorphism and in the geometric structures that they are gluing.

The main application of generalized Luttinger surgery will be in producing
stable generalized complex structures, as made precise in the following
theorem.

######  Theorem 3.8.

Let $(M^{2m},\mathcal{J},H)$ be a stable generalized complex structure and let
$S$ be its type change locus. Let $(\Sigma,\omega_{\Sigma})$ be a symplectic
$(2m-4)$-manifold. Assume that there exists an embedded
$T^{2}\times\Sigma\hookrightarrow M$ such that

  * $(i)$

there exists a neighborhood $N_{T}$ of $T^{2}\times\Sigma$ that is
diffeomorphic to $D^{2}\times T^{2}\times\Sigma$;

  * $(ii)$

$T^{2}\times\Sigma$ is $\mathcal{J}$-symplectic and
$\omega_{|T^{2}\times\Sigma}=d\theta^{12}+\omega_{\Sigma}$, where $\omega$ is
the symplectic form induced by $\mathcal{J}$ on $N_{T}$ up to $B$-field
equivalence.

Then performing generalized Luttinger surgery along $T^{2}\times\Sigma$ yields
a stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type changes along
$S\sqcup\left(T^{2}\times\Sigma\right)$.

###### Proof.

Let $N_{T}\cong D^{2}_{\epsilon}\times T^{2}\times\Sigma$ be a neighborhood of
$T^{2}\times\Sigma$. By assumption $\mathcal{J}$ induces on $N_{T}$ a
symplectic form $\omega$ that, when restricted to $T^{2}\times\Sigma$,
coincides with the product one. Applying Weinstein’s symplectic neighborhood
theorem (cf. [38] or [30], Theorem 3.4.10), up to taking a smaller $\epsilon$,
$(N_{T},\omega_{|N_{T}})$ is symplectomorphic to

$(D^{2}_{\epsilon}\times
T^{2}\times\Sigma,\,\tilde{\omega}:=r\,dr\,d\theta^{0}+d\theta^{12}+\omega_{\Sigma}).$

We perform generalized Luttinger surgery along $T^{2}\times\Sigma$ obtaining a
stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$. $T^{2}\times\Sigma$ is
$\mathcal{J}$-symplectic, thus $N_{T}$ can be taken to be disjoint from the
type change locus $S$. The type of $\tilde{\mathcal{J}}$ jumps along $S$,
since $\tilde{\mathcal{J}}$ agrees with $\mathcal{J}$ on $M\setminus N_{T}$,
and also along $T^{2}\times\Sigma$ because the structure that we glued back
has type jump $0\mapsto 2$ on $\\{0\\}\times T^{2}\times\Sigma$. ∎

######  Remark 3.9.

In _[31]_, Theorem 3.2.7, it is implicitly assumed that symplectic forms on
$\mathcal{J}$-symplectic $6$-tori are the product form. Setting $2m=6$,
$\Sigma=T^{2}$ in Theorem 3.8 and precomposing with a permutation of the
coordinates on $\partial D^{2}\times T^{4}$ (cf. Remark 4.5), we recover the
same result with the additional hypothesis made explicit.

### 3.3 Examples and applications.

We apply generalized Luttinger surgery in several instances. Our main
achievements consist in showing the existence of stable generalized complex
structures with non-homotopy-equivalent path-connected components of their
type change locus and in enlarging the class of diffeomorphism types of
manifolds admitting a stable generalized complex structure.

As a first application, we exploit the fact that generalized Luttinger surgery
can be performed along $T^{2}\times\Sigma$ to produce stable generalized
complex structures whose path-connected components of the type change locus
are not pairwise homotopy-equivalent.

######  Theorem 3.10.

There exist stable generalized complex structures with non-homotopy-equivalent
path-connected components of their type change locus.

###### Proof.

Denote by $X$ the manifold described in [10], Theorem 20. Recall that $X$ is a
symplectic $4$-manifold containing an embedded $2$-torus $T$ and an embedded
surface $\Sigma$ of genus $2$ that can be taken to be symplectic and disjoint.
Both have a neighborhood diffeomorphic to $D^{2}\times T$, respectively
$D^{2}\times\Sigma$. Consider on $M:=X\times T^{2}$ the generalized complex
structure induced by the product symplectic form and perform two generalized
Luttinger surgery on the embedded, $\mathcal{J}$-symplectic, disjoint $T\times
T^{2}$ and $\Sigma\times T^{2}$. This provides a stable generalized complex
structure $(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ on a $6$-manifold whose
type change locus has two path connected components that are not homotopy-
equivalent. Stable structures in any even dimension $2m\geq 8$ can be obtained
taking products of $\tilde{M}$ with a generalized complex $(2m-6)$-manifold of
constant type $0$. ∎

######  Remark 3.11.

We point out that in dimension $4$ each path-connected component of the type
change locus of a stable generalized complex structure is diffeomorphic to a
$2$-torus, inheriting the structure of an elliptic curve _([13]_, Theorem
2.1), thus the phenomenon described in Theorem 3.10 is exclusive of stable
structures of dimension $2m\geq 6$.

Ideally, one would be able to determine the diffeomorphism type of the
manifold $\tilde{M}$ obtained via generalized Luttinger surgery, with the
purpose of producing generalized complex structures on manifolds that were not
previously known to admit one. In the rest of this section we obtain some
partial result in this direction.

Perform a single generalized Luttinger surgery on a generalized complex
manifold of the form $M\times T^{2}$. Since the surgery acts as the identity
on the first coordinate of the $2$-torus, the resulting manifold can be
written as a product $\tilde{M}\times\mathbb{S}^{1}$. More precisely, we have
the following

######  Proposition 3.12.

Let $(M^{2m-2},\mathcal{J},H)$ be a stable generalized complex structure. Let
$(\Sigma,\omega_{\Sigma})$ be a symplectic $(2m-4)$-manifold. Assume that
there exists an embedded, $\mathcal{J}$-symplectic $\Sigma\hookrightarrow M$
with a neighborhood diffeomorphic to $D^{2}\times\Sigma$. Fix on $T^{2}$ a
generalized complex structure induced by a symplectic form. Then performing
generalized Luttinger surgery along the submanifold

$\Sigma\times T^{2}\hookrightarrow M\times T^{2},$

endowed with the product structure, yields a stable generalized complex
structure

$(\tilde{M}^{2m-1}\times\mathbb{S}^{1},\tilde{\mathcal{J}},\tilde{H}).$

In dimension $2m=6$, under some assumptions on the manifold $M$ and the
diffeomorphism $\phi$, it is possible to obtain more information on the
diffeomorphism type of $\tilde{M}$.

######  Proposition 3.13.

Let $(M^{4},\mathcal{J},H)$ be a stable generalized complex structure as in
Proposition 3.12. Assume that both $M$ and the complement
$M\setminus\left(D^{2}\times\Sigma\right)$ are simply connected. Let $g$ be
the genus of $\Sigma$ and set $k=b_{2}(M)+2g-1$. Perform a generalized
Luttinger surgery with parameters $p=0$, $a=1$, $q=1$, $b=0$ along the
submanifold

$\Sigma\times T^{2}\hookrightarrow M\times T^{2}$

to obtain a stable generalized complex structure on
$\tilde{M}\times\mathbb{S}^{1}$. The resulting manifold is either
diffeomorphic to

$\left(\\#_{k}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right)\times\mathbb{S}^{1},$
(3.4)

when $\tilde{M}$ is spin, or to

$\left(\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{3}\\#\left(\\#_{k-1}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right)\right)\times\mathbb{S}^{1},$
(3.5)

when $\tilde{M}$ is non-spin.

###### Proof.

Using Seifert-van Kampen’s theorem (cf. Example 3.2), we see that $\tilde{M}$
is simply connected. Using Poincaré duality and a Mayer-Vietoris argument, it
is straightforward to compute

$H_{2}(\tilde{M};\mathbb{Z})\cong
H_{3}(\tilde{M};\mathbb{Z})\cong\mathbb{Z}^{k},$

where $k=b_{2}(X)+2g-1$. We conclude the proof applying classification results
for simply connected $5$-manifolds proved in [32] and [11], that allow to
determine the diffeomorphism type of $\tilde{M}$ knowing its homology, its
linking form and whether or not it is spin. If $\tilde{M}$ is spin, then

$\tilde{M}\cong\\#_{k}\,\mathbb{S}^{2}\times\mathbb{S}^{3}.$

If $\tilde{M}$ is non-spin, then

$\tilde{M}\cong\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{3}\\#\left(\\#_{k-1}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right).$

∎

Using Proposition 3.13, producing stable generalized complex structures on
$6$-manifolds with diffeomorphism type (3.4) or (3.5) reduces to a symplectic
geography problem, as shown in the next example.

######  Example 3.14.

Let $X$ be a simply connected, symplectic $4$-manifold containing an embedded
symplectic surface $\Sigma$ of genus $g\geq 0$ with trivial normal bundle and
such that $X\setminus D^{2}\times\Sigma$ is simply connected. Suppose that
$\Sigma$ intersects transversely a surface $S$ of self-intersection $-1$, at a
single point $x$. In particular, the normal bundle of $S$ is not trivial.
Endow $X\times T^{2}$ with a product symplectic form and perform generalized
Luttinger surgery along $\Sigma\times T^{2}$ as in Proposition 3.13 obtaining
a stable generalized complex structure on $\tilde{M}\times\mathbb{S}^{1}$. The
manifold $\tilde{M}$ is simply connected and non-spin, since the surface
obtained removing from $S$ a disk around $x$ and gluing it back via
generalized Luttinger surgery still has non-trivial normal bundle. This
produces a stable structure on

$\left(\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{3}\\#\left(\\#_{b_{2}(X)+2g-2}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right)\right)\times\mathbb{S}^{1}.$

Taking explicit examples of manifolds satisfying our assumptions we have that:

  * •

the elliptic surface $E(1)$, whit $\Sigma$ given by the generic fiber and $S$
by a cusp neighborhood, yields a stable generalized complex structure on

$\left(\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{3}\\#\left(\\#_{10}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right)\right)\times\mathbb{S}^{1};$

  * •

the manifold $X$ obtained in [10], Theorem 13, with $\Sigma$ given by a
surface of genus $2$ and $S$ by a torus denoted by $H_{1}$, yields a stable
structure on

$\left(\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{3}\\#\left(\\#_{6}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right)\right)\times\mathbb{S}^{1}.$

The construction can be slightly modified asking $\Sigma$ to be disjoint from
$S$, leading to the same conclusion as above. For example, this is the case of
the manifold $B$ obtained in [10], Theorem 18, with $\Sigma$ and $S$ both
given by disjoint tori. This produces a stable structure on

$\left(\mathbb{S}^{2}\tilde{\times}\mathbb{S}^{3}\\#\left(\\#_{8}\,\mathbb{S}^{2}\times\mathbb{S}^{3}\right)\right)\times\mathbb{S}^{1}.$

## 4\. Changing the gluing diffeomorphism.

We show how generalized Luttinger surgery can be modified choosing a gluing
diffeomorphism different from (3.2), that does not necessarily act as the
identity on $\Sigma$. The general procedure is as follows:

  * •

Pick a diffeomorphism $\phi\colon A(r_{0},1)\times
T^{2}\times\Sigma\rightarrow A(\epsilon_{0},\epsilon)\times
T^{2}\times\Sigma$. This gives a lot of freedom of choice. For example, $\phi$
can act on $D^{2}\times T^{2}$ as torus surgery of multiplicity $p$ and on
$\Sigma$ as any self-diffeomorphism.

  * •

Fix on $D^{2}_{\epsilon}\times T^{2}\times\Sigma$ a symplectic form $\omega$.
Assume that we can extend $\phi^{*}(\omega)$ to a stable generalized complex
structure $\mathcal{J}_{0}$ on $D^{2}_{1}\times T^{2}\times\Sigma$.

  * •

Suppose that $(M,\mathcal{J},H)$ is a stable generalized complex manifold
admitting an embedded, $\mathcal{J}$-symplectic $D^{2}\times
T^{2}\times\Sigma$ on which $\mathcal{J}$ is $B$-field equivalent to the
generalized structure induced by $\omega$. Remove $D^{2}\times
T^{2}\times\Sigma$ and glue it back via $\phi$, carrying the generalized
complex structure $\mathcal{J}_{0}$ extending $\phi^{*}(\omega)$, as we did in
Section 3.2. This endows the resulting manifold $\tilde{M}$ with a stable
generalized complex structure $\tilde{J}$.

### 4.1 Generalized Gluck twist

As an example of the general procedure, we pick a specific diffeomorphism and
obtain a surgery that we call _generalized Gluck twist_. Suppose that

$(\Sigma,\,\omega_{\Sigma})\cong(R\times\mathbb{S}^{2},\,\omega_{R}+\omega_{\mathbb{S}^{2}}),$

where $(R,\omega_{R})$ is a symplectic $(2m-6)$ manifold. The gluing
diffeomorphism is obtained combining generalized Luttinger surgery and Gluck
twist as we explain now.
Denote by $(h,\theta^{S})$ cylindrical coordinates on the sphere. Suppose that
$X$ is a $4$-manifold admitting an embedded $2$-sphere with trivial normal
bundle. The manifold obtained from $X$ via Gluck twist is the manifold

$\tilde{X}:=\left(X\setminus
D^{2}\times\mathbb{S}^{2}\right)\cup_{\tau}D^{2}\times\mathbb{S}^{2},$

where $\tau$ is the diffeomorphism of the boundary $\partial
D^{2}\times\mathbb{S}^{2}$ given by

$\tau(\theta^{0},\,h,\,\theta^{S})=(\theta^{0},\,h,\,\theta^{S}+\theta^{0}).$

Suppose now that $(M^{2m},\mathcal{J},H)$ is a stable generalized complex
manifold admitting an embedded, $\mathcal{J}$-symplectic
$D^{2}_{\epsilon}\times T^{2}\times R\times\mathbb{S}^{2}$ such that on it
$\mathcal{J}$ is $B$-field equivalent to the structure induced by the
symplectic form

$\omega:=r\,dr\,d\theta^{0}+d\theta^{12}+\omega_{R}+dh\,d\theta^{S}.$

Consider the manifold

$\tilde{M}:=M\setminus\left(D^{2}_{\epsilon_{0}}\times T^{2}\times
R\times\mathbb{S}^{2}\right)\cup_{\phi}(D^{2}_{1}\times T^{2}\times
R\times\mathbb{S}^{2}),$

obtained removing the embedded submanifold and gluing it back identifying the
annuli via the diffeomorphism $\phi$, defined by

$\phi\left(re^{i\theta^{0}},\,\theta^{1},\,\theta^{2},\sigma,h,\theta^{S}\right)=\left(\sqrt{2\log(r\epsilon)}\,e^{i\left(p\theta^{0}+a\theta^{2}\right)},\,\theta^{1},\,q\theta^{0}+b\theta^{2},\varphi(\sigma),h,\theta^{S}+\theta^{0}\right),$

where $\sigma\in R$, $p,q,a,b$, are as in Section 3.2 and $\varphi$ is any
self-diffeomorphism of $R$. Note that $\phi$ restricts to torus surgery of
multiplicity $p$ on $D^{2}\times T^{2}$ and to Gluck twist on
$D^{2}\times\mathbb{S}^{2}$. The symplectic form $\phi^{*}(\omega)$ can be
extended from $A(r_{0},1)\times T^{2}\times R\times\mathbb{S}^{2}$ to a stable
generalized complex structure on $D^{2}_{1}\times T^{2}\times
R\times\mathbb{S}^{2}$, obtained modifying the pure spinor $\rho_{1}$ defined
in [31], proof of Theorem 3.3.2. We set

$\displaystyle\tilde{\rho}_{1}=z_{1}\,\exp\Big{(}$
$\displaystyle-\frac{p}{4}\xi(\lvert
z_{1}\rvert^{2})\frac{dz^{1\bar{1}}}{\lvert
z_{1}\rvert^{2}}-\frac{b}{2}\,dz^{2\bar{2}}+$
$\displaystyle\frac{dz^{1}}{2z_{1}}\left[(\frac{a}{2}-q)dz^{2}-(\frac{a}{2}+q)dz^{\bar{2}}\right]-$
$\displaystyle\frac{2}{(1+\lvert
z_{3}\rvert^{2})^{2}}\,dz^{2}\left[\bar{z}_{3}dz^{3}+z_{3}dz^{\bar{3}}\right]\Big{)},$

where $z_{1}$, $z_{2}$ are complex coordinates on $D^{2}$, $T^{2}$
respectively, and $z_{3}$ is the complex coordinate on $\mathbb{S}^{2}$ given
by the stereographic projection.

######  Lemma 4.1.

Consider on $D^{2}_{1}\times T^{2}\times R\times\mathbb{S}^{2}$ the stable
generalized complex structure $\mathcal{J}_{0}$ defined by the pure spinor

$\rho_{0}=\tilde{\rho}_{1}\wedge e^{i\phi^{*}(\omega_{R})}.$

The pullback form $\phi^{*}(\omega)$ can be extended to a stable generalized
complex structure on $D^{2}_{1}\times T^{2}\times R\times\mathbb{S}^{2}$ that
is $B$-field equivalent to $\mathcal{J}_{0}$.

Following Section 3.2, we can glue $\mathcal{J}$ and $\mathcal{J}_{0}$ to
endow the manifold $\tilde{M}$ with a stable generalized complex structure
$\tilde{\mathcal{J}}$.

######  Definition 4.2.

We say that the generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ is obtained from
$(M,\mathcal{J},H)$ performing _generalized Gluck twist_ along $T^{2}\times
R\times\mathbb{S}^{2}$.

Several results proved for generalized Luttinger surgery hold also for
generalized Gluck twist with analogous proofs. For instance, Theorem 3.8
corresponds to the following result.

######  Theorem 4.3.

Let $(M^{2m},\mathcal{J},H)$ be a stable generalized complex structure and let
$S$ be its type change locus. Let $(R,\omega_{R})$ be a symplectic
$(2m-6)$-manifold. Assume that there exists an embedded $T^{2}\times
R\times\mathbb{S}^{2}\hookrightarrow M$ such that

  * $(i)$

there exists a neighborhood $N_{T}$ of $T^{2}\times R\times\mathbb{S}^{2}$
that is diffeomorphic to $D^{2}\times T^{2}\times R\times\mathbb{S}^{2}$;

  * $(ii)$

$T^{2}\times R\times\mathbb{S}^{2}$ is $\mathcal{J}$-symplectic and
$\omega_{|T^{2}\times
R\times\mathbb{S}^{2}}=d\theta^{12}+dh\,d\theta^{S}+\omega_{R}$, where
$\omega$ is the symplectic form induced by $\mathcal{J}$ on $N_{T}$ up to
$B$-field equivalence.

Then performing generalized Gluck twist along $T^{2}\times
R\times\mathbb{S}^{2}$ yields a stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type changes along
$S\sqcup(T^{2}\times R\times\mathbb{S}^{2})$.

######  Example 4.4.

Let $(X^{2m},\mathcal{J},H)$ be a stable generalized complex structure. Assume
that $X$ contains an embedded, $\mathcal{J}$-symplectic $T^{2}\times R$ with
trivial normal bundle. Endow $M:=X\times\mathbb{S}^{2}$ with the product
generalized complex structure induced by a symplectic form on
$\mathbb{S}^{2}$. Performing generalized Gluck twist along $T^{2}\times
R\times\mathbb{S}^{2}\hookrightarrow M$ yields a stable generalized complex
structure on a manifold $\tilde{M}$ which has the structure of a fiber bundle
with fiber $\mathbb{S}^{2}$ and base $\tilde{X}$, where $\tilde{X}$ is the
manifold obtained performing the surgery along $T^{2}\times R\hookrightarrow
X$ induced by the restriction of $\phi$ to the boundary $\partial D^{2}\times
T^{2}\times R$.

######  Remark 4.5.

In some special cases, our surgeries recover known techniques. Allowing in
Section 3.1 $\Sigma$ to be a point, generalized Luttinger surgery coincides
with torus surgery of multiplicity $p$. Similarly, allowing in Section 4.1 $R$
to be a point, generalized Gluck twist reduces to the surgery of Theorem 3.3.2
in [31].

## 5\. Branched coverings of generalized complex manifolds

We begin this section recalling the notion of branched coverings and we give
some basic example. Then, we add the geometric structure and show that
branched coverings of stable generalized complex manifolds with
$\mathcal{J}$-symplectic branch locus are still stable generalized complex.
Finally we explain how to obtain stable generalized complex structures
combining the techniques we described.

### 5.1 Branched coverings.

Let $M$ and $\tilde{M}$ be $n$-manifolds and $d\in\mathbb{N}$.

######  Definition 5.1.

A _$d$ -fold branched covering_ of $M$ is a proper, smooth map
$\pi\colon\tilde{M}\longrightarrow M$ with critical set $B\subset M$ such that

  * $(i)$

outside of $B$, the restriction of $\pi$

$\pi|_{\tilde{M}\setminus\pi^{-1}(B)}\colon\tilde{M}\setminus\pi^{-1}(B)\longrightarrow
M\setminus B$

is a covering map of degree $d$;

  * $(ii)$

for each $p\in\pi^{-1}(B)$ there are local coordinate charts $U$,
$V\longrightarrow\mathbb{C}\times\mathbb{R}^{n-2}_{+}$ around $p$ and
$\pi(p)$, respectively, on which $\pi$ has the form

$(z,x)\longmapsto(z^{t},x)$

for some $t\in\mathbb{N}$, $t\geq 2$.

The set $B$ is called the _branch locus_ of $\pi$ and the integer $t$ is the
_branching index_ of $\pi$ at the point $p\in B$.

We give some basic example of branched coverings of smooth manifolds that will
be useful in building explicit examples in generalized complex geometry.

######  Example 5.2 (Branched covering of product of surfaces).

Let $\Sigma$ and $\tilde{\Sigma}$ be closed orientable surfaces of genus $g$,
$\tilde{g}\geq 0$, respectively. If there exists a branched covering

$\pi\colon\tilde{\Sigma}\longrightarrow\Sigma,$ (5.1)

the Riemann-Hurwitz formula implies that $\tilde{g}\geq g$. Viceversa, if
$\tilde{g}\geq g$, $\tilde{\Sigma}$ can always be realized as a branched
covering of $\Sigma$. In such a case, its branch locus consists in a finite
number $k$ of isolated points $\\{x_{1},\dots,x_{k}\\}$. As a consequence, we
can build branched coverings of $4$-manifolds that are products of two
surfaces

$\psi\colon\Sigma_{h}\times\tilde{\Sigma}\longrightarrow\Sigma_{h}\times\Sigma,$

where $\psi$ is obtained extending $\pi$ to act as the identity on
$\Sigma_{h}$. Their branch locus is
$B=\Sigma_{h}\times\\{x_{1},\dots,x_{k}\\}$.

######  Example 5.3 ([23], p. 245).

Let $E(n)$ be a simply connected elliptic surface. There is a 2-fold branched
covering

$E(2n)\rightarrow E(n)$

with branch locus a pair of generic regular fibers.

### 5.2 The geometric structure.

We start from a given stable generalized complex structure on a $2m$-manifold
$M$, with $2m\geq 4$. We use a branched covering
$\pi\colon\tilde{M}\rightarrow M$ to build a stable generalized complex
structure on $\tilde{M}$. This is straightforward when the branch locus is
empty, i.e. for genuine covering maps, as proved in the following

######  Lemma 5.4.

Let $(M,\mathcal{J},H)$ be a stable generalized complex structure whose type
change locus has $k\in\mathbb{N}$ path-connected components. Let
$\pi\colon\tilde{M}\longrightarrow M$ be a $d$-fold covering map of $M$. Then
there exists a stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type change locus has
$d\cdot k$ path-connected components.

###### Proof.

Let $\rho$ be the pure spinor defined by $\mathcal{J}$. Then
$\tilde{\rho}:=\pi^{*}\rho$ is well-defined since $\pi$ is a local
diffeomorphism and defines on $\tilde{M}$ a generalized complex structure
$\tilde{\mathcal{J}}$ integrable with respect to $\tilde{H}:=\pi^{*}H$. If
$\rho$ has type $t$ at $p\in M$, then $\tilde{\rho}$ has type $t$ at
$\pi^{-1}(p)$. Thus the type of $\tilde{\mathcal{J}}$ jumps precisely along
the preimage of the type change locus of $\mathcal{J}$, that has $d\cdot k$
path-connected components. For the same reason, since $\mathcal{J}$ is stable
so is $\tilde{\mathcal{J}}$. ∎

A similar statement can be proved for branched coverings. Having underlined
the role of $\mathcal{J}$-symplectic submanifolds, the proof reduces to using
Lemma 5.4 and to rephrasing a result in [22] in the generalized complex
language.

######  Theorem 5.5 ([22], Lemma 1).

Let $(M^{2m},\mathcal{J},H)$ be a stable generalized complex structure whose
type change locus has $k\in\mathbb{N}$ path-connected components. Let $B$ be a
$\mathcal{J}$-symplectic submanifold of $M$, and let $\pi:\tilde{M}\rightarrow
M$ be a $d$-fold branched covering with branch locus $B$. Then there exists a
stable generalized complex structure
$(\tilde{M},\tilde{\mathcal{J}},\tilde{H})$ whose type change locus has
$d\cdot k$ path-connected components.

###### Proof.

Lemma 1 in [22] establishes that if $(M,\omega)$ is a closed, symplectic
manifold and $\pi\colon\tilde{M}\rightarrow M$ is a $d$-fold branched covering
with symplectic branch locus $B$ of codimension $2$, then there exist a
neighborhood $V$ of $B$ and a symplectic form $\tilde{\omega}$ on $\tilde{M}$
that agrees with $\pi^{*}\omega$ on $\tilde{M}\setminus\pi^{-1}(V)$, with
$[\tilde{\omega}]=\pi^{*}[\omega]$. In other words, $\pi^{*}\omega$ can be
extended from $\tilde{M}\setminus\pi^{-1}(V)$ to the preimage of $V$.
In our case, let $N$ be a neighborhood of $B$ and let $\omega$ be a symplectic
form on $N$ inducing a generalized complex structure equivalent to
$\mathcal{J}$, up to the $B$-transform $e^{B_{0}}$. Since the construction in
[22] is performed on a neighborhood of $B$, it can be applied to $(N,\omega)$
to find a neighborhood $V\subset N$ of $B$ and a symplectic form
$\tilde{\omega}$ extending $\pi^{*}\omega$ from $\pi^{-1}(N\setminus V)$ to
$\pi^{-1}(N)$. Let $\rho_{\mathcal{J}}$ be the pure spinor associated to
$\mathcal{J}$, and consider the pure spinor defined by

$\tilde{\rho}:=\begin{cases}\pi^{*}\rho_{\mathcal{J}}&\text{on
}\tilde{M}\setminus\pi^{-1}(V),\\\ &\\\
e^{\pi^{*}\tilde{B}_{0}+\pi^{*}(B_{\mathcal{J}})+i\tilde{\omega}}&\text{on
}\pi^{-1}(N),\end{cases}$

where $B_{\mathcal{J}}$ is the $2$-form defined by $\rho_{\mathcal{J}}$ and
$\tilde{B}_{0}$ is an extension of $B_{0}$ as in the proof of Theorem 3.8. Due
to Lemma 5.4 and Lemma 1 in [22], $\tilde{\rho}$ is a well-defined stable
generalized complex structure on $\tilde{M}$, that is integrable with respect
to

$\tilde{H}:=\begin{cases}\pi^{*}H&\text{on }\tilde{M}\setminus\pi^{-1}(V),\\\
&\\\ \pi^{*}H-d(\pi^{*}\tilde{B}_{0})&\text{on }\pi^{-1}(N).\end{cases}$

Since $\tilde{\rho}$ has type $0$ on $\pi^{-1}(B)$, the second statement from
Lemma 5.4 completes the proof of the theorem. ∎

### 5.3 Examples and applications.

We use coverings and branched coverings, combined with other techniques, to
exemplify how to obtain stable generalized complex structures.

Our first example uses generalized Luttinger surgery together with coverings
of generalized complex manifolds to produce stable generalized complex
structures on a variety of manifolds.

######  Example 5.6.

Let $2m\geq 6$. Consider a stable generalized complex $2m$-manifold
$(X,\mathcal{J},H)$ admitting an embedded, $\mathcal{J}$-symplectic
$(2m-2)$-manifold $\Sigma$. Fix a generalized complex structure on $T^{2}$
induced by a symplectic form and endow $M=X\times T^{2}$ with the product
structure. Perform generalized Luttinger surgery along $\Sigma\times T^{2}$ to
add torsion to the fundamental group (e.g. Example 3.3). The resulting
manifold will admit a stable generalized complex structure and any of its
coverings will also admit one, according to Lemma 5.4. This construction
expands the possible diffeomorphism types of stable generalized complex
manifolds.

A similar construction works also in the $4$-dimensional case, involving
symplectic sum ([21]) and replacing generalized Luttinger surgery with torus
surgery. We give an example of how this can be done.

######  Example 5.7.

The procedure described in [34] (Lemma 13 and 14) allows to build, using
symplectic sum and torus surgery, symplectic manifolds with fundamental group
$G\in\left\\{\\{1\\},\mathbb{Z}_{p},\mathbb{Z}_{p}\oplus\mathbb{Z}_{q},\mathbb{Z}\oplus\mathbb{Z}_{q},\mathbb{Z},\mathbb{Z}^{2}\right\\}$
(see also [8, 10, 36]). The construction can be repeated verbatim for stable
generalized complex manifolds, as long as the submanifolds along which we
perform the symplectic sum and the torus surgeries are
$\mathcal{J}$-symplectic. Furthermore, taking coverings (cf. [34], Remark 20),
we obtain more stable generalized complex structures thanks to Lemma 5.4.

Thanks to Theorem 5.5, finding generalized complex structures amounts to
finding branched coverings with $\mathcal{J}$-symplectic branch locus. We show
here two examples. However, we point out that in general checking that a
submanifold is $J$-symplectic is not an easy task.

######  Example 5.8 (Generalized complex structures on
$T^{2}\times\Sigma_{g}$).

Fix $g\geq 0$. In [37] it was proved that $T^{2}\times\Sigma_{g}$ admits a
family of stable generalized complex structures $\\{\mathcal{J}_{n}\\}$ whose
type change locus has $n$ path-connected components. In particular, consider
any of such structures $\mathcal{J}$ on $T^{2}\times\mathbb{S}^{2}$ and let
$S$ be its type change locus. As long as $x\notin S$,
$T^{2}\times\mathbb{S}^{2}$ contains $T^{2}\times\\{x\\}$ as a
$\mathcal{J}$-symplectic submanifold. Thus we can use the branched covering
from Example 5.2 to put a stable generalized complex structure on
$T^{2}\times\Sigma_{g}$ for $g\geq 1$. These structures can also be obtained
using only torus surgeries as described in [37].

######  Example 5.9 (Generalized complex structures on elliptic surfaces).

Stable generalized complex structures on elliptic surfaces have been built in
[14, 33, 37, 25, 19, 18]. Branched coverings provide a new way of finding
stable structures on $E(2n)$. Indeed, the elliptic surface $E(n)$ admits a
stable generalized complex structure for which the generic fibers that are
disjoint from the type change locus can be taken to be
$\mathcal{J}$-symplectic ([14], Example 5.3). Taking the branched covering of
Example 5.3, we obtain a stable generalized complex structure on $E(2n)$.

## References

  * [1] M. Abouzaid and M. Boyarchenko  “Local structure of generalized complex manifolds”  In _J. Symplectic Geom._ 4.1, 2006, pp. 43–62
  * [2] A. Akhmedov  “New symplectic $6$-manifolds via coisotropic Luttinger surgery”  In _Adv. Math. Models & Appl._ 3.2, 2018, pp. 106–116
  * [3] A. Akhmedov  “Symplectic Calabi-Yau 6-manifolds”  In _Adv. Math._ 262, 2014, pp. 115–125
  * [4] M. Bailey  “Local classification of generalized complex structures”  In _J. Differ. Geom._ 95.1, 2013, pp. 1–37
  * [5] M. Bailey  “On the local and global classification of generalized complex structures”, 2012
  * [6] M. Bailey, G.. Cavalcanti and M. Gualtieri  “Type one generalized Calabi–Yaus”  In _J. Geom. Phys._ 120, 2017, pp. 89–95
  * [7] M. Bailey, G.. Cavalcanti and J.. Leer Durán  “Blow-ups in generalized complex geometry”  In _Trans. Amer. Nath. Soc._ 371, 2019, pp. 2109–2131
  * [8] S. Baldridge and P. Kirk  “A symplectic manifold homeomorphic but not diffeomorphic to $\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$”  In _Geom. Topol._ 12.2, 2008, pp. 919–940
  * [9] S. Baldridge and P. Kirk  “Coisotropic Luttinger Surgery and Some New Examples of Symplectic Calabi-Yau $6$-manifolds”  In _Indiana Univ. Math. J._ 62.5, 2013, pp. 1457–1471
  * [10] S. Baldridge and P. Kirk  “Constructions of small symplectic $4$-manifolds using Luttinger surgery”  In _J. Differ. Geom._ 82.2 Lehigh University, 2009, pp. 317–361
  * [11] D. Barden  “Simply Connected Five-Manifolds”  In _Ann. Math._ 82.3, 1965, pp. 365–385
  * [12] S. Behrens, G.. Cavalcanti and R.. Klasse  “Classification of Boundary Lefschetz Fibrations over the Disc”  In _Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin_ Oxford University Press, 2018
  * [13] G.. Cavalcanti and M. Gualtieri  “A surgery for generalized complex structures on 4-manifolds”  In _J. Differ. Geom._ 76.1 Lehigh University, 2007, pp. 35–43
  * [14] G.. Cavalcanti and M. Gualtieri  “Blow-up of generalized complex $4$-manifolds”  In _J. Topol._ 2.4, 2009, pp. 840–864
  * [15] G.. Cavalcanti and M. Gualtieri  “Stable generalized complex structures”  In _Proc. London Math. Soc._ 116.5, 2018, pp. 1075–1111
  * [16] G.. Cavalcanti and R.. Klaasse  “Fibrations and log-symplectic structures”  In _J. Symplectic Geom._ 17.3, 2019, pp. 603–638
  * [17] G.. Cavalcanti and R.. Klaasse  “Fibrations and stable generalized complex structures”  In _Proc. London Math. Soc._ 117.3, 2018, pp. 1242–1280
  * [18] G.. Cavalcanti, R.. Klaasse and A. Witte  “Fibrations in semi-toric and generalized complex geometry”  In _Can. J. Math._ Published online Canadian Mathematical Society, 2022, pp. 1–41
  * [19] G.. Cavalcanti, R.. Klaasse and A. Witte  “Self-crossing stable generalized complex structures”  In _J. Symplectic Geom. (To appear)_ , 2020  arXiv:2004.07559 [math.DG]
  * [20] H. Gluck  “The embedding of two-spheres in the four-sphere”  In _Trans. Amer. Math. Soc._ 104.2, 1962, pp. 308–333
  * [21] R.. Gompf  “A New Construction of Symplectic Manifolds”  In _Ann. Math._ 142.3 Annals of Mathematics, 1995, pp. 527–595
  * [22] R.. Gompf  “Symplectically aspherical manifolds with nontrivial $\pi_{2}$”  In _Math. Res. Lett._ 5, 1998, pp. 599–603
  * [23] R.. Gompf and A.. Stipsicz  “4-Manifolds and Kirby Calculus” 20, Graduate Studies in Mathematics  Amer. Math. Soc., 1999, pp. xv + 557
  * [24] R. Goto  “Unobstructed deformations of generalized complex structures induced by $C^{\infty}$ logarithmic symplectic structures and logarithmic Poisson structures, _Geometry and topology of manifolds_ ”  In _Geometry and topology of manifolds_ 154, Springer Proc. Math. Stat.  Springer, 2016, pp. 159–183
  * [25] R. Goto and K. Hayano  “$C^{\infty}$-logarithmic transformations and generalized complex structures”  In _J. Symplectic Geom._ 14.2 International Press, 2016, pp. 341–357
  * [26] M. Gualtieri  “Generalized complex geometry”, 2003
  * [27] M. Gualtieri  “Generalized complex geometry”  In _Ann. Math._ 174.1, 2011, pp. 75–123
  * [28] N. Hitchin  “Generalized Calabi-Yau manifolds”  In _Q. J. Math._ 54.3, 2003, pp. 281–308
  * [29] K.. Luttinger  “Lagrangian tori in $\mathbb{R}^{4}$”  In _J. Differ. Geom._ 42.2, 1995, pp. 220–228
  * [30] D. McDuff and D. Salamon  “Introduction to symplectic topology, Third edition”, Oxford Graduate Texts in Mathematics  Oxford University Press, 2017, pp. xi + 623
  * [31] U.. Mun  “Constructions of stable generalized complex 6-manifolds and their fundamental groups”, 2018
  * [32] S. Smale  “On the structure of 5-manifolds”  In _Ann. Math._ 75.1, 1962, pp. 38–46
  * [33] R. Torres  “Constructions of generalized complex structures in dimension four”  In _Comm. Math. Phys._ 314.2, 2012, pp. 351–371
  * [34] R. Torres  “Geography and botany of irreducible non-spin symplectic $4$-manifolds with abelian fundamental group”  In _Glasg. Math. J._ 56.2, 2014, pp. 261–281
  * [35] R. Torres  “Some examples of vanishing Yamabe invariant and minimal volume, and collapsing of inequivalent smoothings and PL-structures”  In _Ann. Glob. Anal. Geom._ 49.49, 2016, pp. 177–194
  * [36] R. Torres and J. Yazinski  “Geography of symplectic $4$\- and $6$-manifolds”  In _Topology Proc._ 46, 2015, pp. 87–115
  * [37] R. Torres and J. Yazinski  “On the number of type change loci of a generalized complex structure”  In _Lett. Math. Phys._ 104.4, 2014, pp. 451–464
  * [38] A. Weinstein  “Symplectic manifolds and their lagrangian submanifolds”  In _Adv. Math._ 6, 1971, pp. 329–346