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# Obstructions for symplectic Lie algebroids

Ralph L. Klaasse  Department of Mathematics, Utrecht University, 3508 TA
Utrecht, The Netherlands  Département de Mathématique, Université libre de
Bruxelles, Brussels 1050, Belgium<EMAIL_ADDRESS>
###### Abstract.

Several generically-nondegenerate Poisson structures can be effectively
studied as symplectic structures on naturally associated Lie algebroids.
Relevant examples of this phenomenon include log-, elliptic, $b^{k}$-,
scattering and elliptic-log Poisson structures.

In this paper we discuss topological obstructions to the existence of such
Poisson structures (obtained through their symplectic Lie algebroids) of
several different flavors, namely coming from cohomology, characteristic
classes, and Seiberg–Witten theory.

###### 2010 Mathematics Subject Classification:

53D17,53D05

###### Contents

  1. 1 Introduction
  2. 2 Background material
  3. 3 Cohomological obstructions
  4. 4 Homotopical obstructions
  5. 5 Obstructions from Seiberg–Witten theory

##  1\. Introduction

Generically-nondegenerate Poisson structures have recently seen an intense
increase in interest. The main reason for this has been the ability to
effectively study them using Lie algebroids. Namely, in several instances it
is possible to, given a Poisson structure $\pi\in{\rm Poiss}(X)$, define a Lie
algebroid $\mathcal{A}\to X$ adhering to the same mild degeneracies as $\pi$,
such that $\pi$ is in a precise sense dual to a symplectic structure in
$\mathcal{A}$, i.e. a closed nondegenerate $\mathcal{A}$-two-form.

Symplectic Lie algebroids were first considered in [36], and have more
recently been studied especially when the anchor map
$\rho_{\mathcal{A}}\colon\mathcal{A}\to TX$ is generically an isomorphism.
This class includes log- [10, 17, 19, 28, 29], elliptic [8, 11], $b^{k}$\-
[18, 34, 35, 38] and scattering symplectic structures [26]. Through the use of
symplectic Lie algebroids, powerful symplectic techniques can be brought to
bear to study the associated Poisson structures, leading to various results.

In this paper we are interested in obtaining obstructions to the existence of
a symplectic structure on a Lie algebroid, and thus to their underlying
Poisson structures. The obstructions we present here have three different
flavors: they are of cohomological, homotopical, and gauge theoretical nature.
We use the language of divisors as we developed in [22] (see also [8, 11]) to
describe such Poisson structures and their associated symplectic Lie
algebroids. While we focus in this paper primarily on Lie algebroids and their
symplectic structures, these should be thought of as tools to make statements
about interesting classes of Poisson structures.

The remainder of the introduction describes each of these types of
obstructions, including the known cohomological obstructions, and followed by
stating the results of this paper.

###  1.1. Cohomological obstructions

A simple obstruction to the existence of a symplectic structure is of
cohomological nature. There are analogues of this for two symplectic Lie
algebroids [10, 29, 11]. We recall in Section 3 these results (and their
methods of proof) for log- and elliptic symplectic structures. Let $X^{2n}$ be
an oriented compact manifold. Then:

  * •

a symplectic structure leads to a class $c\in H^{2}(X;\mathbb{R})$ such that
$c^{n}\neq 0$.

  * •

a log-symplectic structure leads to classes $a,b\in H^{2}(X;\mathbb{R})$ such
that $a^{n-1}b\neq 0$, $b^{2}=0$;

  * •

elliptic structures give $a\in H^{2}(X\backslash D;\mathbb{R})$ and $b\in
H^{2}_{\rm c}(X\backslash D;\mathbb{R})$ with $a^{n-1}b\neq 0$, $b^{2}=0$.

The first of these is standard, the second combines [29, 10], and the third is
due to [11], and requires that the elliptic symplectic structure has zero
elliptic residue and $D$ is coorientable. Note that the latter cannot be
stated without also specifying the degeneracy locus $D\subseteq X$.

###  1.2. Homotopical obstructions

A symplectic manifold inherits a natural orientation, and is further almost-
complex. An analogous statement holds for any symplectic Lie algebroid
$\mathcal{A}\to X$ (see Section 4), or indeed any symplectic vector bundle,
where there need not be an integrability condition. The existence of an
orientation and complex structure on $\mathcal{A}$ is determined by the
underlying vector bundle, and is obstructed by its characteristic classes.
Indeed, a vector bundle $\mathcal{A}\to X$ is orientable if and only if
$w_{1}(\mathcal{A})=0$. Making use of this we obtain the following
obstructions for $\mathcal{A}$-symplectic structures, for concrete Lie
algebroids:

######  Theorem A.

Let $\mathcal{A}\to X^{n}$ be a symplectic Lie algebroid. Then in
$H_{1}(X;\mathbb{Z}_{2})$ we have:

  * •

$w_{1}(TX)+k{\rm PD}_{\mathbb{Z}_{2}}[Z]=0$ if
$\mathcal{A}=\mathcal{A}_{Z}^{k}$, the $b^{k}$-tangent bundle;

  * •

$w_{1}(TX)+n\,{\rm PD}_{\mathbb{Z}_{2}}[Z]=0$ if
$\mathcal{A}=\mathcal{B}_{Z}$, the zero tangent bundle;

  * •

$w_{1}(TX)+(n+1){\rm PD}_{\mathbb{Z}_{2}}[Z]=0$ if
$\mathcal{A}=\mathcal{C}_{Z}$, the scattering tangent bundle;

  * •

$w_{1}(TX)=0$ if $\mathcal{A}=\mathcal{A}_{|D|}$, the elliptic tangent bundle;

  * •

$w_{1}(TX)+{\rm PD}_{\mathbb{Z}_{2}}[Z]=0$ if $\mathcal{A}=\mathcal{A}_{W}$,
the elliptic-log tangent bundle.

Here $w_{1}$ is the first Stiefel–Whitney class, and ${\rm
PD}_{\mathbb{Z}_{2}}$ is the Poincaré dual with $\mathbb{Z}_{2}$-coefficients.

This result can be found in the main text as Theorem 4.2, and the Lie
algebroids that are mentioned, as well as our notation for them, are discussed
in Section 2.2. Due to explicit use of the Lie algebroid as a vector bundle,
these depend on the hypersurface $Z\subseteq X$. Note moreover that this
result provides the full obstruction for a surface to be
$\mathcal{A}$-symplectic. This latter statement is because the integrability
condition (closedness) is immediate, so that only a nondegenerate
$\mathcal{A}$-two-form is required, which exists if and only if $\mathcal{A}$
satisfies $w_{1}(\mathcal{A})=0$.

We further determine a more intricate obstruction for four-dimensional
$b^{k}$-symplectic manifolds, making use of the required complex structure on
the Lie algebroid (see Theorem 4.4).

######  Theorem B.

Let $(X^{4},Z)$ be a compact oriented $b^{k}$-symplectic four-manifold. Then:

$b_{2}^{+}(X)+b_{1}(X)+kf(X,Z)\text{ is odd}.$

Here $b_{1}(X)$ is the first Betti number of $X$, $b_{2}^{+}(X)$ is the
dimension of a maximal positive definite subspace on $H^{2}(X;\mathbb{R})$,
and $f(X,Z)$ is the discrepancy of the oriented pair $(X,Z)$.

In the above, a $b^{k}$-symplectic structure is a symplectic structure for
$\mathcal{A}_{Z}^{k}$, the $b^{k}$-tangent bundle [38], so that a
$b^{1}$-symplectic structure is log-symplectic. Moreover,
$kf(X,Z)\in\mathbb{Z}$ is (essentially) the _discrepancy_ of
$\mathcal{A}_{Z}^{k}$ and $TX$, which we will introduce here in Section 4.4.
There is a similar result for scattering-symplectic manifolds (see Theorem
4.6).

######  Theorem C.

Let $(X^{4},Z)$ be a compact oriented scattering-symplectic manifold. Then:

$b_{2}^{+}(X)+b_{1}(X)+f(X,Z)\text{ is odd}.$

Here $b_{1}(X)$ is the first Betti number of $X$, $b_{2}^{+}(X)$ is the
dimension of a maximal positive definite subspace on $H^{2}(X;\mathbb{R})$,
and $f(X,Z)$ is the discrepancy of the oriented pair $(X,Z)$.

This obstruction in the scattering-symplectic case is identical to the log-
symplectic case.

###  1.3. Obstructions from Seiberg–Witten theory

A useful remark of Osorno–Torres ([37, Lemma 4.2.6]) allows us to transform
questions regarding oriented log-symplectic manifolds into those for
symplectic manifolds whose boundaries are of cosymplectic-type (see Section
5). Consequently, powerful results from Seiberg-Witten theory can be used
([25, 39, 40], and [5, 3]) to obstruct their existence. We use these
techniques to prove the following (see Theorem 5.1).

######  Theorem D.

Let $(X^{4},Z)$ be an oriented log pair, splitting along
$X_{1}\sqcup_{Z}X_{2}$. If $b_{2}^{+}(X_{1})>0$ and each connected component
of $Z_{i}$ admits a metric of positive scalar curvature (e.g. $S^{1}\times
S^{2}$), then $(X,Z)$ cannot be log-symplectic. Here it is allowed that
$X_{2}=\emptyset$, so that $X$ has boundary.

Similar results obtained using holomorphic curves for log-symplectic manifolds
are in [1].

### Organization of the paper

This paper is built up as follows. In Section 2 we recall some required
background material, namely divisors capturing mild degeneracy, Lie algebroids
built using them, and Poisson and symplectic structures having such
degeneracy. In Section 3 we then discuss cohomological obstructions for these
Lie algebroids to be symplectic. In Section 4 we turn to homotopical
obstructions, computing various characteristic classes. Finally, in Section 5
we discuss obstructions coming from Seiberg–Witten theory in dimension four.

### Acknowledgements

This work is partially based on [21, Chapter 11], and was supported by ERC
grant 646649; and by VIDI grant 639.032.221 from NWO, the Netherlands
Organisation for Scientific Research. The author would like to thank Gil
Cavalcanti for useful discussions.

##  2\. Background material

In this section we very briefly recall required background material. Let $X$
be a manifold.

###  2.1. Divisors on smooth manifolds

We provide a very brief primer on the language of (real) divisors on smooth
manifolds. For more information, see [8, 22].

######  Definition .

A _(real) divisor_ on $X$ is a pair $(U,\sigma)$ consisting of a real line
bundle with a section $\sigma\in\Gamma(U)$ that has nowhere dense zero set
$Z_{\sigma}=\sigma^{-1}(0)$. Evaluation via the map
$\sigma\colon\Gamma(U^{*})\to C^{\infty}(X)$ specifies a _divisor ideal_
$I_{\sigma}\subseteq C^{\infty}(X)$ with _support_ $Z_{\sigma}\subseteq X$.

A divisor ideal determines a divisor up to line bundle isomorphism and
multiplication by a nonvanishing smooth function, allowing us to mostly work
with divisor ideals. In this paper we will use the following three examples of
divisors (see [8, 11, 22]):

  * •

_Log divisors_ $(L,s)$ where $s$ has transverse zeroes. Here $Z:=Z_{s}$ is a
codimension-one hypersurface, and $I_{Z}:=I_{s}$ is its vanishing ideal,
locally $I_{Z}=\langle z\rangle$ with $Z=\\{z=0\\}$

  * •

_Elliptic divisors_ $(R,q)$, where $q$ has definite Hessian ${\rm
Hess}(q)\in\Gamma(D;{\rm Sym}^{2}N^{*}D\otimes R)$ along its smooth
codimension-two zero set $D:=Z_{q}$. Its divisor ideal is $I_{|D|}:=I_{q}$,
and is locally given by $I_{|D|}=\langle r^{2}\rangle$ with $r$ a radial
distance in $ND$;

  * •

_Elliptic-log divisors_ $(L,s)\otimes(R,q)$, obtained as the product of a log
and elliptic divisor such that $D\subseteq Z$. Its divisor ideal is
$I_{W}=I_{Z}\cdot I_{|D|}$, and locally $I_{W}=\langle
r^{3}\cos\theta\rangle$.

We write $(X,Z)$, $(X,|D|)$ and $(X,W)$ for _log_ , _elliptic_ , and
_elliptic-log pairs_ respectively. We denote by $L_{Z}$ the line bundle of the
log divisor associated to the manifold pair $(X,Z)$.

An immediate consequence of Section 2.1 is the following, which we will use
later.

######  Lemma .

Let $(X,Z)$ be a log pair. Then we have $w_{1}(L_{Z})={\rm
PD}_{\mathbb{Z}_{2}}[Z]\in H^{1}(X;\mathbb{Z}_{2})$.

Here $w_{1}$ is the first Stiefel–Whitney class, and ${\rm
PD}_{\mathbb{Z}_{2}}$ is the Poincaré dual with $\mathbb{Z}_{2}$-coefficients.
This follows because the section $s\in\Gamma(L_{Z})$ can be used to determine
the Euler class of $L_{Z}$.

###  2.2. Lie algebroids from divisors

Recall that a _Lie algebroid_ is a vector bundle $\mathcal{A}\to X$ equipped
with an _anchor_ map $\rho_{\mathcal{A}}\colon\mathcal{A}\to TX$ and a Lie
bracket $[\cdot,\cdot]_{\mathcal{A}}$ on $\Gamma(\mathcal{A})$ which satisfies
$[fv,w]_{\mathcal{A}}=f[v,w]_{\mathcal{A}}+\mathcal{L}_{\rho_{\mathcal{A}}(v)}f\cdot
w$ for all $v,w\in\Gamma(\mathcal{A})$ and $f\in C^{\infty}(X)$. Divisor
ideals are an effective tool to construct Lie algebroids generically
isomorphic to $TX$, as we now explain.

Let $I\subseteq C^{\infty}(X)$ be a divisor ideal and
$\Gamma(TX)_{I}=\\{V\in\Gamma(TX):\mathcal{L}_{V}I\subseteq
I\\}\subseteq\Gamma(TX)$ be the involutive submodule of vector fields
preserving $I$. When $\Gamma(TX)_{I}$ is projective it specifies uniquely a
Lie algebroid $\mathcal{A}_{I}\to X$ such that
$\Gamma(\mathcal{A}_{I})\cong\Gamma(TX)_{I}$ by the Serre–Swan theorem.

######  Definition  ([22]).

Let $I\subseteq C^{\infty}(X)$ be a divisor ideal for which $\Gamma(TX)_{I}$
is projective. Then the Lie algebroid $\mathcal{A}_{I}\to X$ with
$\Gamma(\mathcal{A}_{I})\cong\Gamma(TX)_{I}$ is called the _ideal Lie
algebroid_ of $I$.

In [22] these Lie algebroids are denoted by $TX_{I}$. Examples of this
construction include:

  * •

The _log-tangent bundle_ $\mathcal{A}_{Z}=TX(-\log Z)$ associated to $I_{Z}$
(see [31]);

  * •

The _elliptic tangent bundle_ $\mathcal{A}_{|D|}=TX(-\log|D|)$ associated to
$I_{|D|}$ (see [11]);

  * •

The _elliptic-log tangent bundle_ $\mathcal{A}_{W}=TX(-\log W)$ associated to
$I_{W}$ (see [22]).

Note that the latter has natural morphisms onto $\mathcal{A}_{D}$ and
$\mathcal{A}_{Z}$ via the section module inclusion. These Lie algebroids all
have the property that their anchor $\rho_{\mathcal{A}}\colon\mathcal{A}\to
TX$ is an isomorphism on a dense open set, which is the complement of their
_degeneracy locus_. In fact, the anchor map of such a Lie algebroid specifies
a divisor ${\rm
div}(\mathcal{A})=(\det(TX)\otimes\det(\mathcal{A}^{*}),\det(\rho_{\mathcal{A}}))$
with divisor ideal $I_{\mathcal{A}}$. We say that a divisor ideal $I$ is
_standard_ if its ideal Lie algebroid satisfies $I_{\mathcal{A}_{I}}=I$.
Studying the above examples we see that all three given divisor ideals are
standard.

The above Lie algebroids admit residue maps, which can be used to extract
important information along their degeneracy loci (see [11] and the
discussions in [8, 22]). We mention:

  * •

The _logarithmic residue_ ${\rm
Res}_{Z}\colon\Omega^{\bullet}(\mathcal{A}_{Z})\to\Omega^{\bullet-1}(Z)$,
given by $d\log z\wedge\alpha+\beta\mapsto i_{Z}^{*}(\alpha)$;

  * •

The _radial residue_ ${\rm
Res}_{r}\colon\Omega^{\bullet}(\mathcal{A}_{|D|})\to\Omega^{\bullet-1}({\rm
At}(S^{1}ND))$, given by

$d\log r\wedge d\theta\wedge\alpha+d\log
r\wedge\beta+d\theta\wedge\gamma+\eta\mapsto(d\theta\wedge\alpha+\beta)|_{D},$

where ${\rm At}(S^{1}ND)\to D$ is the Atiyah algebroid of $S^{1}ND\to D$;

  * •

The _elliptic residue_ ${\rm
Res}_{q}\colon\Omega^{\bullet}(\mathcal{A}_{|D|})\to\Omega^{\bullet-2}(D;\mathfrak{k}^{*})$,
given by $\iota_{\partial_{\theta}}\circ{\rm Res}_{r}$, i.e.

$d\log r\wedge d\theta\wedge\alpha+d\log
r\wedge\beta+d\theta\wedge\gamma+\eta\mapsto i_{D}^{*}(\alpha),$

where $\mathfrak{k}^{*}\cong\wedge^{2}ND$ is the determinant of the dual of
the isotropy bundle over $D$.

One can also obtain Lie algebroids by modifying a given Lie algebroid using a
Lie subalgebroid supported on a hypersurface $Z$. This process is called
_(lower) elementary modification_ [17, 27] or _rescaling_ [31, 26]. This can
be extended to divisor ideals $I\subseteq C^{\infty}(X)$ supported on smooth
submanifolds other than log ideals $I_{Z}$ (see [22]), but we will not have
use for this here.

######  Definition .

Let $(X,Z)$ be a log pair and $(\mathcal{B},Z)\subseteq(\mathcal{A},X)$ a Lie
subalgebroid. The _lower elementary modification_ or _$(\mathcal{B},Z)$
-rescaling_ of $\mathcal{A}$ along $\mathcal{B}$ is the Lie algebroid
$[\mathcal{A}\text{:}\mathcal{B}]$ defined by

$\Gamma([\mathcal{A}\text{:}\mathcal{B}])\cong\\{v\in\Gamma(\mathcal{A}):v|_{Z}\in\Gamma(\mathcal{B})\\}.$

######  Remark .

Given a Lie algebroid $\mathcal{A}\to X$ one can always perform
$(0,Z)$-rescaling. The resulting Lie algebroid $[\mathcal{A}\text{:}0]$ is
isomorphic to the tensor product $\mathcal{A}\otimes L_{Z}$ as a vector
bundle.

######  Example .

Let $(X,Z)$ be a log pair. The following are examples of modifications ([31,
32]).

  * •

The log-tangent bundle $\mathcal{A}_{Z}=[TX\text{:}TZ]$, locally given by
$\Gamma(\mathcal{A}_{Z})=\langle z\partial_{z},\partial_{x_{i}}\rangle$;

  * •

The _zero tangent bundle_ $\mathcal{B}_{Z}=[TX\text{:}0]$, locally given by
$\Gamma(\mathcal{B}_{Z})=\langle z\partial_{z},z\partial_{x_{i}}\rangle$;

  * •

The _scattering tangent bundle_ $\mathcal{C}_{Z}=[\mathcal{A}_{Z}\text{:}0]$,
locally given by $\Gamma(\mathcal{C}_{Z})=\langle
z^{2}\partial_{z},z\partial_{x_{i}}\rangle$.

Finally, given $k\geq 1$, by fixing a $(k-1)$-jet
$j\in\Gamma(X;\iota_{Z}^{-1}(C^{\infty}(X)/I_{Z}^{k}))$, where
$\iota_{Z}\colon Z\hookrightarrow X$ is the inclusion, we can define a Lie
algebroid $\mathcal{A}_{Z}^{k}\to X$ by
$\Gamma(\mathcal{A}_{Z}^{k})\cong\\{V\in\Gamma(TX):\mathcal{L}_{V}j\in
I_{Z}^{k}\\}$. This is the _$b^{k}$ -tangent bundle_ [38], and is locally
given by $\Gamma(\mathcal{A}_{Z}^{k})=\langle
z^{k}\partial_{z},\partial_{x_{i}}\rangle$ for a local $z\in j$. Note that
when $k=1$ the jet data is vacuous, so that
$\mathcal{A}_{Z}^{1}=\mathcal{A}_{Z}$, the log-tangent bundle.

###  2.3. Poisson structures on Lie algebroids

Poisson structures are readily linked to divisors and the Lie algebroids built
from them. Let $\pi\in{\rm Poiss}(X^{2n})$ be a Poisson structure, and
consider its _Pfaffian_ , $\wedge^{n}\pi\in\Gamma(\det(TX))$. If $\pi$ is
generically nondegenerate this defines a divisor $(\det(TX),\wedge^{n}\pi)$
and hence a divisor ideal $I_{\pi}$. We say $\pi$ is of _$I$ -divisor-type_ if
$I_{\pi}=I$ (see [22]). We thus obtain, for example, the following classes of
Poisson structures:

  * •

_Log-Poisson structures_ , which are of $I_{Z}$-divisor-type ([19, 29], and
see below);

  * •

_Elliptic Poisson structures_ , which are of $I_{|D|}$-divisor-type ([11],
also [8, 23, 24]);

  * •

_Elliptic-log Poisson structures_ , which are of $I_{W}$-divisor-type ([22,
23]).

Poisson structures on a Lie algebroid $\mathcal{A}\to X$ are defined as those
sections $\pi_{\mathcal{A}}\in\Gamma(\wedge^{2}\mathcal{A})$ such that
$[\pi_{\mathcal{A}},\pi_{\mathcal{A}}]_{\mathcal{A}}=0$. These specify
underlying Poisson structures
$\pi=\rho_{\mathcal{A}}(\pi_{\mathcal{A}})\in{\rm Poiss}(X)$. In [22] we
showed that if $\pi\in{\rm Poiss}(X)$ is of $I$-divisor-type, and $I$ is such
that its ideal Lie algebroid $\mathcal{A}_{I}$ exists, then $\pi$ admits an
_$\mathcal{A}_{I}$ -lift_: there exists a (unique) $\mathcal{A}_{I}$-Poisson
structure $\pi_{\mathcal{A}_{I}}$ such that
$\pi=\rho_{\mathcal{A}_{I}}(\pi_{\mathcal{A}_{I}})$. Moreover, if the divisor
ideal $I$ is standard, then $\pi_{\mathcal{A}_{I}}$ is nondegenerate.

###  2.4. Symplectic Lie algebroids

A Lie algebroid $\mathcal{A}\to X$ of even rank is _symplectic_ if it carries
a nondegenerate closed $\mathcal{A}$-two-form $\omega_{\mathcal{A}}$ (after
[36]). Such an _$\mathcal{A}$ -symplectic structure_ corresponds to a
nondegenerate $\mathcal{A}$-Poisson structure $\pi_{\mathcal{A}}$ via the
relation $\pi_{\mathcal{A}}^{\sharp}=(\omega_{\mathcal{A}}^{\flat})^{-1}$. Due
to this, in order to study Poisson structures of $I$-divisor-type, we must
study $\mathcal{A}_{I}$-symplectic geometry. For clarity, the Lie algebroids
of Section 2.2 allow us to define:

  * •

_Log-symplectic structures_ , associated to $\mathcal{A}_{Z}$ ([19, 29], also
[10, 17, 28] and others);

  * •

_Elliptic symplectic structures_ , associated to $\mathcal{A}_{|D|}$ ([11],
also [8, 23, 24]);

  * •

_Elliptic-log symplectic structures_ , associated to $\mathcal{A}_{W}$ ([22,
23]);

  * •

_Zero symplectic structures_ , associated to $\mathcal{B}_{Z}$ (c.f. [26], and
Section 4.2);

  * •

_Scattering symplectic structures_ , associated to $\mathcal{C}_{Z}$ ([26]);

  * •

_$b^{k}$ -symplectic structures_, associated to $\mathcal{A}_{Z}^{k}$ ([38],
also [18]).

Each of these has an underlying Poisson structure, which can often be
characterized intrinsically. While the remainder of this note uses Lie
algebroids and Lie algebroid objects, these are viewed as tools to make
statements about generically-nondegenerate Poisson structures.

##  3\. Cohomological obstructions

In this section we discuss cohomological obstructions for the existence of
$\mathcal{A}$-symplectic structures. Such results can be found in the
literature, but are included here to provide contrast. These results are
analogous to the fact that a symplectic structure on a compact
$2n$-dimensional manifold forces the existence of a cohomology class $c\in
H^{2}(X;\mathbb{R})$ with $c^{n}\neq 0$.

For log-symplectic manifolds, cohomological obstructions were obtained by
Mărcuţ–Osorno Torres [29] (also [37]) and Cavalcanti [10] (for simplicity we
assume $X$ is compact oriented).

######  Theorem 3.1 ([29, 10]).

Let $(X^{2n},Z)$ be compact oriented log-symplectic. Then there exists:

  * •

a class $a\in H^{2}(X;\mathbb{R})$ such that $a^{n-1}\neq 0$;

  * •

a class $b\in H^{2}(X;\mathbb{R})$ such that $b^{2}=0$ and $a^{n-1}\wedge
b\neq 0$, if $n\geq 2$ and $Z\neq\emptyset$.

Consequently, if $\dim X=4$ and $Z\neq\emptyset$, then $X$ must have
indefinite intersection form.

###### Proof..

We reproduce the proofs of [37, Theorem 4.3.1] and [10, Theorem 4.2]
respectively.

Let $\omega_{Z}\in{\rm Symp}(\mathcal{A}_{Z})$ be a log-symplectic structure.
If $Z=\emptyset$, then set $a=[\omega_{Z}]$. If $Z\neq\emptyset$, consider the
residue one-form $\alpha={\rm Res}_{Z}(\omega_{Z})\in\Omega^{1}_{\rm cl}(Z)$.
By the semi-global normal form for $\omega_{Z}$ around $Z$, we have that
$\omega_{Z}=d\log|z|\wedge p^{*}(\alpha)+p^{*}(\beta)$, where $p\colon NZ\to
Z$ is a tubular neighbourhood, $\beta\in\Omega^{2}_{\rm cl}(Z)$, and $|z|$ is
a distance function on $NZ$. Now let $\lambda\colon X\to[0,1]$ be such that
$\lambda(z)=1$ for $z\in NZ$ with $|z|\leq\frac{1}{2}$, and
$\lambda(z)=\log|z|$ on $X\backslash\\{z\in NZ:|z|<1\\}$. Then
$\omega:=d\lambda(z)\wedge p^{*}(\alpha)+p^{*}(\beta)$ is a smooth closed two-
form agreeing with $\omega_{Z}$ away from $Z$, hence can be extended to a
global two-form on $X$, and whose pullback to $Z$ is $\beta$. Note that:

$\omega_{Z}^{n}\neq 0\text{ on $X$, so that }{\rm
Res}_{Z}(\omega_{Z}^{n})=\alpha\wedge\beta^{n-1}\neq 0\text{ on $Z$}.$

Consequently, the class $a:=[\omega]\in H^{2}(X;\mathbb{R})$ satisfies
$a^{n-1}\neq 0$, as on the level of forms we have
$i_{Z}^{*}(\omega^{n-1})=\beta^{n-1}\neq 0$, and the two-form $\beta$ cannot
be exact due to compactness of $Z$.

For the second point, if $Z\neq\emptyset$, by compactness of $X$ we can
perturb $\omega_{Z}$ slightly to ensure that $\alpha$ has a cohomology class
that is a real multiple of an integer class [10, 37]. Then $\alpha$ exhibits
$Z$ as a symplectic mapping torus with fiber $F$, and $\omega_{Z}$ is said to
be _proper_. The two-form $\omega$ constructed above can then be chosen such
that its restriction to $F$ is symplectic. Consider $[F]\in
H_{2n-2}(X;\mathbb{R})$ which is necessarily nonzero, and set $b:={\rm
PD}[F]\in H^{2}(X;\mathbb{R})$. As $F$ is the fiber of a fibration, we have
$b^{2}=0$. Moreover, $\langle a^{n-1}\wedge b,[X]\rangle=\langle
a^{n-1},[F]\rangle\neq 0$. ∎

There is a similar result for elliptic symplectic manifolds with zero elliptic
residue by Cavalcanti–Gualtieri, contained in a preliminary version of [11]
(we state the compact case).

######  Theorem 3.2 ([11]).

Let $(X^{2n},D)$ be a compact elliptic symplectic manifold with zero elliptic
residue and cooriented degeneracy locus $D$. Then there exists:

  * •

a class $a\in H^{2}(X\backslash D;\mathbb{R})$ such that $a^{n-1}\neq 0$;

  * •

a class $b\in H^{2}_{\rm c}(X\backslash D;\mathbb{R})$ such that $b^{2}=0$ and
$a^{n-1}\wedge b\neq 0$.

###### Proof..

We reproduce the proof by Cavalcanti–Gualtieri [11] (found in its preliminary
version).

Let $\omega\in{\rm Symp}(\mathcal{A}_{|D|})$ have zero elliptic residue. By
[11, Theorem 1.8], decompose $[\omega]=a+c$ for $a=i^{*}[\omega]\in
H^{2}(X\setminus D;\mathbb{R})$ with $i\colon X\backslash D\hookrightarrow X$
the inclusion of the divisor complement, and $c={\rm Res}_{r}([\omega])\in
H^{1}(S^{1}ND;\mathbb{R})$. As $\omega^{n}\neq 0$, we see that ${\rm
Res}_{r}(\omega^{n})\neq 0$ is a volume form in $\Omega^{2n-1}(D;S^{1}ND)$, so
that ${\rm Res}_{r}([\omega]^{n})\neq 0$ in cohomology by compactness of $D$.
Using [11, Theorem 1.9] we compute (with $r\colon H^{2}(X\setminus
D;\mathbb{R})\to H^{1}(S^{1}ND;\mathbb{R})$ the restriction):

$0\neq{\rm Res}_{r}([\omega^{n}])={\rm
Res}_{r}([\omega]^{n})=nr(i^{*}[\omega]^{n-1})\cup{\rm
Res}_{r}[\omega]=nr(a^{n-1})\cup c.$

We conclude that $a^{n-1}\neq 0$ as desired. Next choose a small tubular
neighbourhood for $D$ and identify $S^{1}ND$ with its boundary, so that it
includes via a map $j\colon S^{1}ND\to X\backslash D$. Above we showed that
$j^{*}(a^{n-1})\cup c\neq 0$ in $S^{1}ND$. Define $b:=j_{*}(c)\in H^{2}_{\rm
c}(X\backslash D)$, and note that $a^{n-1}\cup b\neq 0\in H^{2n-1}_{\rm
c}(X\backslash D;\mathbb{R})$. As cochains, $b$ and $c$ are related by wedging
with a Thom form for the normal bundle $N_{X}(S^{1}ND)$, which is thus seen to
square to zero, hence $b^{2}=0$. ∎

######  Remark .

Theorems 3.1 and 3.2 are related as follows (due to Cavalcanti–Gualtieri, in a
preliminary version of [11]; see [24] for more details): given an elliptic
pair $(X,|D|)$, one can perform real oriented blow-up along $D$ and obtain a
log pair with boundary, $(X^{\prime},S^{1}ND)$. The blow-down map $p\colon
X^{\prime}\to X$ satisfies $p^{*}I_{|D|}=I_{Z}$, where $Z=S^{1}ND$, and
induces a Lie algebroid morphism
$(\varphi,p)\colon\mathcal{A}_{Z}\to\mathcal{A}_{|D|}$ where $\varphi\equiv
dp$ on sections. The elliptic-symplectic structure on $(X,D)$ pulls back via
$\varphi$ to a log-symplectic structure on $(X^{\prime},Z)$. One can then
readily relate the cohomology classes from Theorem 3.1 to those of Theorem
3.2.

One can also obtain obstructions on log and elliptic pairs $(X,Z)$ and
$(X,|D|)$ to admit Lie algebroid symplectic structures (i.e. log-, or zero
elliptic residue elliptic) by considering the geometric structures induced on
$Z$ and $D$ respectively. Namely, $Z$ becomes cosymplectic ([19]), while $D$
inherits a $2$-cosymplectic structure if it is coorientable (see [23]).111A
_$k$ -cosymplectic structure_ on a manifold $Z^{2\ell+k}$ is a tuple
$(\alpha_{i},\beta)$ of $k$ closed one-forms and a single closed two-form such
that $\alpha_{1}\wedge\dots\wedge\alpha_{k}\wedge\beta^{\ell}\neq 0$. These
have a foliation of rank $2\ell$ with tangencies
$\ker\langle\alpha_{i}\rangle$. See [37]. Both structures can be after slight
perturbation assumed to be proper ([10, 28], and [2, 8]), resulting in induced
fibration maps $Z\to S^{1}$ and $D\to T^{2}$ (which themselves also form
obstructions).

If we consider the cohomology classes of a $k$-cosymplectic structure, we see
that:

######  Proposition  (c.f. [18]).

Let $(X,Z)$ be a proper log-symplectic manifold. Then $Z^{2n-1}$ has classes
$a\in H^{1}(Z;\mathbb{R})$ and $b\in H^{2}(Z;\mathbb{R})$ such that $a\wedge
b^{n-1}\neq 0$;

######  Proposition  (c.f. [23, 2]).

Let $(X,|D|)$ be a proper zero residue elliptic symplectic manifold. Then
$D^{2n}$ has classes $a,b\in H^{1}(D;\mathbb{R})$ and $c\in
H^{2}(D;\mathbb{R})$ such that $a\wedge b\wedge c^{n-1}\neq 0$.

We stress again that our notion of being proper implies that $D$ is
coorientable.

######  Remark .

It is natural to ask whether the above results can be extended to other
symplectic Lie algebroids. Based on the above proofs, this would require an
understanding of the ring structure on Lie algebroid cohomology, or by having
semi-global normal form results.

######  Remark .

Instead of adapting the strategies used in Theorem 3.1 and Theorem 3.2 to
other Lie algebroids $\mathcal{A}\to X$, one can try to prove directly that
given $\omega_{\mathcal{A}}\in{\rm Symp}(\mathcal{A}^{n})$, the class
$[\omega_{\mathcal{A}}^{n}]\in H^{n}(\mathcal{A})$ is nontrivial, by
understanding the canonical Evens–Lu–Weinstein pairing [13], i.e.
$H^{n}(\mathcal{A})\times H^{0}(\mathcal{A};Q_{\mathcal{A}})\to\mathbb{R}$,
where $Q_{A}=\det(\mathcal{A})\otimes\det(T^{*}X)$ is the canonical
$\mathcal{A}$-module. Note that the divisor ${\rm div}(\mathcal{A})$ of
$\mathcal{A}$ uses the dual bundle $Q_{\mathcal{A}}^{*}$, which is also an
$\mathcal{A}$-module.

##  4\. Homotopical obstructions

In this section we discuss homotopical obstructions to the existence of
$\mathcal{A}$-symplectic structures on a given closed manifold $X$. More
precisely, we focus on the following simple facts.

######  Proposition .

Let $\mathcal{A}\to X$ be a symplectic Lie algebroid. Then:

  * •

$\mathcal{A}$ must be orientable, i.e. it must satisfy
$w_{1}(\mathcal{A})=0\in H_{1}(X;\mathbb{Z}_{2})$;

  * •

$\mathcal{A}$ must be complex, i.e. there must exist a $J_{\mathcal{A}}\in{\rm
End}(\mathcal{A})$ with $J_{\mathcal{A}}^{2}=-{\rm id}$.

These properties both follow from the linear algebra of having a nondegenerate
$\mathcal{A}$-two-form. Indeed, they hold for any symplectic vector bundle
(e.g. [30]), as they do not use integrability.

###### Proof..

Let $\omega_{\mathcal{A}}$ be an $\mathcal{A}$-symplectic structure. Then
${\rm rank}(\mathcal{A})=2m$ is necessarily even, and
$\omega_{A}^{m}\in\Gamma(\det(\mathcal{A}^{*}))$ is nonvanishing. Thus
$\det(\mathcal{A}^{*})$ is trivial, and
$w_{1}(\mathcal{A})=w_{1}(\det(\mathcal{A}^{*}))=0$. To see $\mathcal{A}$ must
admit a complex structure, follow the standard proof for $\mathcal{A}=TX$
(e.g. [30]). ∎

Note that when both $\mathcal{A}$ and $X$ are four-dimensional, a classical
result by Wu [41] (see also [20]) can be used, characterizing when an oriented
vector bundle admits a complex structure.

######  Theorem 4.1 ([41]).

Let $E^{4}\to X^{4}$ be an oriented Euclidean rank-four vector bundle over a
compact oriented four-manifold. Then $E$ admits a complex structure if and
only if there exists a class $c\in H^{2}(X;\mathbb{Z})$ such that $c\\!\\!\mod
2\equiv w_{2}(E)\in H^{2}(X;\mathbb{Z}_{2})$ and $c^{2}=p_{1}(E)+2e(E)$.

To make effective use of these observations, it is clear that we must
determine the relevant characteristic classes of the bundle $\mathcal{A}\to
X$. We do this via stable bundle isomorphisms.

###  4.1. Stable bundle isomorphisms

We determine how a vector bundle changes when performing lower elementary
modification. Denote by ${\underline{\mathbb{R}}}\to X$ the trivial real line
bundle.

######  Proposition .

Let $\mathcal{A}\to X$ be a Lie algebroid and $Z\subseteq X$ a hypersurface.
Consider a $(\mathcal{B},Z)$-rescaling $[\mathcal{A}\text{:}\mathcal{B}]$ of
$\mathcal{A}$ with ${\rm corank}(\mathcal{B})=k$. Then using $k$ copies of
$L_{Z}$ and $\underline{\mathbb{R}}$ we have

$[\mathcal{A}\text{:}\mathcal{B}]\oplus L_{Z}\oplus\dots\oplus
L_{Z}\cong\mathcal{A}\oplus\underline{\mathbb{R}}\oplus\dots\oplus\underline{\mathbb{R}}.$

We emphasize that is a vector bundle isomorphism, and not one of Lie
algebroids.

######  Remark .

When $\mathcal{A}=TX$ and $\mathcal{B}=TZ$, Section 4.1 reduces to the
statement that

$\mathcal{A}_{Z}\oplus L_{Z}\cong TX\oplus\underline{\mathbb{R}},$

as $\mathcal{B}$ has corank one. This was noted without proof in [6, 7] when
$X$ is orientable, although they inaccurately state that in general one has
$\mathcal{A}_{Z}\oplus\underline{\mathbb{R}}\cong TX\oplus L_{Z}$.222To see
that their assertion cannot be correct in general, restrict to $Z$ so that it
(non-canonically) reads $\mathbb{L}_{Z}\oplus
TZ\oplus\underline{\mathbb{R}}\cong TZ\oplus NZ\oplus NZ$, with
$\mathbb{L}_{Z}=\ker(\rho_{\mathcal{A}_{Z}}\colon\mathcal{A}_{Z}\to TX)$ the
isotropy, and using that $L_{Z}|_{Z}\cong NZ$. The line bundle
$\mathbb{L}_{Z}\to Z$ is canonically trivialized ([19, Proposition 4]),
leading to a contradiction. However, when $Z$ is separating so that $L_{Z}$ is
trivial (see Section 2.1), these two above statements are equivalent. In
particular this is the case when $(X,Z)$ is log-symplectic and $X$ is
orientable (see Section 4.3).

###### Proof..

We first consider the case when $k=1$. Set $n={\rm rank}(\mathcal{A})$. Over
$X\backslash Z$, the bundle $L_{Z}$ is trivial and
$[\mathcal{A}\text{:}\mathcal{B}]\cong\mathcal{A}$ via the morphism induced by
the inclusion on sections, so that there we have an obvious isomorphism. Near
$Z$ we define a bundle isomorphism as follows.

Choose a tubular neighbourhood embedding of $Z$ and let
$\\{U_{\alpha}\\}_{\alpha}$ be a simultaneous trivializing cover of $L_{Z}$
and of $\mathcal{B}$ extended to $\mathcal{A}$ (and hence of
$[\mathcal{A}\text{:}\mathcal{B}]$ and $NZ$, as $L_{Z}|_{Z}\cong NZ$), with
transverse vanishing sections $s_{\alpha}\in\Gamma(U_{\alpha};L_{Z})$ and
transition maps $g_{\alpha}^{\beta}$, and moreover

$\Gamma(U_{\alpha};\mathcal{A})=\langle
v_{\alpha,1},v_{\alpha,i}\rangle,\qquad\text{and}\qquad\Gamma(U_{\alpha};[\mathcal{A}\text{:}\mathcal{B}])=\langle
z_{\alpha}v_{\alpha,1},v_{\alpha,i}\rangle,$

with $z_{\alpha}$ the associated normal bundle coordinates. Choose a metric on
$NZ$ and consider the disk bundle $\mathbb{D}_{NZ}$ in $NZ$ with radius
$\pi/2$. We now explicitly define a bundle isomorphism
$\varphi\colon[\mathcal{A}\text{:}\mathcal{B}]\oplus
L_{Z}\to\mathcal{A}\oplus\underline{\mathbb{R}}$ on $\mathbb{D}_{NZ}\cap
U_{\alpha}$.

Generic sections of $[\mathcal{A}\text{:}\mathcal{B}]\oplus L_{Z}$ and
$\mathcal{A}\oplus\underline{\mathbb{R}}$ can be expressed respectively as the
tuples

$\left(\lambda_{1}\cdot z_{\alpha}v_{\alpha,1}+\sum_{2\leq i\leq
n}\lambda_{i}\cdot v_{\alpha,i},\,\,\lambda_{n+1}\cdot
s_{\alpha}\right)\qquad\text{and}\qquad\left(\mu_{1}\cdot
v_{\alpha,1}+\sum_{2\leq i\leq n}\mu_{i}\cdot
v_{\alpha,i},\mu_{n+1}\cdot\mathbf{1}\right),$

where $\lambda_{i},\mu_{i}\in C^{\infty}(U_{\alpha})$. The map $\varphi$ is
defined to be in matrix form given by

$\begin{pmatrix}\mu_{1}&\mu_{n+1}&\mu_{2}&\dots&\mu_{n}\end{pmatrix}^{\rm
t}=\begin{pmatrix}z_{\alpha}\sin|z_{\alpha}|&\cos|z_{\alpha}|&0\\\
-\cos|z_{\alpha}|&z_{\alpha}\sin|z_{\alpha}|&0\\\
0&0&I_{n-1}\end{pmatrix}\begin{pmatrix}\lambda_{1}&\lambda_{n+1}&\lambda_{2}&\dots&\lambda_{n}\end{pmatrix}^{\rm
t},$

where $I_{n-1}$ is the $(n-1)\times(n-1)$ identity matrix. We see the top left
part of $\varphi$ is given by:

$z_{\alpha}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\text{ when
$|z_{\alpha}|=\pi/2$},\qquad\text{and}\qquad\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\text{ when $|z_{\alpha}|=0$, i.e.\ on $Z$}.$

Thus $\varphi$ can be extended smoothly to $U_{0}$ as being $\pi/2$ times the
identity after introducing a radial bump function. It is well-defined because
on the intersections $U_{\alpha}\cap U_{\beta}$ we have that:

$z_{\alpha}v_{\alpha,1}=z_{\beta}v_{\beta,1},\qquad\text{so that}\qquad
v_{\alpha,1}=\frac{z_{\beta}}{z_{\alpha}}v_{\beta,1}=g^{\beta}_{\alpha}v_{\beta,1},\qquad\text{and
also}\qquad
s_{\alpha}=\frac{z_{\beta}}{z_{\alpha}}s_{\beta}=g^{\beta}_{\alpha}s_{\beta}.$

To see that $\varphi$ is an isomorphism, we merely note that in matrix form it
has determinant equal to
$z_{\alpha}^{2}\sin^{2}|z_{\alpha}|+\cos^{2}|z_{\alpha}|$, which is positive
for all values of $z_{\alpha}$, showing invertibility.

In the general case when $k\geq 2$, repeat this argument for each generator of
the complement of $\mathcal{B}$ inside $\mathcal{A}|_{Z}$. This gives a bundle
isomorphism with block-diagonal matrix form. ∎

A similar result holds for the $b^{k}$-tangent bundles $\mathcal{A}_{Z}^{k}$
constructed using a $(k-1)$-jet $j$ at $Z$.

######  Proposition .

Let $(X,Z)$ be a log pair with $(k-1)$-jet $j$ at $Z$. Then
$\mathcal{A}_{Z}^{k}\oplus L_{Z}^{k}\cong TX\oplus\underline{\mathbb{R}}$.

###### Proof..

Follow the same strategy as for Section 4.1, except take $z_{\alpha}\in j$,
replace $\lambda_{1}z_{\alpha}v_{\alpha,1}$ by
$\lambda_{1}z_{\alpha}^{k}v_{\alpha,1}$ (where
$v_{\alpha,1}=\partial_{z_{\alpha}}$), and define the map $\varphi$ near $Z$
to be given in matrix form by

$\begin{pmatrix}\mu_{1}&\mu_{n+1}&\mu_{2}&\dots&\mu_{n}\end{pmatrix}^{\rm
t}=\begin{pmatrix}z_{\alpha}^{k}\sin|z_{\alpha}|&\cos|z_{\alpha}|&0\\\
-\cos|z_{\alpha}|&z_{\alpha}^{k}\sin|z_{\alpha}|&0\\\
0&0&I_{n-1}\end{pmatrix}\begin{pmatrix}\lambda_{1}&\lambda_{n+1}&\lambda_{2}&\dots&\lambda_{n}\end{pmatrix}^{\rm
t}.$

This map has determinant
$z_{\alpha}^{2k}\sin^{2}|z_{\alpha}|+\cos^{2}|z_{\alpha}|$, again showing
invertibility of the map $\varphi$. ∎

###  4.2. Computing characteristic classes

In this section we compute relevant characteristic classes of the Lie
algebroids we have introduced. We will mainly be interested in the first and
second Stiefel–Whitney classes $w_{1},w_{2}\in H^{i}(X;\mathbb{Z}_{2})$, and
in the first Pontryagin class $p_{1}\in H^{4}(X;\mathbb{Z})$. We recall
several properties of these characteristic classes (see e.g. [33]).

######  Proposition .

Let $E^{m},F^{n}\to X$ be real vector bundles. Denote the full Stiefel–Whitney
and Pontryagin classes by $w\colon{\rm Vect}(X)\to
H^{\bullet}(X;\mathbb{Z}_{2})$ and $p\colon{\rm Vect}(X)\to
H^{\bullet}(X;\mathbb{Z})$. Then:

  * i)

$w(E\oplus F)=w(E)\cup w(F)$, and $w_{1}(E\otimes F)=nw_{1}(E)+mw_{1}(F)$;

  * ii)

$w_{2}(E\otimes F)=w_{2}(E)+w_{1}(F)\cup w_{1}(E)$ if $m=4$ and $n=1$.

  * iii)

$2p(E\oplus F)=2p(E)\cup p(F)$, and $p(E\otimes F)=p(E)$ if $n=1$;

We now determine the relevant characteristic classes for the Lie algebroids
$\mathcal{A}_{Z}^{k}$, $\mathcal{B}_{Z}$, and $\mathcal{C}_{Z}$.

######  Proposition .

Let $(X^{n},Z)$ be a log pair with Lie algebroids $\mathcal{A}_{Z}^{k}$,
$\mathcal{B}_{Z}$, and $\mathcal{C}_{Z}$. Then:
For $\mathcal{A}^{k}_{Z}$: $\bullet$
$w_{1}(\mathcal{A}^{k}_{Z})=w_{1}(TX)+kw_{1}(L_{Z})$, $\bullet$
$w_{2}(\mathcal{A}^{k}_{Z})=w_{2}(TX)+kw_{1}(L_{Z})\cup w_{1}(TX)$, $\bullet$
$p_{1}(\mathcal{A}^{k}_{Z})=p_{1}(TX)$ if $X$ is orientable and four-
dimensional; For $\mathcal{B}_{Z}$: $\bullet$
$w_{1}(\mathcal{B}_{Z})=w_{1}(TX)+nw_{1}(L_{Z})$, $\bullet$
$w_{2}(\mathcal{B}_{Z})=w_{2}(TX)+w_{1}(L_{Z})\cup w_{1}(TX)$ if $X$ is four-
dimensional; For $\mathcal{C}_{Z}$: $\bullet$
$w_{1}(\mathcal{C}_{Z})=w_{1}(TX)+(n+1)w_{1}(L_{Z})$, $\bullet$
$w_{2}(\mathcal{C}_{Z})=w_{2}(TX)$ if $X$ is four-dimensional, $\bullet$
$p_{1}(\mathcal{C}_{Z})=p_{1}(TX)$ if $X$ is orientable and four-dimensional.

######  Remark .

The fact that $w(\mathcal{A}_{Z})=w(TX)(1+{\rm PD}_{\mathbb{Z}_{2}}[Z])$ can
be found in [7], as this would also be what follows from the stable
isomorphism relation noted by them, see Section 4.1.

######  Remark .

We will have no direct use for $w_{2}(\mathcal{B}_{Z})$ and
$p_{1}(\mathcal{B}_{Z})$ (when $X$ is four-dimensional), as by [26,
Proposition 2.21] we know that $\mathcal{B}_{Z}$ does not admit Lie algebroid
symplectic structures when $\dim X\geq 4$. This follows from studying the ring
structure of the space $\Omega^{\bullet}(\mathcal{B}_{Z})$.

###### Proof..

For $\mathcal{A}_{Z}^{k}$: By Section 4.1 we have $\mathcal{A}_{Z}^{k}\oplus
L_{Z}^{k}\cong TX\oplus\underline{\mathbb{R}}$, hence due to Section 4.2.i) we
get $w(\mathcal{A}_{Z}^{k})=(1+kw_{1}(L_{Z}))\cup w(TX)$. In degree one this
gives $w_{1}(\mathcal{A}_{Z}^{k})=w_{1}(TX)+kw_{1}(L_{Z})$ as desired. In
degree two it follows that
$w_{2}(\mathcal{A}_{Z}^{k})=w_{2}(TX)+kw_{1}(L_{Z})\cup w_{1}(TX)$. We see
that $2p_{1}(\mathcal{A}_{Z}^{k})=2p_{1}(TX)$, as $p\equiv 1$ for line
bundles. If $X$ is orientable and four-dimensional we know that
$H^{4}(X;\mathbb{Z})\cong\mathbb{Z}$, which in particular has no two-torsion,
so that $p_{1}(\mathcal{A}_{Z}^{k})=p_{1}(TX)$.

For $\mathcal{B}_{Z}$: This follows from Section 4.2 after using Section 2.2
that $\mathcal{B}_{Z}\cong TX\otimes L_{Z}$.

For $\mathcal{C}_{Z}$: By Section 2.2 we have
$\mathcal{C}_{Z}\cong\mathcal{A}_{Z}\otimes L_{Z}$, so that from Section 4.1
for $\mathcal{A}=\mathcal{A}_{Z}$ we obtain $\mathcal{C}_{Z}\oplus
L_{Z}^{2}\cong\mathcal{B}_{Z}\oplus L_{Z}$. As $L_{Z}^{2}$ is canonically
trivial, using Section 4.2.i) this gives
$w(\mathcal{C}_{Z})=(1+w_{1}(L_{Z}))\cup w(\mathcal{B}_{Z})$. In degree one
this results in (using the case of $\mathcal{B}_{Z}$ above):

$w_{1}(\mathcal{C}_{Z})=w_{1}(\mathcal{B}_{Z})+w_{1}(L_{Z})=w_{1}(TX)+(n+1)w_{1}(L_{Z}).$

In degree two we see similarly that if $X$ is four-dimensional that

$\displaystyle w_{2}(\mathcal{C}_{Z})$
$\displaystyle=w_{2}(\mathcal{B}_{Z})+w_{1}(L_{Z})\cup w_{1}(\mathcal{B}_{Z})$
$\displaystyle=w_{2}(TX)+w_{1}(L_{Z})\cup
w_{1}(TX)+w_{1}(L_{Z})\cup(w_{1}(TX)+4w_{1}(L_{Z}))=w_{2}(TX).$

Assuming also orientability of $X$, Section 4.2 and the case of
$\mathcal{A}_{Z}$ determine $p_{1}(\mathcal{C}_{Z})$. ∎

We can compute these characteristic classes somewhat more generally for
rescalings.

######  Proposition .

Let $[\mathcal{A}\text{:}\mathcal{B}]\to X$ be a corank-$k$
$(\mathcal{B},Z)$-rescaling of $\mathcal{A}\to X$. Then:

  * •

$w_{1}([\mathcal{A}\text{:}\mathcal{B}])=w_{1}(\mathcal{A})+kw_{1}(L_{Z})$;

  * •

$w_{2}([\mathcal{A}\text{:}\mathcal{B}])=w_{2}(\mathcal{A})+kw_{1}(L_{Z})\cup
w_{1}(\mathcal{A})+\frac{k(k-1)}{2}w_{1}(L_{Z})^{2}$;

  * •

$p_{1}([\mathcal{A}\text{:}\mathcal{B}])=p_{1}(\mathcal{A})$, if $X$ is
orientable and four-dimensional.

###### Proof..

By Section 4.1 we have that $[\mathcal{A}\text{:}\mathcal{B}]\oplus
k\mathbb{R}\cong\mathcal{A}\oplus kL_{Z}$ using the shorthand notation
$kL=L\oplus\dots\oplus L$ with $k$ copies. This implies using Section 4.2.i)
by $k$-fold induction that $w([\mathcal{A}\text{:}\mathcal{B}])\cup
1^{k}=w(\mathcal{A})\cup(1+w_{1}(L_{Z}))^{k}$. We have
$(1+w_{1}(L_{Z}))^{k}\equiv 1+kw_{1}(L_{Z})+\frac{k(k-1)}{2}w_{1}(L_{Z})^{2}$
up to degree two. In degree one this gives
$w_{1}([\mathcal{A}\text{:}\mathcal{B}])=w_{1}(\mathcal{A})+kw_{1}(L_{Z})$ as
desired, while in degree two it instead gives
$w_{2}([\mathcal{A}\text{:}\mathcal{B}])=w_{2}(\mathcal{A})+kw_{1}(L_{Z})\cup
w_{1}(\mathcal{A})+\frac{k(k-1)}{2}w_{1}(L_{Z})^{2}$. The last property
follows because by Section 4.2.iii) we have
$2p_{1}([\mathcal{A}\text{:}\mathcal{B}])=2p_{1}(\mathcal{A})\in
H^{4}(X;\mathbb{Z})$, and the hypothesis ensures that
$H^{4}(X;\mathbb{Z})\cong\mathbb{Z}$ has no two-torsion (c.f. Section 4.2). ∎

We can further determine the first Stiefel–Whitney class of the bundles
$\mathcal{A}_{|D|}$ and $\mathcal{A}_{W}$.

######  Proposition .

Let $(X,|D|)$ and $(X^{\prime},W)$ be an elliptic and elliptic-log pair. Then:

  * •

$w_{1}(\mathcal{A}_{D})=w_{1}(TX)$;

  * •

$w_{1}(\mathcal{A}_{W})=w_{1}(\mathcal{A}_{Z})=w_{1}(TX)+w_{1}(L_{Z})$, if
$I_{W}=I_{Z}\otimes I_{|D^{\prime}|}$.

To prove this we first turn to an auxilliary lemma regarding triviality of
line bundles.

######  Lemma .

Let $L\to X$ be a real line bundle with a section vanishing only on a
submanifold of codimension at least two. Then $L$ is trivial, i.e.
$w_{1}(L)=0\in H_{1}(X;\mathbb{Z}_{2})$. Consequently, if $(\varphi,{\rm
id}_{X})\colon E\to F$ is a base-preserving vector bundle morphism which is an
isomorphism outside a submanifold of codimension at least two in $X$, then
$w_{1}(E)=w_{1}(F)$.

###### Proof..

Let $N\subseteq X$ be that submanifold and consider the Mayer–Vietoris
sequence in cohomology for $(X,N)$, giving (for $\mathbb{D}(N)$ the unit disk
bundle of the normal bundle to $N$):

$\displaystyle H^{0}(X)$ $\displaystyle\to H^{0}(X\backslash N)+H^{0}(N)\to
H^{0}(\mathbb{D}(N))\to$ $\displaystyle\to H^{1}(X)\to H^{1}(X\backslash
N)\oplus H^{1}(N)\to H^{1}(\mathbb{D}(N))\to\dots.$

Assume that $X$ is connected, so that $X\backslash N$ is still connected by
the codimension assumption. Then the map $H^{0}(\mathbb{D}(N))\to H^{1}(X)$ is
zero, so that the map $H^{1}(X)\hookrightarrow H^{1}(X\setminus N)\oplus
H^{1}(N)$ is injective. As $L$ is trivial on $X\backslash N$ by hypothesis, we
have $w_{1}(L|_{X\setminus N})=0$, so that $w_{1}(L)=0$.

The condition on $\varphi$ being generically an isomorphism implies that ${\rm
rank}(E)={\rm rank}(F)$. Equivalently, using
$\det(\varphi)\colon\det(E)\to\det(F)$, the pair
$(\det(F)\otimes\det(E)^{*},\det(\varphi))$ is a divisor, and $\det(\varphi)$
vanishes only on a submanifold of codimension at least two by hypothesis. The
first part then implies that $w_{1}(\det(F)\otimes\det(E)^{*})=0$, from which
the conclusion follows. ∎

###### Proof of Section 4.2..

The natural maps $\rho_{\mathcal{A}_{|D|}}\colon\mathcal{A}_{|D|}\to TX$ and
$\varphi_{\mathcal{A}_{W}}\colon\mathcal{A}_{W}\to\mathcal{A}_{Z}$ are both
isomorphisms outside of $D$ and $D^{\prime}$ respectively, both of which are
of codimension two. Consequently Section 4.2 applies, hence the result follows
(using Section 4.2 for $\mathcal{A}_{Z}$). ∎

###  4.3. Orientability of Lie algebroids

In this section we discuss orientability for the Lie algebroids
$\mathcal{A}_{Z}^{k}$, $\mathcal{B}_{Z}$ and $\mathcal{C}_{Z}$ associated to
log pairs $(X,Z)$, and for the Lie algebroids $\mathcal{A}_{|D|}$ and
$\mathcal{A}_{W}$ given elliptic and elliptic-log pairs $(X,|D|)$ and $(X,W)$.
This further settles when these Lie algebroids admit symplectic structures in
dimension two, and gives an obstruction to their existence in arbitrary
dimensions, noting Section 4. They moreover characterize the existence of
$\mathcal{A}$-Nambu structures of highest degree (i.e. nonvanishing sections
$\Pi\in\Gamma(\det(\mathcal{A})$).

Given a Lie algebroid $\mathcal{A}\to X$, we say $X$ is _$\mathcal{A}$
-orientable_ if the vector bundle $\mathcal{A}$ admits an orientation. As this
does not depend on the Lie algebroid structure, it is clear that this happens
if and only if $w_{1}(\mathcal{A})=0$. Consequently, Section 4.2 shows the
following (where we use Section 2.1 for the fact that $w_{1}(L_{Z})={\rm
PD}_{\mathbb{Z}_{2}}[Z]\in H^{1}(X;\mathbb{Z}_{2})$).

######  Proposition .

Let $(X^{n},Z)$ be a log pair. Then $X$ is $\mathcal{A}$-orientable if and
only if:

  * •

$w_{1}(TX)+k{\rm PD}_{\mathbb{Z}_{2}}[Z]=0$, in case
$\mathcal{A}=\mathcal{A}_{Z}^{k}$;

  * •

$w_{1}(TX)+n\,{\rm PD}_{\mathbb{Z}_{2}}[Z]=0$, in case
$\mathcal{A}=\mathcal{B}_{Z}$;

  * •

$w_{1}(TX)+(n+1){\rm PD}_{\mathbb{Z}_{2}}[Z]=0$, in case
$\mathcal{A}=\mathcal{C}_{Z}$.

For later convenience, let us make explicit what happens in the
$\mathcal{A}$-orientable case.

######  Corollary .

Let $(X,Z)$ be an $\mathcal{A}$-orientable log pair. Then $X$ is orientable if
and only if:

  * •

$k$ is even or $[Z]=0$, in case $\mathcal{A}=\mathcal{A}_{Z}^{k}$ (c.f. [34,
35]);

  * •

$n$ is even or $[Z]=0$, in case $\mathcal{A}=\mathcal{B}_{Z}$;

  * •

$n$ is odd or $[Z]=0$, in case $\mathcal{A}=\mathcal{C}_{Z}$.

We can further use Section 4.2 to see that $\mathcal{A}_{|D|}\to X$ is
orientable if and only if $X$ is, and $\mathcal{A}_{W}\to X$ is orientable if
and only if the associated log-tangent bundle $\mathcal{A}_{Z}\to X$ is.
Consequently, elliptic-log Poisson structures exist on any surface $X$, as is
true for log-Poisson.

Thus, the consequences of Section 4 and $\mathcal{A}$-orientability are as
follows (Theorem A):

######  Theorem 4.2.

Let $\mathcal{A}\to X^{n}$ be a symplectic Lie algebroid. Then in
$H_{1}(X;\mathbb{Z}_{2})$ we have:

  * •

If $\mathcal{A}=\mathcal{A}_{Z}^{k}$, then $w_{1}(TX)+k{\rm
PD}_{\mathbb{Z}_{2}}[Z]=0$;

  * •

If $\mathcal{A}=\mathcal{B}_{Z}$, then $w_{1}(TX)+n\,{\rm
PD}_{\mathbb{Z}_{2}}[Z]=0$;

  * •

If $\mathcal{A}=\mathcal{C}_{Z}$, then $w_{1}(TX)+(n+1){\rm
PD}_{\mathbb{Z}_{2}}[Z]=0$;

  * •

If $\mathcal{A}=\mathcal{A}_{|D|}$, then $w_{1}(TX)=0$;

  * •

If $\mathcal{A}=\mathcal{A}_{W}$, then $w_{1}(TX)+{\rm
PD}_{\mathbb{Z}_{2}}[Z]=0$.

###### Proof..

If $\mathcal{A}$ is symplectic, by Section 4 we must have that $\mathcal{A}$
is $\mathcal{A}$-orientable, so that we see that $w_{1}(\mathcal{A})=0$. The
result then follows from Section 4.2 and Section 4.2. ∎

###  4.4. Existence of $\mathcal{A}$-almost-complex structures

In this section we discuss when some Lie algebroids of interest can admit a
complex structure. For this we use Theorem 4.1 together with our earlier
computations of characteristic classes (see Section 4.2). Due to Section 4
this provides obstructions to when these Lie algebroids can be symplectic.

Let $(X^{4},Z)$ be a four-dimensional log pair, and assume that $X$ is
oriented. Consider a $(k-1)$-jet for $Z$ and its $b^{k}$-tangent bundle
$\mathcal{A}_{Z}^{k}$, which recall includes the log-tangent bundle if $k=1$.
Assume further that an orientation for $\mathcal{A}_{Z}^{k}$ is given. Then we
can define the following:

######  Definition .

Given orientations on the bundles $\mathcal{A}_{Z}^{k}$ and $TX$, the _$k$
-discrepancy_ $f_{k}(X,Z)$ of $Z$ is defined as the difference
$2f_{k}(X,Z):=e(\mathcal{A}_{Z}^{k})-e(TX)\in
H^{4}(X;\mathbb{Z})\cong\mathbb{Z}$.

######  Lemma .

In the situation above, the $k$-discrepancy for $Z$ is well-defined, i.e. the
difference in Euler classes of $\mathcal{A}_{Z}^{k}$ and $TX$ is even.
Further, we have $f_{k}(X,Z)\equiv kf_{1}(X,Z)\pmod{2}$.

Write $f(X,Z):=f_{1}(X,Z)$, so that the second statement of the lemma is
shorthand for the facts that $f_{k}(X,Z)\equiv 0\pmod{2}$ if $k$ is even, and
$f_{k}(X,Z)\equiv f(X,Z)\pmod{2}$ if $k$ is odd.

###### Proof..

Recall that the Euler class of an oriented vector bundle reduces mod $2$ to
its top Stiefel–Whitney class. Because both $\mathcal{A}_{Z}^{k}$ and $TX$ are
oriented, we have $w_{1}(\mathcal{A}_{Z}^{k})=w_{1}(TX)=0$, hence
$w_{1}(L_{Z}^{k})=kw_{1}(L_{Z})=0$ by Section 4.2. This means that $L_{Z}^{k}$
is trivial, so that by Section 4.1 we have that
$\mathcal{A}_{Z}^{k}\oplus\underline{\mathbb{R}}\cong
TX\oplus\underline{\mathbb{R}}$. Using Section 4.2.i) this implies that
$w(\mathcal{A}_{Z}^{k})=w(TX)$, so that in particular
$e(\mathcal{A}_{Z}^{k})\equiv e(TX)\pmod{2}$ as desired.

For the second statement, we remark that there is a more geometric description
of the $k$-discrepancy. If $\mathcal{A}^{k}_{Z}$ is oriented and $X$ is
orientable, any choice of orientation for $TX$ does not agree with the
orientation on the isomorphism locus $X\backslash Z$ induced by
$\mathcal{A}_{Z}$ if and only if $k$ is odd. This follows from the local
description of the bundle $\mathcal{A}_{Z}^{k}$ (c.f. [9] for when $k=1$). If
$k$ is odd, because $Z$ is then separating due to Section 4.2, after a choice
of orientation on $TX$, we can write $X\backslash Z=X_{+}\sqcup X_{-}$, where
$X_{\pm}$ denote the subsets where the orientations from $TX$ and
$\mathcal{A}_{Z}^{k}$ do or do not agree. Then the $k$-discrepancy is given by

$f_{k}(X,Z)=-\langle e(TX),[X_{-}]\rangle=-\chi(X_{-}).$

We see here that the right-hand side does not depend on $k$, nor does the
decomposition of $X\backslash Z$ into $X_{\pm}$. It follows that in fact
$f_{k}(X,Z)=f_{1}(X,Z)$ if $k$ is odd. On the other hand, if $k$ is even, the
orientation on $X\backslash Z$ induced by $\mathcal{A}_{Z}^{k}$ can be made
compatible with the one on $TX$, so that their Euler classes are equal: any
generically vanishing section of $TX$ can be assumed to not vanish on $Z$, so
that its zero set can compute both $e(TX)$ and $e(\mathcal{A}_{Z}^{k})$ (we
use here that $Z$ has codimension one in $X$), and its zeros are counted for
$TX$ and $\mathcal{A}_{Z}^{k}$ with equal signs. ∎

We can now state our obstruction to the existence of an
$\mathcal{A}^{k}_{Z}$-almost-complex structure.

######  Theorem 4.3.

Let $(X^{4},Z)$ be a compact oriented $\mathcal{A}^{k}_{Z}$-almost-complex log
pair. Then we have
$\langle[c_{1}^{2}(\mathcal{A}^{k}_{Z})],[X]\rangle=3\sigma(X)+2\chi(X)+4f_{k}(X,Z)$,
and $b_{2}^{+}(X)+b_{1}(X)+f_{k}(X,Z)$ is odd.

Here $\chi(X)$ is the Euler characteristic, and
$\sigma(X)=b_{2}^{+}(X)-b_{2}^{-}(X)$ is the signature of $X$. The following
proof is similar to the case when $Z=\emptyset$, see [16, Theorem 1.4.13].

###### Proof..

By Theorem 4.1 we have, using the definition of the $k$-discrepancy, that

$c_{1}^{2}(\mathcal{A}_{Z}^{k})=p_{1}(\mathcal{A}_{Z}^{k})+2e(\mathcal{A}_{Z}^{k})=p_{1}(\mathcal{A}_{Z}^{k})+2e(TX)+4f_{k}(X,Z).$

Using the other part of Theorem 4.1 together with Section 4.2 we get (as
$kw_{1}(L_{Z})=0$)

$c_{1}(\mathcal{A}_{Z}^{k})\\!\\!\\!\\!\mod 2\equiv
w_{2}(\mathcal{A}_{Z}^{k})=w_{2}(TX)\in H^{2}(X;\mathbb{Z}_{2}),$

so that $c_{1}(\mathcal{A}_{Z}^{k})$ is _characteristic_ , i.e. it reduces
modulo two to $w_{2}(TX)$. By Van der Blij’s lemma ([16, Lemma 1.2.20]) we
obtain $c_{1}^{2}(\mathcal{A}_{Z}^{k})\equiv\sigma(X)\\!\\!\mod 8$. Using
Section 4.2 again we have $p_{1}(\mathcal{A}_{Z}^{k})=p_{1}(TX)$, which
integrates to $3\sigma(X)$ by the Hirzebruch signature theorem. This implies
that $\sigma(X)+\chi(X)+2f_{k}(X,Z)\equiv 0\\!\\!\mod 4$, from which the
conclusion follows. ∎

Out of this result we can immediately draw the following consequence (Theorem
B).

######  Theorem 4.4.

Let $(X^{4},Z)$ be a compact oriented $b^{k}$-symplectic manifold. Then:

$b_{2}^{+}(X)+b_{1}(X)+kf(X,Z)\text{ is odd}.$

###### Proof..

Because $\mathcal{A}_{Z}^{k}$ is a symplectic Lie algebroid, it is also
complex by Section 4. Due to this $(X,Z)$ has an $\mathcal{A}_{Z}^{k}$-almost-
complex structure, so that the result follows from Theorem 4.3 and applying
Section 4.4 to replace $f_{k}(X,Z)$ by $kf(X,Z)$ after reducing modulo two. ∎

We would like to stress again that this obstruction, if $k$ is even, agrees
with that for $X$ to admit an almost-complex structure in the usual sense (see
[16, Theorem 1.4.13]).

######  Remark .

Theorem 4.4 is an obstruction for the log pair $(X,Z)$ (with a $(k-1)$-jet),
and not just $X$ itself. This is unlike the cohomological obstructions for
$\mathcal{A}_{Z}$ discussed in Section 3. As we make direct use of the Lie
algebroid $\mathcal{A}_{Z}^{k}$ associated to $(X,Z)$, whose characteristic
classes depend on $Z$, such a dependence seems inevitable for obstructions of
this type.

A similar thing can be done for the scattering tangent bundle
$\mathcal{C}_{Z}$, as we now discuss. Consider again a four-dimensional
oriented log pair $(X^{4},Z)$, and assume that an orientation for
$\mathcal{C}_{Z}$ is given. There is an analogous definition of discrepancy
here, similar to Section 4.4.

######  Definition .

Given orientations on the bundles $\mathcal{C}_{Z}$ and $TX$, the _scattering
discrepancy_ $f_{\rm sc}(X,Z)$ of $Z$ is defined as the difference
$2f_{k}(X,Z):=e(\mathcal{C}_{Z})-e(TX)\in H^{4}(X;\mathbb{Z})\cong\mathbb{Z}$.

In fact, we can quickly relate the scattering discrepancy to the usual
discrepancy of $Z$.

######  Lemma .

Let $(X^{2n},Z)$ be a log pair, and choose orientations on $\mathcal{C}_{Z}$
and $TX$. Then $\mathcal{A}_{Z}$ is naturally oriented, and we have the
equality $f_{\rm sc}(X,Z)\equiv f(X,Z)\pmod{2}$.

###### Proof..

The natural Lie algebroid morphism
$\varphi\colon\mathcal{C}_{Z}\to\mathcal{A}_{Z}$ can be used to orient
$\mathcal{A}_{Z}$. Note that because the dimension of $X$ is even, we have
that $w_{1}(\mathcal{C}_{Z})=w_{1}(\mathcal{A}_{Z})$ by Section 4.2. From the
fact that the divisor ideal of $\varphi$ is given by $I_{\varphi}=I_{Z}^{2n}$,
or by the local description of $\mathcal{C}_{Z}$, it readily follows that as
for $\mathcal{A}_{Z}$, the orientation on $X\backslash Z$ induced by
$\mathcal{C}_{Z}$ similarly cannot match the one induced from $TX$ everywhere,
from which the result follows. ∎

Using this we can obtain an obstruction to the existence of a
$\mathcal{C}_{Z}$-almost-complex structure.

######  Theorem 4.5.

Let $(X^{4},Z)$ be a compact oriented $\mathcal{C}_{Z}$-almost-complex log
pair. Then we have
$\langle[c_{1}^{2}(\mathcal{C}_{Z})],[X]\rangle=3\sigma(X)+2\chi(X)+4f_{\rm
sc}(X,Z)$, and $b_{2}^{+}(X)+b_{1}(X)+f(X,Z)$ is odd.

###### Proof..

The proof follows the along the same lines as that of Theorem 4.3. The first
statement follows by definition of the scattering discrepancy. Further, here
Section 4.2 provides that $c_{1}(\mathcal{C}_{Z})\\!\\!\mod 2\equiv
w_{2}(\mathcal{C}_{Z})=w_{2}(TX)$, showing again that $c_{1}(\mathcal{C}_{Z})$
is characteristic. Section 4.2 also gives $p_{1}(\mathcal{C}_{Z})=p_{1}(TX)$,
which implies that $\sigma(X)+\chi(X)+2f_{\rm sc}(X)\equiv 0\\!\\!\mod 4$. By
Section 4.4 we can replace $f_{\rm sc}(X,Z)$ by $f(X,Z)$ after reduction
modulo two. ∎

As for Theorem 4.4, we can draw the following immediate consequence (Theorem
C).

######  Theorem 4.6.

Let $(X^{4},Z)$ be a compact oriented scattering-symplectic manifold. Then:

$b_{2}^{+}(X)+b_{1}(X)+f(X,Z)\text{ is odd}.$

In other words, we see that the obstruction for scattering symplectic
structures obtained via the existence of almost-complex structures is
identical to that of log-symplectic structures.

###### Proof..

Because $\mathcal{C}_{Z}$ is a symplectic Lie algebroid, it is also complex by
Section 4. Due to this $(X,Z)$ has a $\mathcal{C}_{Z}$-almost-complex
structure, hence the result follows from Theorem 4.5. ∎

######  Remark .

One wonders whether Section 4 can be used effectively for other symplectic Lie
algebroids in dimension four, for example the elliptic tangent bundle
$\mathcal{A}_{|D|}$. Note that elliptic symplectic structures (of zero
elliptic residue) can exist on $\mathcal{A}_{|D|}$ both in cases when $X$ is
and is not almost-complex (c.f. [4]), depending on the coorientability of $D$
as measured by $w_{1}(ND)\in H^{1}(X;\mathbb{Z}_{2})$. We see there is
nontrivial dependence on the locus $D$ in this case.

To illustrate Theorem 4.3, we determine the parity of $f(X,Z)$ in the
following situation. As in explained in the proof of Section 4.4, if both $X$
and $\mathcal{A}_{Z}$ are oriented, then $Z$ must be separating and decompose
$X\backslash Z=X_{+}\sqcup X_{-}$ according to whether the orientations agree.

######  Corollary .

Let $(X,Z)$ be a compact oriented four-dimensional log pair which is
$\mathcal{A}_{Z}$-almost-complex, such that $X$ is not almost-complex. Then
$f(X,Z)$ is odd, and the log pair $(X_{-}\cup(X_{+}\\#\mathbb{C}P^{2}),Z)$
after connected summing does not admit an $\mathcal{A}_{Z}$-symplectic
structure.

###### Proof..

If $X$ is not almost-complex, then $b_{2}^{+}(X)+b_{1}(X)\equiv 0\pmod{2}$,
while because $(X,Z)$ is $\mathcal{A}_{Z}$-almost-complex we obtain from
Theorem 4.3 that $b_{2}^{+}(X)+b_{1}(X)+f(X,Z)\equiv 1\pmod{2}$. We conclude
that $f(X,Z)\equiv 1\pmod{2}$. If we perform a connected sum with
$\mathbb{C}P^{2}$ in the subset $X_{+}$ to form the manifold
$X^{\prime}=X_{-}\cup(X_{+}\\#\mathbb{C}P^{2})$, we see that
$b_{2}^{+}(X^{\prime})=b_{2}^{+}(X)+1$ while $f(X^{\prime},Z)=f(X,Z)$. Hence
then $b_{2}^{+}(X^{\prime})+b_{1}(X^{\prime})+f(X^{\prime},Z)\equiv
0\pmod{2}$, so that by applying Theorem 4.4 we see that $(X^{\prime},Z)$ does
not admit an $\mathcal{A}_{Z}$-symplectic structure. ∎

We finish by giving a simple example of how to apply the above results.

######  Example  ($3\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}$).

The manifold $X=2\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}$ admits a log-
symplectic structure with $Z=S^{1}\times S^{2}$ (see [10]), and
$b_{2}^{+}(X)=2$ and $b_{1}(X)=0$. Hence $X$ is not almost-complex while
$(X,Z)$ is $\mathcal{A}_{Z}$-almost-complex, and $f(X,Z)$ is odd. By Section
4.4 the manifold $X^{\prime}=3\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}$
does not admit a log-symplectic structure with the given $Z=S^{1}\times
S^{2}$.

######  Remark .

It seems somewhat nontrivial to determine the discrepancy $f(X,Z)$ of a
separating log pair, even just its parity. We are only able to do so
indirectly, c.f. Section 4.4. Nevertheless, the question of how $Z$
distributes the Euler characteristic of $X$ seems classical.

##  5\. Obstructions from Seiberg–Witten theory

In this section we discuss obstructions for $\mathcal{A}$-symplectic four-
manifolds coming from more sophisticated techniques. Namely, they rely on the
fact that their existence implies that $X$ symplectically fills $Z$ in a
certain sense, which is then obstructed by Seiberg–Witten theory.

Given a log pair $(X,Z)$ with a symplectic structure $\omega$, we say it is
_of cosymplectic-type at $Z$_ if the pair $(\iota_{v}\omega,i^{*}\omega)$ is a
cosymplectic structure on $Z$, where $v\in\Gamma(TX)$ is transverse to $Z$,
and $i\colon Z\hookrightarrow X$ the inclusion. The following is a remark of
Osorno Torres ([37, Lemma 4.2.6]).

######  Lemma  ([37]).

Let $X$ be a manifold with boundary. Then the log pair $(X,\partial X)$ admits
a log-symplectic structure if and only if it admits a symplectic structure of
cosymplectic-type.

###### Proof..

Assume first that $(X,Z=\partial X)$ admits a log-symplectic structure
$\omega_{Z}\in{\rm Symp}(\mathcal{A}_{Z})$. Fix a collar neighbourhood
$U\cong\partial X\times[0,1)$ where $\omega_{Z}=d\log z\wedge
p^{*}(\alpha)+p^{*}(\beta)$. Choose a positive function
$\lambda\colon[0,1]\to\mathbb{R}$ such that $\lambda(t)=1/t$ near $t=1$, and
$\lambda(t)=1$ near $t=0$. Then $\omega:=\lambda(z)dz\wedge
p^{*}(\alpha)+p^{*}(\beta)$ is a smooth closed nondegenerate two-form on $U$
which can be extended to $X$ as it agrees with $\omega_{Z}$ away from $Z$. It
readily follows that $\omega$ is a symplectic structure of cosymplectic-type
at $\partial X$ given by $(\alpha,\beta)$. The converse is similar and
omitted. ∎

By combining this result with Section 4.3, which gives that $Z$ is separating,
we get:

######  Corollary .

Let $(X,Z)$ be log-symplectic with $X$ oriented, so that $X\setminus
Z=X_{+}\sqcup X_{-}$. Then the log pairs with boundary $(X_{\pm},Z)$ admit
symplectic structures of cosymplectic-type.

This result can obstruct the existence of log-symplectic structures, as
specific symplectic fillings of $Z$ as a cosymplectic manifold need not always
exist. This contrasts with results of [14] and [15, Theorem 2], [12, Theorem
3.1], which describe which cosymplectic manifolds admit (log-)symplectic
fillings. We are crucially fixing the diffeomorphism type of the filling.

######  Remark .

Note that Section 5 does not say that $(X,Z)$ admits a symplectic structure of
cosymplectic-type. Often $X$ cannot admit symplectic structures at all (e.g.
$2\mathbb{C}P^{2}\\#\overline{\mathbb{C}P}^{2}$).

Let us say a manifold is _psc_ if it admits a positive scalar curvature
metric. Then an almost direct consequence of results by Taubes [39, 40] and
Kronheimer–Mrowka [25] is the following:

######  Proposition  ([3]).

Let $(X^{4},Z)$ be an oriented log pair, splitting into
$X_{1}\sqcup_{Z}X_{2}$, where $b_{2}^{+}(X_{i})>0$. Then if each component
$Z_{i}$ of $Z$ is psc (e.g. $S^{1}\times S^{2}$), then $X$ cannot be
symplectic.

######  Remark .

This result appeared first as [5, Proposition 1] and then [3, Lemma 3.4],
where the authors remark that any psc three-manifold is a connected sum of
spherical three-manifolds and copies of $S^{1}\times S^{2}$, and show how to
allow $Z$ to be disconnected (by performing tubing). Note here that the total
space of any spherical symplectic mapping torus is given by $S^{2}\times
S^{1}$.

We now state our obstruction result for orientable log-symplectic manifolds
(Theorem D).

######  Theorem 5.1.

Let $(X^{4},Z)$ be an oriented log pair, splitting along
$X_{1}\sqcup_{Z}X_{2}$. If $b_{2}^{+}(X_{1})>0$ and each connected component
of $Z_{i}$ is psc (e.g. $S^{1}\times S^{2}$), then $(X,Z)$ cannot be log-
symplectic.

###### Proof..

If it can, then by Section 5, the log pair with boundary $(X_{1},Z)$ is
symplectic of cosymplectic-type. In dimension three any cosymplectic manifold,
such as $Z$, automatically has a taut foliation. Hence by [25, Theorem 41.3.1]
we can find another symplectic manifold $X^{\prime}$ with
$b_{2}^{+}(X^{\prime})>0$ such that $X_{1}\cup_{Z}X^{\prime}$ is symplectic
and splits along $Z$ (compare this with [10, Theorem 6.1]). This contradicts
Section 5, so that $(X,Z)$ cannot be log-symplectic. ∎

######  Remark .

This result is similar to those in [1, Section 5.2], where instead of
Seiberg–Witten theory, techniques from pseudo-holomorphic curves adapted to
log pairs are used.

######  Remark .

Conceivably something similar can be done for other symplectic Lie algebroids,
such as the scattering tangent bundle $\mathcal{C}_{Z}$, and the
$b^{k}$-tangent bundles $\mathcal{A}_{Z}^{k}$ (in light of [18]).

## References

  * [1] D. Alboresi, _Holomorphic curves in log-symplectic manifolds_ , arXiv:1711.10978.
  * [2] M. Bailey, G. R. Cavalcanti, and M. Gualtieri, _Type one generalized Calabi-Yaus_ , J. Geom. Phys. 120 (2017), 89–95.
  * [3] R. I. Baykur and S. Friedl, _Virtually symplectic fibered 4-manifolds_ , Indiana Univ. Math. J. 64 (2015), no. 4, 983–999.
  * [4] S. Behrens, G. R. Cavalcanti, and R. L. Klaasse, _Classification of boundary Lefschetz fibrations over the disc_ , Geometry and Physics: Volume 2, Chapter 15, Oxford University Press, Oxford, 2018.
  * [5] J. Bowden, _The topology of symplectic circle bundles_ , Trans. Amer. Math. Soc. 361 (2009), no. 10, 5457–5468.
  * [6] A. Cannas da Silva, _Fold-forms for four-folds_ , J. Symplectic Geom. 8 (2010), no. 2, 189–203.
  * [7] A. Cannas da Silva, V. Guillemin, and C. Woodward, _On the unfolding of folded symplectic structures_ , Math. Res. Lett. 7 (2000), no. 1, 35–53.
  * [8] G. R. Cavalcanti and R. L. Klaasse, _Fibrations and stable generalized complex structures_ , Proc. Lond. Math. Soc. 0 (2018), no. 0, 1–39.
  * [9] G. R. Cavalcanti and R. L. Klaasse, _Fibrations and log-symplectic structures_ , arXiv:1606.00156. To appear in J. Symplectic Geom.
  * [10] G. R. Cavalcanti, _Examples and counter-examples of log-symplectic manifolds_ , J. Topol. 10 (2017), no. 1, 1–21.
  * [11] G. R. Cavalcanti and M. Gualtieri, _Stable generalized complex structures_ , Proc. Lond. Math. Soc. 116 (2018), no. 5, 1075–1111.
  * [12] Y. Eliashberg, _A few remarks about symplectic filling_ , Geom. Topol. 8 (2004), 277–293.
  * [13] S. Evens, J.-H. Lu, and A. Weinstein, _Transverse measures, the modular class and a cohomology pairing for Lie algebroids_ , Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 417–436.
  * [14] P. Frejlich, D. Martínez Torres, and E. Miranda, _A note on symplectic topology of $b$-manifolds_, arXiv:1312.7329. To appear in J. Symplectic Geom.
  * [15] H. Geiges and K. Zehmisch, _Cobordisms between symplectic fibrations_ , Manuscripta Math. 153 (2017), no. 3-4, 331–340.
  * [16] R. E. Gompf and A. I. Stipsicz, _$4$ -manifolds and Kirby calculus_, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999.
  * [17] M. Gualtieri and S. Li, _Symplectic groupoids of log symplectic manifolds_ , Int. Math. Res. Not. IMRN (2014), no. 11, 3022–3074.
  * [18] V. Guillemin, E. Miranda, and J. Weitsman, _Desingularizing $b^{m}$-symplectic structures_, Int. Math. Res. Not. IMRN (2017), rnx126.
  * [19] V. Guillemin, E. Miranda, and A. R. Pires, _Symplectic and Poisson geometry on $b$-manifolds_, Adv. Math. 264 (2014), 864–896.
  * [20] F. Hirzebruch and H. Hopf, _Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten_ , Math. Ann. 136 (1958), 156–172.
  * [21] R. L. Klaasse, _Geometric structures and Lie algebroids_ , Ph.D. thesis, Utrecht University, 2017.
  * [22] R. L. Klaasse, _Poisson structures of divisor-type_ , arXiv:1811.04226.
  * [23] R. L. Klaasse and M. Lanius, _Poisson cohomology of almost-regular Poisson manifolds_. In preparation.
  * [24] R. L. Klaasse and S. Li, _Symplectic groupoids of elliptic Poisson manifolds_. In preparation.
  * [25] P. Kronheimer and T. Mrowka, _Monopoles and three-manifolds_, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007.
  * [26] M. Lanius, _Symplectic, Poisson, and contact geometry on scattering manifolds_ , arXiv:1603.02994.
  * [27] S. Li, _Constructions of Lie groupoids_ , Ph.D. thesis, University of Toronto, 2013.
  * [28] I. Mărcuţ and B. Osorno Torres, _Deformations of log-symplectic structures_ , J. Lond. Math. Soc. (2) 90 (2014), no. 1, 197–212.
  * [29] I. Mărcuţ and B. Osorno Torres, _On cohomological obstructions for the existence of log-symplectic structures_ , J. Symplectic Geom. 12 (2014), no. 4, 863–866.
  * [30] D. McDuff and D. Salamon, _Introduction to symplectic topology_, third ed., Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017.
  * [31] R. B. Melrose, _The Atiyah–Patodi–Singer index theorem_ , Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993.
  * [32] R. B. Melrose, _Geometric scattering theory_ , Stanford Lectures, Cambridge University Press, Cambridge, 1995.
  * [33] J. W. Milnor and J. D. Stasheff, _Characteristic classes_ , Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76.
  * [34] E. Miranda and A. Planas, _Classification of $b^{m}$-Nambu structures of top degree_, C. R. Math. Acad. Sci. Paris 356 (2018), no. 1, 92–96.
  * [35] E. Miranda and A. Planas, _Equivariant Classification of $b^{m}$-symplectic Surfaces_, Regular and Chaotic Dynamics 23 (2018), no. 4, 355–371.
  * [36] R. Nest and B. Tsygan, _Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems_ , Asian J. Math. 5 (2001), no. 4, 599–635.
  * [37] B. Osorno Torres, _Codimension-one symplectic foliations: constructions and examples_ , Ph.D. thesis, Utrecht University, 2015.
  * [38] G. Scott, _The geometry of $b^{k}$ manifolds_, J. Symplectic Geom. 14 (2016), no. 1, 71–95.
  * [39] C. H. Taubes, _The Seiberg-Witten invariants and symplectic forms_ , Math. Res. Lett. 1 (1994), no. 6, 809–822.
  * [40] C. H. Taubes, _More constraints on symplectic forms from Seiberg-Witten invariants_ , Math. Res. Lett. 2 (1995), no. 1, 9–13.
  * [41] W.-T. Wu, _Sur les classes caractéristiques des structures fibrées sphériques_ , Actualités Sci. Ind., no. 1183, Hermann & Cie, Paris, 1952. Publ. Inst. Math. Univ. Strasbourg 11, pp. 5–89, 155–156.