# Topology of almost complex structures on six–manifolds

Gustavo Granja  Center for Mathematical Analysis, Geometry and Dynamical
Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais,
1049-001 Lisboa<EMAIL_ADDRESS>and  Aleksandar Milivojević
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn
<EMAIL_ADDRESS>
###### Abstract.

We study the space of (orthogonal) almost complex structures on closed
six–dimensional manifolds as the space of sections of the twistor space for a
given metric. For a connected six–manifold with vanishing first Betti number,
we express the space of almost complex structures as a quotient of the space
of sections of a seven–sphere bundle over the manifold by a circle action, and
then use this description to compute the rational homotopy theoretic minimal
model of the components that satisfy a certain Chern number condition. We
further obtain a formula for the homological intersection number of two
sections of the twistor space in terms of the Chern classes of the
corresponding almost complex structures.

###### Key words and phrases:

Almost complex structure, twistor space, space of almost complex structures

###### 2020 Mathematics Subject Classification:

32Q60, 53C27, 53C28, 55P62

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Twistor spaces of four– and six–manifolds
  4. 4 On the homotopy type of the space of almost complex structures on six–manifolds
  5. 5 Intersections of almost complex structures on six–manifolds as sections of the twistor space

##  1\. Introduction

A major open problem in differential geometry is to determine whether a closed
almost complex manifold of dimension at least six always admits an integrable
complex structure. By a celebrated theorem of Newlander and Nirenberg, an
almost complex structure is integrable if and only if it satisfies a certain
system of first order partial differential equations codified by the vanishing
of the Nijenhuis tensor. An understanding of the topology of the space of all
almost complex structures may be useful in the search for those which are
integrable. We take this as motivation for our study of the topology of the
space of almost complex structures on a six–manifold. The present work is a
continuation of [FGM21] which focused on the six–sphere and touched upon the
case of almost complex structures on six–manifolds with vanishing first Chern
class.

Let $M$ denote an oriented six–manifold and $\mathcal{J}(M)$ be the space of
all almost complex structures on $M$ inducing the given orientation. Fixing a
Riemannian metric on $M$, the inclusion $J(M)\hookrightarrow\mathcal{J}(M)$ of
the subspace of almost complex structures which are orthogonal with respect to
the metric is a homotopy equivalence, allowing us to analyze $\mathcal{J}(M)$
via a space of sections of a fiber bundle with compact fibers. Generally, over
an oriented Riemannian $2n$–manifold $M$ one can consider the $SO(2n)/U(n)$
bundle of pointwise linear orthogonal complex structures $Z_{+}(M)\to M$
compatible with the orientation on $M$; the space $Z_{+}(M)$ is known as the
_(positive) twistor space_ of $M$. Sections of this bundle correspond to
orthogonal almost complex structures on $M$ compatible with the orientation.

The twistor space construction can be done in any dimension, but six–manifolds
enjoy the special property that the fiber of the twistor space is likewise
six–dimensional. Hence two orthogonal almost complex structures give
embeddings of the manifold into its twistor space which generically intersect
at a finite number of points. One of our main results (5.10) computes this
intersection number in terms of the Chern classes of the two almost complex
structures. Employing a theorem of Michelsohn and Salamon, we observe that
negative self-intersection restricts the possible deformations of an
orthogonal complex structure (5.6).

In order to study the homotopy type of $J(M)$, in Section 3 we use the
existence of spinc structures on oriented four–manifolds and almost complex
six–manifolds to describe the integral cohomology of their twistor spaces.
This will include both the positive and the _negative_ twistor space
$Z_{-}(M)$, i.e. the total space of the bundle of linear complex structures
inducing the opposite orientation on the tangent space. Presumably these
results are well known, but we were not able to find them in the literature in
this generality.

We take particular care with orientations throughout, due to varying
historical conventions (e.g. the original [AHS78], and [Hi81], refer to the
negative twistor space as the “twistor space”, while this terminology is
reserved for the positive twistor space for instance in [LM89]111Despite this,
the statement of [AHS78, Theorem 4.1] in [LM89, Theorem IV.9.14] uses the
convention of [AHS78]. and [Sa84]), and to the authors’ difficulty in
verifying some formulas in [Hi81, p.135]; see Remark 3.11.

The twistor spaces $Z_{+}(M)$ and $Z_{-}(M)$ (whose homotopy types generally
differ when the dimension of $M$ is divisible by four) carry two natural
almost complex structures, differing by the choice of induced orientation on
the fiber (see e.g. [Sa84, Proposition 3.1]), known as the
_Atiyah–Hitchin–Singer_ and _Eells–Salamon_ almost complex structures. Even
though the standard complex structure and its negative on the fibers
$SO(4)/U(2)\cong\mathbb{CP}^{1}$ and $SO(6)/U(3)\cong\mathbb{CP}^{3}$ are
biholomorphic via complex conjugation, the Eells–Salamon almost complex
structure is never integrable [Sa95, Proposition 3.4], in contrast to the
Atiyah–Hitchin–Singer structure (this phenomenon is not unique to dimensions
four and six).

Our second main result is a description of the rational homotopy type of (a
given component of) the space of almost complex structures on six–manifolds,
under the additional assumption that $b_{1}(M)=0$ and
$\int_{M}c_{1}c_{2}-c_{3}\neq 0$ (4.4). For this we use an expression of
$J(M)$ as the quotient of the space of sections of an $S^{7}$–bundle by an
$S^{1}$–action (4.1), which one can compare with the result obtained for
$c_{1}=0$ in [FGM21, Proposition 5.1].

Throughout we provide examples for all of the mentioned results.

### Notation and conventions

The projectivization of a complex vector bundle $E$ is denoted by
$\mathbb{P}(E)$. We denote the Chern classes of a complex vector bundle by
$c_{i}$, and the Pontryagin classes of a real vector bundle by $p_{i}$; if the
vector bundle is complex, by its Pontryagin classes we mean those of the
underlying real bundle. The Euler class is denoted by $\mathrm{e}$. The Euler
characteristic of a space is denoted by $\chi$, and $\sigma$ is the signature
of an oriented closed manifold. All manifolds are assumed connected unless
otherwise stated.

We denote by $\Gamma(E)$ the space of sections of a bundle $E\to M$ (where $E$
is not necessarily a vector bundle). If $s\colon M\to E$ is a section,
$\Gamma(E)_{s}$ denotes the connected component of $s$ in $\Gamma(E)$. For
$E\to M$ a vector bundle with a metric, we write $S(E)\to M$ for the
associated sphere bundle. We denote by $J(M)$ the connected component of a
specific point in $\Gamma(Z_{+}(M))$, which will be clear from context. For a
vector space $V$ with an inner product, $J(V)$ will denote the space of all
orthogonal (linear) complex structures on $V$. If $V$ is oriented, $J_{+}(V)$
(respectively $J_{-}(V)$) denotes the connected component of $J(V)$ consisting
of elements which induce the given orientation on $V$ (respectively the
opposite orientation). By $\operatorname{Map}(X,Y)$ we denote the space of
unbased maps $X\to Y$ with the compact-open topology.

Cohomology is singular cohomology with integral coefficients unless another
choice of coefficients is indicated. The evaluation of a cohomology class
$\alpha$ on a homology class $A$ is denoted by $\langle\alpha,A\rangle$ or by
$\int_{A}\alpha$. In the context of rational homotopy theoretic models,
$\Lambda(x_{i})$ denotes the free graded–commutative algebra on generators
$x_{i}$.

We will make frequent use of the projective bundle formula, which states that
for a complex vector bundle $E\to M$ of complex rank $k$, the cohomology of
the projectivized bundle $\mathbb{P}(E)\xrightarrow{p}M$ is the free
$H^{*}(M)$–module on one generator $x$ subject only to the relation
$x^{k}+c_{1}(E)x^{k-1}+\cdots+c_{k-1}(E)x+c_{k}(E)=0$, where $x$ is the first
Chern class of the line bundle $\mathcal{O}_{E^{*}}(1)$ (see [D12, Definition
15.13]). This line bundle is dual to the tautological (Hopf) line sub-bundle
of $p^{*}(E)$ whose fiber over $\ell\in\mathbb{P}(E_{x})$ is $\ell\subset
E_{x}$. It restricts to $\mathcal{O}(1)$ on each fiber $\mathbb{CP}^{k-1}$.

### Acknowledgements

We thank Luis Fernandez and Scott Wilson for numerous helpful discussions. The
first author was partially supported by FCT/Portugal through CAMGSD, IST-ID,
projects UIDB/04459/2020 and UIDP/04459/2020. The second author would like to
thank the Max Planck Institute for Mathematics in Bonn for its support, along
with the Mittag-Leffler Institute in Djursholm for its hospitality during a
visit to the “Higher algebraic structures in algebra, topology and geometry”
program, where part of this work was carried out.

##  2\. Preliminaries

In this section we review some of the models for the space of orthogonal
complex structures on a euclidean space following [LM89, Section IV.9], to
which we refer for further details.

Let $(V,(\cdot,\cdot))$ be a $2n$–dimensional real inner product space. We
will write

$J(V)=\\{J\in\operatorname{End}(V)\colon J^{2}=-I,J\text{ is orthogonal}\\}$

for the space of orthogonal complex structures on $V$. This space naturally
has the structure of a complex algebraic variety, given by the isomorphism

$J(V)\xrightarrow{\phi}\operatorname{Gr}_{n}^{Iso}(V\otimes\mathbb{C})$

with the Grassmannian of $n$–dimensional complex subspaces of
$V\otimes\mathbb{C}$ which are isotropic with respect to the complex bilinear
form on $V\otimes\mathbb{C}$ obtained from the inner product on $V$ by linear
extension. The map $\phi$ is defined by

$\phi(J)=\\{v\otimes 1+Jv\otimes i\colon v\in V\\}$

(it assigns to $J$ the plane $V_{J}^{0,1}\subset V\otimes\mathbb{C}$, which is
the $-i$ eigenspace of the complex linear extension of $J$ to
$V\otimes\mathbb{C}$).

There is a tautologous complex vector bundle on $J(V)$ defined by

$J(V)\times V\xrightarrow{\pi_{1}}J(V),$

where the fiber over $J\in J(V)$ is given the complex structure determined by
$J$. The isomorphism $\phi$ is covered by a bundle map to the dual (or
conjugate) tautological bundle over the Grassmannian.

######  Lemma 2.1.

Consider the universal bundle

$J(\mathbb{R}^{2n})=O(2n)/U(n)\xrightarrow{i}BU(n)\to BO(2n).$

Then

  1. (i)

$i$ classifies the tautologous complex vector bundle on $J(\mathbb{R}^{2n})$.

  2. (ii)

$i^{*}(c_{1})$ generates a subgroup of index $2$ in the infinite cyclic second
cohomology group of each of the two components of $J(\mathbb{R}^{2n})$.

  3. (iii)

$i^{*}(c_{1})$ is positive. In particular, $i^{*}(c_{1})^{\frac{n^{2}-n}{2}}$
evaluates on the orientation class of each of the two components of
$J(\mathbb{R}^{2n})$ to a positive integer.

###### Proof.

  1. (i)

The universal bundle can be realized by taking $BU(n)=EO(2n)/U(n)$. The
universal bundle over $BU(n)$ is then given by
$EO(2n)\times_{U(n)}\mathbb{R}^{2n}\to EO(2n)/U(n)$, and pulls back along the
inclusion of the fiber $i$ to $O(2n)\times_{U(n)}\mathbb{R}^{2n}\to
O(2n)/U(n)$, which is the homogeneous expression of the tautologous bundle.

  2. (ii)

The inclusion-induced map $BU(n)\to BO(2n)$ factors through the double cover
$BSO(2n)\to BO(2n)$, and the two components of $O(2n)/U(n)$ are, respectively,
the fiber $SO(2n)/U(n)$ of $BU(n)\to BSO(2n)$ and the image of this fiber
under a lift to $BU(n)$ of the non-trivial deck transformation of $BSO(2n)$.
Therefore it suffices to prove the statement for the component $SO(2n)/U(n)$.
This follows from the Serre spectral sequence of the bundle $BU(n)\to BSO(2n)$
(see for instance [MiTo, Theorem III.6.11]).

  3. (iii)

$i^{*}(c_{1})$ is the pullback of $c_{1}$ of the dual tautological bundle
under the embedding

$J(\mathbb{R}^{2n})\xrightarrow{\cong}\operatorname{Gr}_{n}^{Iso}(\mathbb{C}^{2n})\subset\operatorname{Gr}_{n}(\mathbb{C}^{2n}).$

The $n^{\mathrm{th}}$ exterior power of the dual tautological bundle on
$\operatorname{Gr}(\mathbb{C}^{2n})$ gives the Plücker embedding of the
Grassmannian in projective space. Hence $i^{*}(c_{1})$ is the first Chern
class of a line bundle on $J(\mathbb{R}^{2n})$ whose sections embed
$J(\mathbb{R}^{2n})$ in projective space and is therefore positive.∎

The above identification of orthogonal complex structures with isotropic
planes in $V\otimes\mathbb{C}$ leads to another description of the space
$J(V)$ which will play an important role in this paper.

Let $\mathcal{C}l(V)$ denote the Clifford algebra determined by the quadratic
form $q(v)=-\|v\|^{2}$, and let
$\mathbb{C}l(V)=\mathcal{C}l(V)\otimes_{\mathbb{R}}\mathbb{C}$ denote its
complexification. Let $S_{\mathbb{C}}$ denote an irreducible
($2^{n}$–dimensional) $\mathbb{C}l(V)$-module.

For each element, i.e. _spinor_ , $\sigma\in S_{\mathbb{C}}$ we have the map
$V\otimes_{\mathbb{R}}\mathbb{C}\to S_{\mathbb{C}}$ given by right Clifford
multiplication by $\sigma$. A spinor is said to be _pure_ if the associated
kernel is half–dimensional. Let $PS_{\mathbb{C}}\subset S_{\mathbb{C}}$ denote
the subset of pure spinors. As the subspace $\ker(\cdot\sigma)\subset
V\otimes\mathbb{C}$ is isotropic there is a natural map from the
projectivization of the set of pure spinors to the isotropic Grassmannian

$\mathbb{P}(PS_{\mathbb{C}})\xrightarrow{\psi}\operatorname{Gr}^{Iso}_{2n}(V\otimes\mathbb{C})$

defined by $[\sigma]\mapsto\ker(\cdot\sigma)$. This is a map of algebraic
varieties and is in fact an $SO(2n)$-equivariant isomorphism [LM89,
Proposition IV.9.7].

The space $J(V)$ has two connected components corresponding to the two
possible orientations induced by the complex structure. A choice of
orientation for $V$ leads to a decomposition

$J(V)=J_{+}(V)\coprod J_{-}(V)$

with $J_{+}(V)$ the component corresponding to the given orientation. In turn
this decomposes the Grassmannian of isotropic subspaces into the components of
positive and negative isotropic subspaces.

On the level of spinors this decomposition takes the following form. The
complex volume element in $\mathbb{C}l(V)$ is the element
$\omega_{\mathbb{C}}=i^{n}e_{1}\cdots e_{2n}$, where $\\{e_{i}\\}$ is any
oriented orthonormal basis. As $\omega_{\mathbb{C}}^{2}=1$, the volume element
decomposes the module $S_{C}$ into a direct sum of $+1$ and $-1$ eigenspaces

$S_{\mathbb{C}}=S_{\mathbb{C}}^{+}\oplus S_{\mathbb{C}}^{-}.$

The elements of the summands are called the positive and negative spinors,
respectively. Note that, since $\omega_{\mathbb{C}}$ anti-commutes with the
action of $V\subset\mathbb{C}l(V)$, both $S_{\mathbb{C}}^{+}$ and
$S_{\mathbb{C}}^{-}$ are spinc representations, called the positive and
negative spinor representations respectively.

Every pure spinor is necessarily either positive or negative and the
isomorphisms $\phi,\psi$ give rise to isomorphisms of algebraic varieties

$\mathbb{P}(PS_{\mathbb{C}}^{\pm})\cong J_{\pm}(V).$

In dimensions $2n=4,6$ all non-zero spinors are pure [LM89, Remark IV.9.12]
and the isomorphisms identify the spaces of orthogonal complex structure with
the disjoint union of the projectivization of positive and negative spinors.

A point $J\in J(V)$ gives rise to a useful description (cf. [LM89, Chapter I
(5.27)]) of the irreducible module $S_{\mathbb{C}}$. We will write
$\langle\cdot,\cdot\rangle$ for the Hermitian inner product on the complex
vector space $(V,J)$ whose real part is $(\cdot,\cdot)$. Let
$\Lambda_{\mathbb{C}}^{*}(V)$ denote the exterior algebra of the complex
vector space $(V,J)$. Contraction of a vector $v\in V$ with
$\omega\in\Lambda_{\mathbb{C}}^{k}(V)$ is determined by the expression

$v\lrcorner(w_{1}\wedge\cdots\wedge w_{k})=\sum_{j=1}^{k}(-1)^{j+1}\langle
v,w_{j}\rangle w_{1}\wedge\cdots\wedge\widehat{w_{j}}\wedge\cdots\wedge
w_{k}.$

The action
$V\otimes_{\mathbb{R}}\Lambda_{\mathbb{C}}^{*}(V)\to\Lambda_{\mathbb{C}}^{*}(V)$
defined by

$v\cdot\omega=v\wedge\omega-v\lrcorner\,\omega$

satisfies

$v\cdot(v\cdot\omega)=-\|v\|^{2}\omega,$

as one easily checks on an orthonormal basis for $\Lambda_{\mathbb{C}}^{*}$
having a real multiple of a non-zero $v$ as its first element. The complex
linear extension of this action gives $\Lambda_{\mathbb{C}}^{*}(V)$ the
structure of a $\mathbb{C}l(V)$-module of complex dimension $2^{n}$. This is
the dimension of the irreducible complex $\mathbb{C}l(V)$-module, so
$\Lambda_{\mathbb{C}}^{*}(V)$ is a convenient description of this module.

######  Lemma 2.2.

Let $(V,J)$ be an oriented $2n$–dimensional euclidean real vector space with
an orthogonal complex structure $J$ compatible with the orientation and let
$S_{\mathbb{C}}=\Lambda^{*}_{\mathbb{C}}(V)$ be the irreducible
$\mathbb{C}l(V)$-module described above. Then

$S_{\mathbb{C}}^{+}=\bigoplus_{k\text{ even
}}\Lambda_{\mathbb{C}}^{k}(V),\quad\quad S_{\mathbb{C}}^{-}=\bigoplus_{k\text{
odd }}\Lambda_{\mathbb{C}}^{k}(V).$

###### Proof.

Let $e_{1},\ldots,e_{n}$ be an orthonormal basis for $(V,J)$. Then we can
write the complex volume element as

$\omega_{\mathbb{C}}=i^{n}e_{1}Je_{1}\cdots e_{n}Je_{n}.$

Consider the basis $e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$ for
$\Lambda^{k}_{\mathbb{C}}$ with $1\leq i_{1}<\cdots<i_{k}\leq n$. Let $1\leq
m\leq n$ and assume that $m\not\in\\{i_{1},\ldots,i_{k}\\}$. Then

$\displaystyle(e_{m}Je_{m})\cdot e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$
$\displaystyle=$ $\displaystyle e_{m}\cdot\left(Je_{m}\wedge
e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}\right)$ $\displaystyle=$
$\displaystyle-\langle e_{m},Je_{m}\rangle e_{i_{1}}\wedge\cdots\wedge
e_{i_{k}}=-i\ e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}.$

On the other hand, if $m\in\\{i_{1},\ldots,i_{k}\\}$, let $l$ be such that
$m=i_{l}$. Then

$\displaystyle(e_{m}Je_{m})\cdot e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}$
$\displaystyle=$ $\displaystyle e_{m}\cdot\left(-(-1)^{l-1}\langle
Je_{m},e_{i_{l}}\rangle
e_{i_{1}}\wedge\cdots\wedge\widehat{e_{i_{l}}}\wedge\cdots\wedge
e_{i_{k}}\right)$ $\displaystyle=$ $\displaystyle
e_{m}\wedge\left((-1)^{l}(-i)e_{i_{1}}\wedge\cdots\wedge\widehat{e_{i_{l}}}\wedge\cdots\wedge
e_{i_{k}}\right)$ $\displaystyle=$
$\displaystyle(-1)^{l-1}(-1)^{l}(-i)e_{i_{1}}\wedge\cdots\wedge
e_{i_{k}}=ie_{i_{1}}\wedge\cdots\wedge e_{i_{k}}.$

Hence

$\omega_{\mathbb{C}}\cdot e_{i_{1}}\wedge\cdots\wedge
e_{i_{k}}=i^{n}i^{k}(-i)^{n-k}e_{i_{1}}\wedge\cdots\wedge
e_{i_{k}}=(-1)^{2n-k}e_{i_{1}}\wedge\cdots\wedge e_{i_{k}},$

which completes the proof. ∎

######  Remark 2.3.

For $(V,J)$ an oriented $2n$–dimensional euclidean real vector space with an
orthogonal complex structure $J$ compatible with the orientation, the line in
$S^{+}_{\mathbb{C}}$ corresponding to $J\in J_{+}(V)$ is
$\Lambda^{0}_{\mathbb{C}}(V)\subset S_{\mathbb{C}}^{+}$. Indeed,

$(v\otimes 1+w\otimes i)\cdot 1=v+Jw=0\ \Leftrightarrow\ w=Jv.$

##  3\. Twistor spaces of four– and six–manifolds

In this section we describe the integral cohomology of the twistor space of
(Riemannian) almost complex six–manifolds, along with the integral cohomology
of the twistor space of arbitrary oriented (Riemannian) four–manifolds.
Finally we compute the Chern classes of the Atiyah–Hitchin–Singer almost
complex structure on the latter.

###  3.1. Spinors on almost complex four– and six–manifolds

Let $M$ be an oriented Riemannian manifold and let $Z(M)$ denote the space of
orthogonal almost complex structures on $M$, called the _twistor space_. The
space $Z(M)$ has two components corresponding to the structures which induce
the given orientation on $M$ or the opposite orientation. Recall, we denote
these by $Z_{\pm}(M)$, and refer to them as the _positive_ and _negative_
twistor spaces.

The Atiyah–Hitchin–Singer almost complex structure on $Z(M)$ is defined as
follows: the Levi–Civita connection induces a connection on the bundle
$Z(M)\to M$ splitting $TZ(M)$ into a direct sum of horizontal and vertical
subspaces

$TZ(M)=T^{h}Z(M)\oplus T^{v}Z(M).$

The almost complex structure on $T^{v}Z(M)$ is determined by the algebraic
variety structure of the fibers $J(T_{x}M)$ of $Z(M)\to M$ explained in
Section 2. At a point $J_{x}$ in the fiber over $x\in M$ the complex structure
on $T^{h}_{J_{x}}Z(M)=T_{x}M$ is given222The Eells–Salamon almost complex
structure is obtained replacing the vertical component of the
Atiyah–Hitchin–Singer almost complex structure by its negative; see [Sa84,
Section 3] for details. by $J_{x}$.

A spinc structure on $M$ [LM89, Appendix D] yields a complex spinor bundle
$S_{\mathbb{C}}(M)\to M$ together with an action

$\mathbb{C}l(TM)\otimes S_{\mathbb{C}}(M)\to S_{\mathbb{C}}(M).$

The discussion in the previous section then provides a one-to-one
correspondence between projectivized pure spinors on $M$ and orthogonal
complex structures inducing a given orientation:

$Z_{\pm}(M)\cong\mathbb{P}(PS_{\mathbb{C}}^{\pm}(M)).$

Now suppose $M$ is given an orthogonal almost complex structure $J$ compatible
with the orientation. Then $J$ gives rise to a canonical spinc structure on
$M$ via the group homomomorphism
$U(n)\xrightarrow{\kappa}\operatorname{Spin}^{c}(2n)$, which is the unique
lift as a group homomorphism in the following diagram

${\operatorname{Spin}^{c}(2n)=(\operatorname{Spin}(2n)\times
S^{1})/\\{\pm(1,1)\\}}$${U(n)}$${SO(2n)\times
S^{1}}$$\scriptstyle{\pi\times\delta}$$\scriptstyle{\kappa}$$\scriptstyle{i\times\det}$

Here $i$ denotes the inclusion, $\pi([h,e^{i\theta}])=[h]$, and
$\delta([h,e^{i\theta}])=e^{2i\theta}$.

An element $g\in U(n)$ is diagonalized by some orthonormal basis
$(e_{1},Je_{1},\ldots,e_{n},Je_{n})$. If

$g=\mathrm{diag}(e^{i\theta_{1}},\ldots,e^{i\theta_{n}}),$

then
$\kappa(g)=\left[\prod_{k=1}^{n}\left(\cos\frac{\theta_{k}}{2}+i\sin\frac{\theta_{k}}{2}e_{k}Je_{k}\right),\
e^{i\frac{\sum\theta_{k}}{2}}\right]$ (cf. [LM89, (D.10)]). Using the formulas
for the action of $e_{i}Je_{i}$ in the proof of Lemma 2.2 we see that the
spinor action of $\kappa(g)$ on $\Lambda^{*}_{\mathbb{C}}(\mathbb{R}^{2n})$,
where $\mathbb{R}^{2n}$ is equipped with the complex structure given by
$\mathrm{diag}\left(\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right),\ldots,\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\right)\right)$, agrees with the standard action of $g$ on
this space. We obtain the following result:

######  Proposition 3.2.

Let $M$ be an oriented Riemannian manifold with an orthogonal almost complex
structure $J$ compatible with the orientation. Then we have the following
isomorphisms of smooth fiber bundles over $M$:

  1. (1)

If $\dim(M)=4$, then

$Z_{+}(M)\cong\mathbb{P}(\mathbb{C}\oplus\Lambda^{2}_{\mathbb{C}}(TM)),\quad\quad
Z_{-}(M)\cong\mathbb{P}(TM).$

  2. (2)

If $\dim(M)=6$, then

$Z_{+}(M)\cong\mathbb{P}(\mathbb{C}\oplus\Lambda^{2}_{\mathbb{C}}(TM)),\quad\quad
Z_{-}(M)\cong\mathbb{P}(TM\oplus\Lambda_{\mathbb{C}}^{3}(TM)).$

######  Remark 3.3.

The Atiyah–Hitchin–Singer and Eells–Salamon almost complex structures on
(either) twistor space do not agree with the natural almost complex structure
on the projectivization. Indeed, the projection map from the projectivization
is complex linear whilst this is not the case for the twistor projection.

The projective bundle formula now gives the following formulas for the
cohomology rings of the components of the twistor space in terms of the Chern
classes of $(TM,J)$.

######  Corollary 3.4.

Let $M$ be an oriented Riemannian manifold with an orthogonal almost complex
structure $J$ compatible with the orientation. We have the following
isomorphisms as algebras over $H^{*}(M)$:

  1. (1)

If $\dim(M)=4$, then

$\displaystyle H^{*}(Z_{+}(M))$ $\displaystyle\cong
H^{*}(M)[x]/(x^{2}+c_{1}x),$ $\displaystyle H^{*}(Z_{-}(M))$
$\displaystyle\cong H^{*}(M)[x]/(x^{2}+c_{1}x+c_{2}).$

  2. (2)

If $\dim(M)=6$, then

$\displaystyle H^{*}(Z_{+}(M))$ $\displaystyle\cong
H^{*}(M)[x]/(x^{4}+2c_{1}x^{3}+(c_{1}^{2}+c_{2})x^{2}+(c_{1}c_{2}-c_{3})x),$
$\displaystyle H^{*}(Z_{-}(M))$ $\displaystyle\cong
H^{*}(M)[x]/(x^{4}+2c_{1}x^{3}+(c_{1}^{2}+c_{2})x^{2}+(c_{1}c_{2}+c_{3})x).$

######  Remark 3.5.

  * •

The formulas above in dimension four agree with [EeSa85, Theorem 11.2], where
it is established that over a Kähler surface $M$, the positive twistor space
is the projectivization of $T^{2,0}M\oplus\mathbb{C}$ (note that
$T^{2,0}M\cong\Lambda_{\mathbb{C}}^{2}(TM)$), and the negative twistor space
is the projectivization of the holomorphic tangent bundle.

  * •

In [Ev14, Section 6], Evans describes the cohomology ring of the positive
twistor space of a (not necessarily almost complex) six–manifold, with complex
coefficients. The description is in terms of the first Chern class of
$T^{h}(Z_{+}(M))$. Using 3.10, one can check that Evans’ description agrees
with the one in 3.4 in the case of almost complex manifolds. We remark that
Evans’ computation can quickly be reproduced using naturality in the pullback
diagram

${Z_{+}(M)}$${BU(3)}$${M}$${BSO(6)}$

and expressing $H^{*}(BU(3);\mathbb{Q})$ as an
$H^{*}(BSO(6);\mathbb{Q})$-algebra using the known behavior of the right-hand
side vertical map on cohomology.

######  Remark 3.6.

In order for the formulas of 3.4 (1) to hold, it is crucial to have an almost
complex structure _inducing the given orientation_ on the base manifold. For
instance, the positive twistor space of $\overline{\mathbb{CP}^{2}}$ (i.e. the
complex projective plane with the opposite orientation) with the Fubini–Study
metric is the full flag variety $U(3)/\left(U(1)\times U(1)\times U(1)\right)$
[Sa84, p.217, Example 1], whose cohomology is given by

$\mathbb{Z}[x,y,z]/(x+y+z,xy+xz+yz,xyz)\cong\mathbb{Z}[x,y]/(x^{2}+xy+y^{2},x^{2}y+xy^{2}),$

where $x,y,z$ are in degree two [B53, Proposition 31.1]. This ring is not
isomorphic to

$\mathbb{Z}[x,y]/(y^{3},x^{2}+kyx)$

for any integer $k$, so the cohomology of $Z_{+}(\overline{\mathbb{CP}^{2}})$
can not be expressed as in the first item of 3.4 (1). This corresponds to the
fact that $\overline{\mathbb{CP}^{2}}$ does not admit an almost complex
structure compatible with its orientation (which can of course be seen by
other means as well, e.g. Hirzebruch’s congruence $\chi+\sigma\equiv 0\bmod 4$
for closed almost complex four–manifolds).

###  3.7. $\operatorname{Spin}^{c}(4)$-bundles and twistor spaces of
four–dimensional manifolds

Twistor spaces of oriented Riemannian four–manifolds give interesting examples
of almost complex six–manifolds which we will consider in the following
sections. In this sub-section we will compute their integral cohomology as
well as the Chern classes of their Atiyah–Hitchin–Singer almost complex
structures.

Our computation requires knowledge of the integral cohomology of
$B\operatorname{Spin}^{c}(4)$. The cohomology of $B\operatorname{Spin}^{c}(n)$
is described in detail in [Du18]. We need only the simple case when $n=4$, so
we include an elementary treatment of this case.

Our aim is to compute the cohomology ring of the “universal twistor space”
over $B\operatorname{Spin}^{c}(4)$, namely the bundle of orthogonal complex
structures on the fibers of the $4$–plane bundle over
$B\operatorname{Spin}^{c}(4)$ classified by the canonical map
$B\operatorname{Spin}^{c}(4)\to BSO(4)$. Since every oriented four–manifold
admits a spinc structure, naturality will yield the cohomology ring of the
twistor space.

We will use unit quaternions to describe the four–dimensional spin groups and
orthogonal complex structures. We consider the quaternions $\mathbb{H}$ as a
vector space over $\mathbb{C}$ via right multiplication by
$\mathbb{C}\subset\mathbb{H}$, i.e. we consider the identification of
$\mathbb{C}^{2}$ with $\mathbb{H}$ given by

$(z_{1},z_{2})\mapsto z_{1}+jz_{2}.$

The standard complex structure on $\mathbb{H}=\mathbb{R}^{4}$ is then right
multiplication by $i$, and $(1,i,j,-k)$ is an oriented basis.

The unit quaternion $q=w_{1}+jw_{2}$ acts on $\mathbb{H}$ by left
multiplication as

$(w_{1}+jw_{2})(z_{1}+jz_{2})=(w_{1}z_{1}-\overline{w_{2}}z_{2})+j(w_{2}z_{1}+z_{2}\overline{w_{1}})$

which corresponds via the identification with $\mathbb{C}^{2}$ to the matrix

$\left[\begin{array}[]{cc}w_{1}&-\overline{w_{2}}\\\
w_{2}&\overline{w_{1}}\end{array}\right]$

with $|w_{1}|^{2}+|w_{2}|^{2}=1$. This will be our identification of $Sp(1)$
with $SU(2)$.

The universal cover of $SO(4)$ is modelled by the map $Sp(1)\times Sp(1)\to
SO(4)$ given by

$(q_{1},q_{2})\mapsto\left(v\mapsto q_{1}v\overline{q_{2}}\right)$

and we will identify

$SO(4)=(Sp(1)\times Sp(1))/\\{\pm(1,1)\\}.$

In these terms, the subgroup $U(2)\subset SO(4)$ is

$U(2)=(Sp(1)\times S^{1})/\\{\pm(1,1)\\}.$

The group $\operatorname{Spin}^{c}(4)$ is defined as

$\operatorname{Spin}^{c}(4)=(Sp(1)\times Sp(1)\times S^{1})/\\{\pm(1,1,1)\\}.$

There are two canonical homomorphisms

(1) $SO(4)\xleftarrow{\pi}\operatorname{Spin}^{c}(4)\xrightarrow{\delta}S^{1}$

given by

$\pi([q_{1},q_{2},e^{i\theta}])=[q_{1},q_{2}],\quad\quad\delta([q_{1},q_{2},e^{i\theta}])=e^{2i\theta}.$

The canonical map $U(2)\xrightarrow{\kappa}\operatorname{Spin}^{c}(4)$ is the
unique lift as a group homomorphism in the diagram

${\operatorname{Spin}^{c}(4)}$${U(2)}$${SO(4)\times
S^{1}}$$\scriptstyle{\pi\times\delta}$$\scriptstyle{\kappa}$$\scriptstyle{i\times\det}$

In terms of the coordinates above, this is given by
$\kappa([q_{1},e^{i\theta}])=[q_{1},e^{i\theta},e^{-i\theta}]$.

Let $x\in H^{2}(BS^{1})=H^{2}(\mathbb{CP}^{\infty})$ denote the standard
generator (the Chern class of the dual tautological line bundle on
$\mathbb{CP}^{\infty}$) and

$\alpha=(B\delta)^{*}(x)\in H^{2}(B\operatorname{Spin}^{c}(4)).$

We will write $p_{1}$ and $\mathrm{e}$ respectively for the universal
Pontryagin class and Euler class in $H^{*}(BSO(4))$.

######  Lemma 3.8.

Let $\pi\colon\operatorname{Spin}^{c}(4)\to SO(4)$ be as in (1). There are
unique classes $S_{1},S_{2}\in H^{4}(B\operatorname{Spin}^{c}(4))$ such that

$\displaystyle 4S_{1}$
$\displaystyle=B\pi^{*}p_{1}-2B\pi^{*}\mathrm{e}-\alpha^{2},$ $\displaystyle
4S_{2}$ $\displaystyle=B\pi^{*}p_{1}+2B\pi^{*}\mathrm{e}-\alpha^{2}.$

Moreover the cohomology ring of $B\operatorname{Spin}^{c}(4)$ is a polynomial
algebra,

$H^{*}(B\operatorname{Spin}^{c}(4))\cong\mathbb{Z}[\alpha,S_{1},S_{2}].$

###### Proof.

The short exact sequence of groups

$Sp(1)\times
Sp(1)\xrightarrow{\iota}\operatorname{Spin}^{c}(4)\xrightarrow{\delta}S^{1},$

where $\iota(q_{1},q_{2})=[q_{1},q_{2},1]$, leads to a fiber sequence

(2) $BSp(1)\times
BSp(1)\xrightarrow{B\iota}B\operatorname{Spin}^{c}(4)\xrightarrow{B\delta}BS^{1}.$

Let $A,B\in H^{4}(BSp(1)\times BSp(1))$ be the generators which map to
$x^{2}\in H^{4}(BS^{1})$ under the maps induced by the natural inclusions
$e^{i\theta}\mapsto(e^{i\theta},1)$ and $e^{i\theta}\mapsto(1,e^{i\theta})$ of
$S^{1}$ in $Sp(1)\times Sp(1)$. By the Serre spectral sequence of the
fibration (2), any elements in $H^{4}(B\operatorname{Spin}^{c}(4))$ mapping to
$A,B$ under $B\iota^{*}$ together with $\alpha$ will freely generate the
cohomology ring of $B\operatorname{Spin}^{c}(4)$.

The effect of the composition

$BSp(1)\times
BSp(1)\xrightarrow{B\iota}B\operatorname{Spin}^{c}(4)\xrightarrow{B\pi}BSO(4)$

on degree $4$ cohomology is determined by the homomorphism

$S^{1}\times S^{1}\xrightarrow{\begin{bmatrix}1&-1\\\
-1&-1\end{bmatrix}}S^{1}\times S^{1}$

induced by $\pi\circ\iota$ on the standard maximal tori. Writing $a,b\in
H^{2}(BS^{1}\times BS^{1})$ for the standard generators corresponding to the
maximal torus of $SO(4)$, and $x,y$ for the corresponding generators for
$Sp(1)\times Sp(1)$, we have

$a^{2}+b^{2}\mapsto(x-y)^{2}+(-x-y)^{2}=2x^{2}+2y^{2},\quad\quad
ab\mapsto-x^{2}+y^{2},$

and therefore $p_{1},\mathrm{e}\in H^{4}(BSO(4))$ map to $2A+2B$ and $-A+B$ in
$H^{4}(BSp(1)\times BSp(1))$ respectively. Therefore

$B\iota^{*}B\pi^{*}(p_{1}-2\mathrm{e})=4A,\quad
B\iota^{*}B\pi^{*}(p_{1}+2\mathrm{e})=4B.$

It follows from the short exact sequence

$0\to\mathbb{Z}\alpha^{2}\to
H^{4}(B\operatorname{Spin}^{c}(4))\xrightarrow{B\iota^{*}}H^{4}(BSp(1)\times
BSp(1))\to 0$

that there exists $\lambda\in\mathbb{Z}$ such that
$B\pi^{*}(p_{1}-2\mathrm{e})+\lambda\alpha^{2}$ is (uniquely) divisible by
$4$.

The composition

$\mathbb{CP}^{2}\xrightarrow{\tau_{\mathbb{CP}^{2}}}BU(2)\xrightarrow{B\kappa}B\operatorname{Spin}^{c}(4)\xrightarrow{B\delta}BS^{1}$

classifying the second exterior power of the tangent bundle of
$\mathbb{CP}^{2}$ induces multiplication by $3$ on $H^{2}$, and hence
$(B\kappa\circ\tau_{\mathbb{CP}^{2}})^{*}$ provides a $2$–local splitting of
the map $B\delta^{*}$ on $H^{4}$.

The pullback to $H^{4}(\mathbb{CP}^{2})$ of
$B\pi^{*}(p_{1}-2\mathrm{e})+\lambda\alpha^{2}\in
H^{4}(B\operatorname{Spin}^{c}(4))$ is $(3-6+9\lambda)$ times the orientation
class, and the smallest value of $\lambda$ for which this number is a multiple
of $4$ is $\lambda=-1$.

Since $B\pi^{*}(p_{1})-2B\pi^{*}(\mathrm{e})-\alpha^{2}\in
H^{4}(B\operatorname{Spin}^{c}(4))$ is divisible by $4$, we conclude that

$B\iota^{*}\left(\frac{B\pi^{*}(p_{1})-2B\pi^{*}(\mathrm{e})-\alpha^{2}}{4}\right)=A,$

and similarly

$B\iota^{*}\left(\frac{B\pi^{*}(p_{1})+2B\pi^{*}(\mathrm{e})-\alpha^{2}}{4}\right)=B,$

which completes the proof. ∎

For four–dimensional $M$, a choice of spinc structure gives us the following
diagram of pullback squares:

${Z(M)}$${Z(B\operatorname{Spin}^{c}(4))}$${Z(BSO(4))}$${M}$${B\operatorname{Spin}^{c}(4)}$${BSO(4)}$$\scriptstyle{B\pi}$

where $Z(BG)$ denotes the bundle of orthogonal complex structures on the
universal oriented $4$–plane bundle over $BG$ for $G=SO(4)$ and its pullback
for $G=\operatorname{Spin}^{c}(4)$.

We will see that each of the two components of
$Z(B\operatorname{Spin}^{c}(4))$ is the projectivization of a complex plane
bundle over $B\operatorname{Spin}^{c}(4)$. The projective bundle formula
together with naturality will give us the following description of the
cohomology ring of $Z(M)$ as an $H^{*}(M)$-algebra.

######  Proposition 3.9.

Let $M$ be an oriented Riemannian four–manifold and $\alpha\in H^{2}(M)$ be an
integral lift of $w_{2}(M)$ (classifying the complex line bundle associated to
a spinc-structure on $M$). Then there is an isomorphism of $H^{*}(M)$-algebras

(3) $H^{*}(Z_{\pm}(M))\cong H^{*}(M)[x]/\left(x^{2}+\alpha x-\frac{p_{1}\pm
2\mathrm{e}-\alpha^{2}}{4}\right).$

###### Proof.

Under our identification of $\mathbb{R}^{4}$ with $\mathbb{H}$, the orthogonal
complex structures on $\mathbb{R}^{4}$ compatible with the orientation
(respectively, opposite orientation) are given by right (respectively, left)
multiplication by a unit imaginary quaternion. The element $[q_{1},q_{2}]\in
SO(4)$ acts on $J_{+}(\mathbb{R}^{4})$ by

$[q_{1},q_{2}]u=q_{2}u\overline{q}_{2},$

and on $J_{-}(\mathbb{R}^{4})$ by

$[q_{1},q_{2}]u=q_{1}u\overline{q}_{1}.$

That is, $SO(4)$ acts on orthogonal complex structures via the two projections
to $SO(3)$. The commutative diagrams

${\operatorname{Spin}^{c}(4)}$${U(2)}$${\left[q_{1},q_{2},e^{i\theta}\right]}$${\left[q_{2},e^{-i\theta}\right]}$${SO(4)}$${SO(3)}$${\left[q_{1},q_{2}\right]}$${\left[q_{2}\right]}$$\scriptstyle{\pi}$$\scriptstyle{\chi_{+}}$$\scriptstyle{\pi_{\mathrm{right}}}$

and

${\operatorname{Spin}^{c}(4)}$${U(2)}$${\left[q_{1},q_{2},e^{i\theta}\right]}$${\left[q_{1},e^{-i\theta}\right]}$${SO(4)}$${SO(3)}$${\left[q_{1},q_{2}\right]}$${\left[q_{1}\right]}$$\scriptstyle{\pi}$$\scriptstyle{\chi_{-}}$$\scriptstyle{\pi_{\mathrm{left}}}$

allow us to express each component of the bundle of orthogonal complex
structures associated to an oriented $4$–plane bundle with a spinc structure
as the projectivization of a complex plane bundle. The formula (3) will follow
from the projective bundle formula once we compute the Chern classes of these
plane bundles. By naturality it suffices to compute the images of
$c_{1},c_{2}$ under $B\chi_{\pm}^{*}$.

As the composition
$\operatorname{Spin}^{c}(4)\xrightarrow{\chi_{\pm}}U(2)\xrightarrow{\det}S^{1}$
equals $\delta$, we have $B\chi_{\pm}^{*}(c_{1})=\alpha$.

We will now show that

$\quad\quad B\chi_{+}^{*}(c_{2})=-S_{2},\quad\quad
B\chi_{-}^{*}(c_{2})=-S_{1}.$

In view of Lemma 3.8, this will complete the proof. The maps

$Sp(1)\times
Sp(1)\xrightarrow{\iota}\operatorname{Spin}^{c}(4)\xrightarrow{\chi_{\pm}}U(2)$

are given respectively by $(q_{1},q_{2})\mapsto[q_{2},1]$ and
$(q_{1},q_{2})\mapsto[q_{1},1]$ (i.e. they correspond respectively to the
inclusions of the right and left copies of $SU(2)$ in $U(2)$). Therefore, with
respect to the standard basis for $H^{4}(BSp(1)\times BSp(1))$ used in the
proof of Lemma 3.8, we have $B(\iota\circ\chi_{+})^{*}c_{2}=-B$ and
$B(\iota\circ\chi_{-})^{*}c_{2}=-A$.

Hence

$B\chi_{+}^{*}(c_{2})=-S_{2}+\lambda\alpha^{2},\quad\quad
B\chi_{-}^{*}(c_{2})=-S_{2}+\mu\alpha^{2},\quad\text{ for some
}\lambda,\mu\in\mathbb{Z}.$

We can determine the coefficients $\lambda$ and $\mu$ by mapping to
$H^{4}(\mathbb{CP}^{2})$ under
$\mathbb{CP}^{2}\xrightarrow{B\kappa\circ\tau_{\mathbb{CP}^{2}}}B\operatorname{Spin}^{c}(4)$.
The compositions

$U(2)\xrightarrow{\kappa}\operatorname{Spin}^{c}(4)\xrightarrow{\chi_{\pm}}U(2)$

are given by

$[q,e^{i\theta}]\mapsto[e^{i\theta},e^{i\theta}]\quad\text{ and
}\quad[q,e^{i\theta}]\mapsto[q,e^{i\theta}]$

respectively. The second map is the identity, while the first map is the
representation $U(2)\xrightarrow{1\oplus\det}U(2)$ (cf. 3.2 (1)). It follows
that the composition

$\mathbb{CP}^{2}\to B\operatorname{Spin}^{c}(4)\xrightarrow{B\chi_{+}}BU(2)$

classifies the bundle $\mathbb{C}\oplus\Lambda^{2}T\mathbb{CP}^{2}$, and

$\mathbb{CP}^{2}\to B\operatorname{Spin}^{c}(4)\xrightarrow{B\chi_{-}}BU(2)$

classifies the tangent bundle of $\mathbb{CP}^{2}$. Thus $c_{2}\in
H^{4}(BU(2))$ must go to $0\in H^{4}(\mathbb{CP}^{2})$ under the composition
$\mathbb{CP}^{2}\to B\operatorname{Spin}^{c}(4)\xrightarrow{B\chi_{+}}BU(2)$,
and to $3\in H^{4}(\mathbb{CP}^{2})$ under the composition $\mathbb{CP}^{2}\to
B\operatorname{Spin}^{c}(4)\xrightarrow{B\chi_{-}}BU(2)$, i.e.

$\displaystyle-(3+6-9)/4+9\lambda=0\quad\Leftrightarrow\quad\lambda=0,\quad\mathrm{and}\quad-(3-6-9)/4+9\mu$
$\displaystyle=3\quad\Leftrightarrow\quad\mu=0.\qed$

Recall that $T^{v}Z_{\pm}(M)$ and $T^{h}Z_{\pm}(M)$ denote the vertical and
horizontal subbundles of $TZ_{\pm}(M)$ which are complex vector bundles via
the Atiyah–Hitchin–Singer almost complex structure.

######  Proposition 3.10.

Let $M$ be an oriented Riemannian four–manifold and $\alpha\in H^{2}(M)$ a
choice of lift of $w_{2}(M)$. Then, in terms of the expression for
$H^{*}(Z_{\pm}(M))$ in 3.9, the total Chern class of the standard almost
complex structure on $Z_{\pm}(M)$ is

(4) $1+(4x+2\alpha)+(p_{1}\pm 3\mathrm{e})\pm(\alpha+2x)\mathrm{e}.$

Moreover, $c_{1}(T^{v}Z_{\pm}(M))=c_{1}(T^{h}Z_{\pm}(M))=\alpha+2x$.

###### Proof.

It suffices to compute the total Chern classes of the vertical and horizontal
sub-bundles of $TZ_{\pm}(M)$. We saw in the proof of 3.9 that $Z_{\pm}(M)$ is
the projectivization of a rank two complex vector bundle $E$ with total Chern
class

$1+\alpha-\frac{p_{1}\pm 2\mathrm{e}-\alpha^{2}}{4}.$

As $T^{v}(\mathbb{P}(E))\oplus\mathbb{C}\cong\mathcal{O}_{E^{*}}(1)\otimes
p^{*}E$ (where $\mathbb{P}(E)\xrightarrow{p}M$ is the projection and
$\mathcal{O}_{E^{*}}(1)$ is the canonical line bundle restricting to
$\mathcal{O}(1)$ on each fiber [D12, Definition 15.13]) we have that

$c_{1}(T^{v}(Z_{\pm}(M)))=c_{1}(E)+2c_{1}(\mathcal{O}_{E^{*}}(1))=\alpha+2x.$

As for the horizontal bundle, consider the pullback diagrams

${Z_{+}(M)}$${Z_{+}(B\operatorname{Spin}^{c}(4))}$${Z_{+}(BSO(4))=BU(2)}$${M}$${B\operatorname{Spin}^{c}(4)}$${BSO(4)}$$\scriptstyle{\tau}$$\scriptstyle{Bi}$$\scriptstyle{B\pi}$

Note that $Z_{+}(BSO(4))=ESO(4)\times_{SO(4)}SO(4)/U(2)=BU(2)$, and under this
identification the projection $Z_{+}(BSO(4))\to BSO(4)$ is the map $Bi$
induced by the inclusion $U(2)\subset SO(4)$. Moreover, the universal bundle
$ESO(4)\times_{U(2)}\mathbb{R}^{4}\to ESO(4)/U(2)$ pulls back to
$T^{h}Z_{+}(M)$ under the composite map $Z_{+}(M)\to
Z_{+}(B\operatorname{Spin}^{c}(4))\xrightarrow{\tau}BU(2)$ (cf. Lemma 2.1).

We saw in the proof of 3.9 that

$H^{*}(Z_{+}(B\operatorname{Spin}^{c}(4)))\cong\mathbb{Z}[\alpha,S_{1},S_{2},x]/(x^{2}+\alpha
x-S_{2}).$

In terms of this identification we have
$\tau^{*}(c_{2})=B\pi^{*}\mathrm{e}=-S_{1}+S_{2}$ and

$\tau^{*}(c_{1})=\gamma x+\lambda\alpha\text{ for some
}\gamma,\lambda\in\mathbb{Z}.$

As the bundle classified by $\mathbb{CP}^{1}\cong SO(4)/U(2)\to BU(2)$ has
$c_{1}$ equal to two times the orientation class (see Lemma 2.1), we have
$\gamma=2$. Since $p_{1}\in H^{4}(BSO(4))$ maps to $c_{1}^{2}-2c_{2}$ in
$H^{4}(BU(2))$, and to $2S_{1}+2S_{2}+\alpha^{2}$ in
$H^{4}(B\operatorname{Spin}^{c}(4))$ we see that

$(2x+\lambda\alpha)^{2}+2S_{1}-2S_{2}=2S_{1}+2S_{2}+\alpha^{2}\quad\Leftrightarrow\quad
4x^{2}+4\lambda\alpha x+\lambda^{2}\alpha^{2}=4S_{2}+\alpha^{2}.$

Hence $\lambda=1$, yielding the formula (4) in the case of $Z_{+}(M)$.

Let $k\in O(4)$ be an element with determinant $-1$ and let $\varphi\colon
SO(4)\to SO(4)$ denote conjugation by $k$. As a homogeneous space of $SO(4)$
we have $J_{-}(\mathbb{R}^{4})=SO(4)/\varphi(U(2))$. Hence
$Z_{-}(BSO(4))=ESO(4)\times_{SO(4)}\varphi(U(2))=B(\varphi(U(2))$ and we have
a commutative diagram

${Z_{-}(M)}$${Z_{-}(B\operatorname{Spin}^{c}(4))}$${B(\varphi(U(2))}$${BU(2)}$${M}$${B\operatorname{Spin}^{c}(4)}$${BSO(4)}$${BSO(4)}$$\scriptstyle{\tau^{\prime}}$$\scriptstyle{\tau}$$\scriptstyle{B\varphi^{-1}}$$\scriptstyle{B\pi}$$\scriptstyle{B\varphi^{-1}}$

with the middle and left-hand squares both pullback squares and the right hand
square an isomorphism of fiber bundles induced by the automorphism
$\varphi^{-1}$. Note that $B\varphi^{-1}$ is covered by an isomorphism between
the (complex) universal bundles $ESO(4)\times_{\varphi(U(2))}\mathbb{R}^{4}\to
ESO(4)\times_{U(2)}\mathbb{R}^{4}$, so that the composition $Z_{-}(M)\to
Z_{-}(B\operatorname{Spin}^{c}(4))\xrightarrow{\tau}BU(2)$ classifies
$T^{h}Z_{-}(M)$.

Considering the action of $B\varphi$ on the maximal torus we see that
$B\varphi\colon BSO(4)\to BSO(4)$ has the following effect on cohomology:

$B\varphi^{*}(p_{i})=p_{i},\quad B\varphi^{*}(\mathrm{e})=-\mathrm{e}.$

Arguing as before, we conclude that
$\tau^{*}(c_{2})=-B\pi^{*}(\mathrm{e})=S_{1}-S_{2}$ and
$\tau^{*}(c_{1})=2x+\alpha$, leading to the formula (4) in the case of
$Z_{-}(M)$. ∎

######  Remark 3.11.

  * •

The formulas in 3.10 for $c_{2}$ and $c_{3}$ of $Z_{-}(M)$ differ by a sign
from those given in [Hi81, p.135]; however, the fundamental class being used
in loc. cit. seems to also differ by a sign from the one induced by the
Atiyah–Hitchin–Singer almost complex structure on $Z_{-}(M)$. Hence the values
of the Chern numbers given in [Hi81, (1.5)] coincide with those obtained with
the Chern classes in 3.10.

  * •

In [Hi81, (1.4)], there is a description of the real cohomology ring of the
negative twistor space of a four–manifold $X$, as the free
$H^{*}(X;\mathbb{R})$–module generated by $h=\frac{1}{2}c_{1}(T^{v}Z_{-}(X))$
subject only to the relation
$h^{2}=\tfrac{1}{2}\mathrm{e}(X)-\tfrac{1}{4}p_{1}(X)$. However, this
description is at odds with the identification of the negative twistor space
of $\mathbb{CP}^{2}$ as the full flag variety $U(3)/\left(U(1)\times
U(1)\times U(1)\right)$ [Hi81, p.133], [Sa84, p.217, Example 1]. Namely,
$H^{*}(\mathbb{CP}^{2};\mathbb{R})\cong\mathbb{R}[x]/(x^{3})$, where $\langle
x^{2},[\mathbb{CP}^{2}]\rangle=1$, so
$\mathrm{e}(\mathbb{CP}^{2})=p_{1}(\mathbb{CP}^{2})=3x^{2}$. Therefore the
real cohomology of the negative twistor space is, according to [Hi81, (1.4)],
$\mathbb{R}[x,h]/(x^{3},h^{2}-\tfrac{3}{4}x^{2})$. One can check directly,
however, that this (graded) ring is not isomorphic to the cohomology of the
flag variety, i.e. to $\mathbb{R}[x,h]/(x^{2}+xh+h^{2},x^{2}h+hx^{2})$.

In terms of 3.9 and 3.10, we have $h=x+\frac{\alpha}{2}$ and therefore
$h^{2}=\frac{1}{4}p_{1}(x)-\frac{1}{2}\mathrm{e}(X)$.

######  Remark 3.12.

Let $M$ be an almost complex Riemannian six–manifold. The fiberwise
diffeomorphism $a\colon Z_{+}(M)\to Z_{-}(M)$ sending $J_{x}\mapsto-J_{x}\in
Z(T_{x}M)$ induces an anti-holomorphic map between the fibers and hence

$a^{*}(c_{1}(T_{v}(Z_{-}(M))))=-c_{1}(T_{v}(Z_{+}(M))).$

The spaces $Z_{\pm}(M)$ are projectivizations of rank 4 complex vector bundles
with first Chern class $2c_{1}(M)$ (see 3.4), and so as in 3.10 we see that
$c_{1}(T_{v}Z_{\pm}(M))=2c_{1}(M)+4x$. Therefore

$a^{*}(2c_{1}+4x)=-2c_{1}-4x\Rightarrow a^{*}(4x)=-4x-4c_{1}.$

As there is no torsion in $H^{2}$ of the universal example $Z(BU(3))$, it
follows that $a^{*}(x)=-x-c_{1}$. One can check that $a^{*}$ yields an
isomorphism between the $H^{*}(M)$-algebras of 3.4.

In view of the previous remark, from now on we will restrict our attention to
the positive twistor space of almost complex Riemannian six–manifolds.

##  4\. On the homotopy type of the space of almost complex structures on
six–manifolds

In this section we study the homotopy type of the components of the space of
almost complex structures on connected six-manifolds $M$, complementing our
previous treatment [FGM21] of the case when $c_{1}=0$.

######  Theorem 4.1.

Let $M$ be an oriented Riemannian manifold of dimension four or six, $J$ an
orthogonal almost complex structure on $M$ compatible with the orientation,
and $J(M)$ its component in the space of orthogonal almost complex structures.
Let $S_{\mathbb{C}}^{+}(M)$ be the positive spinor bundle on $M$ determined by
$J$. Then the map

$\Gamma(S(S_{\mathbb{C}}^{+}(M)))\to J(M)$

is surjective, and for any lift $s$ of $J$,

$\operatorname{Map}(M,S^{1})\to\Gamma(S(S_{\mathbb{C}}^{+}))_{s}\to J(M)$

is a fiber sequence.

###### Proof.

Identifying $S_{\mathbb{C}}^{+}(M)$ with $\oplus_{k\text{
even}}\Lambda^{k}_{\mathbb{C}}(TM)$, Remark 2.3 shows that the constant
section $s\colon M\to\Lambda^{0}_{\mathbb{C}}(TM)\subset
S_{\mathbb{C}}^{+}(M)$ defined by $s(x)=1$ lifts the section $J$. As
$S(S_{\mathbb{C}}^{+})\to Z_{+}(M)$ is a fiberwise fibration over $M$, the map
induced on sections is a fibration. Since the fiber over $J$ is non-empty, any
choice of lift of $J$ identifies the fiber with $\operatorname{Map}(M,S^{1})$.
∎

######  Remark 4.2.

The above argument remains valid in all dimensions if one replaces
$S(S_{\mathbb{C}}^{+}(M))$ with $S(S_{\mathbb{C}}^{+}(M))\cap
PS^{+}_{\mathbb{C}}(M)$.

Now let $(M,J)$ be a closed almost complex Riemannian six–manifold with
$H^{1}(M;\mathbb{Z})=0$ satisfying $c_{1}c_{2}-c_{3}\neq 0$. By 4.1, we have
the fibration $\operatorname{Map}(M,S^{1})\to\Gamma(S(S_{C}^{+}(M)))_{s}\to
J(M)$. Since $H^{1}(M;\mathbb{Z})=0$, evaluation at a chosen basepoint gives a
homotopy equivalence $\operatorname{Map}(M,S^{1})\xrightarrow{ev}S^{1}$, so we
have a principal fiber sequence $S^{1}\to\Gamma(S(S_{C}^{+}(M)))_{s}\to J(M)$.

We can thus consider instead the fiber sequence

(5) $\Gamma(S(S_{\mathbb{C}}^{+}))_{s}\to J(M)\to
BS^{1}\simeq\mathbb{CP}^{\infty}.$

Now, $S(S_{\mathbb{C}}^{+})$ is an oriented fiber bundle over $M$ with fiber
$S^{7}$, and hence it is classified by a map $M\to BAut^{+}(S^{7})$ to the
classifying space of orientation-preserving homotopy automorphisms of $S^{7}$.
It is known that $BAut^{+}(S^{7})_{\mathbb{Q}}\simeq K(\mathbb{Q},8)$, where
the subscript of $\mathbb{Q}$ denotes (Sullivan) rationalization [S77, §11].
By [M87, Theorem 5.3], the rationalization of
$\Gamma(S(S_{\mathbb{C}}^{+}))_{s}$ is homotopy equivalent to the space of
sections of the fiberwise rationalized $S^{7}$ bundle over $M$. Since
$H^{8}(M;\mathbb{Q})=0$, this latter bundle is trivial, and hence
$\Gamma(S_{\mathbb{C}}^{+})_{\mathbb{Q}}\simeq\operatorname{Map}(M,S^{7}_{\mathbb{Q}})$.
Denoting the Betti numbers of $M$ by $b_{i}$, by Thom’s theorem on the space
of maps into an Eilenberg–Maclane space, the latter has the homotopy type of
$S^{1}_{\mathbb{Q}}\times S^{7}_{\mathbb{Q}}\times
K(\mathbb{Q},3)^{b_{4}}\times K(\mathbb{Q},4)^{b_{3}}\times
K(\mathbb{Q},5)^{b_{2}}$.

We can describe the fundamental group of $J(M)$ by using a theorem of Crabb
and Sutherland:

######  Theorem 4.3.

[CS84, Theorem 2.12(i)] Let $M$ be a closed connected $2n$–manifold and $\xi$
a complex $(n+1)$–plane bundle over $X$. Denote by $N\xi$ any component of the
space of sections of $\mathbb{P}\xi$ whose elements lift to sections of $\xi$.
Then $\pi_{1}(N\xi)$ is a central extension

$0\to\mathbb{Z}/\left(\langle c_{n}(\xi),[M]\rangle\right)\to\pi_{1}(N\xi)\to
H^{1}(M)\to 0.$

By 4.1 we can apply this to
$\xi=S_{\mathbb{C}}^{+}(M)\cong\mathbb{C}\oplus\Lambda_{\mathbb{C}}^{2}TM$,
which satisfies $N\xi=J(M)$. Therefore, the fundamental group of $J(M)$ is
given by $\mathbb{Z}$ modulo $\int_{M}c_{1}c_{2}-c_{3}$ (where
$c_{i}=c_{i}(TM)$), see 3.4. By [M87], $J(M)$ is a nilpotent space as it is
the space of sections of a fibration with nilpotent fiber (namely
$\mathbb{CP}^{3}$) over a finite–dimensional base. We may thus rationalize the
fibration (5) to obtain the fibration

$(\Gamma(S(S_{\mathbb{C}}^{+}))_{s})_{\mathbb{Q}}\to J(M)_{\mathbb{Q}}\to
K(\mathbb{Q},2).$

Consider the degree one rational class corresponding to the factor
$S^{1}_{\mathbb{Q}}$ in the fiber. If $c_{1}c_{2}-c_{3}\neq 0$, then since
$\pi_{1}(J(M)_{\mathbb{Q}})=0$, this class must hit (a non-zero multiple of)
the degree two generator in $K(\mathbb{Q},2)$ in the Serre spectral sequence.
This tells us that a model for $J(M)$ is of the form

$(\Lambda(x_{2},z_{1},z_{7},z_{3}^{i},z_{4}^{j},z_{5}^{k}),dz_{1}=x_{2},dz_{7}\in(x_{2}),dz_{3}^{i}\in(x_{2}),dz_{4}^{j}\in(x_{2}),dz_{5}^{k}\in(x_{2})),$

where $(x_{2})$ denotes the ideal generated by $x_{2}$, and $i,j,k$ range over
sets of size $b_{4},b_{3},b_{2}$ respectively.

Notice that such a model is not minimal, and in fact a minimal model is
obtained by quotienting out the differential ideal generated by $z_{1}$.
Indeed, by the argument in [VS76, Proposition 2] we see that the map
$(\Lambda,d)\to(\Lambda/(z_{1},dz_{1})\Lambda,\bar{d})$ induces an isomorphism
on cohomology. Here by $(\Lambda,d)$ we denote the differential graded algebra
displayed above, and by $\bar{d}$ the induced differential on the quotient. We
see that

$(\Lambda/(z_{1},x_{2})\Lambda,\bar{d})\cong(\Lambda(z_{7},z_{3}^{i},z_{4}^{j},z_{5}^{k}),\bar{d}=0),$

where the right-hand side is minimal. To summarize, we have the following:

######  Theorem 4.4.

Let $M$ be a connected closed six–manifold with $b_{1}=0$ equipped with an
almost complex structure $J$ such that $\int_{M}c_{1}c_{2}-c_{3}\neq 0$. Then
the space of almost complex structures on $M$ in the component of $J$ is a
nilpotent space with finite cyclic fundamental group and minimal model given
by $(\Lambda(z_{7},z_{3}^{i},z_{4}^{j},z_{5}^{k}),d=0)$; here $i,j,k$ range
over sets of size $b_{4}(M),b_{3}(M),b_{2}(M)$ ($=b_{4}(M)$) respectively. In
particular, this space is formal. ∎

We emphasize that the above makes no nilpotency assumption on $M$.
Alternatively, a model for (a given component of) the space of almost complex
structures can be obtained with the Haefliger–Sullivan model for the space of
sections of a fibration [S77, §11], [H82], applied to $\mathbb{CP}^{3}\to
Z_{+}(M)\to M$. Our approach above lets one quickly describe the minimal model
of $J(M)$ using the geometry of the setup.

######  Example 4.5.

Using 4.4, we immediately obtain the following:

  1. (1)

Observe that the space of almost complex structures inducing a given
orientation on the connected sum $g(S^{3}\times S^{3})$ is connected. By 4.4,
for $g\neq 1$, the rationalization of this space is $K(\mathbb{Q},7)\times
K(\mathbb{Q},4)^{2g}$. In particular, the space of almost complex structures
on $S^{6}$ has the same rational homotopy type as $S^{7}$, i.e.
$K(\mathbb{Q},7)$; cf. [FGM21] where it is shown that a certain natural
inclusion $\mathbb{RP}^{7}\hookrightarrow J(S^{6})$ induces an isomorphism on
rational homotopy groups and fundamental groups. (Note that the covering
$S^{7}\to\mathbb{RP}^{7}$ induces an isomorphism on rational homotopy groups.)

For $g=1$, one can calculate directly using the Haefliger–Sullivan model that
the space of almost complex structures on $S^{3}\times S^{3}$ has the same
rational homotopy type as
$S^{1}\times\mathbb{CP}^{3}\times(\mathbb{HP}^{\infty})^{2}$, cf. [FGM21,
Example 5.4].

  2. (2)

The components of almost complex structures on $\mathbb{CP}^{3}$ are
parametrized by $c_{1}=2kx$, $c_{2}=(2k^{2}-2)x^{2}$, where $x$ is the
generator of $H^{2}(\mathbb{CP}^{3})$ such that $\langle
x^{3},[\mathbb{CP}^{3}]\rangle=1$ (i.e. $x=c_{1}(\mathcal{O}(1))$ for the
standard complex structure). Hence $c_{1}c_{2}-c_{3}=4k^{3}-4k-4$, which is
never zero. Hence every component of almost complex structures has
rationalization $K(\mathbb{Q},7)\times K(\mathbb{Q},5)\times K(\mathbb{Q},3)$.
Contrast this with the trivial $\mathbb{CP}^{3}$ bundle over
$\mathbb{CP}^{3}$, whose (infinitely many) components of sections exhibit two
distinct rational homotopy types [MR85, Example 3.4]; the space of maps
$\mathbb{CP}^{3}\to\mathbb{CP}^{3}$ homotopic to any fixed essential map has
rationalization $K(\mathbb{Q},7)\times K(\mathbb{Q},5)\times K(\mathbb{Q},3)$.

##  5\. Intersections of almost complex structures on six–manifolds as
sections of the twistor space

We will now use the results of Section 3 to calculate the homological
intersection of two almost complex structures on a given six–manifold $M$,
after identifying them with the image of $M$ under the corresponding section
of the twistor bundle.

######  Lemma 5.1.

Let $E\xrightarrow{\pi}M$ be a complex vector bundle of rank $n$ over a closed
oriented manifold $M$. Let $s\colon M\to\mathbb{P}(E\oplus\mathbb{C})$ denote
the canonical section given pointwise by $[0:\cdots:0:1]$. Then the Poincaré
dual of $s_{*}[M]$ in $H^{*}(\mathbb{P}(E\oplus\mathbb{C}))\cong
H^{*}(M)[x]/(x^{n+1}+c_{1}x^{n}+\cdots+c_{n}x)$ is

$x^{n}+c_{1}x^{n-1}+\cdots+c_{n},$

where $c_{i}$ denote the Chern classes of $E$.333Here the orientation on
$\mathbb{P}(E\oplus\mathbb{C})$ is understood to be the one induced by the
orientation on $M$ and the canonical orientation on the fiber
$\mathbb{CP}^{n}$ corresponding to $\langle x^{n},[\mathbb{CP}^{n}]\rangle=1$.

###### Proof.

The Poincaré dual to $s_{*}[M]$ is the image of the Thom class of the normal
bundle to the submanifold $s(M)\subset\mathbb{P}(E\oplus\mathbb{C})$ under the
first map below in the long exact sequence of a pair,

$H^{2n}(\mathbb{P}(E\oplus\mathbb{C}),\mathbb{P}(E\oplus\mathbb{C})\setminus
s(M))\to H^{2n}(\mathbb{P}(E\oplus\mathbb{C}))\to
H^{2n}(\mathbb{P}(E\oplus\mathbb{C})\setminus s(M)),$

where
$H^{2n}(\mathbb{P}(E\oplus\mathbb{C}),\mathbb{P}(E\oplus\mathbb{C})\setminus
s(M))$ has been identified with $H^{2n}(U,U\setminus s(M))$ for some tubular
neighborhood $U$ of $s(M)$ via excision; see [Br97, §6.11]. This is a non-zero
class, as $s_{*}[M]$ is non-zero since it projects to $[M]$.

Now notice that $\mathbb{P}(E\oplus\mathbb{C})\setminus s(M)$ deformation
retracts to $\mathbb{P}(E)$. Therefore the effect of the second map on
cohomology is the natural quotient

$H^{2n}(M)[x]/(x^{n+1}+c_{1}x^{n}+\cdots+c_{n}x)\to
H^{2n}(M)[x]/(x^{n}+c_{1}x^{n-1}+\cdots+c_{n}),$

and thus the kernel is generated by $x^{n}+c_{1}x^{n-1}+\cdots+c_{n}$. Hence
the Thom class maps to $x^{n}+c_{1}x^{n-1}+\cdots+c_{n}$ up to sign. To check
the sign, we restrict to a fiber and evaluate against the fundamental class
$[\mathbb{CP}^{n}]$. Since the Thom class is the orientation of the normal
bundle, which is given by the orientation of the fibers, and $\langle
x^{n},[\mathbb{CP}^{n}]\rangle=1$, we see that the sign is positive. ∎

######  Theorem 5.2.

Let $s$ be a section of the positive twistor bundle over a closed Riemannian
six–manifold $M$, and give $M$ the orientation determined by $s$. Then the
homological self-intersection number $s_{*}[M]\cdot s_{*}[M]$ of $s(M)$ in
$Z_{+}(M)$ is given by $\int_{M}c_{1}c_{2}-c_{3}$, where the $c_{i}$ are the
Chern classes of the almost complex structure determined by $s$.

###### Proof.

In terms of the identification in 3.2, $s$ is the canonical section of
$\mathbb{P}(S_{\mathbb{C}}^{+})=\mathbb{P}(\Lambda_{\mathbb{C}}^{2}TM\oplus\mathbb{C})$
(see the proof of 4.1), where $TM$ is equipped with the complex structure
determined by $s$. To compute the homological self-intersection of $s(M)$, we
integrate the square of the expression obtained in Lemma 5.1 for its Poincaré
dual in $Z_{+}(M)$. We use 3.4, and observe that since
$x(x^{3}+c_{1}(S_{\mathbb{C}}^{+})x^{2}+c_{2}(S_{\mathbb{C}}^{+})x+c_{3}(S_{\mathbb{C}}^{+}))=0$
in $H^{*}(Z_{+}(M))$, we have

$\displaystyle(x^{3}+c_{1}(S_{\mathbb{C}}^{+})x^{2}+c_{2}(S_{\mathbb{C}}^{+})x+c_{3}(S_{\mathbb{C}}^{+}))^{2}$
$\displaystyle=c_{3}(S_{\mathbb{C}}^{+})(x^{3}+c_{1}(S_{\mathbb{C}}^{+})x^{2}+c_{2}(S_{\mathbb{C}}^{+})x+c_{3}(S_{\mathbb{C}}^{+}))$
$\displaystyle=c_{3}(S_{\mathbb{C}}^{+})x^{3}=(c_{1}c_{2}-c_{3})x^{3}.\qed$

Using 5.2, we obtain the following:

######  Example 5.3.

  1. (1)

For any closed oriented Riemannian four–manifold $M$, by 3.10, we have the
following Chern numbers of the Atiyah–Hitchin–Singer almost complex structure
on $Z_{+}(M)$:

$\displaystyle\int_{Z_{+}(M)}c_{1}(Z_{+}(M))c_{2}(Z_{+}(M))$
$\displaystyle=\int_{Z_{+}(M)}(4x+2\alpha)(p_{1}+3\mathrm{e})$
$\displaystyle=\left(\int_{\mathbb{CP}^{1}}4x\right)\left(\int_{M}(p_{1}(M)+3\mathrm{e}(M))\right)=12(\sigma(M)+\chi(M)),$
$\displaystyle\int_{Z_{+}(M)}c_{3}(Z_{+}(M))$
$\displaystyle=\int_{Z_{+}(M)}(2x+\alpha)\mathrm{e}=2\chi(M).$

Hence the self-intersection of the Atiyah–Hitchin–Singer almost complex
structure on $Z_{+}(M)$ in $Z_{+}(Z_{+}(M))$ is $12\sigma(M)+10\chi(M)$.

By [Ar97, Lemma 3], closed oriented four–manifolds admitting a maximally non-
integrable almost complex structure satisfy $5\chi+6\sigma=0$. Hence for these
manifolds, the corresponding self-intersection number is zero.

  2. (2)

By a theorem of Taubes [T92, Theorem 1.1], given any closed oriented
four–manifold $M$, for every large enough $k$ the manifold
$N=\overline{M}\\#k\mathbb{CP}^{2}$ carries a self-dual metric. Hence its
negative twistor space $X=Z_{-}(N)$ is a complex manifold [AHS78, Theorem
4.1]. It satisfies

$\displaystyle\int_{X}c_{1}(X)c_{2}(X)=12(\chi(N)-\sigma(N)),\quad\int_{X}c_{3}(X)=2\chi(N).$

Indeed, we obtain this again from 3.10; namely

$\displaystyle\int_{X}c_{1}(X)c_{2}(X)$
$\displaystyle=\int_{X}2(\alpha+2x)(p_{1}(N)-3\mathrm{e}(N))=\left(\int_{\mathbb{CP}^{1}}4x\right)\left(\int_{\overline{N}}(p_{1}(N)-3\mathrm{e}(N))\right)$
$\displaystyle=-12(\sigma(N)-\chi(N)).$

This agrees with [Hi81, (1.5)]. Alternatively, from [EaSi93, Section 4] we see
that the holomorphic Euler characteristic of $X$ is given by
$1-b_{1}(N)+b_{2}^{+}(N)$. Since
$\chi(N)=2+b_{2}^{+}(N)+b_{2}^{-}(N)-2b_{1}(N)$ and
$\sigma(N)=b_{2}^{+}(N)-b_{2}^{-}(N)$, we have
$\tfrac{1}{2}(\chi(N)-\sigma(N))=1-b_{1}+b_{2}^{-}$, which equals the Todd
genus $\int_{X}\tfrac{1}{24}c_{1}c_{2}(X)$. The Euler characteristic of $X$ is
twice that of $N$, as seen immediately e.g. using the Leray–Hirsch theorem.

From here we obtain

$\int_{X}c_{1}(X)c_{2}(X)-c_{3}(X)=10\chi(M)+12\sigma(M)-2k.$

Hence for any $M$, for large enough $k$ we obtain compact complex threefolds
$X$ with negative homological self-intersection in their own positive twistor
space.

######  Example 5.4.

  1. (1)

For the connected sum $g(S^{3}\times S^{3})$, the self-intersection number of
any almost complex structure in $Z_{+}(g(S^{3}\times S^{3}))$ is $2g-2$. In
particular, for $S^{6}$ we have $-2$.

  2. (2)

$S^{2}\times S^{4}$ admits a unique homotopy class of almost complex
structures for every choice of $c_{1}=2k\alpha$, where $\alpha$ generates
$H^{2}$. Since $p_{1}=0$ we see that $c_{2}=0$ for any $k\in\mathbb{Z}$, and
so the self-intersection is $-4$ for any homotopy class of almost complex
structure. Likewise, for $g\left(S^{2}\times S^{4}\right)$, the self-
intersection of any almost complex structure is $-2g-2$.

  3. (3)

Recall that the homotopy classes of almost complex structures on
$\mathbb{CP}^{3}$ are parametrized by $c_{1}=2kx$, $c_{2}=(2k^{2}-2)x^{2}$.
The self-intersection numbers are given by $4k^{3}-4k-4$; in particular, for
the standard complex structure it is 20.

  4. (4)

To obtain more examples of compact complex threefolds with negative self-
intersection, one can blow up any given compact complex threefold $X$ at a
point sufficiently many times. Indeed, the blowup at one point, diffeomorphic
to $X\\#\overline{\mathbb{CP}^{3}}$, satisfies
$\int_{X\\#\overline{\mathbb{CP}^{3}}}c_{1}c_{2}(X\\#\overline{\mathbb{CP}^{3}})=\int_{X}c_{1}c_{2}(X)$
by invariance of the Todd genus under bimeromorphisms, and the Euler
characteristic is two larger than that of $X$. Note that if $X$ is Kähler,
this gives examples of Kähler threefolds with negative self-intersection.

  5. (5)

In [Le99], LeBrun builds examples of compact complex threefolds $X_{m}$,
$m>0$, each diffeomorphic to $K3\times S^{2}$, with
$\int_{X_{m}}c_{1}c_{2}(X_{m})=48m$. Hence the homological self-intersection
number is given by $48(m-1)$.

  6. (6)

By [GS14, Corollary 3.1], for a connected symplectic six–manifold $M$ with a
Hamiltonian circle action with isolated fixed points, we have
$\int_{M}c_{1}c_{2}=24$ and so $\int_{M}c_{1}c_{2}-c_{3}=24-\chi(M)$. In
particular, this formula applies to toric six–manifolds.

Recall the following theorem of Michelsohn, Salamon, Atiyah–Hitchin–Singer:

######  Theorem 5.5.

([LM89, Theorem 9.11], [Sa95, Theorem 1]) A section $s$ of the positive
twistor space $Z_{+}(M)\to M$ is pseudoholomorphic, with respect to the almost
complex structure $J$ corresponding to $s$ on $M$ and the
Atiyah–Hitchin–Singer almost complex structure on $Z_{+}(M)$, if and only if
$J$ is integrable.

Now suppose we have an integrable orthogonal $J$ on a Riemannian six–manifold
$M$. Then $s$ gives us an embedding of $M$ into $Z_{+}(M)$ as an almost
complex submanifold. If we were able to perturb $s(M)$ to another almost
complex submanifold that intersects $s(M)$ transversally, then the homological
intersection number $s_{*}[M]\cdot s_{*}[M]$ would have to be $\geq 0$. A way
to obtain perturbations of $s(M)$ to almost complex submanifolds is by
translating $J$ by an isometry of $(M,g)$, but any perturbation to a
transverse representative would do. Note that it is unique to dimension six
that $s_{*}[M]\cdot s_{*}[M]$ inside $Z_{+}(M)$ gives an integer.

######  Corollary 5.6.

Suppose a closed Riemannian six–manifold $M$ carries an integrable complex
structure, corresponding to a section $s$ of $Z_{+}(M)$, with
$\int_{M}c_{1}c_{2}-c_{3}<0$. Then there is no representative of the homology
class $s_{*}[M]$ given by an almost complex submanifold of $Z_{+}(M)$ which
intersects $s(M)$ transversally.

######  Example 5.7.

Consider $S^{6}$ with the round metric. Take the canonical almost complex
structure induced by the octonions, and consider its orbit under the action of
the isometry group $SO(7)$, diffeomorphic to $\mathbb{RP}^{7}$. After
identifying the twistor space of round $S^{6}$ with the Grassmannian of
oriented 2–planes in $\mathbb{R}^{8}$, Calabi and Gluck [CG93] show that these
“octonion” $J$’s send $S^{6}$ to the family of 2–planes of the form $(o,v)$,
where $o$ is a fixed unit octonion depending on $J$ (so, the image of $S^{6}$
is the unit sphere in the hyperplane orthogonal to a given line in
$\mathbb{R}^{8}$.) Two such nearby families intersect transversally.
Therefore, the $-2$ homological self-intersection obtained in Example 5.4
shows the octonion $J$’s are not integrable. Of course, this also follows by a
direct computation of the Nijenhuis tensor, and the result is a special case
of the classical theorem of Blanchard [Bl53] and LeBrun [Le87] that there is
no integrable complex structure on $S^{6}$ orthogonal with respect to the
round metric.

In order to study intersections of almost complex structures in different
components of the space of sections of the twistor space we will now analyze
the dependence of our formula for the cohomology (3.4) on the choice of almost
complex structure.

Consider the map of fibrations

(6)
${\mathbb{CP}^{3}}$${\mathbb{CP}^{\infty}}$${BU(3)}$${B\operatorname{Spin}^{c}(6)}$${BSO(6)}$${BSO(6)}$$\scriptstyle{j}$$\scriptstyle{B\kappa}$$\scriptstyle{=}$

As the canonical map $\kappa\colon U(3)\to\operatorname{Spin}^{c}(6)$ is an
isomorphism on $\pi_{1}$, we see that $j$ is an isomorphism on $\pi_{2}$ and
hence the $5$–lemma implies that $B\kappa$ is a $7$–equivalence. Therefore the
set of isomorphism classes of almost complex structures on six–manifolds is in
natural bijective correspondence with the set of isomorphism classes of spinc
structures via the canonical map from the former to the latter.

The set of isomorphism classes of spinc-structures on a manifold $M$ is a
torsor over the group $H^{2}(M)$ of isomorphism classes of complex line
bundles (corresponding to the action of the fiber on the principal fibration
along the right-hand column of (6)). Letting $[s]$ denote the spinc structure
associated to the almost complex structure $s\colon M\to Z_{+}(M)$, and
writing $a\cdot[s]$ for the action of the cohomology class $a\in H^{2}(M)$ on
$[s]$, we have the following result:

######  Lemma 5.8.

Let $M$ be an oriented Riemannian six–manifold and let $s,s^{\prime}\colon
M\to Z_{+}(M)$ be two almost complex structures on $M$ compatible with the
orientation of $M$. Let $a\in H^{2}(M)$ be (the unique cohomology class) such
that $[s^{\prime}]=a\cdot[s]$. Then the canonical isomorphism between the
expressions for $H^{*}(Z_{+}(M))$ given in 3.4 in terms of $s,s^{\prime}$ is
given by

$x^{\prime}\mapsto x-a.$

###### Proof.

Let $E$ and $E^{\prime}$ be the positive spinor bundles associated to the
spinc structures $[s]$ and $[s^{\prime}]$ respectively, and let $L$ denote a
complex line bundle with $c_{1}(L)=a$.

The relation $a\cdot[s]=[s^{\prime}]$ implies that there is an isomorphism

$\psi\colon L\otimes E\to E^{\prime}.$

The kernel in $T_{x}M\otimes\mathbb{C}$ of right multiplication by a positive
spinor $v\in E_{x}$ is the same as the kernel of right multiplication by
$\lambda\otimes v\in L_{x}\otimes E_{x}$ for any $\lambda\neq 0$, so the
canonical isomorphism given by the composition

$\mathbb{P}(L\otimes
E)\xrightarrow{\phi^{\prime}}Z_{+}(M)\xrightarrow{\phi^{-1}}\mathbb{P}(E)$

is the obvious one sending the line $L\otimes(\mathbb{C}v)$ to $\mathbb{C}v$.

We conclude that on $Z_{+}(M)$ the tautological (Hopf) line bundles $H$ and
$H^{\prime}$ coming respectively from the identifications with $\mathbb{P}(E)$
and $\mathbb{P}(E^{\prime})$ satisfy the relation

$H^{\prime}\cong H\otimes L.$

Since $x^{\prime}=c_{1}(H^{\prime*})$ and $x=c_{1}(H^{*})$, this completes the
proof. ∎

Given a spinc structure $[s]$ we will write $c_{1}([s])$ for the Chern class
of the line bundle classified by the projection
$B\operatorname{Spin}^{c}(2n)\xrightarrow{B\delta}BS^{1}$. Note that
$c_{1}(a\cdot[s])=2a+c_{1}([s])$ so the difference class $a\in H^{2}(M)$
between two almost complex structures $s,s^{\prime}$ is a specific “square
root” of the difference $c_{1}^{\prime}-c_{1}$. We can suggestively write

(7) $x^{\prime}=x+\frac{c_{1}-c_{1}^{\prime}}{2},$

which is unambiguous on rational cohomology (or when there is no $2$–torsion
in $H^{2}(M)$).

######  Remark 5.9.

If we use $s\colon M\to Z_{+}(M)$ to write

$H^{*}(Z_{+}(M))=H^{*}(M)[x]/(x^{4}+2c_{1}x^{3}+(c_{1}^{2}+c_{2})x^{2}+(c_{1}c_{2}-c_{3})x),$

then $s^{*}(x)=0$. This follows from Remark 2.3 which implies that the
pullback by $s$ of the Hopf bundle over $\mathbb{P}(E)$ is trivial. Using the
isomorphism $H^{2}(Z_{+}(M))\xrightarrow{i^{*}\oplus
s^{*}}H^{2}(\mathbb{CP}^{3})\oplus H^{2}(M)$ one sees that the first Chern
class of the horizontal plane bundle on $Z_{+}(M)$ is444This computation also
follows from the arguments used in the proof of 3.10. $c_{1}(M)+2x$ (for any
choice of almost complex structure) leading again to (7).

######  Theorem 5.10.

Let $J$ and $J^{\prime}$ be two orthogonal almost complex structures inducing
the same orientation on a closed Riemannian six–manifold $M$, corresponding to
sections $s$ and $s^{\prime}$ of the twistor bundle. Denote by
$c_{i},c_{i}^{\prime}$ their respective Chern classes. Then the intersection
number $s_{*}[M]\cdot s^{\prime}_{*}[M]$ in $Z_{+}(M)$ is given by

$\int_{M}\tfrac{1}{8}\left(c_{1}^{3}+c_{1}^{2}c_{1}^{\prime}-c_{1}c_{1}^{\prime
2}-c_{1}^{\prime
3}\right)+\tfrac{1}{2}\left(c_{1}c_{2}^{\prime}+c_{1}^{\prime}c_{2}^{\prime}\right)-c_{3}.$

###### Proof.

By 3.4, the presentation of $H^{*}(Z_{+}(M))$ with reference to $J$ is given
by

$H^{*}(M)[x]/(x^{4}+2c_{1}x^{3}+(c_{1}^{2}+c_{2})x^{2}+(c_{1}c_{2}-c_{3})x),$

and with reference to $J^{\prime}$ it is given by

$H^{*}(M)[x^{\prime}]/((x^{\prime})^{4}+2c_{1}^{\prime}(x^{\prime})^{3}+((c_{1}^{\prime})^{2}+c_{2}^{\prime})(x^{\prime})^{2}+(c_{1}^{\prime}c_{2}^{\prime}-c_{3}^{\prime})x^{\prime}).$

Combining 3.4 with Lemma 5.1, we have that the Poincaré dual of $s_{*}[M]$ is
given by

$x^{3}+2c_{1}x^{2}+(c_{1}^{2}+c_{2})x+(c_{1}c_{2}-c_{3}),$

and the Poincaré dual of $s^{\prime}_{*}[M]$ is given by

$(x^{\prime})^{3}+2c_{1}^{\prime}(x^{\prime})^{2}+((c_{1}^{\prime})^{2}+c_{2}^{\prime})x^{\prime}+c_{1}^{\prime}c_{2}^{\prime}-c_{3}^{\prime}.$

Recall from (7) that $x^{\prime}=x+\tfrac{1}{2}(c_{1}-c_{1}^{\prime})$. (Our
calculation will be insensitive to torsion, so we may in fact work in the
rational cohomology ring where there is no ambiguity in writing
$\tfrac{1}{2}(c_{1}-c_{1}^{\prime})$.) Using this to rewrite the expression
for $s^{\prime}_{*}[M]$, and multiplying with $s_{*}[M]$, yields

$x^{3}\left(\tfrac{1}{8}\left(c_{1}^{3}+c_{1}^{2}c_{1}^{\prime}-c_{1}c_{1}^{\prime
2}-c_{1}^{\prime
3}\right)+\tfrac{1}{2}\left(c_{1}c_{2}^{\prime}+c_{1}^{\prime}c_{2}^{\prime}\right)-c_{3}^{\prime}\right).$

Pairing with the fundamental class of $Z_{+}(M)$ gives the result. ∎

######  Remark 5.11.

The expression for the intersection number in 5.10 is symmetric in $J$ and
$J^{\prime}$ as expected, which can be verified by using
$c_{1}^{2}-2c_{2}=p_{1}=c_{1}^{\prime 2}-2c_{2}^{\prime}$ and
$c_{3}=c_{3}^{\prime}$.

######  Example 5.12.

Let $J$ and $J^{\prime}$ be two almost complex structures on
$\mathbb{CP}^{3}$, with Chern classes $c_{i}$ and $c_{i}^{\prime}$. Then
$c_{1}=2kx,c_{2}=(2k^{2}-2)x^{2},c_{3}=4x^{3}$ and $c_{1}^{\prime}=2\ell
x,c_{2}^{\prime}=(2\ell^{2}-2)x^{2},c_{3}^{\prime}=4x^{3}$, where $k,\ell$ are
integers. From 5.10 we get that $J$ and $J^{\prime}$, thought of as sections
of the twistor bundle, have homological self-intersection given by

$k^{3}+k^{2}\ell+k\ell^{2}+\ell^{3}-2k-2\ell-4.$

In particular, taking $J$ to be the standard complex structure, lying in the
component determined by $k=2$, we have that the homological intersection with
any $J^{\prime}$ with $\ell<0$ is negative.

## References

  * [Ar97] Armstrong, J., _On four-dimensional almost Kähler manifolds_. Quart. J. Math. Oxford Ser. (2) 48 (1997), no. 192, 405–415.
  * [AHS78] Atiyah, M.F., Hitchin, N.J. and Singer, I.M., _Self-duality in four-dimensional Riemannian geometry_. Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461.
  * [Bl53] Blanchard, A. _Recherche de structures analytiques complexes sur certaines variétés_ , C. R. Acad. Sci., Paris 236 (1953) 657–659.
  * [B53] Borel, A. _Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts_. Ann. of Math. (2) 57 (1953), 115–207.
  * [Br97] Bredon, G. E., _Topology and geometry_. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1997. xiv+557 pp.
  * [CG93] Calabi, E. and Gluck, H., _What are the best almost-complex structures on the 6–sphere?_. Differential geometry: geometry in mathematical physics and related topics (Los Angeles, CA, 1990), 99–106, Proc. Sympos. Pure Math., 54, Part 2, Amer. Math. Soc., Providence, RI, 1993.
  * [CS84] Crabb, M.C. and Sutherland, W.A., 1984. _Function spaces and Hurwitz-Radon numbers_. Math. Scand. 55 (1984), no. 1, 67–90.
  * [D12] Demailly, J.P., _Complex Analytic and Differential Geometry_. available at https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf
  * [Du18] Duan, H., _The characteristic classes and Weyl invariants of Spinor groups_. arXiv preprint arXiv:1810.03799.
  * [EaSi93] Eastwood, M.G. and Singer, M.A., _The Frölicher spectral sequence on a twistor space_. J. Differential Geom. 38 (1993), no. 3, 653–669.
  * [EeSa85] Eells, J. and Salamon, S., _Twistorial construction of harmonic maps of surfaces into four–manifolds_. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 4, 589–640 (1986).
  * [Ev14] Evans, J.D., _Quantum cohomology of twistor spaces and their Lagrangian submanifolds_. J. Differential Geom. 96 (2014), no. 3, 353–397.
  * [FGM21] Ferlengez, B., Granja, G., and Milivojević, A., _On the topology of the space of almost complex structures on the six sphere_ , New York J. Math. 27 (2021), 1258–1273.
  * [GS14] Godinho, L. and Sabatini, S., _New tools for classifying Hamiltonian circle actions with isolated fixed points_. Found. Comput. Math. 14 (2014), no. 4, 791–860.
  * [H82] Haefliger, A., _Rational homotopy of the space of sections of a nilpotent bundle_. Trans. Amer. Math. Soc. 273 (1982), no. 2, 609–620.
  * [Hi81] Hitchin, N.J., _Kählerian twistor spaces_. Proc. London Math. Soc. (3) 43 (1981), no. 1, 133–150.
  * [LM89] Lawson, H.B. and Michelsohn, M.-L., _Spin Geometry_ , Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. xii+427 pp.
  * [Le87] LeBrun, C., 1987. _Orthogonal complex structures on $S^{6}$_. Proceedings of the American Mathematical Society, 101(1), pp.136-138.
  * [Le99] LeBrun, C., _Topology versus Chern numbers for complex 3–folds_. Pacific J. Math. 191 (1999), no. 1, 123–131.
  * [MiTo] Mimura, M. and Toda, H., _Topology of Lie groups, I and II_ Translations of Mathematical Monographs, 91. American Mathematical Society, Providence, RI, 1991. iv+451 pp.
  * [MR85] Møller, J.M. and Raussen, M., _Rational homotopy of spaces of maps into spheres and complex projective spaces_. Trans. Amer. Math. Soc. 292 (1985), no. 2, 721–732.
  * [M87] Møller, J.M., _Nilpotent spaces of sections_. Trans. Amer. Math. Soc. 303 (1987), no. 2, 733–741.
  * [Sa84] Salamon, S.M., _Harmonic and holomorphic maps_. Geometry seminar Luigi Bianchi II-1984, 161–224, Lecture Notes in Math., 1164, Springer, Berlin, 1985.
  * [Sa95] Salamon, S.M., _Orthogonal complex structures_. Differential geometry and applications (Brno, 1995), 103–117, Masaryk Univ., Brno, 1996.
  * [S77] Sullivan, D., _Infinitesimal computations in topology_. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 269–331 (1978).
  * [T92] Taubes, C.H., _The existence of anti-self-dual conformal structures_. J. Differential Geom. 36 (1992), no. 1, 163–253.
  * [VS76] Vigué-Poirrier, M. and Sullivan, D., _The homology theory of the closed geodesic problem_. J. Differential Geometry 11 (1976), no. 4, 633–644.