2.1 GKM manifolds

For an action of a compact torus $T$ on a connected, compact manifold $M$, we consider its fixed point set $M^T=\{p\in M\mid T\cdot p =\{p\}\}$ as well as its one-skeleton $M_1=\{p\in M\mid \dim T\cdot p\leq 1\}$. In GKM theory, named after Goresky, Kottwitz, and MacPherson [Reference Goresky, Kottwitz and MacPhersonGKM98], one puts certain assumptions on the action that allow us to encode the structure of the one-skeleton in a labelled graph. More precisely, we say that the action satisfies the GKM conditions, or that it is a GKM action, if $M$ is orientable, $M^T$ is a finite set of points, and $M_1$ a finite union of $T$-invariant $2$-spheres. A manifold together with a GKM ($T$-)action will also be referred to as a GKM ( $T$-)manifold. Note that a finite non-empty fixed point set forces the manifold to be even-dimensional.

In this setting, the orbit space of the one-skeleton $M_1/T$ has the structure of a graph $\Gamma$, with one vertex for each fixed point, and one edge for each invariant $2$-sphere. The vertex set of a graph $\Gamma$ will be denoted by $V(\Gamma )$, and the set of edges by $E(\Gamma )$. Formally, we include each edge twice, once for each orientation, in the edge set $E(\Gamma )$. For an oriented edge $e\in E(\Gamma )$ we denote its initial vertex by $i(e)$ and its terminal vertex by $t(e)$; the edge $e$ with the opposite orientation will be denoted $\bar {e}$. We write $E(\Gamma )_v$ for the set of edges $e\in E(\Gamma )$ emanating from $v$. At each fixed point $p$ of the action the isotropy representation decomposes into $n$ two-dimensional summands, where $2n$ is the dimension of $M$. The $n$ weights of these irreducible submodules are elements of $\mathbb {Z}_\mathfrak {t}^*/\pm 1$, where $\mathbb {Z}_\mathfrak {t}^*\subset \mathfrak {t}^*$ is the weight lattice of $T$. Any such weight corresponds uniquely to an invariant $2$-sphere containing $p$, and we put it as a label of the corresponding edge of $\Gamma$. In total, we obtain a map $\alpha :E(\Gamma )\to \mathbb {Z}_\mathfrak {t}^*/\pm 1$ which we call an axial function, following [Reference Guillemin and ZaraGZ99]. The graph $\Gamma$, together with the axial function $\alpha$, will be called the GKM graph of the $T$-action.

Independent of this geometric setting, one can define abstract GKM graphs [Reference Guillemin and ZaraGZ01]. The graphs one considers have finite vertex and edge sets; we allow multiple edges between vertices, but no loops, i.e. edges that connect a vertex to itself.

Note that the linear independence of $\alpha (e)$ and $\alpha (f)$ in the first condition, which is defined via preimages under $\mathbb {Z}^m\to \mathbb {Z}^m/\pm 1$, is well defined.

Following [Reference Franz and YamanakaFY19] we make the following definition.

We remark that any equivariant homeomorphism between two GKM manifolds induces such an isomorphism of GKM graphs, with the automorphism $\varphi$ equal to the identity.

The terminology is justified by the fact that a GKM $T^m$-action on a manifold is effective if and only if the corresponding GKM graph is effective: for any vertex $v$ of the GKM graph, the kernel of the action is given as the intersection of the kernels of the $\alpha (e)$, where $e$ varies over all edges emanating from $v$. Here, we interpret the $\alpha (e)$ as homomorphisms $T^m\to S^1$.

2.2 Almost complex structures

Recall that an almost complex structure on a smooth manifold $M$ is a smooth endomorphism $J$ of $TM$ satisfying $J^2=-\mathrm {Id}$. If such an endomorphism is present on the bundle $TM\oplus \mathbb {R}^k$ for some $k$, then we speak of a stable almost complex structure. If an almost complex structure is integrable, then we refer to it as a complex structure.

Given a $T$-invariant almost complex structure on $M$, the weights of the isotropy representation have a well-defined sign, i.e. are elements of $\mathbb {Z}_\mathfrak {t}^*$. We will speak of the signed GKM graph of the action when we consider the graph $\Gamma$ with these weights as labels. Formally, the axial function becomes a map $\alpha :E(\Gamma )\to \mathbb {Z}_\mathfrak {t}^*$, by associating to $e$ the weight of the corresponding summand of the isotropy representation at $i(e)$. Abstractly, we make the following definition.

Remark 2.10 By composing the axial function of a signed GKM graph with the projection $\mathbb {Z}^m\rightarrow \mathbb {Z}^m/\pm$ one obtains an (unsigned) GKM graph. Note that a connection which is compatible with the signed GKM graph is, in particular, compatible with the (unsigned) GKM graph. We also call the signed graph a compatible signed structure of the underlying GKM graph. There might be different signed structures compatible with a single GKM graph. If these were induced by an almost complex GKM action as explained previously, then different signed structures would correspond to different homotopy classes of compatible almost complex structures. The existence of a compatible signed structure is clearly an obstruction to the existence of an invariant almost complex structure. For example, it is easy to check that the GKM graph

of the standard $T^2$-action on $S^4\subset \mathbb {C}^2\oplus \mathbb {R}$ does not carry a compatible signed structure.

The definitions of GKM subgraphs and isomorphisms of GKM graphs carry over to the signed setting in an obvious fashion. Moreover, any equivariant diffeomorphism between two GKM manifolds that respects $T$-invariant almost complex structures induces an isomorphism of the corresponding signed GKM graphs.

2.3 Symplectic structures

Just as the existence of invariant almost complex structures impacts the GKM graph of a GKM manifold, the existence of a compatible symplectic structure forces certain properties onto the GKM graph. Recall that an almost complex structure $J$ is called compatible with a symplectic form $\omega$ if the two-form $v\otimes w\mapsto \omega (v, Jw)$ defines a Riemannian metric. For any $T$-invariant symplectic form there exists a $T$-invariant compatible almost complex structure, hence combinatorial consequences of symplectic structures contain those from the almost complex case.

More precisely, we consider for a Hamiltonian GKM action of a torus $T$ on a symplectic manifold $M$ a moment map $\mu \colon M\rightarrow \mathfrak {t}^*$ into the dual of the Lie algebra of $T$. An invariant two sphere $S^2\subset M$ gets mapped under $\mu$ to an affine linear interval in $\mathfrak {t}^*$ whose boundary points are the images of the fixed points. By piecing those together we obtain a linear realization of the underlying graph of the GKM graph, i.e. an edge-wise affine linear map from the topological realization of $\Gamma$ to $\mathfrak {t}^*$ (not an embedding), whose image is $\mu (M_1)$. If $(\Gamma,\alpha )$ is the signed GKM graph associated to an almost complex structure which is compatible with the symplectic form, then the labels are encoded in the linear realization as follows: if $e$ is an oriented edge, then the slope of the corresponding affine linear segment in $\mathfrak {t}^*$ is given by $\alpha (e)\in \mathbb {Z}_\mathfrak {t}^*\subset \mathfrak {t}^*$. Thus, the linear realization determines $(\Gamma,\alpha )$ up to multiples of the weights. The labels are uniquely determined if we add the assumption that they are primitive.

Remark 2.11 Having such a linear realization is a non-trivial obstruction for a signed GKM graph to come from a Hamiltonian action. For example, the signed structures of type III in Theorem 7.1 do not since the cone spanned by the weights of the outgoing edges at every point is all of $\mathfrak {t}^*$.

The convexity theorem, which implies that the image $\mu (M)$ of the moment map is the convex hull of $\mu (M_1)$, gives additional obstructions. The preimage of an outer edge of the resulting polytope is contained in $M_1$. Thus, every outer edge of the convex hull $\mu (M)$ must be an edge in $\mu (M_1)$. As a counterexample, the linear realization

does not have this property (the grid represents the standard basis of $\mathbb {Z}^2$) and it follows from the results in § 7.3 that, more generally, the signed GKM graph which is uniquely defined by having this realization and primitive weights does not admit any linear realization with the above convexity property. Hence, it cannot come from a Hamiltonian action.

Note that the convexity obstruction from the above remark does not concern the inner edges of the linear realization. However, in order for a GKM graph to come from a Hamiltonian $T$-action additional obstructions for inner edges do exist: for any subtorus $T'\subset T$, the action of $T$ on the submanifold $M^{T'}$ is again Hamiltonian, so certain subgraphs of the original GKM graph do again satisfy the convexity criterion explained in Remark 2.11. These subtleties will not play a role in our low-dimensional considerations, so we refrain from introducing a precise definition outside of the case below. Note that GKM graphs of $4$-dimensional Hamiltonian manifolds do indeed satisfy the following.

2.4 Kähler structures

As a last step in the hierarchy we state a certain property of signed GKM graphs coming from invariant Kähler structures which goes beyond the previous obstructions for Hamiltonian actions. Recall that a symplectic form is called Kähler if it admits a compatible complex structure. Note that unlike for $T$-invariant symplectic forms and $T$-invariant almost complex structures, the $T$-invariance of a Kähler form does not immediately imply the existence of a compatible $T$-invariant complex structure. In fact this is false in general as follows from the result of Tolman [Reference TolmanTol98] mentioned in the introduction as well as from the main results of this article.

In [Reference TolmanTol98, Lemma 3.5] the following is shown. Consider a Hamiltonian $T$-action on $(M,\omega )$ with $\omega$ a Kähler form admitting a $T$-invariant compatible complex structure. We also refer to this as a $T$-invariant Kähler structure. Let $V\subset T_pM$ be a sum of irreducible summands of the isotropy representation at a fixed point $p$ such that the weights of those summands form a minimal generating set of a convex cone in $\mathfrak {t}^*$. Then there is a Hamiltonian submanifold of $M$ containing $p$ whose tangent space at $p$ is $V$. As a special case, this implies the following.

An example where this fails while the obstructions for general Hamiltonian actions hold is given by the GKM graph with primitive labels and the linear realization

since the only signed GKM subgraph containing the edges between the three inner vertices is the graph from Remark 2.11. See also § 7.2 where we will consider this obstruction to the Kähler property for a large class of graphs.

5.1 Realization in dimension 4

As a starting point, we need to geometrically realize the base graph of the fibration which corresponds to a $2$-dimensional torus action on a $4$-manifold. Actions of tori of dimension half the dimension of the manifold are quite well studied so we can draw on the existing theory of quasitoric manifolds. Recall that a quasitoric manifold is a smooth compact $2n$-dimensional manifold $X$ with a locally standard smooth action of the $n$-dimensional torus $T^n$ which is such that its orbit space $X/T^{n}$ is homeomorphic, as a manifold with corners, to a simple convex polytope $P$. Via the homeomorphism $X/T^{n}\cong P$, the facets of $P$ correspond to the so-called characteristic submanifolds of $X$ which are the preimages of the facets under the projection. The characteristic function of a quasitoric manifold $X$ is the assignment sending a facet of $P\cong X/T^{n}$ to the isotropy circle of the corresponding characteristic submanifold.

Let us describe the inverse construction to this [Reference Davis and JanuszkiewiczDJ91], see also [Reference Buchstaber and PanovBP15, § 7.3]: we start with a characteristic pair $(P,\lambda )$ consisting of a simple convex polytope $P$ with facets $F_1,\ldots,F_m$, together with a map $\lambda$ sending a facet $F_j$ to a subcircle $\lambda (F_j)\subset T^n$, with the property that the multiplication map $\lambda (F_{j_1})\times \cdots \times \lambda (F_{j_k})\to T^{n}$ is injective whenever the intersection $F_{j_1}\cap \cdots \cap F_{j_k}$ is non-empty. We extend $\lambda$ to all faces of $P$, i.e. we denote by $\lambda (F)$ the subtorus of $T^{n}$ generated by all $\lambda (F_i)$, where $F_i$ is a facet containing $F$. Moreover, we denote by $F(x)$, for $x\in P$, the smallest face of $P$ containing $x$. In this situation, we define $X:=P\times T^n/\sim$, where the equivalence relation $\sim$ is generated by $(x,s)\sim (x,t) \Leftrightarrow s^{-1}t\in \lambda (F(x))$. Then $X$, equipped with the $T^n$-action defined as the multiplication on the second factor of $P\times T^n$, is a quasitoric manifold with orbit space $P$ and characteristic function $\lambda$; it is called the canonical model of $(P,\lambda )$.

Furthermore, we recall that an omniorientation of a quasitoric manifold is an orientation of $X$ and a choice of orientations of its characteristic submanifolds. Note that given an orientation of $X$, an orientation of a characteristic submanifold is equivalent to an orientation of the normal bundle. For a fixed point $p\in X^{T^{n}}$ we have a decomposition $T_p X=V_1\oplus \cdots \oplus V_n$ where the $V_i$ are the irreducible $2$-dimensional subrepresentations and, in fact, normal spaces to the characteristic submanifolds containing $p$. An omniorientation thus induces an orientation on all the $V_i$ and, hence, on $T_p X$. The sign of the fixed point is said to be $1$ if this orientation agrees with that coming from the orientation on $X$ and $-1$ if it does not. A positive omniorientation is an omniorientation such that the signs of all fixed points are positive. Now any omniorientation induces a complex structure on the tangent spaces of the fixed points as explained below and Kustarev proved in [Reference KustarevKus09] that this extends to a $T^n$-invariant almost complex structure on $X$ if and only if the omniorientation is positive (see also [Reference Buchstaber and PanovBP15, Theorem 7.3.24]).

To define the complex structure on $T_p X=V_1\oplus \cdots \oplus V_n$ induced by an omniorientation recall that $V_i$ is normal to a unique characteristic submanifold containing $p$ which corresponds to some facet $F_i$. Let $\lambda (F_i)\subset T^{n}$ denote the kernel of the action on the characteristic submanifold belonging to $F_i$. Then $\lambda (F_i)$ acts effectively on $V_i$. There is now a unique way to identify $\lambda (F_i)\cong S^1$ such that there is an orientation-preserving $S^1$-equivariant isomorphism $V_i\cong \mathbb {C}$ to the standard $S^1$-representation on $\mathbb {C}$ with the orientation given by the standard complex structure. This complex structure on $\mathbb {C}$ then induces a $T^n$-invariant complex structure on all the $V_i$ and, hence, on $T_p X$.

Proof. Part (iii) is Delzant's theorem. In part (i), the statement holds for $n=2$ since any such graph can be realized by a $T$-action on $S^4\subset \mathbb {C}^2\oplus \mathbb {R}$ which acts on the $\mathbb {C}^2$ factor as a pullback of the standard representation along some automorphism of $T$; for $n\geq 3$, we translate the canonical model of $4$-dimensional quasitoric manifolds, which we recalled above in arbitrary dimension, to GKM graphs. As $B$ is an $n$-gon with $n\geq 3$, the underlying graph is realized as the boundary of a convex polytope $P\subset \mathbb {R}^2$. For an edge $e$ of $P$ we set $\lambda (e) :=\ker \alpha \subset T$, where $\alpha$ is the weight (up to sign) associated to $e$ in $B$, interpreted as a homomorphism $T\to S^1$. Then $(P,\lambda )$ is a characteristic pair, and we consider the corresponding canonical model, which is a $4$-dimensional quasitoric manifold $X$. By construction, the GKM graph of $X$ is $B$.

Regarding part (ii), the existence of invariant almost complex structures on quasitoric manifolds is already well understood, as explained above the statement of the proposition, and it only remains to draw the connection to the signed GKM graph. Assume $B$ admits a signed structure. Then necessarily $n\geq 3$, and by part (i) we may choose $X$ as a quasitoric $4$-manifold. In view of the discussion above, it suffices to find a positive omniorientation of $X$ such that the induced complex structures on the tangent spaces of all fixed points yield the weights prescribed by the signed structure.

As a first step we show that the signed structure gives rise to canonical parametrizations of the isotropy circles of all characteristic submanifolds. Consider $p\in X^T$, let $e,e'$ be the edges emanating from $p$, and let $\alpha (e),\alpha (e')\in \mathbb {Z}_\mathfrak {t}^*$ be the corresponding weights. Let $\alpha (e)^*,\alpha (e')^*\in \mathbb {Z}_\mathfrak {t}$ be the associated dual basis. Note that a facet of $P$ is just an edge and these correspond to the edges of the GKM graph. Let $S_e$ denote the two-sphere corresponding to $e$, i.e. $S_e$ is the characteristic submanifold belonging to $e$. Now the kernel $T_e\subset T$ of the action on $S_{e}$ is a circle and its Lie algebra is generated by $\alpha (e')^*$. The element $\alpha (e')^*$, hence, fixes an identification $T_e\cong S^1$. We claim that this is well defined for the characteristic submanifold $S_e$ and does not depend on the initial choice of fixed point $p$ in the characteristic submanifold. Indeed, let $q$ be the terminal vertex of $e$ and let $\overline {e},e''$ be the emanating edges at $q$. As before, we have weights $\alpha (\overline {e}),\alpha (e'')\in \mathbb {Z}_\mathfrak {t}^*$ and a dual basis $\alpha (\overline {e})^*,\alpha (e'')^*\in \mathbb {Z}_\mathfrak {t}$. The compatible connection implies $\alpha (e'')\equiv \alpha (e')\mod \alpha (e)$ and, thus, $\alpha (e)=-\alpha (\overline {e})$ implies $\alpha (e')^*=\alpha (e'')^*$. Hence, the latter is indeed a canonical generator of the Lie algebra of $T_e$ arising from the signed structure and it will be denoted by $\lambda (e)\in \mathbb {Z}_\mathfrak {t}$.

We now fix an orientation of the polytope $P$. Together with this orientation the datum of the $\lambda (e)$ for all characteristic submanifolds $e$ is called a combinatorial quasitoric pair in [Reference Buchstaber and PanovBP15]. It carries the information of the characteristic pair as well as orientations for the polytope and all isotropy circles. It is shown in [Reference Buchstaber and PanovBP15, Proposition 7.3.11] that there is an associated omniorientation on the canonical model for which, by [Reference Buchstaber and PanovBP15, Proposition 7.3.18], the weights at a fixed point $p$, with signs given by the complex structure on $T_p M$ associated to the omniorientation, is precisely the dual basis of $\lambda (e),\lambda (e')$ where $e,e'$ are the edges adjacent to $p$. By construction, this dual basis is precisely the weights $\alpha (e),\alpha (e')$ provided by the signed structure. Hence, there is an omniorientation inducing the weights prescribed by the signed graph.

It remains to show that the omniorientation is positive. Let $p,q,e,e',e''$ be as above. By [Reference Buchstaber and PanovBP15, Lemma 7.3.21] as well as the subsequent remark, we may compute the sign at the fixed point $p$ as follows: let $e,e'$ be the order of the edges coming out of $p$ such that the orientation provided by vectors along $e,e'$ agrees with the fixed orientation of $P$. Then the sign at $p$ is $\det (\alpha (e),\alpha (e'))$ where the determinant is understood via the identification $\mathbb {Z}_\mathfrak {t}^*\cong \mathbb {Z}^2$ using the dual of the standard basis. If the signs of all fixed points are negative, then by reversing all orientations in the omniorientation, we obtain a positive omniorientation which produces the same complex structure in the tangent spaces of the fixed points. Hence, it suffices to show that the above determinants agree for all fixed points. The orientation on $P$ provided by $e,e'$ agrees with that of $e'',\overline {e}$. We compute

\[ \det(\alpha(e''),\alpha(\overline{e}))=-\det(\alpha(e''),\alpha(e))= \det(\alpha(e),\alpha(e''))=\det(\alpha(e),\alpha(e')), \]

where the last equation is again due to the congruence $\alpha (e'')\equiv \alpha (e')\mod \alpha (e)$ provided by the compatible connection.

5.2 A vector bundle over the one-skeleton

Having realized the 2-regular graph $B$ by a 4-dimensional manifold $X$, it is now our intermediate goal to construct an equivariant complex vector bundle over $X_1$ such that the set of $0$- and $1$-dimensional orbits in its projectivization is precisely the $T^2$-space encoded in the graph $\Gamma$. In this section, we will use the theory of cohomogeneity one actions [Reference MostertMos57, Reference Alekseevsky and AlekseevskyAA93], see also [Reference Galaz-García and ZareiGZ18], i.e. actions of compact Lie groups on closed manifolds whose principal orbits have codimension one. Let us briefly recall the relevant ingredients from this theory. A group diagram is a quadruple $(G,K^+,K^-,H)$ of compact Lie groups $H\subset K^\pm \subset G$ such that $K^+/H$ and $K^-/H$ are spheres. Given a cohomogeneity one $G$-action on $M$ with orbit space homeomorphic to the closed interval $[0,1]$, we can (non-canonically) associate a group diagram to it in the following way: we choose an auxiliary $G$-invariant Riemannian metric, as well as a geodesic $\gamma :[0,1]\to M$ orthogonal to the $G$-orbits, which meets every $G$-orbit exactly once. Then $G\cdot \gamma (0)$ and $G\cdot \gamma (1)$ are the unique non-regular orbits of the action, and their stabilizers $K^+=G_{\gamma (1)}$ and $K^- = G_{\gamma (0)}$, together with the pointwise stabilizer $H$ of the geodesic $\gamma$ yields the group diagram $(G,K^+,K^-,H)$.

Conversely, a group diagram $(G,K^+,K^-,H)$ yields a cohomogeneity one action, as we now explain. By [Reference BredonBre72, Theorem 10.1, Chapter II], any transitive Lie group action on a sphere is linear; in particular, this is the case for the $K^+$- and $K^-$-actions on $K^+/H$ and $K^-/H$. Hence, we find orthogonal $K^\pm$-representations on Euclidean vector spaces $V^\pm$ with principal isotropy group $H$. Denoting the unit disks in $V^\pm$ by $D^\pm$, we obtain a cohomogeneity one manifold by gluing the disk bundles $G\times _{K^\pm } D^\pm$ along their boundaries $G\times _{K^+} K^+/H = G/H = G\times _{K^-} K^-/H$. As emphasized above, the group diagram associated to a cohomogeneity one manifold is not unique, as it depends on a certain choice. However, two cohomogeneity one manifolds with the same group diagram are equivariantly diffeomorphic.

We make use of the notation from Remark 4.6. We also return to the notation from the proof of Proposition 4.5 which we shall recall here: choose an orientation for the edge in the fiber over $v_1$ with associated weight $\alpha _1$. Using a compatible connection, this inductively defines orientations on the edges in the fibers over all the $v_i$ such that $\alpha _i\equiv \alpha _{i+1}\ {\rm mod}\ \gamma _i$. We extend this notation with this property to all $i\in \mathbb {Z}$. Transporting an edge around a lift of the path which goes around the base graph once reverses its orientation if $\Gamma$ is of twisted type and preserves the orientation if it is of product type. Hence, we have $\alpha _i=(-1)^{\eta }\alpha _{i+n}$, where $\eta =1$ if $\Gamma$ is of twisted type and $\eta =0$ if it is of product type. Furthermore, let $k_i\in \mathbb {Z}$ be as in Remark 4.6, i.e. the numbers uniquely determined by the equation

\[ \alpha_i=k_i\gamma_{i-1}-k_{i-1}\gamma_i. \]

We now come to the construction of the vector bundle: for each invariant $2$-sphere in $X$ corresponding to some edge $e_i$ ($i=1,\ldots,n$) we want to construct a $T$-equivariant $\mathrm {U}(2)$-principal bundle $P_i\to S^2$ such that the projectivization of the associated $\mathbb {C}^2$-bundle $P_i\times _{\mathrm {U}(2)}\mathbb {C}^2\to S^2$ with respect to the standard representation of $\mathrm {U}(2)$ on $\mathbb {C}^2$ has as GKM graph exactly $\pi ^{-1}(e_i)$, as depicted in Lemma 5.4 below. We will construct $P_i$ as a $(T\times \mathrm {U}(2))$-cohomogeneity one manifold such that the $\mathrm {U}(2)$-subaction is free. In this situation, we can divide out the $\mathrm {U}(2)$-action and obtain a smooth manifold, to which the $T$-subaction on $P_i$ descends. This quotient action will again be of cohomogeneity one. Using the observation recalled above that a group diagram of a cohomogeneity one action determines it up to equivariant diffeomorphism, this will allow us to identify the quotient as $S^2$ with the correct action. In this way, $P_i$ will be seen to be a $T$-equivariant $\mathrm {U}(2)$-principal bundle. For some arbitrary integers $a_i,b_i,c_i$ (which we will specify below), define the homomorphisms

\[ A_i^+\colon T\rightarrow \mathrm{U}(2),\quad t\mapsto\begin{pmatrix} a_i\gamma_{i-1}(t)-b_i\gamma_i(t) & \\ & (a_i-k_i)\gamma_{i-1}(t) + (k_{i-1}-b_i)\gamma_i (t) \end{pmatrix} \]

and

\[ A_i^-\colon T\rightarrow \mathrm{U}(2),\quad t\mapsto\begin{pmatrix} c_i\gamma_i(t) - a_i\gamma_{i+1}(t) & \\ & (c_i-k_{i+1})\gamma_i(t) + (k_{i}-a_i)\gamma_{i+1}(t) \end{pmatrix}, \]

where the entries of the matrices are to be interpreted as the homomorphisms $T\rightarrow S^1$ corresponding to the respective elements of $\mathbb {Z}_\mathfrak {t}^*$.

Let $P_i$ be the cohomogeneity one manifold defined by the group diagram $(G,K^+_i,K^-_i,H_i)$; here,

\begin{align*} G &= T\times \mathrm{U}(2) \\ K^+_i&=\{(t,A_i^+(t))\mid t\in T\},\\ K^-_i&= \{(t,A_i^-(t))\mid t\in T\},\\ \quad H_i&=\{(t,A_i^+(t))\mid t\in \ker \gamma_i\}= \{(t,A_i^-(t))\mid t\in \ker \gamma_i\}. \end{align*}

The last equation holds since, by definition, $\gamma _{i-1}=-\gamma _{i+1}\mod \gamma _i$. Consequently $H_i$ is contained both in $K^+_i$ and in $K^-_i$. Furthermore, $K_i^\pm /H_i\cong S^1$, so that this really defines a valid group diagram. We fix identifications of the outer (non-principal) orbits with $G/K_i^+$ and $G/K_i^-$.

We observe that $\mathrm {U}(2)\cong \{e\}\times \mathrm {U}(2)$ intersects $K^\pm _i$ and all its $G$-conjugates trivially, so that the $\mathrm {U}(2)$-subaction on $P_i$ is free; dividing out this subaction we obtain a $T$-manifold of cohomogeneity one. Let us identify the group diagram of this action. To this end, we choose a $G$-invariant Riemannian metric on $P_i$ as well as a geodesic perpendicular to the orbits, which induce the above group diagram. We introduce on the quotient the Riemannian metric turning the projection map into a Riemannian submersion. Then the geodesic projects onto a geodesic perpendicular to the orbits. The corresponding group diagram is given by the projection of the respective subgroups to the $T$-factor, i.e. $(T, T, T, \ker \gamma _i)$. Thus, the quotient manifold is $U$-equivariantly diffeomorphic to the $S^2\subset X$ corresponding to the edge $e_i$.

Associated to $P_i\to S^2$, we obtain an associated $T$-equivariant $\mathbb {C}^2$-bundle

\[ E_i:=P_i\times_{\mathrm{U}(2)}\mathbb{C}^2\longrightarrow S^2, \]

where $\mathrm {U}(2)$ acts on $\mathbb {C}^2$ by the standard representation. Let us compute the $T^2$-representation on the fibers over the fixed points in $S^2$ through the identifications

\[ \mathbb{C}^2 \cong ((T\times \mathrm{U}(2))/K^\pm_i) \times_{\mathrm{U}(2)} \mathbb{C}^2,\quad v\mapsto [(e,I_2)K_i^\pm,v]. \]

It is given by pulling back the standard $\mathrm {U}(2)$-representation along $A_i^\pm$ since

\[ t\cdot [(e,I_2)K^\pm_i,v] = [(t,I_2)K^\pm_i,v] = [(e,(A_i^\pm(t))^{-1})K^\pm_i,v] = [(e,I_2)K^\pm_i,A_i^\pm(t)v]. \]

Next we pass to the projectivized bundle $\mathbb {P}(E_i)\to S^2$, which is a $T$-equivariant $\mathbb {C} \mathbb {P}^1$-bundle over $S^2$.

Lemma 5.4 The projectivization $\mathbb {P}(E_i)\rightarrow S^2$ is a fibration of GKM manifolds. On (unsigned) GKM graphs it is given by

where (with respect to the identifications above) $p_i$ and $p_{i+1}$ correspond to $[1:0]$ while $q_i$ and $q_{i+1}$ correspond to $[0:1]$.

Proof. We have a fibration

\[ \mathbb{CP}^1\rightarrow \mathbb{P}(E_i)\rightarrow S^2 \]

with structure group in $\mathrm {U}(2)$. Thus, fixing the standard complex structure on $\mathbb {CP}^1$ and some $T^2$-invariant complex structure on $S^2$, we obtain an almost complex structure on $\mathbb {P}(E_i)$ such that $\mathbb {P}(E_i)\rightarrow S^2$ respects almost complex structures. We argue via the corresponding signed GKM graph.

The identifications and orientations are such that the weights at $q_i$ are given by $\pm \gamma _i$ (the sign depending on the chosen almost complex structure on $S^2$) and the upper left entry of $A_i^+$ minus the lower right entry of $A_i^+$ (coming from the fiber over $v_i$). By the definition of $A_i^+$, the latter difference is precisely $k_{i}\gamma _{i-1}-k_{i-1}\gamma _{i}=\alpha _i$. Similarly, one computes the weights at $q_{i+1}$ to be $\pm \gamma _i$ and $\alpha _{i+1}$. As $\alpha _i=\alpha _{i+1}\mod \gamma _i$, the existence of a connection that is compatible with the signed structure implies that $q_i$ and $q_{i+1}$ are adjacent. Thus, in particular, the unsigned GKM graph has the form as claimed in the lemma.

Having constructed the squares in $\Gamma$ over every single edge $e_i$ it remains to glue the $E_i$ in an appropriate and $T$-equivariant manner. We start by gluing $E_{i}$ to $E_{i+1}$ for $i=1,\ldots,n-1$, by identifying the fibers over $v_{i+1}$. From the side of $E_i$ this is the representation of $\mathbb {C}^2$ defined by the homomorphism $A_i^-$, while from the side of $E_{i+1}$ it is defined by $A_{i+1}^+$. Recall that there are unspecified parameters $a_i,b_i,c_i\in \mathbb {Z}$ in the construction. If we choose them such that $a_{i+1}=c_i$ and $b_{i+1}=a_i$, then $A_i^-=A_{i+1}^+$ and we glue the two fibers in the canonical way.

At this point the one-skeleton of the action on the projectivization is a ladder formed by gluing the individual squares from the lemma above in the obvious way:

In order to obtain the graph $\Gamma$ it remains to glue $E_{n+1}$ to $E_1$. This needs to be done with respect to the graph theoretical structure of $\Gamma$. If $\Gamma$ is of product type, we obtain $\Gamma$ from the ladder by identifying $p_{n+1}=p_1$ as well as $q_{n+1}=q_1$. In the twisted case we need to identify $p_{n+1}=q_1$ and $q_{n+1}=p_1$.

In any case we wish to identify the representations defined by $A_{n+1}^-$ and $A_1^+$. Recall that $\gamma _0=(-1)^{\varepsilon _0}\gamma _n$, $\gamma _1=(-1)^{\varepsilon _1}\gamma _{n+1}$, and $\alpha _1=(-1)^\eta \alpha _{n+1}$ for $\varepsilon _i,\eta \in \{0,1\}$. We have

\[ k_1\gamma_0-k_0\gamma_1=(-1)^\eta(k_{n+1}\gamma_n-k_n\gamma_{n+1})= (-1)^{\eta+\varepsilon_0}k_{n+1}\gamma_0-(-1)^{\eta+\varepsilon_1}k_n\gamma_1 \]

which implies $k_0=(-1)^{\eta +\varepsilon _1}k_n$ and $k_1=(-1)^{\eta +\varepsilon _0} k_{n+1}$. Consequently the matrices defining $A_{n+1}^-$ and $A_1^+$ can be written as

\[ \begin{pmatrix} (-1)^{\varepsilon_0} c_n\gamma_0 - (-1)^{\varepsilon_1} a_n\gamma_{1} & \\ & ((-1)^{\varepsilon_0}c_n-(-1)^{\eta}k_{1})\gamma_0 + ((-1)^{\eta}k_{0}-(-1)^{\varepsilon_1}a_n)\gamma_{1} \end{pmatrix}, \]

and

\[ \begin{pmatrix} a_1\gamma_{0}-b_1\gamma_1 & \\ & (a_1-k_1)\gamma_{0} + (k_{0}-b_1)\gamma_1 \end{pmatrix}. \]

Thus, in the product case, if we have $a_1=(-1)^{\varepsilon _0}c_n$ and $b_1=(-1)^{\varepsilon _1} a_n$, then $A^-_{n+1}=A^+_1$ and the two naturally glue, finishing the construction. Observe that an arbitrary choice of either the $a_i$, the $b_i$, or the $c_i$ for $i=1,\ldots,n$ uniquely defines the respective other coefficients.

In the twisted case we set $c_n=(-1)^{\varepsilon _0}(a_1-k_1)$ and $a_n=(-1)^{\varepsilon _1}(b_1-k_0)$. Again, an arbitrary choice of the $a_i$, the $b_i$, or the $c_i$ defines the other coefficients uniquely such that this and the previous gluing conditions hold. Now $A_{n+1}^-$ and $A_1^+$ do not agree but arise from one another by swapping the diagonal entries. It follows that the automorphism of $\mathbb {C}^2$ which swaps both factors is equivariant with respect to the actions defined by $A_{n+1}^-$ and $A_1^+$. We use this automorphism to glue $E_n$ to $E_1$ along the fiber over $v_1$. Since the induced automorphism of $\mathbb {CP}^1$ swaps the fixed points $[0:1]$ and $[1:0]$ it follows that in the graph encoding the one-skeleton of the projectivization, $p_{n+1}$ gets glued to $q_1$ and $q_{n+1}$ gets glued to $p_1$ and is, thus, precisely $\Gamma$.

5.3 Obstruction theory for quasitoric manifolds

Let $X$ be a quasitoric manifold of dimension $4k$. Denote by $\pi \colon X \to P$ the projection to the associated simple $2k$-polytope $P$. The preimage under $\pi$ of the interior of $P$ is diffeomorphic to $T^{2k} \times \mathring D^{2k}$ where $\mathring D^{2k}$ denotes the interior of the unit closed ball in $\mathbb {R}^{2k}$. Furthermore, this space can be regarded as the interior of the equivariant free top cell of $X$: recall from the canonical model of a quasitoric manifold (see the beginning of § 5.1) that $X$ is equivariantly homeomorphic to a quotient $T^{2k}\times P/\sim$. Identifying $P$ with $D^{2k}$, the natural projection $T^{2k}\times D^{2k}\rightarrow T^{2k}\times P/\sim$ becomes a characteristic map for a relative CW structure on $(X,A)$, where $A$ is the preimage of the boundary of $P$ under $\pi$.

Note that, although it is not quasitoric, we have the same kind of cellular structure for the $T^2$-action on $X=S^4\subset \mathbb {C}^2\oplus \mathbb {R}$ which acts in standard fashion on the $\mathbb {C}^2$ factor. The space $X$ arises from $A=X_1=\{(v,w,z)\in S^4\,|\,v=0\text { or }w=0\}$ by attaching a single free cell. To see this consider the space $D=\{(x,y,z)\in S^2\,|\,x,y\geq 0\}$, which is a $2$-disk. The map $D\times T^2\rightarrow S^4$, defined as the equivariant extension of the map $D\times \{e\}\cong D\rightarrow S^4$ induced by the inclusion $\mathbb {R}\rightarrow \mathbb {C}$ on the first two components, is a characteristic map for the relative CW structure. Of course, this CW decomposition induces an analogous decomposition for any pullback of the above action along a group automorphism.

Proof. We will use equivariant obstruction theory to prove this lemma, cf. [Reference tom DiecktDi87, Chapter 2, § 3]. Since $E \to A$ is a $T^{2k}$-equivariant complex vector bundle, we have a map $\varepsilon \colon A \to B(T^{2k},U(r))$, where $B(T^{2k},U(r))$ is the classifying space of $T^{2k}$-equivariant $U(r)$-principal bundles, see [Reference tom DiecktDi69, § 3.1]. We would like to extend $\varepsilon$ over the top cell $e^{T^{2k}}_{2k}$ of $X$.

Now $\varepsilon$ has an equivariant extension to $X = A \cup _{\varphi } e^T_{2k}$ if the composition of maps

\[ S^{2k-1} \longrightarrow T^{2k} \times S^{2k-1} \overset{\varphi}{\longrightarrow} A \overset{\varepsilon}{\longrightarrow}B(T^{2k},U(r)) \]

is nullhomotopic, see [Reference tom DiecktDi87, p. 115]. The composition gives an element of $\pi _{2k-1}(B(T^{2k},U(r)))$, but the classifying space $B(T^{2k},U(r))$ is homotopy equivalent to $BU(r)$ (see [Reference tom DiecktDi69, p. 142]) and since $r > k-1$ we have by Bott periodicity $\pi _{2k-1}(B(T^{2k},U(r)))=0$. This implies that $\varepsilon$ can be extended equivariantly to a map $\varepsilon \colon X \to B(T^{2k},U(r))$.

Applying Lemma 5.5 and Remark 5.6 to the vector bundle constructed in § 5.2 we arrive at the following theorem, which is an intermediate step in the proof of Theorem 5.1.

We conclude this section with the proof of the statement on the isotropy groups of the $T$-action on $\mathbb {P}(E)$ in part (i) of Theorem 5.1. Recall first that the weights of the isotropy representation at any of the two fixed points over a vertex $v_i$ of $B$ are (up to sign) $\gamma _{i-1}$, $\gamma _i$, and $\alpha _i = k_i\gamma _{i-1} - k_{i-1}\gamma _i$. If any $k_j\neq \pm 1$, we can therefore find a fixed point in $\mathbb {P}(E)$ such that two of the adjacent weights, considered as characters $T\to S^1$, have a non-trivial common kernel, which then occurs as an isotropy group near the fixed point. Conversely, we assume that all $k_i=\pm 1$. As $\mathbb {P}(E)$ fibers equivariantly over the $T$-manifold $X$, and in $X$ non-trivial stabilizers occur only in the one-skeleton $X_1$, any non-trivial stabilizer in $\mathbb {P}(E)$ necessarily occurs in $\pi ^{-1}(X_1)$. However, this space is, in the notation of § 5.2, the union of 4-dimensional $T^2$-manifolds $\mathbb {P}(E_i)$ that fiber over $S^2$. By assumption on the $k_i$ the $T$-action on $\mathbb {P}(E_i)$ is effective. Hence, the $\mathbb {P}(E_i)$ are torus manifolds with vanishing odd-degree cohomology, which by [Reference Masuda and PanovMP06, Theorem 4.1] have only connected isotropy groups. See Remark 9.6 for an example of a linear realization of a total space of a GKM fibration where not all $k_i=\pm 1$.

6.1 (Stable) almost complex structures

The realization

\[ \mathbb{CP}^1\longrightarrow \mathbb{P}(E)\longrightarrow X \]

constructed in the previous sections is a fibration with structure group $\mathrm {U} (2)$. Let $V_F\subset T\mathbb {P}(E)$ be the subbundle consisting of the tangent spaces of all the fibers. Since each fiber can be identified with $\mathbb {CP}^1$ uniquely up to elements from $\mathrm {U}(2)$, we see that a $\mathrm {U}(2)$-invariant almost complex structure on $\mathbb {CP}^1$ induces on $V_F$ the structure of a complex $T$-vector bundle. A complement of $V_F$ in $\mathbb {P}(E)$ with respect to a $T$-invariant metric can be identified with the pullback $V_X$ of the tangent bundle of $X$ along the projection $\mathbb {P}(E)\rightarrow X$. As $X$ is quasitoric or $S^4$, it has a $T$-invariant stable almost complex structure (see [Reference Buchstaber and PanovBP15, Corollary 7.3.15]). It follows that $T\mathbb {P}(E)=V_F\oplus V_X$ carries a $T$-invariant stable almost complex structure. This finishes the proof of part (i) of Theorem 5.1.

In case the graph fibration is one of the signed graphs, $X$ carries a $T$-invariant almost complex structure by Proposition 5.3. As above, it follows that $\mathbb {P}(E)$ carries an invariant almost complex structure compatible with the map to $X$. For the proof of part (ii) of Theorem 5.1, it remains to check that the signed GKM graphs of $\mathbb {P}(E)\rightarrow X$ with respect to the almost complex structures agree with the given fibration of signed graphs $\Gamma \rightarrow B$. The construction of $X$ and its almost complex structure (see Proposition 5.3) can be carried out such that its signed GKM graph is precisely $B$. Since $\mathbb {P}(E)\rightarrow X$ is compatible with the complex structures, it follows that the given labelling function on oriented edges of ${\Gamma }$ agrees with that induced by the almost complex structure on $\mathbb {P}(E)$ when applied to oriented horizontal edges. On (oriented) vertical edges they a priori only agree up to sign. However, we observe that, since both labelling functions on oriented edges admit a compatible connection, the signs in the vertical edges either agree everywhere or nowhere. In the first case, we are done. In the second case, we may apply the automorphism of $\Gamma$ that interchanges the vertices in each fiber. This transforms one labelling function into the other so the proof of part (ii) is complete.

6.2 Symplectic and Kähler structures

Suppose $E \to X$ is a complex vector bundle of rank $2$ and $X$ a toric $4$-manifold. First note that $X$ is a smooth projective variety [Reference Buchstaber and PanovBP15, Proposition 5.2.2] and, thus, $X$ admits in particular a Kähler structure. We denote by $\omega _X$ the corresponding Kähler form. Since $X$ is projective we deduce from [Reference SchwarzenbergerSch61, Theorem 9] that $E$ is algebraic if the determinant bundle $\det E$ is algebraic. However, since $X$ is toric, $H^2(X) = H^{1,1}(X)$, so that $\det E$ has a holomorphic structure by the Lefschetz theorem on $(1,1)$-classes, cf. [Reference LefschetzLef50]. With the GAGA principle [Reference SerreSer55] we infer that $\det E$ is algebraic, hence $E$. Since $E$ is algebraic, the manifold $\mathbb {P}(E)$ is a smooth projective variety.

From the discussion above we obtain the following result.

In the remaining part of this section we will construct the symplectic and Kähler structure from Theorem 5.1.

Now let $E\rightarrow X$ be a complex, equivariant $T$-bundle. Furthermore let $h$ be a $T$-invariant Hermitian metric on $E$ and let $\omega _X$ be the $T$ invariant Kähler form on $X$. Let $L \to \mathbb {P}(E)$ be the tautological bundle (i.e. $L$ restricted to each fiber of $\mathbb {P}(E)$ is the tautological bundle of $\mathbb {C}\mathbb {P}^{1}$). Since, by construction, $L$ is a subbundle of $\pi ^\ast (E)$, the metric $h$ can be restricted to $L$ and it also induces a metric $h^*$ on the dual bundle $\overline L$. The Chern form $\omega _V$ of the Hermitian line bundle $(\overline L,h^*)$ has the property that it restricts to the Fubini–Study form (associated to the metric $h$) on every fiber of $\mathbb {P}(E) \to X$ and that, furthermore,

\[ \omega_{K} := \omega_V + C \cdot \pi^{\ast}(\omega_X) \]

is a Kähler form on $\mathbb {P}(E)$ for sufficiently large $C$ (see [Reference VoisinVoi07, § 3.3.2]).

The pullback $\pi ^{\ast }(\omega _X)$ is $T$-invariant but $\omega _V$ and, hence, also $\omega _K$ in general are not. However, since $T$ is compact, we can average $\omega _V$ over $T$, obtaining another form $\omega _F$ which is $T$-invariant. As $h$ is $T$-invariant, the $T$ action preserves the Fubini–Study form on the fibers, so for any fiber of $\mathbb {P}(E)\rightarrow X$ and $t\in T$ we have $i^*(t^*\omega _V)=i^*\omega _V$, where $i$ is the fiber inclusion. This implies that also the averaged form $\omega _F$ restricts to the Fubini–Study form on every fiber. The latter is, in particular, symplectic, thus after possibly enlarging the constant $C$, the form

\[ \omega_S := \omega_F + C \cdot \pi^{\ast}(\omega_X) \]

is a $T$-invariant symplectic form on $\mathbb {P}(E)$. Having chosen a fixed sufficiently large $C$, we note that by the same reasoning $t\omega _F+ C\cdot \pi ^{\ast }(\omega _X)$ is symplectic for $t\in (0,1]$. On the symplectic complement $V$ of a fiber, this defines a homotopy between $\omega _S|_V$ and $\pi ^{\ast }(\omega _X)|_V$ through non-degenerate $2$-forms on $V$. Hence, the symplectic structures $(\mathbb {P}(E),\omega _S)$ and $(X,\omega _X)$ are compatible with the fiber bundle in the sense of the definition above Theorem 5.1. Since $H^1(\mathbb {P}(E))=0$ we infer from [Reference Guillemin and SternbergGS84, Addendum to Theorem 28.1] that the action is, in fact, Hamiltonian.