Global properties of toric nearly Kähler manifolds

We study toric nearly Kähler manifolds, extending the work of Moroianu and Nagy. We give a description of the global geometry using multi-moment maps. We then investigate polynomial and radial solutions to the toric nearly Kähler equation.


Introduction
A nearly Kähler manifold is an almost Hermitian manifold (M, g, J) such that ∇J is skew symmetric: (∇ X J)X = 0 for every vector field X on M. Each of these can be decomposed as a Riemannian product of nearly Kähler manifolds which are either Kähler, 6-dimensional, homogeneous, or twistor spaces over quaternionic Kähler manifolds of positive scalar curvature [14]. We will focus on the case of 6-dimensional nearly Kähler manifolds that are strict in the sense that they are not Kähler. These are characterized by being the links of metric cones with holonomy G 2 , which makes them Einstein with positive scalar curvature [2].
A main challenge is to construct complete examples of 6-dimensional strictly nearly Kähler manifolds (which will be referred to simply as nearly Kähler manifolds in the rest of the paper). There are homogeneous nearly Kähler structures on S 6 , S 3 × S 3 , ℂP 3 , and the flag manifold SU(3)∕T 2 . In [3], these are shown to be the only homogeneous examples. In [6], cohomogeneity one examples are constructed on S 6 and S 3 × S 3 . No other complete examples are known. The cohomogeneity two case has been studied in [9], which shows that the infinitesimal symmetry group must be (2).
We will skip to cohomogeneity three in exchange for having an abelian symmetry group by studying nearly Kähler manifolds which are toric in the sense that the automorphism group contains a 3-torus. The homogeneous nearly Kähler structure on S 3 × S 3 is the only known example. The general case has been studied in [12], where the local theory is shown to be equivalent to a Monge-Ampère type equation which we will refer to as the toric nearly Kähler equation. The current paper represents the author's efforts to build on this work. One of the main ingredients is the idea of a multi-moment map, which was introduced in [11] as a multi-symplectic generalization of the moment map associated to a symplectic form. A toric nearly Kähler manifold (M, g, J, T) is equipped with two multi-moment maps, which are defined in Sect. 2, where is the Lie algebra of the torus T acting on M. A foundational result in (symplectic) toric geometry [1,7] states that the moment map of a compact toric manifold has connected fibres, and its image is a convex polytope. The first condition implies that the moment map induces a homeomorphism from the space of orbits onto its image. We prove a nearly Kähler analogue of this:

Theorem 1 Let M be a complete toric nearly Kähler manifold with the action of a torus T 3 with Lie algebra . Then the multi-moment maps induce a homeomorphism
The T action is free away from a finite number of orbits in the equator −1 (0) . Moreover the two orbits in −1 (0) are Lagrangian.
This theorem generalizes previous work by the author in [5], which describes multimoment maps of the homogeneous nearly Kähler structure on S 3 × S 3 . In that case, −1 (0) is Cayley's nodal cubic surface, whose 4 nodal singular points correspond to the singular T orbits. By studying the topological consequences of this theorem, we prove the following: Corollary 1 Any complete toric nearly Kähler manifold has at least 4 torus orbits where the action is not free.
As a consequence of this, radial solutions to the toric nearly Kähler equation cannot give complete metrics. By studying the corresponding ODE, we see that the singularity that forms must occur at the Lagrangian orbit.
We also study the case when a hypothetical solution to the toric nearly Kähler equation is polynomial in the natural multi-moment map coordinates. The homogeneous nearly Kähler structure on S 3 × S 3 corresponds to a cubic solution 0 shown in Equation (4). Using an old theorem of Hesse [8], we prove: Theorem 2 Every polynomial solution of the toric nearly Kähler equation with degree at most 5 is equivalent to the cubic solution 0 up to coordinate transformation.
The toric nearly Kähler equation restricted to the space of polynomials is overdetermined for polynomials of degree greater than three, so it is unlikely that there will be other polynomial solutions. However, to show this explicitly is computationally difficult even in the quintic case.

Local theory
In this section we review the local theory of toric nearly Kähler manifolds from [12], although we will use a coordinate invariant treatment in order to make clear the invariance properties of the expressions. First we introduce SU(3) structures, which are a convenient framework for studying nearly Kähler manifolds: Definition 1 An SU(3) structure ( , = + + i − ) on a 6-manifold M is a pair of forms ∈ 2 (M) and ∈ 3 (M, ℂ) satisfying We will refer to these equations as the SU(3) structure equations.

Theorem 3 ([4])
A nearly Kähler structure is equivalent to an SU(3) structure ( , + ) satisfying We will refer to these equations as the nearly Kähler structure equations. Since + i gives a framing of (1,0)M , we can write = i 3 ( + i ) , so that Similarly, the rest of the structures can be given in terms of the multi-moment maps ( , ) , the frame ( , ) , and a matrix For example, where = ⌟C ∈ 1 (M, * ).
Here, by det C ∈ M , ( 3 * ) ⊗2 , we mean the square of the volume form on induced by C. This agrees with the usual determinant in coordinates.
Since the functions , V, and C are T-invariant, they descend to M ∕T , which can be locally identified with 2 * via . Since we can think of as giving coordinates on 2 * , we can think of , V, and C as functions locally given in these coordinates on some U ⊂ 2 * .
These coordinates allow explicit computations of several expressions in terms of a potential function: There exists a function ∶ U → ( 3 * ) 2 whose Hessian in coordinates is C. We also have where r is the Euler vector field for 2 * (so that in coordinates r = i i ).
Combining this with Equation (1) gives the Monge-Ampère type equation which we will refer to as the toric nearly Kähler equation or just (⋆).
Note that with respect to the frame ( , ) , g is represented by the matrix where ∈ (U, 2 * ) is the inclusion (identity) map.
The above theorem has a partial converse:

Theorem 5 ([12]) Every solution of the toric nearly Kähler equation on some open set U of 2 * defines in a canonical way a nearly Kähler structure with 3 linearly independent commuting infinitesimal automorphisms on
Note that if is given by a toric nearly Kähler structure, then (1 − r ) is proportional to the 2 , and D is the expression of g in the frame ( , ) . Now consider the following set with an a priori weaker constraint than U 0 : However, this constraint is not weaker: Proof Since D being positive definite implies that Hess is positive definite, we find that U 0 ⊆Û 0 . It remains to show that D has no null vectors in Û 0 , which implies the reverse inclusion.
Let C = Hess and 2 = 8 3 (1 − r ) . Defining j = C −1 , we find that any null vector for D is of the form (v, w) ∈ ⊕ at some point p ∈Û 0 with Thus v and w are eigenvectors of j 2 at p with eigenvalue −1 . Thus it suffices to show that j 2 ∈ U 0 , End ( ) never attains an eigenvalue −1. Choosing a basis for so that C p is diagonal at any chosen p ∈Û 0 allows one to verify , where we abuse notation by using ♮ to also denote the isomor- where throughout this computation we've been using juxtaposition to denote 'matrix multiplication', or contraction of a single , * index pair. Since V ∈ −1 (0) ⊆ j −1 (0) , we find that j 2 has eigenvalues 0 with multiplicity 1 and − C(V,V) det C with multiplicity 2. By the toric nearly Kähler equation, −1 is an eigenvalue only when = 0 , which is impossible on Û 0 by definition. ◻ This lemma can be used to interpret what goes wrong when trying to find a completion of a local toric nearly Kähler manifold. If some connected M is maximal in the sense that it is not properly contained in a toric nearly Kähler manifold where doesn't vanish, what is happening at the boundary? Using , we can interpret this boundary as a set of points in 2 * . By (1), C is going to remain positive definite as long as doesn't vanish. Thus the previous lemma shows that if does not limit to 0 at the boundary point, then the local solution to the toric nearly Kähler equation cannot be extended to the boundary point. In Sect. 6, we show that local radial solutions can be extended to have the radius defined between 0 and some finite r 0 . The differential equation is singular at 0, while vanishes when the radius is r 0 .

Relation to toric G 2 manifolds
For a strict nearly Kähler manifold (M, , + + i − ) with metric g, consider the Riemannian cone N = M × (0, ∞), g N = r 2 g + d r 2 , where r ∈ (0, ∞) is the radial coordinate. It is well known that N admits a parallel G 2 structure given bŷ If M is toric, then the torus action lifts to a multi-Hamiltonian action on N with respect to the forms and * . This makes N a toric G 2 manifold as studied in [10]. The corresponding multi-moment maps for and * , respectively, are From [10], N ⊕ N maps the set of singular orbits S of N to a graph in 2 * ⊕ 3 * ≅ ℝ 3 ⊕ ℝ . Moreover, N is constant on each connected component of S. In the case when N = M × (0, ∞) is the cone over a toric nearly Kähler manifold, then the radial symmetries of (2) imply that N vanishes on the graph, and moreover each edge of the graph is a radial ray shining out from the origin in 2 * . Since points on the edge of the graph correspond to torus orbits where a single circle collapses, we immediately find Corollary 2 On a toric nearly Kähler manifold, the torus action is free away from a finite set of orbits where a single circle collapses and vanishes.

Global properties
Let (M, , , T) be a connected complete toric nearly Kähler 6-manifold. In this section we will prove the properties of the multi-moment maps claimed in Theorem 1. Recall that we define M = M� −1 (0).

Lemma 3 |M is a submersion.
Proof Lemma 4.1(i) in [12] gives d |M = −4 ⋅ . The result follows since has full rank and does not vanish on M . (2) Since and are T-invariant, they induce maps ̄∶ M∕T → 2 * and ̄∶ M∕T → 3 * , which are called orbital multi-moment maps.

Lemma 5 For any connected component
Proof For any p ∈ M 0 , let be the path between p and some p � ∈ −1 (0) guaranteed by the previous lemma. The map is clearly well defined and continuous. Since M 0 ∕T is connected and ̄− 1 (0) is discrete, the image of F is a single orbit which we will denote by o 0 ∈ M 0 ∕T. In particular, ̄∶ M → U is a double cover ramified over U , with the sign of ̄ distinguishing the points in each ̄ fibre. Thus (̄,̄) is injective. Since U is diffeomorphic to a 3-ball, the image of (̄,̄) is a 3-sphere.
By relabelling if necessary, we can assume that ± is positive on M ± . The component of ( 2 * ⊕ 3 * )�( , )(M) containing 0 can be written as Now each D ± is star-shaped around 0, since ̄(M ± ) is and ± is decreasing in radial directions. Thus D + ∪ D − is star-shaped around 0 as required. ◻ We can now wrap up the proof of the main theorem: Proof of Theorem 1 Corollary 2 combined with Theorem 6 gives most of the claim. T orbits in −1 (0) must be Lagrangian by definition, and there are two of them, since ̄ is a double cover ramified at −1 (0) . ◻

Some topology
We apply the results from the previous section to prove Corollary 1. The obstruction we use to prove this comes from Myers' theorem [13], which asserts that if a complete Riemannian manifold has Ricci curvature positive and bounded away from zero, then the diameter must be bounded. Since the same must be true for the universal cover, the fundamental group must be finite. In particular, the first Betti number must vanish.

Proposition 1 Let (M, , , T) be a connected complete toric nearly Kähler 6-manifold. Then the action of T is not free.
Proof Assume that the action of T is free, so that M is a T 3 bundle over S 3 . It follows that we have the Wang long exact sequence [16] which shows that H 1 (M) ≅ H 1 (T 3 ) ≅ ℤ 3 has positive rank. This contradicts Myers' theorem. ◻ Consider the set S ∈ M∕T of T orbits where the action is not free. By Corollary 2, each orbit s ∈ S has a one-dimensional isotropy group whose Lie algebra is given by a line (s) ∈ ℙ . We will need the following lemma:

Lemma 7 The map ∶ S → ℙ is at most two to one.
Proof Since C is the metric on the torus orbits, S is the vanishing locus of det C . Thus (1) implies that the non-negative functions 2  To see that such a decomposition exists, note that if H is a hyperplane in 2 * ⊕ 3 * disjoint from ( , )(S) , then D ± = (x, y) ∈ 2 * ⊕ 3 * ∶ x ∈̄ M ± , ±y ∈ 0, ± (y) .
there exists a neighbourhood U of H also disjoint from ( , )(S) . Now it is clear that we can find A and B as claimed with ( , )(A ∩ B) = U . Moreover, we see that no two orbits in S ∩ A (respectively S ∩ B ) correspond to the same element of , since by the previous lemma, they would correspond to antipodal points in −1 (0) , which are avoided by this construction. Now we have A ≅ D i and B ≅ D j where i + j = k and D i is a T 3 fibration over the three-ball D 3 with i orbits where circles collapse. Moreover, these circles are different, in the sense that the collapsing directions correspond to different vectors in . Using ≃ to denote homotopy equivalence, we will compute these for the first few D i : D 1 is a neighbourhood of the collapsed orbit. Thus D 1 ≅ T 2 × ℝ 4 by identifying ℝ 4 with D 1 ∕T and T 2 the quotient of T with the circle that collapses. D 2 ∕T is a neighbourhood of a curve C connecting the two collapsing orbits. Thus D 2 retracts to some D 2 such that D 2 ∕T ≅ C . Since the circles that collapse are different, we can write D 2 ≅ (S 1 × S 1 × [0, 1])∕ ∼, where ∼ collapses the first circle at 0 and the second circle at 1. Now Before we proceed to applying Mayer-Vietoris to the decomposition M = A ∪ B , we still need to understand the equatorial region E ∶= A ∩ B.
Proof E must be a T 3 bundle over U ≃ S 2 . Thus E retracts to a T 3 bundle Ē over S 2 . Since Ē is compact, we have the duality h 1 (Ē) = h 4 (Ē) . Part of the Wang sequence is We can now work with the Mayer-Vietoris sequence with respect to the decomposi-

this sequence is
We are now ready to prove the main result of this section:

Proof of Corollary 1
The Mayer-Vietoris sequence at • = 1 gives where the second inequality uses the previous lemma and Myers' theorem. But by

Radial solutions
In this section, we study solutions of the form ( ) = x(t) , where t = 1 2 ‖ ‖ 2 is a radial coodinate, and ‖ ⋅ ‖ is the Euclidean metric on 2 * . These were studied in [12], where they show that the nearly toric equation simplifies to the ODE subject to the constraint where the derivatives are taken with respect to t. The main result is that such a radial solution cannot be complete: Proof Assume that is radially symmetric. Combining this symmetry with Theorem 6, (M) must be a closed 3-disc centred at the origin. Now consider the set of points S in M where the torus action is not free. By Corollary 2, (S) is a discrete set of points in . But by radial symmetry, (S) must be either empty or all of . But (S) is a discrete set, so it can't be . Thus (S) , and hence S is empty. This contradicts Proposition 1. ◻ We now investigate what goes wrong with the ODE to prevent completeness. Local existence of solutions to ODE's will give a local solution x(t) to D(x) = 0 near any prescribed initial 1-jet t 0 , x(t 0 ), x � (t 0 ) satisfying the constraints (3). Let (t − , t + ) be the maximal open interval on which the solution can be extended while satisfying the constraints. t ± must be either a point where x(t) blows up or a boundary point of the constraints. By Lemma 2, 2 = 8 3 (x − 2tx � ) > 0 implies the other constraint x ′ > √ 2t . Thus the boundary condition is simply 2 = 0.
Lemma 10 x(t) does not blow up at t + < ∞.
Proof First note that 2 > 0 implies that (log x) � < 1 2t . Integrating this implies that . Since x(t) is also positive, it cannot blow up in finite time.
On the other hand, integrating . This lower bound for x grows faster as t increases than the upper bound for x in the previous paragraph. Thus t + is finite. ◻

We compute
Note that by Lemma 2, the constraints can be rewritten as 0 Thus the constraints imply that x �2 − 2t > 0 , so the ODE is regular when the constraints hold and t > 0.
Proof Since t − is the boundary point of a maximal domain of an ODE subject to the constraint 2 > 0 , at t − either the ODE is singular and the solution x(t) becomes unbounded or 2 vanishes. Since the ODE is singular at t = 0 , we must have t − ≥ 0 . Since x is positive and increasing, it must be bounded in (t − , t + ) . Since 2 is decreasing, it cannot vanish at t − . Thus t − must a singular point of the ODE, in particular the only one: 0. ◻ By Theorem 7, there must be some singularity for x(t) in [0, t + ] , and by the previous two lemmas it must be at t = 0.
Note that the estimate in Lemma 10 doesn't essentially require radial symmetry: it only uses 2 ∝ − r > 0 . In particular, continuing the discussion following Lemma 2, should not become unbounded as one tries to extend solutions in radial directions away from the origin.

Polynomial solutions
In this section we will try to understand polynomial solutions to the toric nearly Kähler equation. As described in [12], the toric nearly Kähler structure on S 3 × S 3 corresponds to the solution of the toric nearly Kähler equation where j 3 j=1 are coordinates on 2 * induced by the multi-moment map . We will prove Theorem 2 by treating each degree of polynomial separately. First we will introduce some notation. If E is an equation or expression, and m is a monomial in ℝ 1 , 2 , 3 , then [m]E and (m)E will refer, respectively, to the coefficient of m in E, and the part of E which is a multiple of m. We will use ∇ to denote the gradient in j 3 j=1 coordinates, and abuse notation by not distinguishing it from its transpose, or a restricted gradient to an context-appropriate subset of the coordinates. Similarly, ∇ 2 will denote the Hessian, where the set of coordinates may depend on context.
In particular, 6 √ 3b = 3 xxy = − 3 yyy , where the second inequality comes from 3 being harmonic. Now we compute In particular, y 2 (⋆) gives ◻ Combining the previous three propositions gives Theorem 2. are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.