Common Singularities of Commuting Vector Fields

We study the singularities of commuting vector fields of a real submanifold of a K\"ahler manifold $Z$.


Introduction
Let (Z, ω) be a connected Kähler manifold with an holomorphic action of a complex reductive group U C , where U C is the complexification of a compact connected Lie group U with Lie algebra u.We also assume ω is U -invariant and that there is a U -equivariant momentum map µ : Z → u * .
By definition, for any ξ ∈ u and z ∈ Z, dµ ξ = i ξ Z ω, where µ ξ (z) := µ(z)(ξ) and ξ Z denotes the fundamental vector field induced on Z by the action of U, i.e., ξ Z (z) := d dt t=0 exp(tξ)z (see, for example, [14] for more details on the momentum map).Since U is compact we may identify u ∼ = u * by an Ad(U )-invariant scalar product on u.Hence, we consider a momentum map as a u-valued map, i.e., µ : Z → u.Recently, the momentum map has been generalized to the following settings [11,12].
We say that a subgroup G of U C is compatible if G is closed and the Cartan decomposition The inclusion ip ֒→ u induces by restriction, a K-equivariant map µ ip : Z → (ip) * by composing the momentum map µ with the restriction map u * → (ip) * .Using a Ad(U )invariant scalar product on iu requiring multiplication by i to be an isometry between u and iu, µ ip can be viewed as the orthogonal projection of iµ(z) onto p given as µ p : Z → p.Let for any β ∈ p and z ∈ Z. Then grad µ β p = β Z where grad is computed with respect to the Riemannian metric induced by the Kähler structure.The map µ p is called the gradient map associated with µ.In this paper, a G-invariant compact connected locally closed real submanifold X of Z is fixed and the restriction of µ p to X is denoted by µ p .Then µ p : X → p is a K-equivariant map such that gradµ β p = β X , where the gradient is computed with respect to the induced Riemannian metric on X denoted by (•, •).By the Linearization Theorem [12,15], µ β p is a Morse-Bott function [3,12] and the limit exists and it belongs to X β := {z ∈ X : β X (z) = 0} for any x ∈ X.The Linearization Theorem [12,15] also proves that any connected component of X β is an embedded submanifold, see for instance [3,12].
Let C 1 , . . ., C k be the connected components of X β and W i := {x ∈ X : lim t→+∞ exp(tβ)x ∈ C i }.Then µ β p (C i ) = c i and applying again the Linearization Theorem [12,15], the submanifold C i is a connected component of (µ β p ) −1 (c i ).One of the most important Theorem of Morse theory proves that W i is an embedded submanifold, which is called unstable manifold of the critical submanifold C i , and Let T be a torus of U .This means that T is a connected compact Abelian subgroup of U [1].By a Theorem of Koszul, [10], the connected components of Z T := {x ∈ Z : T • x = x} are embedded Kähler submanifolds of Z.Let t be the Lie algebra of T .It is well-known that the set β ∈ t : exp(Rβ) = T , contains a dense subset [1].Hence, (1) for some β ∈ t.This means Z T is the set of the singularities of the vector field β Z .Moreover, Z T is the image of the gradient flow ϕ β ∞ defined by µ β .In this paper, we investigate the fixed point set of the action of an Abelian compatible subgroup of U C acting on a real submanifold of Z.Let a ⊂ p be an Abelian subalgebra and A = exp(a).Then the A-gradient map on X is given by µ a = π a • µ p , where π a : p −→ a denotes the orthogonal projection of p onto a.Since A is Abelian, then by Lemma 2.2, for any p ∈ X, the stabilizer A p := {a ∈ A : ap = p} = exp(a p ), where a p is the Lie algebra of A p .Therefore then X A is the set of the common singularities of the commuting vector fields (α 1 ) X , . . ., (α n ) X .
Our first main result is the following Hence X A is the set of the singularities of a vector field β X for some β ∈ a and so the critical points of the Morse-Bott function µ β p .We point out that X A contains much information of the geometry of both the A-gradient map and the G-gradient map.Indeed, for any [2,5,13], where conv(•) denotes the convex hull of (•).In particular µ a (X A ) is a finte set and conv(µ a (X)) = conv(µ a (X A )) and so a polytope.Moreover, if a ⊂ p is a maximal Abelian subalgebra, then conv(µ p (X)) is given by Kconv(µ a (X)) [4].
The second main result proves that the existence of β ∈ a such that the limit map associated with the gradient flow of µ β p defines a map from X onto X A .Hence, the set X A is the image of the gradient flow of the Morse-Bott function µ β p for some β ∈ a.Let α 1 , . . ., α n ∈ a be a basis of a. Then Theorem 1.2.Let α 1 , . . ., α n ∈ a be a basis of a.There exists δ > 0 such that for any

Proof of the main results
Suppose X ⊂ Z is a G-invariant compact connected real submanifold of Z with the gradient map µ p : If G x acts on a manifold S, then G × Gx S denotes the associated bundle with principal bundle G → G/G x defined as the quotient of G × S by the G x -action h(g, s) = (gh −1 , hs).We recall the Slice Theorem; see [12] for details.
Theorem 2.1.[Slice Theorem [12,Thm. 3.1], [15]] If x ∈ X and µ p (x) = 0, there are a G x - This follows applying the previous theorem to the action of G β on X.Indeed, it is well known that G β = K β exp(p β ) is compatible [7] and the orthogonal projection of iµ onto p β is the G β -gradient map µ p β .The group G β is also compatible with the Cartan decomposition of (U C ) β = (U C ) iβ = (U iβ ) C and iβ is fixed by the U iβ -action on u iβ .This implies that where π u iβ is the orthogonal projection of u onto u iβ , is the U iβ -shifted momentum map.The associated G β -gradient map is given by Hence, if G is commutative, then we have a Slice Theorem for G at every point of X, see [12, p.169] and [15] for more details.
If β ∈ p, then β X is a vector field on X, i.e. a section of the bundle T X.For x ∈ X, Accordingly the differential of β X , regarded as a section of T X, splits into a horizontal and a vertical part.The horizontal part is the identity map.We denote the vertical part by dβ X (x).The linear map dβ X (x) ∈ End(T x X) is indeed the so-called intrinsic differential of β X , regarded as a section in the tangent bundle T X, at the vanishing point x.
Let {ϕ t = exp(tβ)} be the flow of β X .There is a corresponding flow on T X.Since ϕ t (x) = x, the flow on T X preserves T x X and there it is given by dϕ t (x) ∈ Gl(T x X).Thus we get a linear R-action on T x X with infinitesimal generator dβ X (x).
Corollary 2.1.2.If β ∈ p and x ∈ X is a critical point of µ β p , then there are open invariant neighborhoods S ⊂ T x X and Ω ⊂ X and an R-equivariant diffeomorphism Ψ : S → Ω, such that 0 ∈ S, x ∈ Ω, Ψ(0) = x.(Here t ∈ R acts as dϕ t (x) on S and as ϕ t on Ω.) Proof.Since exp : p −→ G is a diffeomorphism onto the image, the subgroup H := exp(Rβ) is closed and so it is compatible.Hence, it is enough to apply the previous corollary to the H-action on X and the value at x of the corresponding gradient map.
Let α, β ∈ p be such that [α, β] = 0 and let a be the vector space in p generated by α and β.
By the above Lemma, it follows that X A = X β ∩ X β , where A = exp(a).Lemma 2.3.Let β, α ∈ p be such that [β, α] = 0.There exists δ > 0 such that for any ǫ ∈ (0, δ) Proof.Let ǫ > 0 and let A = exp(a), where a = span(α, β).Since the exponential map is a diffeomorphism restricted on p, it follows that A is a closed and compatible subgroup of G.
Let X A denote the fixed point set of A, i.e., X A = {z ∈ X : A • x = x}.By Lemma 2.2, X A = X β ∩ X α .By Corollary 2.1.2,both X β ∩ X α and X α+ǫβ are compact submanifolds satisfying X β ∩ X α ⊆ X α+ǫβ .Since X α+ǫβ is A-invariant, and so there exists A-gradient map [12], any connected component of  Let x ∈ X. Set x 1 := lim t→∞ exp(tα 1 )x and x i = lim t→∞ exp(tα i )x i−1 for i = 2, . . ., n.If x n ∈ Ω j , for some j = 1, . . ., k, then there exits δ j > 0 such that any 0 < ǫ 2 , . . ., ǫ n < δ j we have induces a Cartan decomposition of G.This means that the map K × p → G, (k, β) → kexp(β) is a diffeomorphism where K := G ∩ U and p := g ∩ iu; g is the Lie algebra of G.In particular K is a maximal compact subgroup of G with Lie algebra k and that g = k ⊕ p.
be the connected component of x and let C ′ be the connected component of X α+ǫβ containing C. Since x is fixed by A, by the linearization theorem, Corollary 2.1.2,there exists A-invariant open subsets Ω ⊂ X and S ⊂ T x X and a A-equivariant diffeomorphismϕ : S → Ω such that 0 ∈ S, x ∈ Ω, ϕ(0) = x, dϕ 0 = id TxX .Thus we may assume that Ω = R n , α, β are symmetric matrices of order n satisfying [α, β] = 0.Moreover, T x X α+ǫβ = Ker (α + ǫβ) and T x X α ∩ T x X β = Ker α ∩ Ker β.The matrices α and β are simultaneously diagonalizable.Let {e 1 , . . ., e n } be a basis of R n such that αe i = a i e i and βe i = b i e i for i = 1, . . ., n.Let J = {1 ≤ i ≤ n : a i b i = 0}.Pick δ = min{ |a i | |b i | : i ∈ J}.Now, (α + ǫβ)e i = 0 if and only if a i + ǫb i = 0.If a i = 0, then b i = 0 and vice-versa.Therefore, for any ǫ < δ, we get (α + ǫβ)e i = 0, if and only if a i = b i = 0. Therefore, Ker (α + ǫβ) = Ker α ∩ Ker β.Since C ⊂ C ′ and T x C = T x C ′ , keeping in mind that both C and C ′ are compact, it follows that C = C ′ .Since X α+ǫβ has finitely many connected components, it follows that there exists δ > 0 such that for any 0 < ǫ < δ, we have X α ∩ X β = X α+ǫβ , concluding the proof.

Theorem 2 . 4 .
Let a ⊂ p be an Abelian subalgebra and let A = exp(a).Then the set α ∈ a : X A = X α is dense.