A Remark on the Gradient Map

Leonardo Biliotti, Alessandro Ghigi, and Peter Heinzner^{1}

Received: February 7, 2014

Communicated by Thomas Peternell

Abstract. For a Hamiltonian action of a compact group U of isometries on a compact Kähler manifold Z and a compatible subgroup G of U^{ℂ}, we prove that for any closed G–invariant subset Y ⊂ Z the image of the gradient map μ_{𝔭}(Y) is independent of the choice of the invariant Kähler form ω in its cohomology class [ω].

2010 Mathematics Subject Classification: 53D20

1. Introduction

Let (Z, ω) be a compact Kähler manifold and let U be a compact connected semisimple Lie group such that U^{ℂ} acts holomorphically on Z, U preserves ω and there is a momentum map μ : Z → 𝔲^{∗}. Let G ⊂ U^{ℂ} be a compatible subgroup. By this we mean a subgroup which is compatible with the Cartan involution Θ of U^{ℂ} which defines U, i.e. if 𝔭 = 𝔤 ∩ i𝔲 and K = U ∩ G, then G = K · exp 𝔭. Let μ_{𝔭} : Z → 𝔭 be the associated gradient map (see [4, 5] or section 2). In this note we prove the following.

Theorem 1. Let Y ⊂ Z be a closed G-stable subset. Then up to translation the set μ_{𝔭}(Y) is independent of the choice of the invariant Kähler form ω in the cohomology class [ω].

1The first two authors were partially supported by a grant of Max-Plank Institute für Mathematik, Bonn and by FIRB 2012 MIUR “Geometria differenziale e teoria geometrica delle funzioni”. author was partially supported also by PRIN 2009 MIUR ”Moduli, strutture geometriche e loro applicazioni”. The third author was partially supported by DFG-priority program SPP 1388 (Darstellungstheorie).

Since Z is compact and G is compatible there is a stratification of Z analogous to the Kirwan stratification, see [4]. This gives a stratification of any closed G– invariant subset Y of Z, by intersecting the strata in Z with Y. It follows from Theorem 1 that when the momentum map is properly normalized (see Lemma 2) this stratification does not depend on the choice of ω in its cohomology class. When Z is a projective manifold and ω is the pull-back of a Fubini-Study form via an equivariant embedding of Z in ℙ^{N}, Kirwan [6, §12] proved that the stratification in terms of a properly normalized μ can be defined purely in terms of algebraic geometry. In the present note we give a proof of this fact for a general compact Kähler manifold Z in the more general setting of gradient maps for actions of compatible subgroups on closed G–invariant subsets of Z. Another consequence of the above is the following. Assume that Z is a projective manifold and that [ω] is an integral class. Let Y ⊂ Z be a closed G-invariant real semi-algebraic subset whose real algebraic Zariski closure is irreducible. Let 𝔞 ⊂ 𝔭 be a maximal subalgebra and let 𝔞_{+} be a closed Weyl chamber in 𝔞. Then A(Y)_{+} := μ_{𝔭}(Y) ∩ 𝔞_{+} is convex (see [2], which deals with the case when ω is the restriction of a Fubini-Study metric). Acknowledgements. The first two authors are grateful to the Fakultät für Mathematik of Ruhr-Universität Bochum for the wonderful hospitality during several visits. They also wish to thank the Max-Planck Institut für Mathematik, Bonn for excellent conditions provided during their visit at this institution, where part of this paper was written.

2. Background

Let (Z, ω) be a compact Kähler manifold and let U be a compact Lie group. Assume that U acts on Z by holomorphic Kähler isometries. Since Z is compact the U-action extends to a holomorphic action of the complexified group U^{ℂ}. Assume also that there is a momentum map μ : Z → 𝔲^{∗} ≅ 𝔲, where 𝔲^{∗} is identified with 𝔲 using a fixed U-invariant scalar product on 𝔲 that we denote by ⟨ , ⟩. We also denote by ⟨ , ⟩ the scalar product on i𝔲 such that multiplication by i is an isometry of 𝔲 onto i𝔲. If ξ ∈ 𝔲 we denote by ξ_{Z} the fundamental vector field on Z and we let μ^{ξ} ∈ C^{∞}(Z) be the function μ^{ξ}(z) := ⟨μ(z), ξ⟩. That μ is the momentum map means that it is U-equivariant and that dμ^{ξ} = i_{ξ_{Z}} ω. For a closed subgroup G ⊂ U^{ℂ} let K := G ∩ U and 𝔭 := 𝔤 ∩ i𝔲. The group G is called compatible if G = K · exp 𝔭 [4, 5]. In the following we fix a compatible subgroup G ⊂ U^{ℂ}. If z ∈ Z, let μ_{𝔭}(z) ∈ 𝔭 denote −i times the component of μ(z) in the direction of i𝔭. In other words we require that ⟨μ_{𝔭}(z), β⟩ = −⟨μ(z), iβ⟩ for any β ∈ 𝔭. The map

μ_{𝔭} : Z → 𝔭

is called the gradient map (see [3]) or restricted momentum map. Let μ_{𝔭}^{β} ∈ C^{∞}(Z) be the function μ_{𝔭}^{β}(z) = ⟨μ_{𝔭}(z), β⟩ = μ^{−iβ}(z). Let ( , ) be the Kähler metric associated to ω, i.e. (v, w) = ω(v, Jw). Then β_{Z} is the gradient of μ_{𝔭}^{β} with respect to ( , ).

Example 1. (1) For any compact subgroup K ⊂ U, both K and its complexification G = K^{ℂ} are compatible. In particular G = U^{ℂ} is a compatible subgroup. (2) If G is a real form of U^{ℂ}, then G is compatible. (3) For any ξ ∈ i𝔲, the subgroup G = exp(ℝξ) is compatible.

Next we recall the Stratification Theorem for actions of compatible subgroups. Given a maximal subalgebra 𝔞 ⊂ 𝔭 and a Weyl chamber 𝔞^{+} ⊂ 𝔞 define

η_{𝔭} : X → ℝ η_{𝔭}(x) := \frac{1}{2}||μ_{𝔭}(x)||^{2}

C_{𝔭} := Crit(η_{𝔭})  B_{𝔭} := μ_{𝔭}(C_{𝔭})  B_{𝔭}^{+} := B_{𝔭} ∩ 𝔞^{+}

X(μ) := \{x ∈ X : \overline{G · x} ∩ μ_{𝔭}^{−1}(0) ≠ ∅\}

where X is a compact G-invariant subset of Z. Points lying in X(μ) are called semistable. Using semistability and the function η_{𝔭} one can define a stratification of X in the following way, see [6] and [4]. For β ∈ B_{𝔭}^{+} set

X_{||β||^{2}} := \{x ∈ X : \overline{exp(ℝβ) · x} ∩ (μ^{β})^{−1}(||β||^{2}) ≠ ∅\}
X^{β} := {x ∈ X : β_{X}(x) = 0}
X_{||β||^{2}}^{β} := X^{β} ∩ X_{||β||^{2}}

X_{||β||^{2}}^{β+} := {x ∈ X_{||β||^{2}} : lim_{t→−∞} exp(tβ) · x exists and it lies in X_{||β||^{2}}^{β}}

G^{β+} := {g ∈ G : the limit lim_{t→−∞} exp(tβ)g exp(−tβ) exists in G}.

Set also

G^{β} := {g ∈ G : Ad g(β) = β} 𝔭^{β} := {ξ ∈ 𝔭 : [ξ, β] = 0}.

The group G^{β} = K^{β} ·exp(𝔭^{β}) is a compatible subgroup of U^{ℂ} and the set X_{||β||^{2}}^{β+} is G^{β+}-invariant. Denote by μ_{𝔭^{β}} the composition of μ_{𝔭} with the orthogonal projection 𝔭 → 𝔭^{β}. Then μ_{𝔭^{β}} is a gradient map for the G^{β}-action on X_{||β||^{2}}^{β+}. We set μ̂_{𝔭^{β}} := μ_{𝔭^{β}} − β. Since β lies in the center of 𝔤^{β} and since G^{β} is a compatible subgroup of (U^{β})^{ℂ} = (U^{ℂ})^{β}, it is a gradient map too. We let S^{β+} denote the set of G^{β}-semistable points in X_{||β||^{2}}^{β+} with respect to μ̂_{𝔭^{β}}, i.e.

S^{β+} := \{x ∈ X_{||β||^{2}}^{β+} : \overline{G^{β} · x} ∩ μ_{𝔭^{β}}^{−1}(β) ≠ ∅\}.

The set S^{β+} coincides with the set of semistable points of the group G^{β} in X_{||β||^{2}}^{β+} after shifting. By definition the β-stratum is given by S_{β} := G · S^{β+}.

Stratification Theorem. (See [4, Thm. 7.3]) Assume that X is a compact G-invariant subset of Z. Then B_{𝔭}^{+} is finite and

X = ⨆_{β∈B_{𝔭}^{+}} S_{β}.

Moreover

\overline{S_{β}} ⊂ S_{β} ∪ ⋃_{||γ||>||β||} S_{γ}.

3. Proof of Theorem 1

For a U-invariant function f on Z we set

ω̃ := ω + dd^{c}f

where d^{c}f := −2J^{∗}df. Since Z is compact and U acts by holomorphic transformations, any U-invariant Kähler form ω̃ in the Kähler class [ω] can be written in this way. Since pluriharmonic functions on Z are constant, the function f is unique up to a constant.

Lemma 2. If μ : Z → 𝔲 is a momentum map for the U-action on Z with respect to ω, then the map μ̃ : Z → 𝔲 defined by

μ̃^{ξ} := μ^{ξ} − d^{c}f(ξ_{Z})  (3)

is a momentum map for the U-action on Z with respect to ω̃.

Proof. That μ̃ is a momentum map follows from Cartan formula using that L_{ξ_{Z}} d^{c}f = d^{c}L_{ξ_{Z}} f = 0. This in turn follows from the assumption that the action of U is holomorphic and f is U-invariant. □

A more precise version of Theorem 1 is the following.

Theorem 4. For any closed G-stable subset Y ⊂ Z we have μ_{𝔭}(Y) = μ̃_{𝔭}(Y).

Proof. Let 𝔞 ⊂ 𝔭 be a maximal subalgebra and set A := exp 𝔞. The group A is a compatible subgroup. Let μ_{𝔞} : Z → 𝔞 be the restricted gradient map. Any connected subgroup B ⊂ A is compatible. Given such a B, set Z^{(B)} := {z ∈ Z : A_{z} = B}. A connected component S of Z^{(B)} will be called an A-stratum of type 𝔟. For a given S let C denote the connected component of Z^{B} containing S. Then C is a complex submanifold of Z and the Slice Theorem (see Theorem 14.10 and 14.21 in [3] or Theorem 2.2 in [2]) applied to the A-action on C shows that S is open and dense in C. Let A^{c} be the Zariski closure of A in U^{ℂ}. The group A^{c} is a compatible subgroup of U^{ℂ}, A^{c} ∩ U = T is a torus and A^{c} = T exp(i𝔱), where 𝔱 denotes the Lie algebra of T. Moreover \overline{S} is A^{c}-stable [2, Lemma 3.3 (1)]. Denote by μ_{𝔱} : Z ⟶ 𝔱 the momentum map obtained by projecting μ : Z ⟶ 𝔲 to 𝔱, and denote by Π : i𝔱 → 𝔞 the orthogonal projection. Then μ_{𝔞} = Π ∘ iμ_{𝔱} and μ_{𝔞}(\overline{S}) = Π(iμ_{𝔱}(\overline{S})). By the convexity theorem of Atiyah-Guillemin-Sternberg μ_{𝔱}(\overline{S}) is a convex polytope and its vertices are images of points fixed by A^{c}. It follows that μ_{𝔞}(\overline{S}) is a convex polytope as well. Since Π is linear, any vertex of μ_{𝔞}(\overline{S}) is the projection of at least one vertex of iμ_{𝔱}(\overline{S}). Therefore μ_{𝔞}(\overline{S}) is the convex hull of μ_{𝔞}(\overline{S}^{A}). Now we use Lemma 2: if x ∈ \overline{S}^{A}, then ξ_{Z}(x) = 0, so μ̃^{ξ}(x) = μ^{ξ}(x), for any ξ ∈ 𝔞. Therefore μ̃_{𝔞}(x) = μ_{𝔞}(x) for every A-fixed point x. It follows that both μ_{𝔞}(\overline{S}) and the affine subspace spanned by μ_{𝔞}(S) do not depend on the choice of the Kähler form ω. Let Σ be the collection of affine hyperplanes of 𝔞 that are affine hulls of μ_{𝔞}(\overline{S}) for some A-stratum S. Set P := μ_{𝔞}(Z) and

P_{0} := P − ⋃_{H∈Σ} P ∩ H.

(This construction is similar to the one in [2, §§4-5].) The set P_{0} is an open subset of 𝔞. Let C(P_{0}) denote the set of its connected components. This is a finite set. For γ ∈ C(P_{0}) let P (γ) be the closure of the connected component γ. Then P (γ) is a convex polytope. Since both P and the hyperplanes H are independent of ω, also the polytopes P (γ) do not depend on ω. By [2, Corollary 5.8]

μ_{𝔭}(Y) ∩ 𝔞 = ⋃_{γ∈F (ω)} P (γ),

where F (ω) ⊂ Γ is some subset of C(P_{0}). One can join ω to ω̃ continuously, e.g. by ω_{t} := ω + tdd^{c}f. Then μ̃_{t} := μ − td^{c}f(·_{Z}) also depends continuously on t. So P (γ) ⊂ μ_{𝔭}(Y) ∩ 𝔞 if and only if P (γ) ⊂ μ_{t,𝔭}(Y) ∩ 𝔞. Therefore F (ω_{t}) is independent of t. Thus μ_{𝔭}(Y) ∩ 𝔞 = μ̃_{𝔭}(Y) ∩ 𝔞. Since μ_{𝔭}(Y) = K(μ_{𝔭}(Y) ∩ 𝔞) this implies μ_{𝔭}(Y) = μ̃_{𝔭}(Y). □

Corollary 5. Assume that Z is connected and let ω and ω̃ be two cohomologous Kähler forms with momentum maps μ and μ̃ respectively as in Lemma 2. Then μ̃ is the unique momentum map such that μ(Z) = μ̃(Z).

Proof. Since two momentum maps with respect to ω̃ differ by addition of an element of the center of 𝔲, it is clear that there is at most one such map with the image equal to μ(Z). To complete the proof it is therefore enough to check that μ̃(Z) = μ(Z). This is a special case of the previous theorem. □

Theorem 6. Let ω and ω̃ be two cohomologous Kähler forms on Z, with momentum maps μ and μ̃ respectively as in Lemma 2. Then the set B_{𝔭}^{+} is the same for both momentum maps and the two stratifications of X coincide.

Proof. By [4, Corollary 7.6]

B_{𝔭} = \{β ∈ 𝔭 : there exists x ∈ X : \frac{||β||^{2}}{2} = inf_{G·x} η_{𝔭} and β ∈ μ_{𝔭}(\overline{G · x})\}. (7)

Moreover for β ∈ B_{𝔭}

S_{β} = \{x ∈ X : \frac{||β||^{2}}{2} = inf_{G·x} η_{𝔭} and β ∈ μ_{𝔭}(\overline{G · x})\}.  (8)

For any point x ∈ X, the set \overline{G · x} is closed and G-invariant. Hence by Theorem 4 μ_{𝔭}(\overline{G · x}) = μ̃_{𝔭}(\overline{G · x}). From this it follows that inf_{G·x} η_{𝔭} = inf_{G·x} η̃_{𝔭}, where η̃_{𝔭} := ||μ̃_{𝔭}||^{2}/2. The result follows from (7) and (8). □

¿From the above we obtain the following generalization.

Corollary 9. If Z is a complex projective manifold, U is a compact connected semisimple Lie group acting on Z, ω is a U-invariant Hodge metric and Y ⊂ Z is a closed G-invariant real semi-algebraic subset whose real algebraic Zariski closure is irreducible, then A(Y)_{+} is convex. Moreover if G is semisimple, then X(μ) is dense (if it is nonempty).

Proof. By assumption there is a very ample line bundle L → Z such that [ω] = 2π c_{1}(L)/m for an interger m > 0. Let ω_{F S} be a U-invariant Fubini-Study metric on ℙ(H^{0}(Z, L)^{∗}). Let μ_{F S} be the momentum map with respect to ω_{F S}|_{Z}. In [2] the convexity theorem has been proved for μ_{F S}. A rescaling in the symplectic form yields a corresponding rescaling in the momentum map. Therefore the convexity theorem also holds for the momentum map μ̃ relative to the symplectic form ω̃ := ω_{F S}/m. So it holds also for μ, since μ_{𝔭}(Y) = μ̃_{𝔭}(Y) by Theorem 4. The proof of the last statement is similar: see [2] and Corollary 5. □

Corollary 10. Under the same assumptions, any local minimum of ||μ_{𝔭}||^{2} is a global minimum.

Proof. This follows since ||μ_{𝔭}||^{2} is K-invariant and μ(Z)_{+} is a convex subset of 𝔞_{+}. □

Corollary 11. If ω and ω′ are cohomologous Kähler forms on Z with momentum maps μ and μ̃ as in Lemma 2, then X(μ) = X(μ̃).

Proof. It is enough to observe that X(μ) = S_{0}. □

References

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[3] P. Heinzner and G. W. Schwarz. Cartan decomposition of the moment map. Math. Ann., 337(1):197–232, 2007.

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[5] P. Heinzner and H. Stötzel. Critical points of the square of the momentum map. In Global aspects of complex geometry, pages 211–226. Springer, Berlin, 2006.

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Leonardo Biliotti Università di Parma leonardo.biliotti@unipr.it Alessandro Ghigi Università di Milano Bicocca alessandro.ghigi@unimib.it

Peter Heinzner Ruhr Universität Bochum peter.heinzner@rub.de 1024

Documenta Mathematica 19 (2014)