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Equivariant volumes for linearized actions

Suppose given a linearized action on a polarized complex projective manifold (M,L), and assume that the stable locus is non-empty. We study the leading asymptotics of the dimension of the equivariant summands appearing in the space of global sections of high powers of L. We use Kirwan resolutions and an elementary algebro-geometric argument to reduce the problem to the case of non-singular actions (that is, actions for which the stable and semistable loci coincide and are non-empty).


Introduction
Let M be an n-dimensional complex projective manifold,G a g-dimensional reductive connected complex Lie group, and ν :G × M → M a holomorphic action. AG-line bundle (L,ν) on M will mean the assignment of a holomorphic line bundle L on M together with a lifting (a linearization) ν :G × L → L of the action ofG (to simplify notation, we shall generally leaveν understood, and denote aG-line bundle by A, B, L, . . .).
If L is aG-line bundle, there is for every integer k ≥ 0 an induced representation ofG on the complex vector space of holomorphic sections H 0 (M, L ⊗k ). This implies aG-equivariant direct sum decomposition where Λ + is the set of dominant weights for a given choice of a maximal torus T ⊆G and of a fundamental Weyl chamber. For every dominant weight µ, let us denote the associated irreducible finite dimensional representation of G by V µ . For every µ ∈ Λ + , the summand H 0 µ (M, L ⊗k ) isG-equivariantly isomorphic to a direct sum of copies of V µ .
Suppose that the line bundle L is ample. We shall address in this paper the asymptotic growth of the dimensions of the equivariant spaces of sections H 0 µ (M, L ⊗k ), for µ fixed and k → +∞. This problem has been the object of much attention over the years, and has been approached both algebraically [B], [BD] and symplectically -in the latter sense it is part of the broad and general picture revolving around the quantization commutes with reduction principle [GS1], [GGK], [MS], [S].
In this paper, we shall study this problem under the general assumption that the stable locus of L is non-empty, M s (L) = ∅. Thus the most general result in our setting is now the Riemann-Roch type formula for multiplicities proved in [MS]. This is a deep and fundamental Theorem -with a rather complex symplectic proof. However, we adopt a different, more algebrogeometric approach, taking as point of departure the Guillemin-Sternberg conjecture for regular actions (in the sense of §2 below). Our motivation was partly to understand the leading asymptotics for singular actions by fairly elementary algebro-geometric arguments. Besides the hypothesis that M s (L) = ∅, our arguments need an additional technical assumption, namely that the pair (M, L) admits a Kirwan resolution with certain mildness properties. Roughly, every divisorial component of the unstable locus upstairs should map to the unstable locus downstairs (Definition 1); such resolutions will be called mild.
One can produce examples showing that h 0 µ (M, L ⊗k ) may not be described, in general, by an asymptotic expansion, even if the GIT quotient M//G is nonsingular (but see §2 below and the discussion in [P2]). Inspired by the notion of volume of a big line bundle [DEL], we shall then introduce and study the µ-equivariant volume of aG-line bundle L, defined as Because it leads to a concise and simple statement, we shall focus on the special case where the stabilizer K ⊆G of a general p ∈ M is a (necessarily finite) central subgroup. Our methods can however be applied with no conceptual difficulty to the case of an arbitrary principal type (the conjugacy class of the generic stabilizer). This will involve singling out for each µ the kernel K µ ⊆ K of the action of K on the coadjoint orbit of µ, and considering the contribution coming from each conjugacy class of K µ .
Let us then assume that K is a central subgroup. By restricting the linearization to K, we obtain an induced character χ K,L : K → C * .
Another characterμ K : K → C * is associated to the choice of a dominant weight µ ∈ λ + ⊆ t * . Namely, K lies in the chosen maximal torus, and µ K is the restriction to K of the characterμ :T → C * induced by µ by exponentiation, expG(ξ) → e 2πi<µ,ξ> . We may defineμ K =:μ| K : K → C * (the restriction ofμ to K). An alternative description ofμ K is as follows: Let G ⊆G be a maximal compact subgroup, T ⊆ G a maximal torus, and suppose µ ∈ t * , where t * is the Lie algebra of T . If g is the Lie algebra of G, let O µ ⊆ g * be the coadjoint orbit of µ. Since µ is an integral weight, the natural Kähler structure on O µ is in fact a Hodge form, that is, it represents an integral cohomology class. The associated ample holomorphic line bundle A µ → O µ is a G-line bundle in a natural manner. Since the action of K is trivial on O µ , the linearization induces the characterμ K on K.
Let us begin by considering the case where the stable and semistable loci for theG-line bundle are equal and nonempty: M s (L) = M ss (L) = ∅. First, as we shall make use of the Riemann-Roch formulae for multiplicities for regular actions conjectured by Guillemin and Sternberg, and first proved by Meinrenken [M1], [M2], it is in order to recall how these algebro-geometric hypothesis translate in symplectic terms. Let us choose a maximal compact subgroup G ofG. Thus G is a g-dimensional real Lie group, andG is the complexification of G. Let g denote the Lie algebra of G. We may without loss choose a G-invariant Hermitian metric h L on L, such that the unique covariant derivative on L compatible with h L and the holomorphic structure has curvature −2πiΩ, where Ω is a G-invariant Hodge form on M. The given structure of G-line bundle of L, furthermore, determines (and, up to topological obstructions, is equivalent to) a moment map Φ = Φ L : M → g * for the action of G on the symplectic manifold (M, Ω) [GS1]. The hypothesis that M s (L) = M ss (L) = ∅ may be restated symplectically as follows: 0 ∈ g * is a regular value of Φ, and Φ −1 (0) = ∅ [Ki1]. In this case, P =: Let G and Φ be as above. The action of G on Φ −1 (0) is locally free, and the GIT quotient M//G = M s (L)/G may be identified in a natural manner with the symplectic reduction M 0 =: Φ −1 (0)/G, and is therefore a Kähler orbifold. The quantizing line bundle L descends to a line orbibundle L 0 on M 0 .
Similar considerations apply to symplectic reductions at coadjoint orbits sufficiently close to the origin. If theG-line bundle L is replaced by its tensor power L ⊗k , the Hodge form and the moment map get replaced by their multiples kΩ and Φ k =: kΦ. Given any µ ∈ g * , there exists k 0 such that µ is a regular value of Φ k if k ≥ k 0 . The relevant asymptotic information about the multiplicity of V µ in H 0 (M, L ⊗k ) may then determined by computing appropriate Riemann-Roch numbers on these orbifolds [Ka], [M2].
Let O µ ⊆ g * be the coadjoint orbit through µ; since µ is integral, the Kirillov symplectic form σ µ is a Hodge form on the complex projective manifold O µ . By the Konstant version of the Borel-Bott theorem, there is an ample Let then M (k) µ be the Weinstein symplectic reduction of M at µ with respect to the moment map Φ k = kΦ L (k ≫ 0). Using the normal form description of the symplectic and Hamiltonian structure of (M, Ω) in the neighbourhood of the coisotropic submanifold P = Φ −1 (0) [M2], [G], [GS3], one can verify that M (k) µ is, up to diffeomorphism, the quotient of P × O µ by the product action of G. In other words, M (k) µ is the fibre orbibundle on M 0 = P/G associated to the principal G-orbibundle q : P → M 0 and the G-space O µ (endowed with the opposite Kähler structure); in particular its diffeotype is independent of k for k ≫ 0. Let p µ : M (k) µ → M 0 be the projection.
Let θ be a connection 1-form for q ( [GGK], Appendix B). By the shifting trick, the symplectic structure Ω (k) µ of the orbifold M (k) µ is obtained by descending the closed 2-form kι * (Ω)+ < µ, F (θ) > −σ µ on P × O µ down to the quotient (the symbols of projections are omitted for notational simplicity). The minimal coupling term < µ, F (θ) > −σ µ is the curvature of the line There is a natural inclusionP µ,K ⊆ P µ . Now let Σ µ =:P µ /G, Σ µ,K =: There is a natural orbifold complex immersion Σ µ → M (k) µ , with complex normal orbi-bundle N Σµ , and Σ µ,K ⊆ Σ µ is the union of the |K| connected components mapping dominantly (and isomorphically) onto M (k) µ . The orbifold multiplicity of Σ µ,K is constant and equal to |K|. Let L 0 be the line orbi-bundle on M 0 determined by descending L, and let L 0 be its pull-back to Σ 0 . Let r be the complex dimension of O µ , so that dim M (k) µ = n − g + r. After [M1] and [M2], the multiplicity N (k) (µ) of the irreducible representation V µ in H 0 (M, L ⊗k ) is then given by: Now suppose that χ k K,L ·μ K ≡ 1. Then the action of K on  * (L ⊗k ) ⊠ A µ is not trivial, where  : P ֒→ M is the inclusion. Therefore, the fiber of p * µ (L ⊗k 0 ) ⊗ R µ on the smooth locus of M 0 is a nontrivial quotient of C, and N (k) (µ) = 0 in this case. If there exists k such that χ k K,L ·μ K ≡ 1, on the other hand, the same condition holds with k replaced by k + ℓe, where e is the period of χ K,L and ℓ ∈ Z is arbitrary. Thus k may be assumed arbitrarily large. Passing to the original Kähler structure of O µ in the computation, and recalling that dim(V µ ) = (r!) −1 Oµ σ r µ , we easily obtain: 3 The asymptotics of equivariant volumes.
More precisely, this will hold whenever m = 1 + p e, where e is the period of χ K,L and p ∈ N, p ≫ 0.
As a corollary, we obtain the following equivariant version of Lemma 3.5 of [DEL] (for the case of finite group actions, see Lemma 3 of [P1]).

Corollary 1. Under the same hypothesis, let H be a G-line bundle on M.
Then for any ǫ > 0 there exist arbitrarily large integers m > 0 (how large depending on ǫ and µ) such that υ µ (L ⊗m ⊗ H −1 ) ≥ m n (υ µ (L) − ǫ).

Similarly,
Corollary 3. Under the same hypothesis, Proof of Theorem 2. The proof is inspired by arguments in [DEL]. If υ µ (L) = 0, there is nothing to prove; we shall assume from now on that υ µ (L) > 0, and for simplicity write L for f * (L). Thus there exists 0 ≤ r < e such that χ r K,L ·μ K = 1, where e is the period of χ K,L . If ℓ ≫ 0, by the above Thus there is no loss of generality in replacing H by H ⊗ B ⊗ℓe for some ℓ ≫ 0. In view of the hypothesis on H, we may thus assume without loss the existence of 0 = σ ∈ H 0 (M , H) G Since furthermore the class of B in the G-ample cone introduced in [DH] lies in the interior of some chamber, the class of H ⊗ B ⊗ℓe lies in the same chamber for ℓ ≫ 0. Hence we may as well assume that H is a very ample By definition of υ µ , there exists a sequence s ν ↑ +∞ such that Necessarily s ν ≡ r (mod e), ∀ ν ≫ 0. We shall show that the stated inequality holds if p ≫ 0 and m =: 1 + p e.
Lemma 1. Fix p ≫ 0 and let m =: 1 + p e. There is a sequence k ν ↑ +∞ such that with k ν ≡ r (mod e) and furthermore k ν ≡ e (mod m) for every ν.
Proof. Let x > 0 be an integer. We may assume that there is a non-zero section 0 = τ ∈ H 0 (M, L ⊗xpe ) G . Thus, there are injections and for ν ≫ 0 we have (7) Perhaps after passing to a subsequence, we may assume without loss of generality that s ν ≡ r ′ (mod. m), for a fixed 0 ≤ r ′ ≤ m−1. Thus, s ν = ℓ ν m+ r ′ . Let now x > 0 be an integer of the form x = sm + r ′ − e, s ∈ N. Then xpe = x(m − 1) and Now we need only set k ν =: s ν + xpe.
Proof. If r ≫ 0, the equivalence class of the G-line bundles H ⊗re ⊗ L −e in the G-ample cone lie in the interior of the same chamber as the class of H. Thus, they share the same stable and semistable loci, and determine the same GIT quotientM 0 . The G-line bundles H ′ =: H ⊗e and L ′ =: L ⊗−e descend to genuine line bundles H ′ 0 and L ′ 0 on M 0 , and H ′ is ample. Thus, for some a ≫ 0 the ample line bundles H ′⊗ae and H ′⊗ae ⊗ L ′ are globally generated. Arguing as in [GS1], If R and S are G-line bundles onM , any 0 We may now use the G-equivariant exact sequences of sheaves: for 0 ≤ j < s, to conclude inductively that Proof. Perhaps after replacing H by some appropriate power we may assume that the linear series H 0 (M , H) G is base point free away fromM u (H). The claim then follows from Bertini's Theorem.
We now make use of the mildness assumption on f . If m ≫ 0 is fixed, ν ≫ 0 and 0 ≤ s ≤ ℓ ν + am, then the moment map of the (not necessarily ample) line bundle L ⊗kν ⊗H ⊗−s is bounded away from 0 in the neighbourhood of D u . An adaptation of the arguments in §5 of [GS1] (applied on some resolution of singularities of each D uj ) then shows that h 0 µ (D u , L ⊗kν ⊗ H ⊗(−s) Du ) = 0 (if D u is not reduced, we need only filter O D uj (k ν L − s H) by an appropriate chain of line bundles).
Since furthermore on each D sj we may find a non-vanishing invariant section of H, we obtain: Proof. Given the equivariant injective morphism of structure sheaves O Ds −→ i O D si , we may as well assume that D s is a reduced and irreducible Ginvariant divisor, descending to a Cartier divisor D 0 on the quotient.
Let R =: L ⊗ B, with associated moment map Φ R . By the generality in its choice, we may assume that D s ∈ H 0 (M , H) G is non-singular in the neighbourhood of Φ −1 R (0), and is transversal to it. In fact, the singular locus of H 0 (M , H) G is the unstable locus of H. Furthermore, by the arguments of Lemma 3 in [P2] and compactness one can see the following: There exist a finite number of holomorphic embeddings ϕ i : B →M , where B ⊆ C n−g is the unit ball, satisfying i): ϕ i (B) ⊆ Φ −1 R (0); ii): as a submanifold of Φ −1 R (0), ϕ i (B) is transversal to every G-orbit; iii): the union i ϕ i (B) maps surjectively ontoM //G. In view the local analytic proof of Bertini's theorem in [GH], we may assume that D is transversal to each ϕ i (B). By G-invariance, it is then transversal to all of Φ −1 R (0). Let g :D s → D s be a G-equivariant resolution of singularities [EH], [EV]. For s ≫ 0, g * (R ⊗s )(−F ) is an ample G-line bundle onD s , where F is some effective exceptional divisor. Since 0 ∈ g * is a regular value of g • Φ R :D s → g * and belongs to its image, the same holds for the moment map of g * (R ⊗s )(−F ), for s ≫ 0. Having fixed s ≫ 0, let us also choose By Corollary 3, we have υ µ (L) = lim sup m→+∞ υ µ (f * (L ⊗m ) (−E)) m n−g .
By the results in §2, under the appropriate numerical hypothesis, The statement follows.