Equivariant volumes for linearized actions Alberto Della Vedova and Roberto Paoletti ∗ 6 0 0 2 n a J 1 Introduction 0 1 Let M be an n-dimensional complex projective manifold, G˜ a g-dimensional ] reductive connected complex Lie group, and ν : G˜ × M → M a holomor- G phic action. A G˜-line bundle (L,ν˜) on M will mean the assignment of a A holomorphic line bundle L on M together with a lifting (a linearization) . h ν˜ : G˜ × L → L of the action of G˜ (to simplify notation, we shall generally t a leave ν˜ understood, and denote a G˜-line bundle by A, B, L,...). m ˜ If L is a G-line bundle, there is for every integer k ≥ 0 an induced [ representation of G˜ on the complex vector space of holomorphic sections 2 H0(M,L⊗k). This implies a G˜-equivariant direct sum decomposition v 3 3 H0(M,L⊗k) ∼= H0(M,L⊗k), 4 µ 2 µM∈Λ+ 1 4 where Λ is the set of dominant weights for a given choice of a maximal torus 0 + / T˜ ⊆ G˜ and of a fundamental Weyl chamber. For every dominant weight µ, h let us denote the associated irreducible finite dimensional representation of t a G˜ by V . For every µ ∈ Λ , the summand H0(M,L⊗k) is G˜-equivariantly m µ + µ isomorphic to a direct sum of copies of V . : µ v Suppose that the line bundle L is ample. We shall address in this paper i X the asymptotic growth of the dimensions r a h0(M,L⊗k) =: dim H0(M,L⊗k) µ µ (cid:0) (cid:1) of the equivariant spaces of sections H0(M,L⊗k), for µ fixed and k → +∞. µ This problem has been the object of much attention over the years, and has been approached both algebraically [B], [BD] and symplectically - in the ∗Address. Dipartimento di Matematica e Applicazioni, Universit`a degli Studi di Milano Bicocca, Via R. Cozzi 53, 20125 Milano, Italy; e-mail: al- [email protected], [email protected] 1 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, Ms(L) 6= ∅. 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 algebro- geometric 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 Ms(L) 6= ∅, our arguments need an additional technical assumption, namely that the pair (M,L) admits a Kirwan resolution with certain mildness prop- erties. 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 h0(M,L⊗k) may not be de- µ scribed, 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 a G˜-line bundle L, defined as (n−g)! υ (L) =: limsup h0(X,L⊗k). (1) µ kn−g µ k→+∞ 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 con- ceptual 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 ⊆ K of theaction of K onthe coadjoint orbit ofµ, and considering µ 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 → C∗. K,L Another character µ˜ : K → C∗ is associated to the choice of a dominant K weight µ ∈ λ ⊆ t∗. Namely, K lies in the chosen maximal torus, and + µ˜ is the restriction to K of the character µ˜ : T˜ → C∗ induced by µ by K exponentiation, exp (ξ) 7→ e2πi<µ,ξ>. We may define µ˜ =: µ˜| : K → C∗ G˜ K K (the restriction of µ˜ to K). An alternative description of µ˜ is as follows: K 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 ⊆ g∗ be the coadjoint orbit of µ. Since µ is an integral weight, the µ 2 natural K¨ahler structure on O is in fact a Hodge form, that is, it represents µ an integral cohomology class. The associated ample holomorphic line bundle A → O is a G-line bundle in a natural manner. Since the action of K is µ µ trivial on O , the linearization induces the character µ˜ on K. µ K We then have: Theorem 1. Let M be a complex projective manifold, G˜ a reductive complex Lie group, ν : G˜ × M → M a holomorphic action. Suppose for simplicity ˜ that the stabilizer of a general p ∈ M is a central subgroup K ⊆ G. Let L be an ample G˜-line bundle on M such that Ms(L) 6= ∅ and admitting a mild Kirwan resolution. Let M =: M//G˜ be the GIT quotient with respect to the 0 linearization L. Let µ ∈ Λ be a dominant weight. Let χ ,µ˜ : K → C∗ + K,L K be the characters introduced above. Then: i): If for every r = 1,...,|K| we have χr ·µ˜ 6= 1 (the constant character K K equal to 1), then H0(M,L⊗k) = 0 for every k = 1,2,...; µ ii): Assume that for some r ∈ {1,...,|K|} we have χr ·µ˜ = 1. Then K K υ (L) = dim(V )2 ·vol(M ,Ω ) > 0. µ µ 0 0 c b Here M is an orbifold, ϕ : M → M is a partial resolution of sin- 0 0 0 gularities, L induces on M a natural nef and big line-orbibundle L˜ , f 0 f 0 with first Chern class c (L˜ ) = Ω , and 1 f0 0 h i e ∧(n−g) vol(M ,Ω ) =: Ω . 0 0 Z 0 M0 f e e f 2 The case Ms(L) = Mss(L) 6= ∅. Let us begin by considering the case where the stable and semistable loci for the G˜-line bundle are equal and nonempty: Ms(L) = Mss(L) 6= ∅. 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 of G˜. Thus G is a g-dimensional real Lie group, and G˜ is the complexification of G. Let g denote the Lie algebra of G. We may without loss choose a G-invariant Hermitian metric h on L, such that the unique L covariant derivative on L compatible with h and the holomorphic structure L has curvature −2πiΩ, where Ω is a G-invariant Hodge form on M. The 3 given structure of G-line bundle of L, furthermore, determines (and, up to topological obstructions, is equivalent to) a moment map Φ = Φ : M → g∗ L for the action of G on the symplectic manifold (M,Ω) [GS1]. The hypothesis that Ms(L) = Mss(L) 6= ∅ may be restated symplectically as follows: 0 ∈ g∗ is a regular value of Φ, and Φ−1(0) 6= ∅ [Ki1]. In this case, P =: Φ−1(0) is a connected G-invariant codimension g submanifold of M. Let G and Φ be as above. The action of G on Φ−1(0) is locally free, and the GIT quotient M//G˜ = Ms(L)/G˜ may be identified in a natural manner with the symplectic reduction M =: Φ−1(0)/G, and is therefore a K¨ahler 0 orbifold. The quantizing line bundle L descends to a line orbibundle L on 0 M . 0 Similar considerations apply to symplectic reductions at coadjoint orbits sufficiently close to the origin. If the G˜-line bundle L is replaced by its tensor power L⊗k, theHodgeformandthemoment mapget replaced by their multiples kΩ and Φ =: kΦ. Given any µ ∈ g∗, there exists k such that µ is k 0 a regular value of Φ if k ≥ k . The relevant asymptotic information about k 0 the multiplicity of V in H0(M,L⊗k) may then determined by computing µ appropriate Riemann-Roch numbers on these orbifolds [Ka], [M2]. Let O ⊆ g∗ be the coadjoint orbit through µ; since µ is integral, the Kir- µ illov symplectic form σ is a Hodge form on the complex projective manifold µ O . By the Konstant version of the Borel-Bott theorem, there is an ample µ line bundle A on O such that H0(O ,A ) is the irreducible representation µ µ µ µ of G with highest weight µ. (k) Let then M be the Weinstein symplectic reduction of M at µ with µ respect to the moment map Φ = kΦ (k ≫ 0). Using the normal form k L description of the symplectic and Hamiltonian structure of (M,Ω) in the neighbourhood of the coisotropic submanifold P = Φ−1(0) [M2], [G], [GS3], (k) one can verify that M is, up to diffeomorphism, the quotient of P × O µ µ (k) by the product action of G. In other words, M is the fibre orbibundle µ on M = P/G associated to the principal G-orbibundle q : P → M and 0 0 the G-space O (endowed with the opposite K¨ahler structure); in particular µ (k) its diffeotype is independent of k for k ≫ 0. Let p : M → M be the µ µ 0 projection. Let θ be a connection 1-form for q ([GGK], Appendix B). By the shifting (k) (k) trick, the symplectic structure Ω of the orbifold M is obtained by de- µ µ scending the closed 2-form kι∗(Ω)+ < µ,F(θ) > −σ on P ×O down to the µ µ quotient (the symbols of projections are omitted for notational simplicity). The minimal coupling term < µ,F(θ) > −σ is the curvature of the line µ (k) (k) orbibundle R = (P ×A )/G on M . Thus, Ω is the curvature form of µ µ µ µ the line orbibundle p∗(L⊗k)⊗R . µ 0 µ 4 Let P˜ =: {(p,µ′,g) ∈ P ×O ×G : g ·(p,µ′) = (p,µ′)}, µ µ ˜ P =: P ×O ×K. (2) µ,K µ There is a natural inclusion P˜ ⊆ P . Now let Σ =: P˜ /G, Σ =: µ,K µ µ µ µ,K P˜ /G = M(k) ×K. There is a natural orbifold complex immersion Σ → µ,K µ µ (k) M , with complex normal orbi-bundle N , and Σ ⊆ Σ is the union µ Σµ µ,K µ of the |K| connected components mapping dominantly (and isomorphically) (k) onto M . The orbifold multiplicity of Σ is constant and equal to |K|. µ µ,K Let L be the line orbi-bundle on M determined by descending L, and let 0 0 L˜ be its pull-back to Σ . Let r be the complex dimension of O , so that 0 0 µ dimM(k) = n−g +r. After [M1] and [M2], the multiplicity N(k)(µ) of the µ irreducible representation V in H0(M,L⊗k) is then given by: µ 1 Td(Σ )ChΣµ(p∗(L⊗k)⊗R ) N(k)(µ) = µ µ 0 µ ZΣ0 dΣµ DΣµ(NΣµ) n−g+r 1 kc (L )+c (R ) = kn−g χ (h)kµ˜ (h) 1 0 1 µ |K| hX∈K K,L K ZMµ(k) (cid:0) (n−g +r)! (cid:1) +O(kn−g−1). (3) Now suppose that χk · µ˜ 6≡ 1. Then the action of K on ∗(L⊗k) ⊠ A K,L K µ is not trivial, where  : P ֒→ M is the inclusion. Therefore, the fiber of p∗(L⊗k)⊗R on the smooth locus of M is a nontrivial quotient of C, and µ 0 µ 0 N(k)(µ) = 0 in this case. If there exists k such that χk · µ˜ ≡ 1, on the K,L K other hand, the same condition holds with k replaced by k +ℓe, where e is the period of χ and ℓ ∈ Z is arbitrary. Thus k may be assumed arbitrarily K,L large. Passing to the original K¨ahler structure ofO in the computation, and µ recalling that dim(V ) = (r!)−1 σr, we easily obtain: µ Oµ µ R kn−g N(k)(µ) = dim(V ) c (L )∧(n−g) +O(kn−g−1). (4) µ (n−g)! Z 1 0 M0 3 The asymptotics of equivariant volumes. Let f : M˜ → M be a Kirwan desingularization of the action [Ki2] . This means that f is a G-equivariant birational morphism, obtained as a se- quence of blow-ups along G-invariant smooth centers, such that for all a ≫ 0 the ample G-line bundle B =: f∗(L⊗a)(−E) satisfies Ms(B) = Mss(B) ⊇ f−1(Ms(L)). Here E ⊆ M˜ is an effective exceptional divisor for f. Clearly υ (F) = υ (f∗(F)) for every G-line bundle F on M. µ µ 5 Definition 1. Let M (B) ⊆ M (B) be the divisorial part of the unstable u div u locus of B; in other words, M (B) is the union of the irreducible compo- u div ˜ nents of M (B) having codimension one in M. We shall say that the Kirwan u resolution f is mild if f (M (B) ) ⊆ M (L). u div u We have: Theorem 2. Let L be an ample G-line bundle on M such that Ms(L) 6= ∅. Suppose that f : M˜ → M is a mild Kirwan resolution of (M,L). Let H be a ˜ G-line bundle on M such that χ = 1. Let µ ∈ Λ be a dominant weight. K,H + Then for any ǫ > 0 there exist arbitrarily large positive integers m (how large depending on ǫ and µ) such that υ f∗(L)⊗m ⊗H−1 ≥ mn (υ (f∗(L))−ǫ). µ µ (cid:0) (cid:1) More precisely, this will hold whenever m = 1+pe, where e is the period of χ and p ∈ N, p ≫ 0. K,L 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 (L⊗m ⊗H−1) ≥ mn(υ (L)−ǫ). µ µ By definition, υ (L) ≥ υ (L⊗m)/mn−g for every m > 0. Thus Corollary 1 µ µ with H = O implies: M Corollary 2. Let L be a G-line bundle with Ms(L) 6= ∅. Then υ (L⊗m) µ υ (L) = limsup . µ mn−g m→+∞ Similarly, Corollary 3. Under the same hypothesis, υ (f∗(L⊗m)(−E)) µ υ (L) = limsup . µ mn−g m→+∞ 6 Proof of Theorem 2. The proofis inspired by arguments in [DEL]. If υ (L) = µ 0, there is nothing to prove; we shall assume from now on that υ (L) > 0, µ and for simplicity write L for f∗(L). Thus there exists 0 ≤ r < e such that χr · µ˜ = 1, where e is the period of χ . If ℓ ≫ 0, by the above K,L K K,L H0(M˜,B⊗eℓ)G 6= 0, so that υ (L⊗m ⊗H−1) ≥ υ L⊗m ⊗H−1 ⊗B−⊗ℓe . µ µ Thus there is no loss of generality in replacing H b(cid:0)y H ⊗ B⊗ℓe for som(cid:1)e ℓ ≫ 0. In view of the hypothesis on H, we may thus assume without loss the existence of 0 6= σ ∈ H0(M˜,H)G 6= {0}, with invariant divisor D ∈ H0(M˜,H)G . (cid:12) (cid:12) (cid:12) Since furt(cid:12)hermore the class of B in the G-ample cone introduced in [DH] (cid:12) (cid:12) 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 G-line bundle satisfying Ms(H) = Ms(B) = Mss(H) 6= ∅. By definition of υ , there exists a sequence s ↑ +∞ such that µ ν sn−g ǫ h0(M,L⊗sν) ≥ ν υ (L)− . (5) µ (n−g)!(cid:16) µ 3(cid:17) Necessarily s ≡ r (mode),∀ν ≫ 0. Weshallshowthatthestatedinequality ν holds if p ≫ 0 and m =: 1+pe. Lemma 1. Fix p ≫ 0 and let m =: 1+pe. There is a sequence k ↑ +∞ ν such that kn−g ǫ h0(M,L⊗kν) ≥ ν υ (L)− , (6) µ (n−g)!(cid:16) µ 2(cid:17) with k ≡ r (mod e) and furthermore k ≡ e (mod m) for every ν. ν ν Proof. Let x > 0 be an integer. We may assume that there is a non-zero section 0 6= τ ∈ H0(M,L⊗xpe)G. Thus, there are injections H0(M,L⊗sν) ֒→ H0(M,L⊗(sν+xpe)), µ µ and for ν ≫ 0 we have sn−g ǫ (s +xpe)n−g ǫ h0(X,L⊗(sν+xpe)) ≥ ν υ (L)− ≥ ν υ (L)− . µ (n−g)! (cid:16) µ 3(cid:17) (n−g)! (cid:16) µ 2(cid:17) (7) Perhaps after passing to a subsequence, we may assume without loss of gene- rality that s ≡ r′ (mod. m), for a fixed 0 ≤ r′ ≤ m−1. Thus, s = ℓ m+r′. ν ν ν Let now x > 0 be an integer of the form x = sm + r′ − e, s ∈ N. Then xpe = x(m−1) and s +xpe = (ℓ +x)m+r′ −x = (ℓ +x−s)m+e. ν ν ν 7 Now we need only set k =: s +xpe. ν ν Set ℓ = kν . Thus k = ℓ m+e. ν m ν ν (cid:2) (cid:3) Lemma 2. There exists a > 0 such that H0(M˜,H⊗mae ⊗L⊗−e)G 6= {0} for every m ≥ 1. 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, theysharethesamestableandsemistable loci, anddeterminethesame GIT quotient M˜ . The G-line bundles H′ =: H⊗e and L′ =: L⊗−e descend to 0 genuinelinebundlesH′ andL′ onM ,andH′ isample. Thus,forsomea ≫ 0 0 0 0 the ample line bundles H′⊗ae and H′⊗ae⊗L′ are globally generated. Arguing as in [GS1], H0(M ,H′⊗ma⊗L′) lift to G-invariant sections of H⊗mae⊗L⊗−e. 0 If R and S are G-line bundles on M˜, any 0 6= σ ∈ H0(M˜,R)G induces injections H0(M˜,S) −⊗→σ H0(M˜,R ⊗ S). Thus, if H0(M˜,R)G 6= 0 then µ µ h0(M˜,S) ≤ h0(M˜,R⊗S) for every µ. Now, in additive notation, ℓ (mL− µ µ ν R) = k L−(eL+ℓ R) for any G-line bundle R; therefore, by Lemma 2 ν ν h0 M˜,O (ℓ (mL−H) ≥ h0 M˜,O (k L−(ℓ +am)H . (8) µ M˜ ν µ M˜ ν ν (cid:16) (cid:17) (cid:16) (cid:17) We may now use the G-equivariant exact sequences of sheaves: 0 → O (k L−(j +1)H) → O (k L−jH) → O (k L−jH) → 0, M˜ ν M˜ ν D ν for 0 ≤ j < s, to conclude inductively that h0(M˜,L⊗kν ⊗H⊗(−s)) ≥ h0(M˜,L⊗kν) µ µ − h0(D, L⊗kν ⊗H⊗(−s) ) (9) µ D 0X≤j<s (cid:12) (cid:12) We may decompose D as D = D + D , where D ,D ≥ 0 are effective u s u s divisors onM˜, D is supported on the unstable locus of B, M˜u(B) ⊆ M, and u no irreducible component of D is supported on Mu(B). Let D = D s u j uj and D = D be the decomposition in irreducilbe components. P s i si P Lemma 3. If D ∈ H0(M˜,H)G is general, then D is reduced, and it is s (cid:12) (cid:12) nonsingular away fro(cid:12)m the unstab(cid:12)le locus M˜u(H) = Mu(B). (cid:12) (cid:12) Proof. Perhaps after replacing H by some appropriate power we may assume that the linear series H0(M˜,H)G is base point free away from M˜u(H). The (cid:12) (cid:12) claim then follows fro(cid:12)m Bertini’s (cid:12)Theorem. (cid:12) (cid:12) 8 Wenowmakeuseofthemildnessassumptiononf. Ifm ≫ 0isfixed, ν ≫ 0 and 0 ≤ s ≤ ℓ +am, then the moment map of the (not necessarily ample) ν line bundle L⊗kν⊗H⊗−s is bounded away from0 intheneighbourhood ofD . u An adaptation of the arguments in §5 of [GS1] (applied on some resolution of singularities of each D ) then shows that h0(D , L⊗kν ⊗H⊗(−s) ) = 0 uj µ u Du (if D is not reduced, we need only filter O (k L−sH) by an app(cid:12)ropriate u Duj ν (cid:12) chain of line bundles). Since furthermore on each D we may find a non-vanishing invariant sj section of H, we obtain: h0(M˜,L⊗kν ⊗H⊗(−s)) ≥ h0(M˜,L⊗kν) (10) µ µ − h0(D ,L⊗kν ⊗H⊗(−s) ⊗O ) µ s Ds X 0≤j<s ≥ h0(M˜,L⊗kν)−sh0(D ,L⊗kν ⊗O ) µ µ s Ds ≥ h0(M˜,L⊗kν)−sh0(D ,(L⊗B)⊗kν ⊗O ) µ µ s Ds Proposition 1. There exists C > 0 constant such that if D ∈ H0(M˜,H)G (cid:12) (cid:12) is general then (cid:12) (cid:12) (cid:12) (cid:12) h0(D ,(L⊗B)⊗k ⊗O ) ≤ Ckn−g−1 µ s Ds for every k ≫ 0. Proof. GiventheequivariantinjectivemorphismofstructuresheavesO −→ Ds O , we may as well assume that D is a reduced and irreducible G- i Dsi s iLnvariant divisor, descending to a Cartier divisor D on the quotient. 0 Let R =: L ⊗ B, with associated moment map Φ . By the generality R in its choice, we may assume that D ∈ H0(M˜,H)G is non-singular in the s (cid:12) (cid:12) neighbourhood of Φ−1(0), and is transver(cid:12)sal to it. In(cid:12)fact, the singular locus R (cid:12) (cid:12) of H0(M˜,H)G is the unstable locus of H. Furthermore, by the arguments (cid:12) (cid:12) of L(cid:12) emma 3 in(cid:12)[P2] and compactness one can see the following: There exist (cid:12) (cid:12) a finite number of holomorphic embeddings ϕ : B → M˜, where B ⊆ Cn−g i is the unit ball, satisfying i): ϕ (B) ⊆ Φ−1(0); ii): as a submanifold of i R Φ−1(0), ϕ (B) is transversal to every G-orbit; iii): the union ϕ (B) maps R i i i surjectively onto M˜//G˜. In view the local analytic proof of BeSrtini’s theorem in[GH],wemayassumethatD istransversaltoeachϕ (B). ByG-invariance, i it is then transversal to all of Φ−1(0). R Let g : D˜ → D be a G-equivariant resolution of singularities [EH], s s [EV]. For s ≫ 0, g∗(R⊗s)(−F) is an ample G-line bundle on D˜ , where s F is some effective exceptional divisor. Since 0 ∈ g∗ is a regular value of g ◦Φ : D˜ → g∗ and belongs to its image, the same holds for the moment R s 9 map of g∗(R⊗s)(−F), for s ≫ 0. Having fixed s ≫ 0, let us also choose r ≫ 0 such that H0(M˜,g∗(R⊗sre)(−reF))G 6= 0. The choice of 0 6= σ ∈ H0(D˜ ,g∗(R⊗sre)(−reF))G determines injections s H0(D ,R⊗k ⊗O ) ⊗−σ→⊗k H0(D˜ ,R⊗k(1+sre)(−rekF)⊗O ). µ s Ds µ s D˜s This implies the statement by the arguments in §2, since dim(D˜ ) = n−1. s Given (5), (9) and Proposition 1, we get kn−g ǫ h0(M˜,L⊗kν ⊗H⊗(−s)) ≥ ν υ (L)− −sCkn−g−1. (11) µ (n−g)! (cid:16) µ 2(cid:17) ν Now, in view of (8), we set s = ℓ +am to obtain: µ h0 M˜,O (ℓ (mL−H)) ≥ kνn−g υ (L)− ǫ −C(ℓ +am)kn−g−1 µ M˜ µ (n−g)! µ 2 ν ν (cid:16) (cid:17) ≥ ℓnν−gmn(cid:0)−g υ (L)−(cid:1) ǫ (n−g)! µ 2 −C(ℓ +(cid:0)am)(ℓ +1(cid:1))n−g−1mn−g−1. ν ν (12) The proof of Theorem 2 follows by taking ℓ ≫ m ≫ 1 (see [DEL], Lemma ν 3.5). Remark 3.1. The arguments used in the proof of Theorem 2 may be applica- ble in other situations. For example, suppose that L is a G-ample line bundle with vol (L) > 0; assume that the equivalence class of L in the G-ample cone 0 [DH] lies on a face of measure zero, and that - say - the G-ample line bundles in the interior of an adjacent chamber have unstable locus of codimension ≥ 2. If A is a G-ample line bundle in the interior of the chamber, one may apply the previous arguments to tensor powers of the form L⊗k ⊗A. 4 Proof of Theorem 1. Given the Kirwan resolution f : M˜ → M, for m ≫ 0 the equivalence classes of the ample G-line bundles f∗(L⊗m)(−E) in the G-ample cone of M˜ all lie to the interior of the same chamber. Therefore, they determine the same GIT quotient M˜ = M˜//G˜. The latter is an (n − g)-dimensional complex 0 projective orbifold, which partially resolves the singularities of M//G˜ [Ki2]. Being G-line bundles on M˜, f∗(L) and O (−E) descend to line orbi-bundles M˜ on M˜ . Fixing G-invariant forms Ω˜ and Ω on M˜ representing the first 0 −E Chern class of f∗(L) and O (−E), we obtain forms Ω˜ and Ω on M˜ . M˜ 0 −E0 0 10