Equivariant volumes for linearized actions

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by  Alberto Della Vedova

Equivariant volumes for linearized actions
Alberto Della Vedova and Roberto Paoletti ∗
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J 1 Introduction

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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