Table Of ContentRELATIVE K-POLYSTABILITY OF PROJECTIVE BUNDLES OVER A
CURVE
VESTISLAV APOSTOLOVANDJULIEN KELLER
Abstract. Let P(E) be the projectivization of a holomorphic vector bundle E over a
7 compact complex curve C. We characterize the existence of an extremal K¨ahler metric
1 on P(E) in terms of relative K-polystability and the fact that E decomposes as a direct
0
sum of stable bundles.
2
b
e
F
9 Contents
] 1. Introduction 1
G
2. Preliminaries 3
A
2.1. The Donaldson–Futaki invariant 3
.
h 2.2. Test configurations and K-polystability 4
t 2.3. Relative K-polystability 5
a
m 3. Proof of Theorem 2 and Corollary 1 6
[ 3.1. Constructing a test configuration 7
3.2. Relative Donaldson–Futaki invariant 8
2
v 3.3. Generalized Calabi Ansatz 9
7 3.4. Computing the relative Donaldson–Futaki invariant via g 12
c
0 3.5. Algebraic computation of the relative Donaldson–Futaki invariant 16
5
3.6. The case of an indecomposable bundle 20
8
0 3.7. The case ℓ = 2 21
. 3.8. The general case and the proof of Theorem 2 21
1
0 3.9. Proof of Corollary 1 27
7 4. Appendix 28
1
4.1. Integration over the simplex 28
:
v 4.2. Chern characters of symmetric tensor powers of vector bundles 29
i
X References 30
r
a
1. Introduction
LetM = P(E)bethecomplexmanifoldunderlyingthetotalspaceoftheprojectivization
of a holomorphic vector bundle E C over a compact complex curve C. In this paper,
→
we are interested to understand when P(E) admits an extremal Ka¨hler metric in the sense
of Calabi [8], and if such a special metric does exist, which Ka¨hler classes of P(E) admit
extremal Ka¨hler metrics. A Ka¨hler class Ω endowed with a K¨ahler metric is refereed to as
an extremal class.
Key words and phrases. ExtremalK¨ahlermetrics;stablevectorbundles;projectivebundles;toricfibra-
tions; ruled manifolds.
1
2 V.APOSTOLOVANDJ.KELLER
By the openness of the extremal Ka¨hler classes on M (see [22, 16]) and using the fact
thatH2,0(M,C) = 0,withoutlossofgeneralitywecanrestrictourstudytorationalclasses.
Furthermore, as extremal Ka¨hler classes are invariant under a positive rescaling, we can
even only consider integral classes, i.e. Ω = 2πc ( ) for a positive line bundle on M.
1
L L
Such bundles on M = P(E) C are of the form
→
= = (q) (p) , q,p Z,q > 0,
q,p P(E) C
L L O ⊗O ∈
where (1) denotes (thepull-back toM)ofanyholomorphiclinebundleover C of degree
C
O
1. We note that = becomes positive for p/q 0 and thus defines a polarization on
q,p
L L ≫
M.
Before discussing the case of extremal metrics, let us recall some results about the
particular case of constant scalar curvature Ka¨hler metrics (CSC Ka¨hler for short). In the
caseofM = P(E) C,theexistenceofCSCKa¨hlermetricsanditslinktoK-polystability
→
in the sense of [13, 41] are completely settled thanks to the works [3, 32].
Theorem 1. [3, 32] Let E a holomorphic vector bundle and let M = P(E) C be its
→
projectivisation. The following three conditions are equivalent:
(i) M admits a CSC K¨ahler metric in any class 2πc ( );
1
L
(ii) M is K-polystable for any polarization ;
L
(iii) E is polystable, i.e. decomposes as the sum of stable bundles of same slopes;
Remark 1.1. The notion of stability for bundles refers here to the classical notion of
Mumford–Takemotostability. Theequivalence(iii) (i)isestablishedin[3, Theorem1]
⇐⇒
whenthebaseC hasgenusg 2,andin[32,Theorem5.13]wheng 1(byusingtheresult
≥ ≥
of [33]). This equivalence also holds true for g = 0 as a consequence of the Lichnerowicz–
Matsushima theorem, by noting that in this case E splits as a direct sum of line bundles
over C = CP1. The equivalence (ii) (iii) follows by [32, Theorem. 5.13], by noting
⇐⇒
again thatthecaseg = 0canbetreated apartbyobservingthattheusualFutakiinvariant
associated to 2πc ( ) vanishes if and only if E is the sum of line bundles over CP1 of the
1
L
same degree.
In[3],itwasintroducedthefollowingconjectureinviewoftheclassificationofprojective
bundles over a curve, admitting extremal metrics.
Conjecture 1. [3] Let E a holomorphic vector bundle and let M = P(E) C be its
→
projectivisation. The following three conditions are equivalent:
(a) M admits an extremal K¨ahler metric;
(b) M is relative K-polystable for a certain polarization ;
L
(c) E decomposes as a direct sum of stable bundles.
Wereferto[36]forthenotionofrelativeK-polystability. Ourmainresultisthefollowing
theorem.
Theorem 2. Conjecture 1 is true.
We do now some comments. Firstly, Conjecture 1 is almost optimal in view of the
existence problem of extremal Ka¨hler metric. Actually in the light of Theorem 1, it is
natural to ask if Conjecture 1 could be completed by
(d) M admits an extremal Ka¨hler metric in any Ka¨hler class;
(e) M is relative K-polystable polarization for any polarization ;
L
RELATIVE K-POLYSTABILITY OF PROJECTIVE BUNDLES 3
Butthis does nothold. Ingeneral, with E directsumof stablebundles,M may also admit
polarizations which are not relatively K-polystable, nor extremal, see e.g. [3, Proposition
5]or[2,Theorem6]. TostrengthenConjecture1,itwouldbenaturaltoaskthatconditions
(a) and(b)occur precisely for thesame classes. Thisis precisely theYau–Tian–Donaldson
conjecture extended to the setting of extremal Ka¨hler metrics by Sz´ekelyhidi [36].
Conjecture 2(RelativeYau-Tian-Donaldsonconjecture). For any polarization onM =
L
P(E) C, the following two conditions are equivalent
→
(a’) 2πc ( ) is extremal;
1
L
(b’) (M, ) is relatively K-polystable.
L
By virtue of [35] (see also [10, Theorem 1.2]), we know that (a’) implies (b’) on any
polarized variety. In direction of this conjecture, Theorem 2 combined with some previous
results allows us to establish the following
Corollary 1. Let E C a holomorphic vector bundle over a complex curve C and write
→
E = U U as a direct sum of indecomposable sub-bundles. The relative Yau–Tian–
1 s
⊕···⊕
Donaldson conjecture is true for a polarization on P(E) in the following cases:
q,p
L
(1) s = 1 or s= 2;
(2) s 3 and E is polystable;
≥
(3) s 3 and one of the U is unstable;
i
≥
(4) s 3 and p/q is large enough.
≥
Remark 1.2. Anexamplefrom[2]strongly suggests thatforreaching theremainingcases
(s 3, U stables of different slopes and p/q not large), one would need to enforce the
i
≥
notion of relative K-polystability of (M, ). This would require to consider test configu-
L
rations with “irrational” line bundles (i.e. formal tensor powers of line bundles with real
coefficients). There are two current approaches to this. The first is the notion of Ka¨hler
relative K-stability, whichoriginates in[32]andwas recently developed in [14,12,10]. The
other one is the notion of uniform relative K-stability, as introduced in [39, 6, 11].
Eventually, one would expect that some of the results discussed above can be extended
to projective holomorphic vector bundles M = P(E) B over a base (B,L ) which itself
B
→
is a polarized variety admitting an extremal Ka¨hler metric in 2πc (L ). This is evidenced
1 B
in the works [20, 7, 25, 21].
We sum up now the general structureof the paper. In Section 2 we present the required
material about (relative) Donaldson–Futaki invariant. In Section 3, we construct a test-
configuration and compute by two differentways theassociated relative Donaldson–Futaki
invariant (Sections 3.3, 3.4 and 3.5). Our first approach is based on differential-geometric
ingredients from [3] and has the advantage to apply to any Ka¨hler class (rational or not),
thus evidencing the Ka¨hler feature of the K-stability in line with the recent work [10]. The
second approach is algebro-geometric, following theoriginal arguments in [32], and has the
merit to cover the case when the genus of C equals 1. The proof of our main result is then
given at the end of Section 3.8. The proof of Corollary 1 is obtained in Section 3.9. The
appendix (Section 4) contains certain technical results.
2. Preliminaries
2.1. The Donaldson–Futaki invariant. Suppose(M, ) is a polarized variety endowed
with a C action ρ with a lift to . Let Aρ be the infiniLtesimal generator of the induced
× L k
4 V.APOSTOLOVANDJ.KELLER
linear action on the vector space H := H0(M, k) of holomorphic sections of k, and
k
L L
denote by
ρ
d := dim H , w (ρ) := TrA .
k k k k
It turns out (by using Riemann–Roch) that for k 0, d and w are polynomials
k k
≫
d = a kn +b kn 1+O(kn 2), w (ρ) = a(ρ)kn+1+b(ρ)kn +O(kn 1).
k 0 0 − − k −
We then define
Definition 2.1. The algebraic Donaldson–Futaki invariant of (M, ,ρ) is
L
a b(ρ) a(ρ)b
Falg(ρ) := 0 − 0.
a
0
We shall use this definition in the case when (M, ) is a smooth polarized variety. We
L
notice that there are different sign choices in the literature for the infinitesimal generator
of the induced linear action on H , thus introducing a sign difference in the definition of
k
the (algebraic) Donaldson–Futaki invariant, see e.g. [38, p. 141]. We shall use in this
paper the following convention, which agrees with [19] and, up to a positive constant, with
[38, (7.14)].
Definition 2.2. Let (M, ) be a smooth polarized variety endowed with a C action ρ
×
L
with a lift to , denoted by ρˆ. We let e√ 1t,t R be the circle subgroup of C . Then,
− ×
L ∈
the infinitesimal generator Aρ for the action of ρˆon the space of smooth sections Γ( ) is
L
defined to be
d
(Aρ(s))(x) := √ 1 ρˆ(e√ 1t) s(ρ(e √ 1t)(x)) , s Γ( ),x M.
− − −
− dt ∈ L ∈
t=0
| (cid:16) (cid:17)
(cid:0) (cid:1)
It is shown in [13] (see also [19, 38]) that the above definition agrees, up to a factor of
1 , with the differential geometric definition of the Futaki invariant [17], i.e.
4(2π)n
4(2π)nFalg(ρ) = F (K ) = Scal f v ,
Ω ρ g ρ g
ZM
where n = dim M, K is the (real) holomorphic vector field on M induced by the action
C ρ
of S1 via ρ, Ω = 2πc ( ) is the Ka¨hler class determined by , g is any S1-invariant Ka¨hler
1
L L
metric in Ω, Scal is its scalar curvature, and f denotes the Killing potential of mean
g ρ
value zero for K with respect to g.
ρ
2.2. Test configurations and K-polystability. Recall the following definitions from
[40] and [13].
Definition 2.3. Let (M, ) be a normal polarized variety. A test configuration for (M, )
L L
is a normal variety endowed with a line bundle together with
M P
(i) a C action ρ on with a lift to ;
×
M P
(ii) a C equivariant map π : C where C acts on a standard way on C,
× C ×
M →
such that π : C is a flat family with being relatively ample and, for any t = 0,
C
M → P 6
the fibre (M , ) of π is isomorphic to (M, r) for some fixed r N. The number r is
t PMt C L ∈
called exponent of the test configuration.
A test configuration is said to be a product configuration if = M C and ρ is given
M ×
by a C× action on M (and scalar multiplication on C).
Notice thatforany testconfiguration( , ,ρ) for(M, ), ρinducesaC action onthe
×
M P L
central fibre (M , ) (which we still denote by ρ). With our convention in Definition 2.2,
0 0
L
we then have
RELATIVE K-POLYSTABILITY OF PROJECTIVE BUNDLES 5
Definition 2.4. [13] The Donaldson–Futaki invariant of the test configuration ( , ,ρ)
M P
for (M, ) is the Donaldson–Futaki invariant Falg(ρ) of its central fibre (M , ). The
0 0
L L
variety (M, ) issaidtobeK-polystable (resp. K-stable) iftheDonaldson–Futaki invariant
L
of any normal test configuration for (M, ) is non-negative, and equal to zero if and only
L
if the test configuration is a product configuration (resp a trivial test configuration).
This implies in particular that the Donaldson–Futaki invariant of any C action on
×
(M, ) must be zero, so the notion is adapted to the study of cscK (in particular Ka¨hler–
L
Einstein) metrics.
2.3. Relative K-polystability. In order to account for the obstructions related to the
extremalKa¨hlermetrics,G.Sz´ekelyhidihasintroducedrelativeversionoftheabovenotions
as follows.
Suppose (M, ) is a polarized variety endowed with two commuting C actions ρ and
× 1
L
ρ . We first define an inner product ρ ,ρ for such actions. For that, we take lifts of ρ
2 1 2 1
and ρ to and consider the infinitehsimaligenerators Aρ1 and Aρ2 of the actions on H .
2 L k k k
Then for k sufficiently large,
Tr(Aρ1Aρ) = a(ρ ,ρ )kn+2+O(kn+1)
k 2 1 2
is a polynomial of degree at most n+2 and we let
Definition 2.5. The inner product of two commuting C actions ρ and ρ on (M, ) is
× 1 2
L
defined by
a(ρ )a(ρ )
1 2
ρ ,ρ := a(ρ ,ρ ) .
1 2 1 2
h i − a
0
Notice that ρ ,ρ is the leading coefficient of the expansion in k of Tr(A˚ρ1A˚ρ2) of
the traceless pahrt1s o2fiAρi, so it is independent of the choice of liftings. It is kshowkn in
k
[36] that when M is smooth, the above definition agrees up to a factor of 1/(2π)n, with
the Futaki–Mabuchi bilinear form on Killing potentials, i.e. if for any Ka¨hler metric g in
Ω = 2πc ( ) which is invariant underthe S1 actions correspondingto ρ and ρ we denote
1 1 2
L
by f and f the Killing potentials of zero mean with respect to g, corresponding the
ρ1 ρ2
induced Killing vector fields, then
1
ρ ,ρ = f f v .
h 1 2i (2π)n ρ1 ρ2 g
ZM
We shall next fix a maximal torus Tℓ in the automorphism group Aut(M, ) and denote
L
by ρ ,...,ρ the C corresponding to the S1 generators of Tℓ.
1 ℓ ×
Definition 2.6. The extremal C action ρ of (M, ,Tℓ) is a C subgroup of the com-
× ex ×
L
plexification Tℓ defined by the system of ℓ linear conditions
c
ρ ,ρ = Falg(ρ ).
ex i i
h i
Definition 2.7. Let ρ be a distinguished C action on the polarized manifold (M, ).
0 ×
L
The ρ -relative Donaldson–Futaki invariant of (M, ,ρ ) is defined for any C action ρ
0 0 ×
L
commuting with ρ by
0
ρ ,ρ
Falg(ρ) := Falg(ρ) h 0 i Falg(ρ ).
ρ0 − ρ ,ρ 0
0 0
h i
We now apply the above to a test configuration.
6 V.APOSTOLOVANDJ.KELLER
Definition 2.8. [36, 34] A test configuration ( , ,ρ) for (M, ) is compatible with a
M P L
fixed maximal torus Tℓ Aut(M, ) if there is a Tℓ action on ( , ), commuting with
⊂ L M P
ρ and preserving π : C, which induces the trivial action on C, and restricted to
C
M →
(M , ) for t = 0 coincides with the original Tℓ action under the isomorphism with
t P|Mt 6
(M,L) via ρ.
In this case, we have an induced action of Tℓ on the central fibre M , and we denote by
0
ρM the C action on M corresponding to the extremal C action ρ on (M, ,Tℓ).
ex × 0 × ex L
Now, the relative Donaldson–Futaki invariant of a Tℓ compatible test configuration for
(M, ,Tℓ)) is defined to be the ρM-relative Donaldson–Futaki invariant of (M , ) of the
L ex 0 L0
induced C action ρ.
×
A polarized variety manifold (M, ) is relatively K-polystable (resp. K-stable) with
L
respect to a maximal torus Tℓ Aut(M, ) if the relative Donaldson–Futaki invariant of
⊂ L
anynormaltestconfiguration( , ,ρ)for(M, )compatiblewithTℓ isnon-negative, and
M P L
equal to zero if and only if ( , ) is a productconfiguration (resp a trivial configuration).
M P
Remark 2.1. More recently, following the works of Wang [42] and Odaka [30, 29], a
topological interpretation of the Donaldson–Futaki invariant was given in terms of an
integration over the total space of a given test configuration. Among other applications,
this point of view led to the definition of the stronger notion of Ka¨hler (relative) K-
polystability in [14, 12, 10], where one also takes in consideration the sign of the (relative)
Donaldson–Futaki over “irrational” polarizations of M. We shall not use this point
L
of view explicitly in this paper. However, the Reader could notice that our differential-
geometric approach to the computation of the relative Donaldson–Futaki invariant is well-
adapted to deal with the Ka¨hler relative K-polystability in the sense of [10].
3. Proof of Theorem 2 and Corollary 1
One direction of Theorem 2, namely (c) = (a), follows from the facts that an extremal
⇒
Ka¨hler metric exists in any polarizations = with p/q 0 (see [3, Theorem 3] or
q,p
L L ≫
[7]). Moreover, (a) = (b) is a consequence of the general result of [35], i.e the existence
⇒
of an extremal Ka¨hler metric in 2πc ( ) implies that (M, ) is relative K-polystable. We
1
L L
shall thus focus on establishing (b) = (c). As any vector bundle E over CP1 decomposes
⇒
as the direct sum of line bundles (which are automatically stable), we shall also assume
from now on that the base C has genus g 1.
≥
Assumption 1. C is a compact complex curve of genus g 1.
≥
As thecohomology H2(M,R)of M = P(E)is2-dimensional, uptorescaling, theKa¨hler
cone of M is 1-dimensional. Similarly, it is well-known that any holomorphic line bundle
on M can be written as
L
= := (q) (p) , p,q Z,
q,p P(E) C
L L O ⊗O ∈
where, as usual, (1) denotes the anti-tautological line bundle of E (defined on P(E))
P(E)
O
and (1) standsfor(thepullback toM of)anydegree1holomorphiclinebundleover C,
C
O
see for instance [26, Section 3] for details. If Ω = 2πc ( ) is a Ka¨hler class, evaluation
1 q,p
L
over the fibre of P(E) shows that q > 0, thus any polarization on M = P(E) can be
written as = with q > 0 (notice that becomes positive when p/q 0). Clearly,
q,p q,p
L L L ≫
both properties of existence of extremal Ka¨hler metric and relative K-polystability of the
polarization on M are invariant under taking tensor powers k = k. As it will turn
⊗
L L L
outinourspecificsituation, thesamephenomenahappensunderchangingthepolarization
RELATIVE K-POLYSTABILITY OF PROJECTIVE BUNDLES 7
(1) of the base curve C. It will be useful to normalize the choice of such polarizations,
C
O
by introducing the following
Notation 1. We let := (1) (m) ,m Q denote the class of holomorphic
m P(E) C
L O ⊗O ∈
line bundles over M such that q >0 and p/q = m.
q,p
L
In all of the arguments below involving one can take some (and hence any) line
m
L
bundle as above.
q,p
L
We denote by Autred(M) the reduced automorphism group of M = P(E) C (see
→
e.g. [19]) whose Lie algebra hred(M) consists of all holomorphic vector fields with zero
on M, and let Autred(M) be the subgroup of Autred(M) of elements which preserve C
C
(i.e. act on each fibre), with Lie algebra hred(M). As M is a locally trivial holomorphic
C
CPn 1-fibration over C, we have an exact sequence of Lie algebras
−
(1) 0 hred(M) hred(M) hred(C) 0,
C
→ → → →
wherehred(C) its the Lie algebra of holomorphic vector fields with zeroes on C. Under the
assumption g(C) 1, we have hred(C) = 0, so that hred(M) = hred(M). We let (ℓ 1)
≥ C −
with ℓ 1 denote the rank of Autred(M) (which is also the rank of Autred(M) by the
≥ C
preceding). Thus, ℓ equals the number of summands in the decomposition
ℓ 1
−
(2) E = U .
k
k=0
M
of E as direct sum of indecomposable holomorphic sub-bundles U . We want to show
k
that, in general, each U is stable when M is relative K-polystable with respect to the
k
polarization of M. Without loss, we deal with U and assume rk(U ) > 1.
m 0 0
L
3.1. Constructing a test configuration. This construction follows [32, Remark 5.14]
and [31, Section 3]. For each strict sub-bundle L U E, we consider the exact
0
⊂ ⊂
sequences of holomorphic vector bundles
0 L U F 0,
0 0
(3) → → → →
0 L E F 0,
→ → → →
where F0 = U0/L and F = E/L ∼= F0 ⊕ kℓ−=11Uk . Thus, E is given by an element
e ext1(L,F), coming from an element (still denoted by e) of ext1(L,F ); as U is in-
∈ (cid:0)L (cid:1) 0 0
decomposable, e = 0, and one can consider the smooth family := (M ,t),t C,
t
6 M ∈
where M := P(E ) and E is the extension of (L,F) corresponding to te ext1(L,F ) =
t t t 0
∈
H1(M,L F ) for t C. As explained in [31, Section 3.1], = P( ) C C is itself
⊗ 0∗ ∈ M E → ×
a complex ruled manifold, where is a holomorphic vector bundle whose restriction to
E
C t is E . We denote by π : C the natural holomorphic projection on the
t C
× { } M →
C-factor. As E = E for t =0, we have that π 1(t) = M = M, whereas
t ∼ 6 − t ∼
ℓ
π 1(0) = M = P V C,
C− 0 k →
(cid:16)Mk=0 (cid:17)
where we have set V := U /L, V := L, V := U , k = 2,...,ℓ. As shown in [31,
0 0 1 k k 1
Lemma 3.1.1], there is a natural C action ρ on , m−aking π equivariant with respect
× L C
M
to the standard action on C , and which induces a C action (still denoted by ρ ) on the
× × L
8 V.APOSTOLOVANDJ.KELLER
central fibre M , given by the fibre-wise multiplication with λ C on the factor V = L
0 × 1
∈
in the decomposition
ℓ
V := V .
k
k=0
M
Given a polarization = (1) (m) on M (we can work with any line bundle
m P(E) C
L O ⊗O
representing , see Notation 1), consider on the (class of rational) holomorphic line
m
L M
bundles := (1) (r) which restricts to t = (1) (m) on each
fibre M P=mπ 1(tO). TPh(eE)C⊗ OactioCn ρ on = P( ) coLmmes frOom aPn(Eatc)t⊗ionOpresCerving the
t C− × L M E
vector bundle C C (and acting trivially on C), so ρ naturally lifts to an action
L
E → ×
on C. It thus follows that for any t = 0, (M , t ) is a polarized variety
m t m
P → M → 6 L
isomorphic to (M, ). Furthermore, the holomorphic line bundle 0 induced on the
m m
L L
central fibre M must be at least semi-ample. As the condition for to be ample on
0 m
L
M is relatively open with respect to m Q, it follows that 0 must be ample too. We
m
∈ L
thus conclude that ( ,ρ , ) defines a test-configuration for (M, ) (the flatness of
L m m
M P L
the morphism π : ( , ) C is a direct consequence of the surjectivity of π and
C m C
M P →
the fact that the central fibre is smooth). We finally notice that the rank of the reduced
automorphism group of the central fibre M is at least ℓ, whereas the rank of the same
0
group on M is (ℓ 1) for t = 0, showing that the test configuration is normal and not
t
− 6 M
a product configuration [24, 34]. We thus have established the following
Lemma 3.1. Given a completely decomposable vector bundle E = kℓ−=10Uk → C, a
polarization = (1) (m) on M = P(E) and a sub-bundle L U , the data
m P(E) C 0
L O ⊗O L⊂
(π : C,ρ , )
C L m
M → P
define a normal test configuration for (M = P(E), ) which is not a product configuration
m
L
and with central fibre (M = P(V), 0 ), where
0 m
L
ℓ
V = V , V := U /L, ,V := L, V := U , k = 2,...,ℓ.
k 0 0 1 k k 1
−
k=0
M
The induced C action ρ on M is given by
× L 0
ρ (λ) (x,[e ,e , ,e ]) = (x,[e ,λe ,...,e ]),
L 0 1 ℓ 0 1 ℓ
· ···
where x is a point on C and (v ,v ,...,v ) is a vector in the fibre V of V.
0 1 ℓ x
3.2. Relative Donaldson–Futaki invariant. Thecentral fibreM is a smooth complex
0
variety, endowed with a holomorphic action of the torus Tℓ, coming from the diagonal
action of Tℓ+1 on V = ℓ V . We choose any Ka¨hler metric g on M in the Ka¨hler
k=0 k 0
class Ω = 2πc ( 0 ), which is invariant under the action of Tℓ. The action of the sub-
1 m
L L
torus Tℓ 1 Tℓ by diagonal multiplications on the factors V ,...,V extends to each fibre
− 2 ℓ
⊂
M ,t = 0, and on . As Tℓ 1 is a maximal torus in the connected component of identity
t −
ofAut6 red(M )foraMny t = 0, itfollows thattheextremalvector field K of(M ,Ω)belongs
t ex t
6
to Lie(Tℓ−1) ⊂ Lie(Tℓ) and is independent of t (as Mt ∼= Mt′ via ρL and the action of Tℓ−1
on commutes with ρ ). We shall denote this vector field by KM and let fM be the
M L ex ex
Killing potential of KM of zero mean value with respect to g. As the central fibre M
ex 0
is a smooth variety, the relative Donaldson–Futaki invariant is computed up to a positive
RELATIVE K-POLYSTABILITY OF PROJECTIVE BUNDLES 9
normalization factor by the differential-geometric quantity (see [36] or Section 2)
(f fM)v
F (ρ ) = F (K ) M0 L ex gF (KM)
ρex L Ω L − (fM)2v Ω ex
(4) R M0 ex g
= Scal f Rv M0(fLfeMx)vg Scal fMv ,
g L g − (fM)2v g ex g
ZM0 R M0 ex g ZM0
where K denotes the generator for the inducRed S1 action by ρ (again a subgroup of Tℓ),
L L
f is its Killing potential of zero mean value with respect to g. Of course, the r.h.s. of (4)
L
is independent of the choice of Tℓ-invariant Ka¨hler metric g in Ω.
As explained in the proof of Lemma 3 in [3], one can extend the Tℓ invariant Ka¨hler
metric (g,ω) on M = (M,J ) to a smooth family of Tℓ 1 invariant Ka¨hler metrics (g ,ω )
0 0 − t t
on M = (M,J ) (at least for t < ε) and then use the equivariant Moser lemma in order
t t
| |
to find a Tℓ 1 equivariant family of diffeomorphisms Φ on M, which send the complex
− t
structure J of M = M to a complex structure J˜ on M, compatible with the initial
t ∼ t t
symplectic form ω. As Φ commutes with the action of Tℓ 1 and KM Lie(Tℓ 1), Φ
t − ex ∈ − t
preserves KM. In this symplectic setting, it is shown in [3, Lemma 2] (see also [23])
ex
that fM can be obtained as the L2-projection of the scalar curvature of any Tℓ 1 invariant
ex −
Ka¨hlermetriccompatiblewithωtothefinitedimensionalspaceofnormalizedhamiltonians
for the Tℓ 1 action on (M,ω). In particular, with respect to the initial metric g, we have
−
that fM coincides with the L2-projection ScalTℓ−1 of Scal to the space of normalized
ex g g
hamiltonians of Tℓ 1 Tℓ. In particular, we have
−
⊂
Scal fMv = (fM)2v ,
g ex g ex g
ZM0 ZM0
so that (4) becomes
(5) F (ρ ) = Scal f v (f ScalTℓ−1)v .
ρex L g L g − L g g
ZM0 ZM0
From this point of view, (5) can be entirely computed from the symplectic structure ω on
M , endowed with the hamiltonian action of Tℓ. We thus have
0
Lemma 3.2. The r.h.s of (5) does not depend on the choice of an ω compatible, Tℓ
invariant K¨ahler metric (g,J ) on (M,ω) nor on the choice of a Tℓ invariant Ka¨hler
g
metric g˜ on (M,J ) within the K¨ahler class Ω = [ω].
g
3.3. Generalized Calabi Ansatz. When C is of genus g 2, by using Lemma 3.2 and
≥
the Narasimhan–Ramanan approximation theorem [27], we can compute (5) with respect
to an ω compatible, Tℓ invariant complex structure J˜ on M , corresponding to taking
0
stable holomorphic structures on each V , see [3, Lemma 2]. Furthermore, in this case, we
k
can use the generalized Calabi Ansatz of [3] in order to choose a particularly simple metric
g in the class Ω = [ω] on (M,J˜), which will make the computation of (5) explicit.
c
To simplify the notation, we shall assume throughout this section that M = (M,J ) is
0 0
a ruled complex manifold
ℓ
M = P V C,
0 k
→
(cid:16)Mk=0 (cid:17)
over a compact complex curve of genus g 2, and V are stable vector bundle over C.
k
≥
This is a special case of the semi-simple rigid toric fibre-bundlesconsidered in [3], see Sect.
2.2 loc cit.
10 V.APOSTOLOVANDJ.KELLER
We introduce a family of Ka¨hler metrics (g ,ω ) on M , parametrized by a real constant
c c 0
c, as follows: As each V is a stable and therefore projectively-flat bundleover C, it admits
i
a projectively-flat hermitian metric h whose Chern curvature is µ(V )Id ω , where the
i i C
⊗
topological constant
deg(V )
i
µ(V ):= = c (V )/rk(V )
i 1 i i
rk(V )
i ZC
is the slope of V , and ω is the Ka¨hler form of the metric g on C of constant scalar
k C C
curvature 2(1 g). We denote by z one-half of the square norm function defined by h on
i i
−
V . Thus, z is the fibre-wise momentum map for the standard U(1) action on V by scalar
i i i
multiplication, with respect to the imaginary part of the hermitian product defined by h .
i
We consider the fibre-wise Ka¨hler quotient at moment value z + z = 1 of
0 ℓ
···
ℓ
V = V
k
k=0
M
with respectto thehermitian producth = h h and thediagonal U(1) action on V:
0 ℓ
⊕···⊕
thisgives theFubini–StudymetricgFS ofscalarcurvature2n(n 1)oneach fibreofM =
P(V) − 0
P(V) C. We usetheChernconnection of (V,h) (which inducesa horizontal distribution
→
on TM ) in order to complete trivially (gFS ,ωFS ) in the horizontal direction, and thus
0 P(V) P(V)
define a Ka¨hler metric on M as follows:
0
ℓ
g = c µ(V )L (x) g +gFS ,
c − k k C P(V)
(6) (cid:16) Xk=0 (cid:17)
ℓ
ω = c µ(V )L (x) ω +ωFS ,
c − k k C P(V)
(cid:16) Xk=0 (cid:17)
where:
the function L (x) is the restrictions of z on the level set z + z = 1 and then
j j 0 ℓ
• ···
quotient to M; letting x := z = L (x) for i = 1,...ℓ, we then have L (x) =
i i i 0
1 ℓ x . Thus, (x ,...,x ) is the induced (fibre-wise) moment map for the Tℓ
− j=1 j 1 ℓ
action on (P(V),ω ), taking values in the standard simplex ∆ Rℓ.
P c ⊂
c is a real constant satisfying
•
(7) c > max µ(V ),i = 0,...,ℓ ,
i
{ }
ℓ
or, equivalently, (c µ(V )L (x)) >0 on M .
− k=0 k k 0
(g ,ω ) is the pull back of the Ka¨hler structure on C to M .
C C 0
• P
It is not immediately clear from the above description that ω is a closed form, but for
c
various computational purposes it will be more convenient to describe (g ,ω ) in terms of
c c
its pull-back to the the blow-up Mˆ of M along the sub-manifolds S = P(V ) P(V),
0 0 i i
⊂
which is isomorphic to the total space of the CPℓ fibre-bundle
Mˆ = P ( 1) ( 1) Sˆ,
0 O − P(V0)⊕···⊕O − P(Vℓ) →
(cid:16) (cid:17)
where
Sˆ = P(V ) P(V ) C.
0 C C ℓ
× ···× →
