Semi-stable Higgs sheaves and Bogomolov type inequality

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by  Jiayu Li

SEMI-STABLE HIGGS SHEAVES AND BOGOMOLOV TYPE INEQUALITY
JIAYULI,CHUANJINGZHANG,ANDXIZHANG
6
1 Abstract. Inthispaper,westudysemistableHiggssheavesovercompactK¨ahlermanifolds,
0 we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-
2 stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a
semi-stablereflexiveHiggssheaf.
n
a
J

5
1. Introduction

]
G
Let (M,ω) be a compact K¨ahler manifold, and E be a holomorphic vector bundle on M.
D
Donaldson-Uhlenbeck-Yau theorem states that the ω-stability of E implies the existence of
.
h ω-Hermitian-Einstein metric on E. Hitchin [17] and Simpson [32] proved that the theorem
t holds also for Higgs bundles. We [25] proved that there is an approximate Hermitian-Einstein
a
m structure on a semi-stable Higgs bundle, which confirms a conjecture due to Kobayashi [19]
(also see [18]). There are many interesting and important works related ([21, 17, 32, 4, 6, 12,
[
5, 1, 3, 7, 22, 23, 29, 27, 28], etc.). Among all of them, we recall that, Bando and Siu [6]

1 introduced the notion of admissible Hermitian metrics on torsion-free sheaves, and proved the
v
Donaldson-Uhlenbeck-Yau theorem on stable reflexive sheaves.
9
Let beatorsion-freecoherentsheaf,andΣbethesetofsingularitieswhere isnotlocally
2
E E
7 free. A Hermitian metric H on the holomorphic bundle M\Σ is called admissible if
E|
0 (1) F is square integrable;
H H,ω
0 (2) |Λ F| is uniformly bounded.
ω H H
. | |
1 Here FH is the curvature tensorof ChernconnectionDH with respectto the Hermitianmetric
0 H, and Λ denotes the contraction with the K¨ahler metric ω.
ω
6
Higgs bundle and Higgs sheaf are studied by Hitchin ([17]) and Simpson ([32], [33]), which
1
play animportantrolein many different areasincluding gaugetheory, K¨ahlerand hyperk¨ahler
:
v geometry, group representations, and nonabelian Hodge theory. A Higgs sheaf on (M,ω) is
Xi a pair ( ,φ) where is a coherent sheaf on M and the Higgs field φ Ω1,0(End( )) is a
E E ∈ E
holomorphic section such that φ φ = 0. If the sheaf is torsion-free (resp. reflexive, locally
r ∧ E
a free), then we saythe Higgs sheaf ( ,φ) is torsion-free(resp. reflexive,locally free). A torsion-
E
freeHiggssheaf( ,φ)issaidtobeω-stable(respectively,ω-semi-stable),ifforeveryφ-invariant
E
coherent proper sub-sheaf ֒ , it holds:
F →E
deg ( ) deg ( )
µ ( )= ω F <( )µ ( )= ω E , (1.1)
ω ω
F rank( ) ≤ E rank( )
F E
where µ ( ) is called the ω-slope of .
ω
F F
Given a Hermitian metric H on the locally free part of the Higgs sheaf ( ,φ), we consider
E
the Hitchin-Simpson connection
∂ :=∂ +φ, D1,0 :=D1,0+φ∗H, D =∂ +D1,0 , (1.2)
φ E H,φ H H,φ φ H,φ
1991 Mathematics Subject Classification. 53C07,58E15.
Keywords and phrases. Higgssheaf,approximateHermitian-Einsteinstructure,Bogomolovinequality.
TheauthorsweresupportedinpartbyNSFinChina,No. 11571332, 11131007, 11526212.
1
2 JIAYULI,CHUANJINGZHANG,ANDXIZHANG
where D is the Chern connection with respect to the metric H and φ∗H is the adjoint of φ
H
with respect to H. The curvature of the Hitchin-Simpson connection is
F =F +[φ,φ∗H]+D1,0φ+∂ φ∗H, (1.3)
H,φ H H E
where F is the curvature of the Chern connection D . A Hermitian metric H on the Higgs
H H
sheaf ( ,φ) is said to be admissible Hermitian-Einstein if it is admissible and satisfies the
E
following Einstein condition on M Σ, i.e
\
√ 1Λ (F +[φ,φ∗H])=λId , (1.4)
ω H E
−
where λ is a constant given by λ = 2π µ ( ). Hitchin ([17]) and Simpson ([32]) proved
Vol(M,ω) ω E
that a Higgs bundle admits a Hermitian-Einstein metric if and only if it’s Higgs poly-stable.
Biswas and Schumacher [8] studied the Donaldson-Uhlenbeck-Yau theorem for reflexive Higgs
sheaves.
In this paper, we study the semi-stable Higgs sheaves. We say a torsion-free Higgs sheaf
( ,φ) admits an approximate admissible Hermitian-Einstein structure if for every positive δ,
E
there is an admissible Hermitian metric H such that
δ
sup |√−1Λω(FHδ +[φ,φ∗Hδ])−λIdE|Hδ(x)<δ. (1.5)
x∈M\Σ
The approximate Hermitian-Einstein structure was introduced by Kobayashi ([19]) on a holo-
morphicvectorbundle,itisthedifferentialgeometriccounterpartofthesemi-stability. Kobayashi
[19]provedthere isanapproximateHermitian-Einsteinstructureonasemi-stableholomorphic
vectorbundle overanalgebraicmanifold, whichhe conjecturedshouldbe true overanyK¨ahler
manifold. Theconjecturewasconfirmedin[18,25]. Inthispaper,weprovedourtheoremholds
for a semi-stable reflexive Higgs sheaf over a compact K¨ahler manifold.
Theorem 1.1. A reflexive Higgs sheaf ( ,φ) on an n-dimensional compact K¨ahler manifold
E
(M,ω) is semi-stable, if and only if it admits an approximate admissible Hermitian-Einstein
structure. Specially, forasemi-stablereflexiveHiggssheaf( ,φ)ofrankr,wehavethefollowing
E
Bogomolov type inequality
r 1 ωn−2
(2c ( ) − c ( ) c ( )) 0. (1.6)
2 1 1
Z E − r E ∧ E ∧ (n 2)! ≥
M −
TheBogomolovinequalitywasfirstobtainedbyBogomolov([9])forsemi-stableholomorphic
vector bundles over complex algebraic surfaces, it had been extended to certain classes of
generalized vector bundles, including parabolic bundles and orbibundles. By constructing a
Hermitian-Einstein metric, Simpson provedthe Bogomolovinequality for stable Higgs bundles
on compact K¨ahler manifolds. Recently, Langer ([20]) proved the Bogomolov type inequality
for semi-stable Higgs sheaves over algebraic varieties by using an algebraic-geometric method.
His method can not be applied to the K¨ahler manifold case. We use analytic method to study
theBogomolovinequalityforsemi-stablereflexiveHiggssheavesovercompactK¨ahlemanifolds,
new idea is needed.
We now give an overview of our proof. As in [6], we make a regularization on the reflexive
sheaf , i.e. take blowing up with smooth centers finite times π : M M , where i =
i i i−1
E →
1, ,k and M =M, such that the pull-back of ∗ to M modulo torsion is locally free and
0 k
··· E
π =π π :M M (1.7)
1 k k
◦···◦ →
is biholomorphic outside Σ. In the following, we denote M by M˜, the exceptional divisor
k
π−1Σ by Σ˜, and the holomorphic vector bundle (π∗ ∗/torsion)∗ by E. Since is locally free
E E
outside Σ, and the holomorphic bundle E is isomorphic to on M˜ Σ˜, the pull-back field
E \
SEMI-STABLE HIGGS SHEAVES AND BOGOMOLOV TYPE INEQUALITY 3
π∗φ is a holomorphic section of Ω1,0(End(E)) on M˜ Σ˜. By Hartogs’ extension theorem, the
\
holomorphicsectionπ∗φcanbeextendedtothewholeM˜ asaHiggsfieldofE. Inthefollowing,
wealsodenotetheextendedHiggsfieldπ∗φbyφforsimplicity. SowegetaHiggsbundle(E,φ)
on M˜ which is isomorphic to the Higgs sheaf ( ,φ) outside the exceptional divisor Σ˜.
E
It is well known that M˜ is also K¨ahler ([15]). Fix a K¨ahler metric η on M˜ and set
ω =π∗ω+ǫη (1.8)
ǫ
for any small 0 < ǫ 1. Let K (t,x,y) be the heat kernel with respect to the K¨ahler metric
ǫ
≤
ω . Bando and Siu (Lemma 3 in [6]) obtained a uniform Sobolev inequality for (M˜,ω ), using
ǫ ǫ
Cheng and Li’s estimate ([11]), they gota uniform upper bound of the heat kernels K (t,x,y).
ǫ
Given a smooth Hermitian metric Hˆ on the bundle E, it is easy to see that there exists a
constant Cˆ such that
0
ωn
(Λ F + φ2 ) ǫ Cˆ , (1.9)
ZM˜ | ωǫ Hˆ|Hˆ | |Hˆ,ωǫ n! ≤ 0
for all 0<ǫ≤1. This also gives a uniform bound on M˜ |Λωǫ(FHˆ +[φ,φ∗Hˆ])|Hˆωnǫn!.
WestudythefollowingevolutionequationonHiggsbRundle(E,φ)withthefixedinitialmetric
Hˆ and with respect to the K¨ahler metric ω ,
ǫ
∂H (t)
H (t)−1 ǫ = 2(√ 1Λ (F +[φ,φ∗Hǫ(t)]) λ Id ),
 ǫ ∂t − − ωǫ Hǫ(t) − ǫ E (1.10)
H (0)=Hˆ,
ǫ
where λ = 2π µ (E). Simpson ([32]) proved the existence of long time solution of the
ǫ Vol(M˜,ωǫ) ωǫ
aboveheatflow. Bythestandardparabolicestimatesandtheuniformupperboundoftheheat
kernels Kǫ(t,x,y), we know that |Λωǫ(FHǫ(t) +[φ,φ∗Hǫ(t)])|Hǫ(t) has a uniform L1 bound for
t 0 and a uniform L∞ bound for t t > 0. As in [6], taking the limit as ǫ 0, we have
0
≥ ≥ →
a long time solution H(t) of the following evolution equation on M Σ [0,+ ), i.e. H(t)
\ × ∞
satisfies:
∂H(t)
H(t)−1 = 2(√ 1Λ (F +[φ,φ∗H(t)]) λId ),
 ∂t − − ω H(t) − E (1.11)
H(0)=Hˆ.
HereH(t)canbeseenasaHermitianmetricdefinedonthelocallyfreepartof ,i.e. onM Σ.
E \
Inorder to get the admissibility of Hermitianmetric H(t) for positive time t>0, we should
show that φ L∞ for t > 0. In fact, we can prove that φ has a uniform L∞
H(t),ω H(t),ω
| | ∈ | |
bound for t t > 0. In [24], by using the maximum principle, we proved this uniform L∞
0
≥
bound of φ along the evolution equation for the Higgs bundle case. In the Higgs sheaf
H(t),ω
| |
case, since the equation (1.11) has singularity on Σ, we can not use the maximum principle
directly. So we need new argument to get a uniform L∞ bound of φ , see section 3 for
H(t),ω
| |
details.
The key part in the proof of Theorem 1.1 is to prove the existence of admissible approxi-
mate Hermitian-Einsteinstructureonasemi-stablereflexiveHiggssheaf. TheBogomolovtype
inequality (1.6) is an application. In fact, we prove that if the reflexive Higgs sheaf ( ,φ) is
E
semi-stable, along the evolution equation (1.11), we must have
sup √ 1Λ (F +[φ,φ∗H(t)]) λId (x) 0, (1.12)
ω H(t) E H(t)
| − − | →
x∈M\Σ
as t + . We prove (1.12) by contradiction, if not, we can construct a saturated Higgs
→ ∞
subsheaf such that its ω-slope is greater than µ ( ). Since the singularity set Σ is a complex
ǫ
E
analyticsubsetwithco-dimensionatleast3,itiseasytoshowthat(M Σ,ω)satisfiesallthree
\
assumptions that Simpson ([32]) imposes on the non-compact base K¨ahler manifold. Let’s
4 JIAYULI,CHUANJINGZHANG,ANDXIZHANG
recall Simpson’s argument for a Higgs bundle in the case where the base K¨ahler manifold is
non-compact. Simpson assumes that there exists a good initial Hermitian metric K satisfying
sup Λ F < ,thenhedefinestheanalyticstabilityfor( ,φ,K)byusingtheChern-
M\Σ| ω K,φ|K ∞ E
Weilformulawithrespecttothe metricK (Lemma 3.2in[32]). Under theK-analyticstability
condition, he constructs a Hermitian-Einstein metric for the Higgs bundle by limiting the
evolution equation (1.11).
Here, we have to pay more attention to the analytic stability (or semi-stability) of ( ,φ).
E
Let be a saturatedsub-sheaf of , we know that can be seen as a sub-bundle of outside
asinFgularitysetV =Σ Σ ofcodEimensionatleastF2,thenHˆ inducesaHermitianmEetricHˆ
F F
∪
on . Bruasse (Proposition 4.1 in [10]) had proved the following Chern-Weil formula
F
ωn−1
deg ( )= c ( ,Hˆ ) , (1.13)
ω F Z 1 F F ∧ (n 1)!
M\V −
where c ( ,Hˆ ) is the first Chern form with respect to the induced metric Hˆ . By (1.13),
1 F F
F
we see that the stability (semi-stability ) of the reflexive Higgs sheaf ( ,φ) is equivalent to the
E
analytic stability (semi-stability)with respectto the metric Hˆ in Simpson’s sense. But, we are
not clear whether the above Chern-Weil formula is still valid if the metric Hˆ is replaced by
an admissible metric H(t) (t > 0). So, the stability (or semi-stability) of the reflexive Higgs
sheaf ( ,φ) may not imply the analytic stability (or semi-stability ) with respect to the metric
E
H(t) (t > 0). The admissible metric H(t) (t > 0) can not be chosen as a good initial metric
in Simpson’s sense. On the other hand, the initial metric Hˆ may not satisfy the curvature
finiteness condition (i.e. Λ F may not be L∞ bounded), so we should modify Simpson’s
| ω Hˆ,φ|Hˆ
argument in our case, see the proof of Proposition 4.1 in section 4 for details.
If the reflexive Higgs sheaf ( ,φ) is ω-stable, it is well known that the pulling back Higgs
E
bundle (E,φ) is ω -stable for sufficiently small ǫ. By Simpson’s result ([32]), there exists an
ǫ
ω -Hermitian-Einstein metric H for every small ǫ. In [6], Bando and Siu point out that it is
ǫ ǫ
possible to get an ω-Hermitian-Einstein metric H on the reflexive Higgs sheaf ( ,φ) as a limit
E
of ω -Hermitian-Einstein metric H of Higgs bundle (E,φ) on M˜ as ǫ 0. In the end of this
ǫ ǫ
→
paper, we solve this problem.
Theorem 1.2. Let H be an ω -Hermitian-Einstein metric on the Higgs bundle (E,φ), by
ǫ ǫ
choosing a subsequence and rescaling it, H must converge to an ω-Hermitian-Einstein metric
ǫ
H in local C∞-topology outside the exceptional divisor Σ˜ as ǫ 0.
→
This paper is organizedas follows. In Section 2, we recall some basic estimates for the heat
flow (1.10) and give proofs for local uniform C0, C1 and higher order estimates for reader’s
convenience. In section 3, we give a uniform L∞ bound for the norm of the Higgs field along
the heatflow(1.11). Insection4,we provethe existence ofadmissible approximateHermitian-
Einstein structure on the semi-stable reflexive Higgs sheaf and complete the proof of Theorem
1.1. In section 5, we prove Theorem 1.2.
2. Analytic preliminaries and basic estimates
Let (M,ω) be a compact K¨ahler manifold of complex dimension n, and ( ,φ) be a reflexive
E
Higgs sheaf on M with the singularity set Σ. There exists a bow-up π :M˜ M such that the
pulling back Higgs bundle (E,φ) on M˜ is isomorphic to ( ,φ) outside the→exceptional divisor
E
Σ˜ = π−1Σ. It is well known that M˜ is also K¨ahler ([15]). Fix a K¨ahler metric η on M˜ and
set ω =π∗ω+ǫη for 0 <ǫ 1. Let K (x,y,t) be the heat kernel with respect to the K¨ahler
ǫ ǫ
≤
SEMI-STABLE HIGGS SHEAVES AND BOGOMOLOV TYPE INEQUALITY 5
metric ω . Bando andSiu (Lemma 3 in [6]) obtaineda uniform Sobolev inequalityfor (M˜,ω ).
ǫ ǫ
Combining Cheng and Li’s estimate ([11]) with Grigor’yan’s result (Theorem 1.1 in [16]), we
havethefollowinguniformupperboundoftheheatkernels,furthermore,wealsohaveauniform
lower bound of the Green functions.
Proposition 2.1. (Proposition 2 in [6]) Let K be the heat kernel with respect to the metric
ǫ
ω , then for any τ >0, there exists a constant C (τ) which is independent of ǫ, such that
ǫ K
(d (x,y))2
0 K (x,y,t) C (τ)(t−nexp( ωǫ )+1) (2.1)
ǫ K
≤ ≤ − (4+τ)t
for every x,y M˜ and 0 < t < + , where d (x,y) is the distance between x and y with
∈ ∞ ωǫ
respect to the metric ω . There also exists a constant C such that
ǫ G
G (x,y) C (2.2)
ǫ G
≥−
for every x,y M˜ and 0 < ǫ 1, where G is the Green function with respect to the metric
ǫ
∈ ≤
ω .
ǫ
Let H (t) be the long time solutions of the heat flow (1.10) on the Higgs bundle (E,φ) with
ǫ
the fixed smooth initial metric Hˆ and with respect to the K¨ahler metric ω . By (1.9), there is
ǫ
a constant Cˆ independent of ǫ such that
1
ωn
√ 1Λ (F +[φ,φ∗Hˆ]) λ Id ǫ Cˆ . (2.3)
ZM˜ | − ωǫ Hˆ − ǫ E|Hˆ n! ≤ 1
For simplicity, we set:
Φ(H (t),ω )=√ 1Λ (F +[φ,φ∗Hǫ(t)]) λ Id . (2.4)
ǫ ǫ − ωǫ Hǫ(t) − ǫ E
The followingestimates are essentiallyprovedby Simpson (Lemma 6.1 in [32], see alsoLemma
4 in [25]). Along the heat flow (1.10), we have:
∂
(∆ )tr(Φ(H (t),ω ))=0, (2.5)
ǫ ǫ ǫ
− ∂t
∂
(∆ )Φ(H (t),ω )2 =2D (Φ(H (t),ω ))2 , (2.6)
ǫ− ∂t | ǫ ǫ |Hǫ(t) | Hǫ,φ ǫ ǫ |Hǫ(t),ωǫ
and
∂
(∆ )Φ(H (t),ω ) 0. (2.7)
ǫ− ∂t | ǫ ǫ |Hǫ(t) ≥
Then, for t>0,
ωn ωn
Φ(H (t),ω ) ǫ Φ(Hˆ,ω ) ǫ Cˆ , (2.8)
ZM˜ | ǫ ǫ |Hǫ(t) n! ≤ZM˜ | ǫ |Hˆ n! ≤ 1
ωn
max Φ(H (t),ω ) (x) K (x,y,t)Φ(Hˆ,ω ) ǫ , (2.9)
x∈M˜ | ǫ ǫ |Hǫ(t) ≤ZM˜ ǫ | ǫ |Hˆ n!
and
ωn
max Φ(H (t+1),ω ) (x) K (x,y,1)Φ(H (t),ω ) ǫ . (2.10)
x∈M˜ | ǫ ǫ |Hǫ(t+1) ≤ZM˜ ǫ | ǫ ǫ |Hǫ(t) n!
By the upper bound of the heat kernels (2.1), we have
max Φ(H (t),ω ) (x) C (τ)Cˆ (t−n+1), (2.11)
x∈M˜ | ǫ ǫ |Hǫ(t) ≤ K 1
and
ωn
max Φ(H (t+1),ω ) (x) 2C (τ) Φ(H (t),ω ) ǫ . (2.12)
x∈M˜ | ǫ ǫ |Hǫ(t+1) ≤ K ZM˜ | ǫ ǫ |Hǫ(t) n!
6 JIAYULI,CHUANJINGZHANG,ANDXIZHANG
Set
exp(S (t))=h (t)=Hˆ−1H (t), (2.13)
ǫ ǫ ǫ
where S (t) End(E) is self-adjoint with respect to Hˆ and H (t). By the heat flow (1.10), we
ǫ ǫ
∈
have:
∂ ∂h
logdet(h (t))=tr(h−1 ǫ)= 2tr(Φ(H (t),ω )), (2.14)
∂t ǫ ǫ ∂t − ǫ ǫ
and
ωn ωn
tr(S (t)) ǫ = logdet(h (t)) ǫ =0 (2.15)
ǫ ǫ
ZM˜ n! ZM˜ n!
for all t 0.
≥
In the following, we denote:
B (δ)= x M˜ d (x,Σ)<δ , (2.16)
ω1 { ∈ | ω1 }
where d is the distance function with respect to the K¨ahler metric ω . Since Hˆ is a smooth
ω1 1
Hermitian metric on E, φ Ω1,0(End(E)) is a smooth field, and π∗ω is degenerate only along
∈ M˜
Σ, there exist constants cˆ(δ−1) andˆb (δ−1) such that
k
Λ F + φ2 (y) cˆ(δ−1),
{| ωǫ Hˆ|Hˆ | |Hˆ,ωǫ} ≤ (2.17)
k F 2 + k+1φ2 ˆb (δ−1),
{|∇Hˆ Hˆ|Hˆ,ωǫ |∇Hˆ |Hˆ,ωǫ}≤ k
for all y M˜ B (δ), all 0 ǫ 1 and all k 0.
∈ \ ω1 2 ≤ ≤ ≥
In order to get a uniform local C0-estimate of h (t), We first prove that Φ(H (t),ω )
ǫ | ǫ ǫ |Hǫ(t)
is uniform locally bounded, i.e. we obtain the following Lemma.
Lemma 2.2. There exists a constant C˜ (δ−1) such that
1
Φ(H (t),ω ) (x) C˜ (δ−1) (2.18)
| ǫ ǫ |Hǫ(t) ≤ 1
for all (x,t) (M˜ B (δ)) [0, ), and all 0<ǫ 1.
∈ \ ω1 × ∞ ≤
Proof. Using the inequality (2.9), we have
ωn(y)
Φ(H (t),ω ) (x) + K (x,y,t)Φ(Hˆ,ω ) (y) ǫ . (2.19)
| ǫ ǫ |Hǫ(t) ≤(cid:16)ZM\Bǫ(2δ) ZBǫ(δ2)(cid:17) ǫ | ǫ |Hˆ n!
ωn
Noting M˜ Kǫ(x,y,t)nǫ! =1 and using (2.17), we have
R ωn
K (x,y,t)Φ(Hˆ,ω ) (y) ǫ
ZM˜\Bǫ(δ2) ǫ | ǫ |Hˆ n!
ωn(y) (2.20)
(cˆ(δ−1)+λ √r) K (x,y,t) ǫ
ǫ ǫ
≤ ZM˜ n!
cˆ (δ−1).
1
≤
wherecˆ (δ−1)isaconstantindependentofǫ. Sinceπ∗ω isdegenerateonlyalongΣ,thereexists
1
a constant a˜(δ) such that
a˜(δ)ω <π∗ω <ω <ω (2.21)
1 ǫ 1
on M˜ B (δ), for all 0<ǫ 1. Let x M˜ B (δ) and y ∂(B (δ)), it is clear that
\ ω1 4 ≤ ∈ \ ω1 ∈ ω1 2
δ a˜(δ)
dωǫ(x,y)≥dπ∗ω(x,y)> a˜(δ)dω1(x,y)≥ p2 . (2.22)
p
SEMI-STABLE HIGGS SHEAVES AND BOGOMOLOV TYPE INEQUALITY 7
Let a(δ)= δ√a˜(δ). If x M˜ B (δ) and y B (δ), we have
2 ∈ \ ω1 ∈ ω1 2
d (x,y) a(δ) (2.23)
ωǫ ≥
for all 0 ǫ 1. Then,
≤ ≤
ωn(y)
K (x,y,t)Φ(Hˆ,ω ) (y) ǫ
ZBω1(2δ) ǫ | ǫ |Hˆ n!
d (x,y) ωn(y)
C (τ) (t−nexp( ωǫ )+1)Φ(Hˆ,ω ) (y) ǫ
≤ k ZBω1(δ2) −(4+τ)t | ǫ |Hˆ n!
a(δ) ωn
C (τ) (t−nexp( )+1)Φ(Hˆ,ω ) ǫ (2.24)
≤ k ZBω1(δ2) −(4+τ)t | ǫ |Hˆ n!
C (τ) a(δ) n −nexp( n) Φ(Hˆ,ω ) ωǫn
≤ k (cid:0)4+τ (cid:1) − ZBω1(δ2)| ǫ |Hˆ n!
C (τ)Cˆ a(δ) n −nexp( n),
k 1
≤ 4+τ −
(cid:0) (cid:1)
for all (x,t) (M˜ B (δ)) [0, ). It is obvious that (2.19), (2.20) and (2.24) imply (2.18).
∈ \ ω1 × ∞ ✷
By a direct calculation, we have
∂
log(trh (t)+trh−1(t))
∂t ǫ ǫ
= tr(hǫ(t)·h−ǫ1(t)∂h∂ǫt(t))−tr(h−ǫ1(t)∂h∂ǫt(t) ·h−ǫ1(t)) (2.25)
trh (t)+trh−1(t)
ǫ ǫ
2Φ(H (t),ω ) ,
≤ | ǫ ǫ |Hǫ(t)
and
1
log(2r(trhǫ(t)+trhǫ(t)−1))≤|Sǫ(t)|Hˆ ≤r12 log(trhǫ(t)+trhǫ(t)−1), (2.26)
where r =rank(E). By (2.8) and (2.18), we have
ωn
log(trh (t)+trh−1(t)) log(2r) ǫ Cˆ t, (2.27)
ZM˜ ǫ ǫ − n! ≤ 1
and
log(trh (t)+trh−1(t)) log(2r) 2C˜ (δ−1)T (2.28)
ǫ ǫ − ≤ 1
for all (x,t) (M˜ B (δ)) [0,T]. Then, we have the following local C0-estimate of h (t).
∈ \ ω1 × ǫ
Lemma 2.3. There exists a constant C (δ−1,T) which is independent of ǫ such that
0
S (t) (x) C (δ−1,T) (2.29)
| ǫ |Hˆ ≤ 0
for all (x,t) (M˜ B (δ)) [0,T], and all 0<ǫ 1.
∈ \ ω1 × ≤
In the following lemma, we derive a local C1-estimate of h (t).
ǫ
Lemma 2.4. Let T (t)=h−1(t)∂ h (t). Assume that there exists a constant C such that
ǫ ǫ Hˆ ǫ 0
max S (t) (x) C , (2.30)
(x,t)∈(M˜\Bω1(δ))×[0,T]| ǫ |Hˆ ≤ 0
8 JIAYULI,CHUANJINGZHANG,ANDXIZHANG
for all 0<ǫ 1. Then, there exists a constant C depending only on C and δ−1 such that
1 0
≤
max T (t) C (2.31)
(x,t)∈(M˜\Bω1(32δ))×[0,T]| ǫ |Hˆ,ωǫ ≤ 1
for all 0<ǫ 1.
≤
Proof. By a direct calculation, we have
∂
(∆ )trh (t)
ǫ ǫ
− ∂t
= 2tr( √ 1Λ ∂h (t) h−1(t) ∂ h (t))+2tr(h (t)Φ(Hˆ,ω ))
− − ωǫ ǫ · ǫ · Hˆ ǫ ǫ ǫ
+2√ 1Λ tr h (t) ([φ,φ∗Hǫ(t)] [φ,φ∗Hˆ])
− ωǫ { ǫ ◦ − } (2.32)
= 2tr( √ 1Λ ∂h (t) h−1(t) ∂ h (t))+2tr(h (t)Φ(Hˆ,ω ))
− − ωǫ ǫ · ǫ · Hˆ ǫ ǫ ǫ
+2√ 1Λ tr [φ,h (t)] h−1(t)[h (t),φ∗Hˆ]
− ωǫ { ǫ ∧ ǫ ǫ }
2tr( √ 1Λ ∂h (t) h−1(t) ∂ h (t))+2tr(h (t)Φ(Hˆ,ω )),
≥ − − ωǫ ǫ · ǫ · Hˆ ǫ ǫ ǫ
∂ ∂
T (t)=∂ (h−1(t) h (t))= 2∂ (Φ(H (t),ω )), (2.33)
∂t ǫ Hǫ(t) ǫ ∂t ǫ − Hǫ(t) ǫ ǫ
and
∂
(∆ )T (t)2 2 T (t)2
ǫ− ∂t | ǫ |Hǫ(t),ωǫ ≥ |∇Hǫ(t) ǫ |Hǫ(t),ωǫ
Cˇ (Λ F + F + φ2 + Ric(ω ) )T (t)2 (2.34)
− 1 | ωǫ Hǫ(t)|Hǫ(t) | Hˆ|Hǫ(t),ωǫ | |Hǫ(t),ωǫ | ǫ |ωǫ | ǫ |Hǫ(t),ωǫ
Cˇ (Λ F ) T (t) φ2 ,
− 2|∇Hˆ ωǫ Hˆ |Hǫ(t),ωǫ| ǫ |Hǫ(t),ωǫ −|∇Hˆ |Hǫ(t),ωǫ
where constants Cˇ ,Cˇ depend only on the dimension n and the rank r.
1 2
By the local C0-assumption (2.30), the local estimate (2.18) and the definition of ω , it is
ǫ
easy to see that all coefficients in the right term of (2.34) are uniformly local bounded outside
Σ˜. Then there exists a constant Cˇ depending only on δ−1 and C such that
3 0
∂
(∆ )T (t)2 2 T (t)2
ǫ− ∂t | ǫ |Hǫ(t),ωǫ ≥ |∇Hǫ(t) ǫ |Hǫ(t),ωǫ (2.35)
Cˇ T (t)2 Cˇ
− 3| ǫ |Hǫ(t),ωǫ − 3
on the domain M˜ B (δ) [0,T].
\ ω1 ×
Let ϕ , ϕ be nonnegative cut-off functions satisfying:
1 2
0, x B (5δ),
ϕ (x)= ∈ ω1 4 (2.36)
1 (cid:26) 1, x∈M˜ \Bω1(23δ),
0, x B (δ),
ϕ (x)= ∈ ω1 (2.37)
2 (cid:26) 1, x∈M˜ \Bω1(45δ),
and dϕ 2 8 , c ω √ 1∂∂¯ϕ c ω . By the inequality(2.21), there existsa constant
| i|ω1 ≤ δ2 −δ2 1 ≤ − i ≤ δ2 1
C (δ−1) depending only on δ−1 such that
1
(dϕ 2 + ∆ ϕ ) C (δ−1), (2.38)
| i|ωǫ | ǫ i| ≤ 1
for all 0<ǫ 1.
≤
We consider the following test function
f(,t)=ϕ2 T (t)2 +Wϕ2trh (t), (2.39)
· 1| ǫ |Hǫ(t),ωǫ 2 ǫ
SEMI-STABLE HIGGS SHEAVES AND BOGOMOLOV TYPE INEQUALITY 9
where the constant W will be chosen large enough later. From (2.32) and (2.34), we have
∂
(∆ )f
ǫ
− ∂t
= ϕ2(2 T (t)2 Cˇ T (t)2 Cˇ +∆ ϕ2 T (t)2
1 |∇Hǫ(t) ǫ |Hǫ(t),ωǫ − 3| ǫ |Hǫ(t),ωǫ − 3 ωǫ 1| ǫ |Hǫ(t),ωǫ (2.40)
+4 ϕ ϕ , T (t)2 +W∆ ϕ2trh (t)+4W ϕ ϕ , trh (t)
h 1∇ 1 ∇| ǫ |Hǫ(t),ωǫiωǫ ωǫ 2 ǫ h 2∇ 2 ∇ ǫ iωǫ
+2Wϕ2(tr(√ 1Λ h−1(t)∂ h (t)∂¯h (t)))+tr(h (t)(Φ(Hˆ,ω ))).
2 − ωǫ ǫ Hˆ ǫ ǫ ǫ ǫ
We use
2 ϕ ϕ , T (t)2 4ϕ ϕ T (t) T (t)
h 1∇ 1 ∇| ǫ |Hǫ(t),ωǫiωǫ ≥− 1|∇ 1|ωǫ| ǫ |Hǫ(t),ωǫ|∇Hǫ(t) ǫ |Hǫ(t),ωǫ (2.41)
ϕ2 T (t)2 4 ϕ 2 T (t)2 ,
≥− 1| ǫ |Hǫ(t),ωǫ − |∇ 1|ωǫ| ǫ |Hǫ(t),ωǫ
W ϕ ϕ , trh (t) ϕ2 trh (t)2 W2 ϕ 2 , (2.42)
h 2∇ 2 ∇ ǫ iωǫ ≥− 2|∇ ǫ |Hǫ(t),ωǫ − |∇ 2|ωǫ
and
T (t)2
| ǫ |Hǫ(t),ωǫ
T
= tr(√ 1Λ h−1(t)∂ h (t)H−1(t)(h−1(t)∂ h (t)) H (t))
− ωǫ ǫ Hˆ ǫ ǫ ǫ Hˆ ǫ ǫ (2.43)
= tr(√ 1Λ h−1(t)∂ h (t)h−1(t)∂¯h (t))
− ωǫ ǫ Hˆ ǫ ǫ ǫ
≤ eC0tr(√−1Λωǫh−ǫ1(t)∂Hˆhǫ(t)∂¯hǫ(t)),
and choose
W =(Cˇ +4C (δ−1)+2r)eC0 +1. (2.44)
3 1
Then there exists a positive constant C˜ depending only on C and δ−1 such that
0 0
∂
(∆ )f ϕ2 T (t)2 +ϕ2 T (t)2 C˜ (2.45)
ǫ− ∂t ≥ 1|∇Hǫ(t) ǫ |Hǫ(t),ωǫ 2| ǫ |Hǫ(t),ωǫ − 0
on M˜ [0,T]. Let f(q,t ) = max η, by the definition of ϕ and the uniform local
× 0 M˜×[0,T] i
C0-assumption of h (t), we can suppose that:
ǫ
5
(q,t ) M˜ B ( δ) (0,T].
0 ∈ \ ω1 4 ×
By the inequality (2.45), we have
T (t )2 (q) C˜ . (2.46)
| ǫ 0 |Hǫ(t0),ωǫ ≤ 0
So there exists a constant C depending only on C and δ−1, such that
1 0
T (t)2 (x) C (2.47)
| ǫ |Hǫ(t),ωǫ ≤ 1
for all (x,t) M˜ B (3δ) [0,T] and all 0<ǫ 1.
∈ \ ω1 2 × ≤ ✷
OnecangetthelocaluniformC∞estimatesofh (t)bythestandardSchauderestimateofthe
ǫ
parabolic equation after getting the local C0 and C1 estimates. But by applying the parabolic
Schauderestimates,onecanonlygetthe uniformC∞ estimatesofh (t)onM˜ B (δ) [τ,T],
ǫ \ ω1 ×
where τ > 0 and the uniform estimates depend on τ−1. In the following, we first use the
maximum principle to get a local uniform bound on the curvature F , then we
| Hǫ(t)|Hǫ(t),ωǫ
apply the elliptic estimates to get local uniform C∞ estimates. The benefit of our argument
is that we can get uniform C∞ estimates of h (t) on M˜ B (δ) [0,T]. In the following, for
ǫ \ ω1 ×
simplicity, we denote
Ξ = j (F +[φ,φ∗Hǫ(t)])2 (x)+ j+1 φ2 (2.48)
ǫ,j |∇Hǫ(t) Hǫ(t) |Hǫ(t),ωǫ |∇Hǫ(t) |Hǫ(t),ωǫ
10 JIAYULI,CHUANJINGZHANG,ANDXIZHANG
for j = 0,1, . Here denotes the covariant derivative with respect to the Chern
··· ∇Hǫ(t)
connection D of H (t) and the Riemannian connection of ω .
Hǫ(t) ǫ ∇ωǫ ǫ
Lemma 2.5. Assume that there exists a constant C such that
0
max S (t) (x) C , (2.49)
(x,t)∈(M˜\Bω1(δ))×[0,T]| ǫ |Hˆ ≤ 0
for all 0 < ǫ 1. Then, for every integer k 0, there exists a constant C depending only
k+2
≤ ≥
on C , δ−1 and k, such that
0
max Ξ C (2.50)
ǫ,k k+2
(x,t)∈(M˜\Bω1(2δ))×[0,T] ≤
for all 0 < ǫ 1. Furthermore, there exist constants Cˆ depending only on C , δ−1 and k,
k+2 0
≤
such that
max k+2h Cˆ (2.51)
(x,t)∈(M˜\Bω1(2δ))×[0,T]|∇Hˆ ǫ|Hˆ,ωǫ ≤ k+2
for all 0<ǫ 1.
≤
Proof. By computing, we have the following inequalities (see Lemma 2.4 and Lemma 2.5
in ([24]) for details):
∂
(∆ ) φ2 2 φ2
ǫ− ∂t |∇Hǫ(t) |Hǫ(t),ωǫ − |∇Hǫ(t)∇Hǫ(t) |Hǫ(t),ωǫ
C (F + Rm(ω ) + φ2 ) φ2 (2.52)
≥− 7 | Hǫ(t)|Hǫ(t),ωǫ | ǫ |ωǫ | |Hǫ(t),ωǫ |∇Hǫ(t) |Hǫ(t),ωǫ
C φ Ric(ω ) φ ,
− 7| |Hǫ(t),ωǫ|∇ ǫ |ωǫ|∇Hǫ(t) |Hǫ(t),ωǫ
∂
(∆ )F +[φ,φ∗Hǫ(t)]2 2 (F +[φ,φ∗Hǫ(t)])2
ǫ− ∂t | Hǫ(t) |Hǫ(t),ωǫ − |∇Hǫ(t) Hǫ(t) |Hǫ(t),ωǫ
≥−C8(|FHǫ(t)+[φ,φ∗Hǫ(t)]|2Hǫ(t),ωǫ +|∇Hǫ(t)φ|2Hǫ(t),ωǫ)32 (2.53)
C (φ2 + Rm(ω ) )(F +[φ,φ∗Hǫ(t)]2 + φ2 ),
− 8 | |Hǫ(t),ωǫ | ǫ |ωǫ | Hǫ(t) |Hǫ(t),ωǫ |∇Hǫ(t) |Hǫ(t),ωǫ
then
∂
(∆ǫ )Ξǫ,0 2Ξǫ,1 C8(Ξǫ,0)32
− ∂t ≥ − (2.54)
C (φ2 + Rm(ω ) )(Ξ ) C Ric(ω )2 ,
− 8 | |Hǫ(t),ωǫ | ǫ |ωǫ ǫ,0 − 8|∇ ǫ |ωǫ
where C , C are constants depending only on the complex dimension n and the rank r.
7 8
Furthermore, we have
∂
(∆ )Ξ
ǫ ǫ,j
− ∂t
≥ 2Ξǫ,j+1−C´j(Ξǫ,j)12{ ((Ξǫ,i)21 +|φ|2Hǫ(t),ωǫ +|Rm(ωǫ)|ωǫ +|∇Ric(ωǫ)|ωǫ) (2.55)
i+Xk=j
·((Ξǫ,k)21 +|φ|2Hǫ(t),ωǫ +|Rm(ωǫ)|ωǫ +|∇Ric(ωǫ)|ωǫ)},
where C´ is a positive constant depending only on the complex dimension n, the rank r and j.
j
Direct computations yield the following inequality (see (2.5) in ([24]) for details):
∂
(∆ )φ2 2 φ2
ǫ− ∂t | |Hǫ(t),ωǫ ≥ |∇Hǫ(t) |Hǫ(t),ωǫ (2.56)
+2Λ [φ,φ∗Hǫ(t)]2 2Ric(ω ) φ2 .
| ωǫ |Hǫ(t)− | ǫ |ωǫ| |Hǫ(t),ωǫ