Semistability of Rational Principal $GL_n$-Bundles in Positive Characteristic

0.22 MB·

by  Lingguang Li

SEMISTABILITY OF RATIONAL PRINCIPAL GL -BUNDLES IN
n
POSITIVE CHARACTERISTIC
7
1 LINGGUANGLI
0
2
Abstract. Let k be an algebraically closed field of characteristic p > 0, X
n a smooth projective variety over k with afixed ampledivisor H. Let E be a
a rationalGLn(k)-bundleonX,andρ:GLn(k)→GLm(k)arationalGLn(k)-
J
representation at most degree d such that ρ maps the radical R(GLn(k)) of
1 GLn(k)into the radical R(GLm(k)) of GLm(k). We show that ifFXN∗(E)is
semistableforsomeintegerN ≥ max Cr ·log (dr),thentheinducedrational
] 0<r<m m p
G GLm(k)-bundle E(GLm(k)) is semistable. As an application, if dimX = n,
we get a sufficient condition for the semistability of Frobenius direct image
A FX∗(ρ∗(ΩX1)), where ρ∗(ΩX1 ) is the locally free sheaf obtained from ΩX1 via
. therationalrepresentationρ.
h
t
a
m
[ 1. Introduction

1 Let k be an algebraically closed field of arbitrary characteristic, X a smooth
v projective variety over k with a fixed ample divisor H. Let G and G′ be reductive
2 algebraic groups over k, ρ : G →G′ a homomorphism of algebraic groups. One of
5
the important and essential problem in the studying of G-bundles is to study the
2
0 behaviorof the semistability of G′-bundles under the extension of structure group.
0 Inprecise,letE be asemistablerationalG-bundleonX,doesthe inducedrational
1. G′-bundle E(G′) is also semistable?
0 Suppose that ρ maps the radical of R(G) into the radical R(G′) of G′ (Unless
7 stated otherwise, we always require this condition for any homomorphisms of al-
1
gebraic groups and all representations are rational representations in this paper),
:
v and E is a semistable rational G-bundle on X. If char(k)=0, S. Ramanan and A.
i Ramanathan [9, Theorem 3.18] showedthat the induced rational G′-bundle E(G′)
X
is also semistable on X. If char(k)= p > 0, the induced rational G′-bundle E(G′)
r
a may be not semistable in general. However, S. Ramanan and A. Ramanathan [9,
Theorem 3.23]provedthat strong semistability of rational G-bundle E implies the
strong semistability of rational G′-bundle E(G′). In addition, S. Ilangovan, V. B.
Mehta and A. J. Parameswaran [4] showed that if G′ = GL (k) for some integer
m
m > 0 and p > ht(ρ), then the induced rational GL (k)-bundle E(GL (k)) is
m m
semistable. F. Coiai and Y. I. Holla [2] generalized some results of [9] and showed
that given a representation ρ : G → GL (k), there exists a non-negative inte-
m
ger N, depending only on G and ρ, such that for any rational G-bundle E whose
N-thFrobeniuspullbackFN∗(E)issemistable,thentheinducedrationalGL (k)-
X m
bundle E(GL (k)) is again semistable. S. Gurjar and V. Mehta [3] improved the
m
result of [2] and obtain a explicit bound for N in terms of certain numerical data
attached to ρ. The main ingredient of the proof in [2] and [3] is to give a uniform
bound for the field of definition of the instability parabolics (Kempf’ parabolic)
associated to non-semistable points in related representing space.
1
2 LINGGUANGLI
We now briefly describe the main idea of their proof. Fix a representation
G → GL (k). Let E be rational G-bundle on X, E(G) the group scheme over
m
X associated to E, and E(GL (k)) the induced rational GL (k)-bundle under
m m
the extension of structure group via ρ. Let k(X) be the function field of X, the
generic fiber E(G) of E(G) is a groupscheme overSpec(k(X)). Let P be a maxi-
0
malparabolicsubgroupofGL (k), E(GL (k)/P)theassociatedGL (k)/P-fiber
m m m
space over X, and E(GL (k)/P) the generic fiber of E(GL (k)/P). Then there
m 0 m
is an E(G) -action on the smooth projective variety E(GL (k)/P) over k(X)
0 m 0
which is linearized by a suitable very ample line bundle. If E(GL (k)) adimts a
m
reduction of structure group to this maximal parabolic subgroup P, then we get
a rational section σ : U → E(GL (k)/P), where U is an open subscheme of X
m
with codim (X −U) ≥ 2. Restricting to the generic fiber gives a k(X)-rational
X
point σ of E(GL (k)/P) . In [9], it is shown that if σ is a semistable point
0 m 0 0
in E(GL (k)/P) for the above E(G) -action, then the rational reduction σ does
m 0 0
not violate the semistability of rational GL (k)-bundle E(GL (k)). Also, if σ
m m 0
is not semistable for the above E(G) -action and its instability parabolic P(σ ),
0 0
which is defined over k(X), is actually defined over k(X), then again σ does not
contradictthe semistability of rational GL (k)-bundle E(GL (k)). In the case of
m m
characteristic0, by the uniqueness of instability parabolicand it is invariantunder
the action of Galois group, then P(σ ) is actually defined over k(X). This proves
0
that E(GL (k)) is a semistable rational GL (k)-bundle. However, in the case of
m m
characteristic p >0, P(σ ) may be not defined over k(X), and it is defined over a
0
finite extension of k(X). By the uniqueness of instability parabolic, the Galois de-
scentargumentimplies thatP(σ )isactuallydefinedoverfinite purelyinseparable
0
field extension Kp−N of K for some non-negative integer N.
In[2]and[3],theauthorsshowedthatthereexistsauniformboundN,depending
onlyonG andρ, suchthatfor allpossible rationalreductionsto allmaximalpara-
bolic subgroups the instability parabolics of points corresponding to these rational
reductionsareactuallydefinedoverKp−N viadifferentmethods. Thiscanbeshown
toimplythatifE isasemistablerationalG-bundlesuchthatFN∗(E)issemistable,
X
thentheinducedrationalGL (k)-bundleE(GL (k))isalsosemistable. Themajor
m m
differences between the methods of [2] and that of [3] lie in the approach of esti-
mating the field extension L of K such that a given K-scheme M has a L-rational
point. F. Coiai and Y. I. Holla [2] proved the existence of the uniform bound by
bounding the non-separability of the group action and the non-reducedness of the
stabilizers of various unstable rational points. However, the above estimation does
notseen quantifiable. Onthe other hand, S. Gurjar and V. Mehta [3] directly esti-
mated the field of definition of the instability parabolics which is probably weaker
than the method of [2], but it is quantifiable.
The paper is organizedas follows.
Insection2,werecallsomedefinitionsandresultsaboutgeometricinvariantthe-
oryandrationalprincipalbundles,suchastheinstability1-PS,instabilityparabol-
ics of non-semistable points, etc. These results can be found in [6] and [9].
Insection3,wemainlystudy the rationalityofthe instability parabolicsofnon-
semistable points in GL (k)-representation spaces, and apply these results to the
n
studyofsemistabilityofrationalprincipalbundlesundertheextensionofstructure
groups via a GL (k)-representation ρ:GL (k)→GL (k).
n n m
SEMISTABILITY OF RATIONAL PRINCIPAL GLn-BUNDLES IN POSITIVE CHARACTERISTIC3
Theorem1.1(Theorem3.2). Letk beanalgebraically closedfieldofcharacteristic
p > 0, X a smooth projective variety over k with a fixed ample divisor H, ρ :
GL (k) → GL (k) a GL (k)-representation over k at most degree d. Let E be a
n m n
rational GL (k)-bundle on X such that FN∗(E) is semistable for some integer
n X
N ≥ max Cr ·log (dr).
m p
0<r<m
Then the induced rational GL (k)-bundle E(GL (k)) is also semistable.
m m
In section 4, we study the semistability of truncated symmetric powers Tl(E)
whichisobtainedfromatorsionfreesheafE underaspecialgrouprepresentations.
The main result is to give a sufficient condition for the semistability of Tl(E).
Theorem1.2(Theorem4.4). Letk beanalgebraically closedfieldofcharacteristic
p > 0, X a smooth projective variety over k with a fixed ample divisor H, integer
0 ≤ l ≤ (p−1)·dimX. Let E be a torsion free sheaf of rank n on X such that
FN∗(E) is semistable for some integer
X
N ≥ max Cr ·log (lr),
N(p,n,l) p
1≤r≤N(p,n,l)
l(p)
where N(p,n,l)= (−1)q·Cq·Cl−pq , and l(p) is the unique integer such that
P n n+l−q−1
q=0
0≤l−l(p)·p<p. Then the torsion free sheaf Tl(E) is also a semistable sheaf.
In section 5, we study the semistability of the Frobenius direct image of ρ (Ω1 )
∗ X
which is obtained from Ω1 via the a GL (k)-representation ρ.
X n
Theorem1.3(Theorem5.4). Letk beanalgebraically closedfieldofcharacteristic
p>0, X a smooth projective variety over k of dimension n with a fixed ample di-
visor H such that deg (Ω1 )≥0, ρ:GL (k)→GL (k) a GL (k)-representation
H X n m n
at most degree d. Suppose that FN∗(Ω1 ) is semistable for some integer
X X
N ≥ max Cr log ((d+l)r).
m·N(p,n,l) p
1≤l≤n(p−1)
0<r<m·N(p,n,l)
Then the Frobenius direct image F (ρ (Ω1 )) is a semistable sheaf, where ρ (Ω1 )
X∗ ∗ X ∗ X
is the locally free sheaf obtained from Ω1 via the representation ρ. Moreover, in
X
this case, F (ρ (Ω1 )⊗L) is semistable for any line bundle L ∈Pic(X).
X∗ ∗ X
2. Geometric Invariant Theory and Rational Principal Bundles
2.1. Geometric Invariant Theory. Let K be a field, K the separable closure
s
of K, K the algebraically closure of K. Let G be a connected reductive algebraic
group over K, Then G has a maximal torus T defined over K (See [1, Proposition
7.10]),andT splitsoverK . LetX (T)bethegroupofparametersubgroups(1-PS)
s ∗
ofT,i.e.,grouphomomorphismsofthe multiplicativegroupG intoT. LetN be
m T
thenormalizerofT inG,thenWeylgroupW :=N /T ofGwithrespecttoT acts
T T
on X (T) by conjugation. Fix an inner product h,i on X (T) which is invariant
∗ ∗
under the action of Weyl group W as well as the Galois group Gal(K /K) (See
T s
Section4of[6]). Thenwecandefinenorm kλkof1-PSλ∈X (T)ashλ,λi. LetT′
∗
be another maximal torus of G, T is conjugate to T′ by an element of G, and the
isomorphism T → T′ is well defined up to Weyl group action on T. Therefore the
innerproducth,i onX (T)determinesuniquely one inX (T′). Hence the normof
∗ ∗
any 1-PS in G is well defined.
4 LINGGUANGLI
LetV beafinitedimensionalK-vectorspace,ρ:G(K)→GL (V)arepresenta-
K
tionofG. Avector06=v ∈V issemistable fortheG(K)-actionif0∈/ G(K)·v. One
knows that this is equivalent to existence of a G(K)-invariant element φ∈Sm(V)
for some m>0 such that φ(v)6=0. For a 1-PS λ of G(K), V has a decomposition
V = V , where V ={v ∈V|λ(t)v =tiv}. Define
L i i
m(λ,v):=min{i|v has a non-zero component in V },
i
m(λ,v)
µ(λ,v):= .
kλk
For a 1-PS λ of G(K), the associated subgroup P(λ) of G(K) is defined by
P(λ):{g ∈G(K)|limλ(t)·g·λ−1(t) exists in G(K)},
t→0
which is a parabolic subgroup of G(K).
Foranon-semistablevectorv ∈V,definetheinstability 1-PSofvtobethe1-PS
λ such that µ(λ ,v)=sup{µ(λ,v)|λ∈X (G(K))}, which is not unique.
v v ∗
Wenowrecallsomebasicfactsingeometricinvarianttheory,whichcanbefound
in [6] and [9].
Lemma 2.1. [6][9] Let K be a field, G a connected reductive algebraic group over
K, ρ:G(K)→GL (V) a representation of G(K) on a K-vector space V of finite
K
dimension. Let v ∈V be a non-semistable point for the G(K)-action. Then
(1) The function λ7→µ(λ,v) on thesetX (G(K)) attains themaximumvalue.
∗
(2) Thereis auniqueinstabilityparabolic P(v)ofv suchthatfor anyinstability
1-PS λ of v, we have P(v)=P(λ).
(3) For any maximal torus T ⊆P(v), there is a unique instability 1-PS λ of
T
v such that λ ⊆T.
T
(4) For g ∈ G(K), λ is the instability 1-PS of v, then g · λ(t) · g−1 is the
instability1-PSofg·v,µ(λ,v)=µ(gλ(t)g−1,g·v)andP(g·v)=g·P(v)·g−1.
(5) For an instability 1-PS λ of v, if λ is defined over an extension field L/K,
then the instability parabolic P(v) of v is also defined over L/K.
2.2. Rational Principal G-Bundles. Let K be a field, G a reductive algebraic
groupoverK, X a smoothprojectivevariety overK with a fixed ample divisor H.
A (principal) G-bundle on X is a X-scheme π : E → X with an G-action (acts
on the right) and π is G-invariant and isotrivial, i.e., locally trivial in the ´etale
topology.
If Y is a quasi projective G-scheme over K (on the left), the associated fibre
bundle E(Y) over X is the quotient E× F under the action of G given by
K
g(e,y)=(e·g,g−1·y),g ∈G,e∈E,y ∈Y.
Let M be a projective variety over K with a G-action, which is linearized by
an ample line bundle L on X. Let E(G) := E× G denote the group scheme
G,Int
overX associatedto E by the actionof G on itself by inner automorphisms. Then
X-group scheme E(G) acts naturally on the X-scheme E(M) which is linearized
by line bundle E(L).
Let x ∈ X be a point of X, E(G) , E(M) and E(L) denote the fiber of
x x x
E(G), E(M) and E(L) over x respectively. Then E(G) is a group scheme over
x
Spec(k(x)), and one has the action of E(G) on E(M) linearized by line bundle
x x
E(L) which is defined over Spec(k(x)).
x
SEMISTABILITY OF RATIONAL PRINCIPAL GLn-BUNDLES IN POSITIVE CHARACTERISTIC5
Let P ⊂ G be a closed subgroup of G, a reduction of structure group of E to
P is a pair σ := (E ,φ) with a P-bundle E and an isomorphism of G-bundles
σ σ
φ : E (G) → E. Note that quotient E/P is naturally isomorphic to the fiber
σ
bundle E(G/P) on X and a section σ : X → E/P gives the P-bundle σ∗(E)
with naturalisomorphism σ∗(E)(G)∼=E. This induces a bijection correspondence
betweensectionsofE/P →X andreductionsofstructuregroupofE toP. LetT
P
be the tangentbundle alongthe fibers ofthe mapE/P →X,then T :=σ∗(T ) is
σ P
the vector bundle on X associated to P-bundle E for the natural representation
σ
of P on g/p, where g and p is the Lie algebra of G and P respectively.
Arational G-bundle E onX isaG-bundle onabigopensubschemeU ofX,i.e.
codim (X−U)≥2. Arational reduction of structuregroup ofarationalG-bundle
X
E to a subgroup P ⊂ G is a reduction σ of structure group of E| to P over a
U′
big open subscheme U′ ⊆ U. Then the locally free sheaf T on U′ determine a
σ
reflexivesheafi (T )wherei :U′ →X isthenaturalopenimmersion,denoted
U′∗ σ U′
by T again. The rational G-bundle E is semistable if for any rational reduction σ
σ
of E to any parabolic subgroup P of G over any big open subscheme U′ ⊆ U, the
rationalvector bundle T has deg (T )≥0. If G=GL (K), then there is an one
σ H σ n
to one correspondencebetween rational GL (K)-bundles between reflexive torsion
n
free sheaves. In this case, the semistability of the rational GL (K)-bundle E is
n
equivalentto the semistability of the torsionfree sheafE in the sense of Mumford-
Takemoto, where E is the reflexive torsion free sheaf corresponds to E.
2.3. Frobenius Pull Back of Principal G-Bundles. LetK beafieldofcharac-
teristicp>0,φ:X →Spec(K)aschemeoverk. Theabsolute Frobeniusmorphism
F :X →X isinducedbyO →O ,f 7→fp. Considerthecommutativediagram
X X X
X
❍❍❍ FX
Fg❍❍❍$$X(1) Fa ''//X
φ
F∗(φ) φ
k
(cid:31)(cid:31) (cid:15)(cid:15) (cid:15)(cid:15)
Spec(K) FK //Spec(K).
The morphism F (resp. F ) is called the arithmetic Frobenius morphism (resp.
a g
geometric Frobenius morphism) of φ:X →Spec(K).
IfK isaperfectfield,F andF areisomorphisms. LetGbeanalgebraicgroup
K a
overK,π :E →X aG-bundleoverX. PullingbackbytheabsoluteFrobeniusF∗
X
wegetaG-bundleF∗(π):F∗(E)→X ,whereX istheschemeX endowedwith
X X F F
the K-structure by the composition X →φ Spec(K) F→K Spec(K). If K is a perfect
field, we can change the K-structures of G, X and F∗(E) by composing their
F X
F−1
structure morphisms with Spec(K) →K Spec(K) to get a F∗(G)-bundle F∗(E)→
K X
X. Inthiscase,theF∗(G)-bundle F∗(E)→X isthe sameasthebundle obtained
K X
from G-bundle E by the extension of structure group g :G→F∗(G).
K
Let k be an algebraically closed field of characteristic p > 0, X a smooth pro-
jective variety over k with a fixed ample line bundle H, G a reductive algebraic
group over k. Let E be a rational G-bundle on X, then we can get a rational
F∗(G)-bundle F∗(E) on X. Then E is semistable when F∗(E) is semistable. If
K X X
Fm∗(E) is semistable for any integer m≥0, then E is called strongly semistable.
X
6 LINGGUANGLI
2.4. Semistability of Rational G-bundles under Extension of Structure
Groups. Letk beanalgebraicallyclosedfield, X asmoothprojectivevarietyover
k with a fixed ample line bundle H, G a reductive algebraic group over k with a
rational representation ρ:G→GL (k). Let E be a semistable rational G-bundle
m
on X, we study the semistability of rational GL (k)-bundle E(GL (k)).
m m
Let P be a maximal parabolic subgroup of GL (k), then G acts on GL (k)/P
m m
whichislinearizedbytheveryamplegeneratorL ofPic(GL (k)/P). Thisgivesan
m
G-invariant embedding of GL (k)/P inside projective space P(H0(X,L)). Then
m
X-group scheme E(G) acts naturally on the X-scheme E(GL (k)/P) which is
m
linearized by line bundle E(L).
LetE(G) ,E(GL (k)/P) andE(L) bethefiberofE(G),E(GL (k)/P)and
0 m 0 0 m
E(L)overthegenericpointofX respectively. ThenE(G) isagroupschemeover
0
function field Spec(k(X)), and one has the action of E(G) on E(GL (k)/P)
0 m 0
linearized by line bundle E(L) which is defined over Spec(k(X)). Therefore
0
there is an one to one correspondence between rational reductions of structure
group of rational GL (k)-bundle E(GL (k)) to P and k(X)-rational points of
m m
E(GL (k)/P) .
m 0
Lemma2.2. [9,Proposition3.10,Proposition3.13]Letk beanalgebraically closed
field, X a smooth projective variety over k with an ample line bundle H. Let G
be a reductive algebraic group over k, E a semistable rational G-bundle on X. Let
σ : U′ → E(GL (k)/P) be a rational reduction of structure group of GL (k)-
m m
bundle E(GL (k)) to a maximal parabolic subgroup P of GL (k), where U′ is a
m m
big open scheme of X. The associated k(X)-rational point in E(GL (k)/P) is
m 0
denoted by σ . Then
0
(1) If σ is a semistable point for the action of E(G) on E(GL (k)/P) lin-
0 0 m 0
earized by line bundle E(L) , then deg T ≥0.
0 H σ
(2) If σ is not a semistable point for the action of E(G) on E(GL (k)/P)
0 0 m 0
linearizedbylinebundleE(L) ,anditsinstabilityparabolicP(σ)isdefined
0
over k(X), then deg T ≥0.
H σ
3. The Semistability of Principal Bundles via GL -Representaions
n
LetK beanarbitraryfield,ρ:GL (K)→GL (K)arepresentationofGL (K)
n m n
over K. Then ρ is given by an m×m-matrix of regular functions
f
ij ∈K[T ,det(T )−1] ,
det(Tij)aij ij ij 1≤i,j≤n
where a ∈N and f ∈K[T ] with (det(T ),f )=1. Denote
ij ij ij 1≤i,j≤n ij ij
d:= max {deg(f +n(a−a )}, where a:= max {a }.
ij ij ij
1≤i,j≤n 1≤i,j≤n
We say ρ is a GL (K)-representation over K of dimension m at most degree d.
n
Moreover, if a = 0 for any 1 ≤ i,j ≤ n, i.e., the above regular functions on
ij
GL (K) lie in the subring K[T ] , we say that ρ is a polynomial representa-
n ij 1≤i,j≤n
tion of GL (K) over K.
n
In this section, we use a variant of method of S. Gurjar and V. Mehta [3] to
giveanexplicituniformboundforthe fieldofdefinitionofallinstabilityparabolics
of all non-semistable points in a given GL (k)-representation space V in terms of
n
n, dim V and the maximal degree of regular functions correspond to the given
k
SEMISTABILITY OF RATIONAL PRINCIPAL GLn-BUNDLES IN POSITIVE CHARACTERISTIC7
GL (k)-representation. Moreover, we use these results to the study of semista-
n
bility of principal bundles under the extension of structure groups via GL (k)-
n
representations.
Theorem 3.1. Let K be an arbitrary field of characteristic p > 0, V a K-vector
space of dimension m. Let ρ : GL (K) → GL (V) be a GL (K)-representation
n K n
over K at most degree d. Then the instability parabolic of any unstable K-rational
point in V is defined over Kp−N for some positive integer N ≥m·log (d).
p
Proof. By base change,the representationρ:GL (K)→GL (V) overK induces
n K
arepresentationρ:GL (K)→GL (V )overK atmostdegreed. Fixamaximal
n K K
torusT inGL (K)whichisdefinedoverK. DenoteV :=V ⊗ K,wecanchoose
n K K
a simultaneous eigen basis e ,...,e of V for all 1-PS of GL (k) which lie in T.
1 m K n
Let v ∈ V be a non-semistable K-rational point with respect to the GL (K)-
n
actionρ. Thenthere aref ∈K[T ] ofdeg(f )≤danda∈N, suchthatfor
l ij 1≤i,j≤n l
any K-rational point g ∈GL (K), we have
n
m
f (g)
l
g·v =Xdet(g)ael.
l=1
Let λ(t) ∈ X (GL (K)) be an instability 1-PS of v. Then there exists K-rational
∗ n
point g ∈ GL (K) such that g·λ(t)·g−1 ⊆ T. Then g·λ(t)·g−1 is defined over
n
K , the separable closure of K, and g·λ(t)·g−1 is an instability 1-PS of g·v with
s
µ(λ(t),v)=µ(g·λ(t)·g−1,g·v).
Let f ,...,f (resp. f ,...,f ) denote the set of polynomials which vanish
l1 lr lr+1 lm
at g (resp. non-vanish at g). Consider the K-affine scheme
X :=Spec(K[T ] /(f ,...,f )).
ij 1≤i,j≤n l1 lr
Then g is a K-rational point of X(K)⊆An×n(K) with
m
det(g) Y fl(g)6=0.
l=r+1
Therefore, by [3, Lemma 10], there exists a finite extension field L⊆K of K with
r
[L:K]≤Ydeg(fl)=dr
l=1
such that X has a L-rational point g′ in X and det(g′) m f (g′) 6= 0. Thus
Ql=r+1 l
g′ ∈ GL (L) and g ·v and g′ ·v have the same set of monomials with non-zero
n
coefficients when expanded in terms of the basis e ,...,e . Since e ,...,e is a
1 m 1 m
simultaneous basis for g·λ(t)·g−1, so µ(g·λ(t)·g−1,g·v)=µ(g·λ(t)·g−1,g′·v).
Hence
µ(g′·λ(t)·g′−1,g′·v)=µ(λ(t),v)=µ(g·λ(t)·g−1,g·v)=µ(g·λ(t)·g−1,g′·v).
As g′·λ(t)·g′−1 is an instability 1-PSof g′·v, so g·λ(t)·g−1 is also an instability
1-PSofg′·v. Itfollowsthatg′−1(g·λ(t)·g−1)g′ isaninstability 1-PSofv whichis
defined over L. Thus, by Lemma 2.1, the instability parabolic P(v) of v is defined
over L·K . By the Galois descent argument, any instability parabolic is defined
s
over a purely inseparable extension of K. Suppose that for some positive integer
N with pN ≥ dm ≥ dr. Then P(v) must be defined over L ∩ Kp−N. By the
arbitrariness of non-semistable point v ∈V, this theorem is followed. (cid:3)
8 LINGGUANGLI
Theorem 3.2. Let k be an algebraically closed field of characteristic p > 0, X
a smooth projective variety over k with a fixed ample divisor H, ρ : GL (k) →
n
GL (k) a GL (k)-representation over k at most degree d. Let E be a rational
m n
GL (k)-bundle on X such that FN∗(E) is semistable for some integer
n X
N ≥ max Cr ·log (dr).
m p
0<r<m
Then the induced GL (k)-bundle E(GL (k)) is also semistable.
m m
Proof. LetV beak-vectorspaceofdimensionmwithafixedbasise ,...,e . Then
1 n
there is a natural isomorphism GLn(k)∼=GLk(V) with respect to this basis of V.
LetP be a maximalparabolicsubgroupof GL (V), then GL (V)/P is isomorphic
k k
to the grassmannianGrass(r,V) of r-dimensional subspaces of V for some integer
0<r <dim V. This induces a GL (k)-equivalent embedding
k n
r
GLk(V)/P ֒→P(^V),
which can be lift to a GL (k)-representation
n
r
ρr :GLn(k)→GLk(^V).
Since ρ : GL (k) → GL (V) is a GL (k)-representation over k at most degree d,
n k n
ρr is a GL (k)-representation over k at most degree dr. (In fact, GL (k) acts on
n n
GL (V)/P which is linearized by the very ample generator L of Pic(GL (V)/P).
k k
This gives a GL (k)-invariant embedding of GL (V)/P inside projective space
n k
P(H0(X,L)), and a GLn(k)-equivalent isomorphism VrV ∼=H0(X,L).)
TheX-groupschemeE(GL (k))actsnaturallyontheX-schemeE(GL (V)/P)
n k
which is linearized by line bundle E(L). Let E(G) , E(GL (V)/P) and E(L)
0 k 0 0
denote the fiber of E(G), E(GL (V)/P) and E(L) over the generic point of X
k
respectively. Then E(G) is a group scheme over function field Spec(k(X)), and
0
one has anE(G) -actiononE(GL (V)/P) defined overSpec(k(X)), whichis lin-
0 k 0
earizedbylinebundleE(L) . OneobservesthatE(GL (k)) andE(GL (V)/P)
0 n 0 k 0
became trivial after a finite separable field extension. Hence after change the base
to k(X) , the separable closure of k(X), we get canonical isomorphisms
s
E(GLn(k))0⊗k(X)k(X)s ∼=GLn(k)⊗kk(X)s ∼=GLn(k(X)s),
E(GLk(V)/P)0⊗k(X)k(X)s ∼=(GLk(V)/P)⊗kk(X)s,
E(L)0⊗k(X)k(X)s ∼=L ⊗kk(X)s,
andthisisomorphismsbeingcompatiblewithgroupactions. LetV :=V × k(X) .
s k s
The E(GL (k)) ⊗ k(X) -action on E(GL (V)/P) ⊗ k(X) induces a
n 0 k(X) s k 0 k(X) s
GL (k(X) )-equivalent embedding
n s
r r
(GLm(k)/P)⊗kk(X)s ֒→P(^V)⊗kk(X)s ∼=P(^Vs),
which can be lift to a GL (k(X) )-representation
n s
r
ρrs :GLn(k(X)s)→GLk(X)s(^Vs).
Infact,ρr isjustthebasechangeoftheGL (k)-representationρr fromk tok(X) .
s n s
Hence ρr is a GL (k(X) )-representation over k(X) of dimension Cr at most
s n s s m
degree dr.
SEMISTABILITY OF RATIONAL PRINCIPAL GLn-BUNDLES IN POSITIVE CHARACTERISTIC9
Let σ : U → E(GL (V)/P) be a rational reduction of structure group of E to
k
P, where U is an open subscheme of X with codim (X −U) ≥ 2. It corresponds
X
to a k(X)-rational point σ ∈ E(GL (V)/P) (k(X)). Let T denote the torsion
0 k 0 σ
free sheaf determined by the the locally free sheaf σ∗(T ) on U, where T is the
P P
tangentbundle alongthe fibers ofthe map E(GL (V)/P)→X. By Lemma 2.2, if
k
σ isasemistablepointforthe actionofE(GL (k)) onE(GL (V)/P) linearized
0 n 0 k 0
by line bundle E(L) , then
0
deg T ≥0.
H σ
On the other hand, if σ is not a semistable point for the action of E(GL (k)) on
0 n 0
E(GL (V)/P) linearized by line bundle E(L) , we would like to prove that
k 0 0
deg T ≥0.
H σ
View σ asank(X) -rationalpointin(GL (V)/P)⊗ k(X) , andliftthis point
0 s k k s
r
to a point in V , denote by σ again. Then by Theorem 3.1, we have the
V s 0
instability parabolic P(σ ) of σ is defined over k(X)p−l for any integer l ≥ Cr ·
0 0 s m
log (dr). Then P(σ ) is actually defined over k(X)p−l by the uniqueness of the
p 0
instability parabolic and Galois descent argument.
Pulling back by the Frobenius morphism, the action of the generic fibre
FXl∗(E(GLn(k)))0 ∼=Fkl∗(X)(E(GLn(k))0)
on
FXl∗(E(GLk(V)/P))0 ∼=Fkl∗(X)(E(GLm(V)/P)0)
is the base change by Fl∗ of the E(GL (k)) -action on (GL (V)/P) . The
k(X) n 0 k 0
Frobenius Fl∗ factors through an isomorphism:
k(X)
Fl∗
k(X) k(X) //k(X)
∼= (cid:15)(cid:15) ✉✉✉✉i✉l✉✉✉::
k(X)p−l
where i :Spec(k(X)p−l)→Spec(k(X)) is given by the inclusion k(X)⊆k(X)p−l.
l
Thereforefor this Fl∗ (E(GL (k)) )-action, the instability parabolicof the point
k(X) n 0
Fl∗(σ) is defined over k(X). Since Fl∗(E) is semistable, by Lemma 2.2, we have
X 0 X
deg (T )=deg Fl∗(T )=pl·deg (T )≥0.
H Fl∗(σ) H X σ H σ
X
Thus deg T ≥0. Therefore if FN∗(E) is semistable for some integer
H σ X
N ≥ max Cr ·log (dr),
m p
0<r<m
then for any rational reduction σ of structure group of rational GL (k)-bundle
m
E(GL (k)) to any maximal parabolic subgroup P ⊂GL (k), we have
m m
deg (T )≥0.
H σ
Hence E(GL (k)) is a semistable rational GL (k)-bundle. This completes the
m m
proof of this theorem. (cid:3)
10 LINGGUANGLI
4. The Semistability of Truncated Symmetric Powers
The truncated symmetric powers was first introduced in [10] in order to study
the semistability of Frobenius direct images. L. Li and F. Yu [8] have studied the
instability of Tl(E) and show that Tl(E) is strongly semistable when E is strongly
semistable. In this section, we would like to continue the further study of the
semistability Tl(E), and give a sufficient condition for semistability of Tl(E).
Now, we recall the construction and properties of truncated symmetric powers
of vector spaces (See [10, Section 3]).
Let K be an arbitrary field, V a n-dimensional K-vector space with standard
representationof GL (K). Let l be a positive integer, S the symmetric group of l
n l
elements with a natural action on V⊗l by
(v ⊗···⊗v )·σ =v ⊗···⊗v
1 l σ(1) σ(l)
foranyv ∈V andanyσ ∈S . Lete ,··· ,e beabasisofV. Foranynon-negative
i l 1 n
n
partition (k ,··· ,k ) of l (i.e. l = k , k ≥0, 1≤i≤n), we define
1 n P i i
i=1
v(k1,··· ,kn):= X(e⊗1k1 ⊗···⊗e⊗nkn)·σ.
σ∈Sl
Let Tl(V)⊂V⊗l be the linear subspace generated by all vectors
n
{v(k1,··· ,kn) | l=Xki, ki ≥0, 1≤i≤n}.
i=1
Then Tl(V) is a GL (K)-subrepresentationof V⊗l with
n
l(p)
N(p,n,l):=dimkTl(V)=X(−1)q·Cnq ·Cnl−+plq−q−1,
q=0
where l(p) is the unique integer such that 0≤l−l(p)·p<p.
If char(K) = 0 then we have GLn(K)-equivalent Tl(V) ∼= Syml(V) for any
integer l > 0. On the other hand, if char(K) = p > 0, then we have GL (K)-
n
equivalent Tl(V)∼=Syml(V) when 0<l <p and Tl(V)=0 for l >n(p−1).
Proposition 4.1. For any integer l>0, Tl(V)=(V⊗l)Sl.
Proof. Fix a basis e ,··· ,e of V. Let (k ,··· ,k ) be a non-negative partition of
1 n 1 n
l, W the linear subspace of V⊗l generated by vectors
k1,···,kn
{e ⊗···⊗e | k =♯{i |i =m,1≤j ≤l},1≤m≤n}.
i1 il m j j
Thus W is a S -invariant linear subspace of V⊗l.
(k1,···,kn) l
By definition, it is obvious that Tl(V) ⊆ (V⊗l)Sl. It is easy to see that, any
element in (V⊗l)Sl can be expressed as the form
α= X αk1,···,kn,
n
l=Pki,ki≥0
i=1
where αk1,···,kn ∈ Wk1,···,kn. Then we have αk1,···,kn ∈ (V⊗l)Sl. In order to prove
α ∈ Tl(V), it suffices to show that α ∈ Tl(V). By simple observation, we
k1,···,kn
have αk1,···,kn =a·v(k1,··· ,kn) for some a∈k. It follows that (V⊗l)Sl ⊆Tl(V).
Hence Tl(V)=(V⊗l)Sl. (cid:3)