Semistability of Rational Principal GL_n-Bundles in Positive Characteristic
aa r X i v : . [ m a t h . AG ] J a n SEMISTABILITY OF RATIONAL PRINCIPAL GL n -BUNDLES INPOSITIVE CHARACTERISTIC LINGGUANG LI
Abstract.
Let k be an algebraically closed field of characteristic p > X a smooth projective variety over k with a fixed ample divisor H . Let E be arational GL n ( k )-bundle on X , and ρ : GL n ( k ) → GL m ( k ) a rational GL n ( k )-representation at most degree d such that ρ maps the radical R ( GL n ( k )) of GL n ( k ) into the radical R ( GL m ( k )) of GL m ( k ). We show that if F N ∗ X ( E ) issemistable for some integer N ≥ max Let k be an algebraically closed field of arbitrary characteristic, X a smoothprojective variety over k with a fixed ample divisor H . Let G and G ′ be reductivealgebraic groups over k , ρ : G → G ′ a homomorphism of algebraic groups. One ofthe important and essential problem in the studying of G -bundles is to study thebehavior of the semistability of G ′ -bundles under the extension of structure group.In precise, let E be a semistable rational G -bundle on X , does the induced rational G ′ -bundle E ( G ′ ) is also semistable?Suppose that ρ maps the radical of R ( G ) into the radical R ( G ′ ) of G ′ (Unlessstated otherwise, we always require this condition for any homomorphisms of al-gebraic groups and all representations are rational representations in this paper),and E is a semistable rational G -bundle on X . If char( k )= 0, S. Ramanan and A.Ramanathan [9, Theorem 3.18] showed that the induced rational G ′ -bundle E ( G ′ )is also semistable on X . If char( k )= p > 0, the induced rational G ′ -bundle E ( G ′ )may be not semistable in general. However, S. Ramanan and A. Ramanathan [9,Theorem 3.23] proved that strong semistability of rational G -bundle E implies thestrong semistability of rational G ′ -bundle E ( G ′ ). In addition, S. Ilangovan, V. B.Mehta and A. J. Parameswaran [4] showed that if G ′ = GL m ( k ) for some integer m > p > ht( ρ ), then the induced rational GL m ( k )-bundle E ( GL m ( k )) issemistable. F. Coiai and Y. I. Holla [2] generalized some results of [9] and showedthat given a representation ρ : G → GL m ( k ), there exists a non-negative inte-ger N , depending only on G and ρ , such that for any rational G -bundle E whose N -th Frobenius pull back F N ∗ X ( E ) is semistable, then the induced rational GL m ( k )-bundle E ( GL m ( k )) is again semistable. S. Gurjar and V. Mehta [3] improved theresult of [2] and obtain a explicit bound for N in terms of certain numerical dataattached to ρ . The main ingredient of the proof in [2] and [3] is to give a uniformbound for the field of definition of the instability parabolics (Kempf’ parabolic)associated to non-semistable points in related representing space. We now briefly describe the main idea of their proof. Fix a representation G → GL m ( k ). Let E be rational G -bundle on X , E ( G ) the group scheme over X associated to E , and E ( GL m ( k )) the induced rational GL m ( k )-bundle underthe extension of structure group via ρ . Let k ( X ) be the function field of X , thegeneric fiber E ( G ) of E ( G ) is a group scheme over Spec( k ( X )). Let P be a maxi-mal parabolic subgroup of GL m ( k ), E ( GL m ( k ) /P ) the associated GL m ( k ) /P -fiberspace over X , and E ( GL m ( k ) /P ) the generic fiber of E ( GL m ( k ) /P ). Then thereis an E ( G ) -action on the smooth projective variety E ( GL m ( k ) /P ) over k ( X )which is linearized by a suitable very ample line bundle. If E ( GL m ( k )) adimts areduction of structure group to this maximal parabolic subgroup P , then we geta rational section σ : U → E ( GL m ( k ) /P ), where U is an open subscheme of X with codim X ( X − U ) ≥ 2. Restricting to the generic fiber gives a k ( X )-rationalpoint σ of E ( GL m ( k ) /P ) . In [9], it is shown that if σ is a semistable pointin E ( GL m ( k ) /P ) for the above E ( G ) -action, then the rational reduction σ doesnot violate the semistability of rational GL m ( k )-bundle E ( GL m ( k )). Also, if σ is not semistable for the above E ( G ) -action and its instability parabolic P ( σ ),which is defined over k ( X ), is actually defined over k ( X ), then again σ does notcontradict the semistability of rational GL m ( k )-bundle E ( GL m ( k )). In the case ofcharacteristic 0, by the uniqueness of instability parabolic and it is invariant underthe action of Galois group, then P ( σ ) is actually defined over k ( X ). This provesthat E ( GL m ( k )) is a semistable rational GL m ( k )-bundle. However, in the case ofcharacteristic p > P ( σ ) may be not defined over k ( X ), and it is defined over afinite extension of k ( X ). By the uniqueness of instability parabolic, the Galois de-scent argument implies that P ( σ ) is actually defined over finite purely inseparablefield extension K p − N of K for some non-negative integer N .In [2] and [3], the authors showed that there exists a uniform bound N , dependingonly on G and ρ , such that for all possible rational reductions to all maximal para-bolic subgroups the instability parabolics of points corresponding to these rationalreductions are actually defined over K p − N via different methods. This can be shownto imply that if E is a semistable rational G -bundle such that F N ∗ X ( E ) is semistable,then the induced rational GL m ( k )-bundle E ( GL m ( k )) is also semistable. The majordifferences 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 -rationalpoint. F. Coiai and Y. I. Holla [2] proved the existence of the uniform bound bybounding the non-separability of the group action and the non-reducedness of thestabilizers of various unstable rational points. However, the above estimation doesnot seen quantifiable. On the other hand, S. Gurjar and V. Mehta [3] directly esti-mated the field of definition of the instability parabolics which is probably weakerthan the method of [2], but it is quantifiable.The paper is organized as follows.In section 2, we recall some definitions and results about geometric invariant the-ory and rational principal bundles, such as the instability 1-PS, instability parabol-ics of non-semistable points, etc. These results can be found in [6] and [9].In section 3, we mainly study the rationality of the instability parabolics of non-semistable points in GL n ( k )-representation spaces, and apply these results to thestudy of semistability of rational principal bundles under the extension of structuregroups via a GL n ( k )-representation ρ : GL n ( k ) → GL m ( k ). EMISTABILITY OF RATIONAL PRINCIPAL GL n -BUNDLES IN POSITIVE CHARACTERISTIC3 Theorem 1.1 (Theorem 3.2) . Let k be an algebraically closed field of characteristic p > , X a smooth projective variety over k with a fixed ample divisor H , ρ : GL n ( k ) → GL m ( k ) a GL n ( k ) -representation over k at most degree d . Let E be arational GL n ( k ) -bundle on X such that F N ∗ X ( E ) is semistable for some integer N ≥ max Geometric Invariant Theory. Let K be a field, K s the separable closureof K , K the algebraically closure of K . Let G be a connected reductive algebraicgroup over K , Then G has a maximal torus T defined over K (See [1, Proposition7.10]), and T splits over K s . Let X ∗ ( T ) be the group of parameter subgroups (1-PS)of T , i.e., group homomorphisms of the multiplicative group G m into T . Let N T bethe normalizer of T in G , then Weyl group W T := N T /T of G with respect to T actson X ∗ ( T ) by conjugation. Fix an inner product h , i on X ∗ ( T ) which is invariantunder the action of Weyl group W T as well as the Galois group Gal( K s /K ) (SeeSection 4 of [6]). Then we can define norm k λ k of 1-PS λ ∈ X ∗ ( T ) as h λ, λ i . Let T ′ be another maximal torus of G , T is conjugate to T ′ by an element of G , and theisomorphism T → T ′ is well defined up to Weyl group action on T . Therefore theinner product h , i on X ∗ ( T ) determines uniquely one in X ∗ ( T ′ ). Hence the norm ofany 1-PS in G is well defined. LINGGUANG LI Let V be a finite dimensional K -vector space, ρ : G ( K ) → GL K ( V ) a representa-tion of G . A vector 0 = v ∈ V is semistable for the G ( K )-action if 0 / ∈ G ( K ) · v . Oneknows that this is equivalent to existence of a G ( K )-invariant element φ ∈ S m ( V )for some m > φ ( v ) = 0. For a 1-PS λ of G ( K ), V has a decomposition V = L V i , where V i = { v ∈ V | λ ( t ) v = t i v } . Define m ( λ, v ) := min { i | v has a non-zero component in V i } ,µ ( λ, v ) := m ( λ, v ) k λ k . For a 1-PS λ of G ( K ), the associated subgroup P ( λ ) of G ( K ) is defined by P ( λ ) : { g ∈ G ( K ) | lim t → λ ( t ) · g · λ − ( t ) exists in G ( K ) } , which is a parabolic subgroup of G ( K ).For a non-semistable vector v ∈ V , define the instability v to be the 1-PS λ v such that µ ( λ v , v ) = sup { µ ( λ, v ) | λ ∈ X ∗ ( G ( K )) } , which is not unique.We now recall some basic facts in geometric invariant theory, which can be foundin [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 K ( V ) a representation of G ( K ) on a K -vector space V of finitedimension. Let v ∈ V be a non-semistable point for the G ( K ) -action. Then (1) The function λ µ ( λ, v ) on the set X ∗ ( G ( K )) attains the maximum value. (2) There is a unique instability parabolic P ( v ) of v such that for any instability -PS λ of v , we have P ( v ) = P ( λ ) . (3) For any maximal torus T ⊆ P ( v ) , there is a unique instability -PS λ T of v such that λ T ⊆ T . (4) For g ∈ G ( K ) , λ is the instability -PS of v , then g · λ ( t ) · g − is theinstability -PS of g · v , µ ( λ, v ) = µ ( gλ ( t ) g − , g · v ) and P ( g · v ) = g · P ( v ) · g − . (5) For an instability -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 . Rational Principal G -Bundles. Let K be a field, G a reductive algebraicgroup over K , X a smooth projective variety over K with a fixed ample divisor H .A ( principal ) G -bundle on X is a X -scheme π : E → X with an G -action (actson the right) and π is G -invariant and isotrivial, i.e., locally trivial in the ´etaletopology.If Y is a quasi projective G -scheme over K (on the left), the associated fibrebundle E ( Y ) over X is the quotient E × K F under the action of G given by g ( e, y ) = ( e · g, g − · y ) , g ∈ G, e ∈ E, y ∈ Y. Let M be a projective variety over K with a G -action, which is linearized byan ample line bundle L on X . Let E ( G ) := E × G, Int G denote the group schemeover X associated to E by the action of G on itself by inner automorphisms. Then X -group scheme E ( G ) acts naturally on the X -scheme E ( M ) which is linearizedby line bundle E ( L ).Let x ∈ X be a point of X , E ( G ) x , E ( M ) x and E ( L ) x denote the fiber of E ( G ), E ( M ) and E ( L ) over x respectively. Then E ( G ) x is a group scheme overSpec( k ( x )), and one has the action of E ( G ) x on E ( M ) x linearized by line bundle E ( L ) x which is defined over Spec( k ( x )). EMISTABILITY OF RATIONAL PRINCIPAL GL n -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 σ , φ ) with a P -bundle E σ and an isomorphism of G -bundles φ : E σ ( G ) → E . Note that quotient E/P is naturally isomorphic to the fiberbundle E ( G/P ) on X and a section σ : X → E/P gives the P -bundle σ ∗ ( E )with natural isomorphism σ ∗ ( E )( G ) ∼ = E . This induces a bijection correspondencebetween sections of E/P → X and reductions of structure group of E to P . Let T P be the tangent bundle along the fibers of the map E/P → X , then T σ := σ ∗ ( T P ) isthe vector bundle on X associated to P -bundle E σ for the natural representationof P on g / p , where g and p is the Lie algebra of G and P respectively.A rational G -bundle E on X is a G -bundle on a big open subscheme U of X , i.e.codim X ( X − U ) ≥ 2. A rational reduction of structure group of a rational G -bundle E to a subgroup P ⊂ G is a reduction σ of structure group of E | U ′ to P over abig open subscheme U ′ ⊆ U . Then the locally free sheaf T σ on U ′ determine areflexive sheaf i U ′ ∗ ( T σ ) where i U ′ : U ′ → X is the natural open immersion, denotedby T σ again. The rational G -bundle E is semistable if for any rational reduction σ of E to any parabolic subgroup P of G over any big open subscheme U ′ ⊆ U , therational vector bundle T σ has deg H ( T σ ) ≥ 0. If G = GL n ( K ), then there is an oneto one correspondence between rational GL n ( K )-bundles between reflexive torsionfree sheaves. In this case, the semistability of the rational GL n ( K )-bundle E isequivalent to the semistability of the torsion free sheaf E 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. Let K be a field of charac-teristic p > φ : X → Spec( K ) a scheme over k . The absolute Frobenius morphism F X : X → X is induced by O X → O X , f f p . Consider the commutative diagram X φ (cid:31) (cid:31) F X ' ' F g ❍❍❍ $ $ ❍❍❍ X (1) F ∗ k ( φ ) (cid:15) (cid:15) F a / / X φ (cid:15) (cid:15) Spec( K ) F K / / Spec( K ) . The morphism F a (resp. F g ) is called the arithmetic Frobenius morphism (resp. geometric Frobenius morphism ) of φ : X → Spec( K ).If K is a perfect field, F K and F a are isomorphisms. Let G be an algebraic groupover K , π : E → X a G -bundle over X . Pulling back by the absolute Frobenius F ∗ X we get a G -bundle F ∗ X ( π ) : F ∗ X ( E ) → X F , where X F is the scheme X endowed withthe K -structure by the composition X φ → Spec( K ) F K → Spec( K ). If K is a perfectfield, we can change the K -structures of G , X F and F ∗ X ( E ) by composing theirstructure morphisms with Spec( K ) F − K → Spec( K ) to get a F ∗ K ( G )-bundle F ∗ X ( E ) → X . In this case, the F ∗ K ( G )-bundle F ∗ X ( E ) → X is the same as the bundle obtainedfrom G -bundle E by the extension of structure group g : G → F ∗ K ( G ).Let k be an algebraically closed field of characteristic p > X a smooth pro-jective variety over k with a fixed ample line bundle H , G a reductive algebraicgroup over k . Let E be a rational G -bundle on X , then we can get a rational F ∗ K ( G )-bundle F ∗ X ( E ) on X . Then E is semistable when F ∗ X ( E ) is semistable. If F m ∗ X ( E ) is semistable for any integer m ≥ 0, then E is called strongly semistable . LINGGUANG LI Semistability of Rational G -bundles under Extension of StructureGroups. Let k be an algebraically closed field, X a smooth projective variety over k with a fixed ample line bundle H , G a reductive algebraic group over k with arational representation ρ : G → GL m ( k ). Let E be a semistable rational G -bundleon X , we study the semistability of rational GL m ( k )-bundle E ( GL m ( k )).Let P be a maximal parabolic subgroup of GL m ( k ), then G acts on GL m ( k ) /P which is linearized by the very ample generator L of Pic( GL m ( k ) /P ). This gives an G -invariant embedding of GL m ( k ) /P inside projective space P ( H ( X, L )). Then X -group scheme E ( G ) acts naturally on the X -scheme E ( GL m ( k ) /P ) which islinearized by line bundle E ( L ).Let E ( G ) , E ( GL m ( k ) /P ) and E ( L ) be the fiber of E ( G ), E ( GL m ( k ) /P ) and E ( L ) over the generic point of X respectively. Then E ( G ) is a group scheme overfunction field Spec( k ( X )), and one has the action of E ( G ) on E ( GL m ( k ) /P ) linearized by line bundle E ( L ) which is defined over Spec( k ( X )). Thereforethere is an one to one correspondence between rational reductions of structuregroup of rational GL m ( k )-bundle E ( GL m ( k )) to P and k ( X )-rational points of E ( GL m ( k ) /P ) . Lemma 2.2. [9, Proposition 3.10, Proposition 3.13] Let k be an algebraically closedfield, 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 m ( k ) /P ) be a rational reduction of structure group of GL m ( k ) -bundle E ( GL m ( k )) to a maximal parabolic subgroup P of GL m ( k ) , where U ′ is abig open scheme of X . The associated k ( X ) -rational point in E ( GL m ( k ) /P ) isdenoted by σ . Then (1) If σ is a semistable point for the action of E ( G ) on E ( GL m ( k ) /P ) lin-earized by line bundle E ( L ) , then deg H T σ ≥ . (2) If σ is not a semistable point for the action of E ( G ) on E ( GL m ( k ) /P ) linearized by line bundle E ( L ) , and its instability parabolic P ( σ ) is definedover k ( X ) , then deg H T σ ≥ . The Semistability of Principal Bundles via GL n -Representaions Let K be an arbitrary field, ρ : GL n ( K ) → GL m ( K ) a representation of GL n ( K )over K . Then ρ is given by an m × m -matrix of regular functions f ij det( T ij ) a ij ∈ K [ T ij , det( T ij ) − ] ≤ i,j ≤ n , where a ij ∈ N and f ij ∈ K [ T ij ] ≤ i,j ≤ n with (det( T ij ) , f ij ) = 1. Denote d := max ≤ i,j ≤ n { deg( f ij + n ( a − a ij ) } , where a := max ≤ i,j ≤ n { a ij } . We say ρ is a GL n ( K ) -representation over K of dimension m at most degree d . Moreover, if a ij = 0 for any 1 ≤ i, j ≤ n , i.e., the above regular functions on GL n ( K ) lie in the subring K [ T ij ] ≤ i,j ≤ n , we say that ρ is a polynomial representa-tion of GL n ( K ) over K .In this section, we use a variant of method of S. Gurjar and V. Mehta [3] togive an explicit uniform bound for the field of definition of all instability parabolicsof all non-semistable points in a given GL n ( k )-representation space V in terms of n , dim k V and the maximal degree of regular functions correspond to the given EMISTABILITY OF RATIONAL PRINCIPAL GL n -BUNDLES IN POSITIVE CHARACTERISTIC7 GL n ( k )-representation. Moreover, we use these results to the study of semista-bility of principal bundles under the extension of structure groups via GL n ( k )-representations. Theorem 3.1. Let K be an arbitrary field of characteristic p > , V a K -vectorspace of dimension m . Let ρ : GL n ( K ) → GL K ( V ) be a GL n ( K ) -representationover K at most degree d . Then the instability parabolic of any unstable K -rationalpoint in V is defined over K p − N for some positive integer N ≥ m · log p ( d ) .Proof. By base change, the representation ρ : GL n ( K ) → GL K ( V ) over K inducesa representation ρ : GL n ( K ) → GL K ( V K ) over K at most degree d . Fix a maximaltorus T in GL n ( K ) which is defined over K . Denote V K := V ⊗ K K , we can choosea simultaneous eigen basis e , . . . , e m of V K for all 1-PS of GL n ( k ) which lie in T .Let v ∈ V be a non-semistable K -rational point with respect to the GL n ( K )-action ρ . Then there are f l ∈ K [ T ij ] ≤ i,j ≤ n of deg( f l ) ≤ d and a ∈ N , such that forany K -rational point g ∈ GL n ( K ), we have g · v = m X l =1 f l ( g )det( g ) a e l . Let λ ( t ) ∈ X ∗ ( GL n ( K )) be an instability 1-PS of v . Then there exists K -rationalpoint g ∈ GL n ( K ) such that g · λ ( t ) · g − ⊆ T . Then g · λ ( t ) · g − is defined over K s , the separable closure of K , and g · λ ( t ) · g − is an instability 1-PS of g · v with µ ( λ ( t ) , v ) = µ ( g · λ ( t ) · g − , g · v ).Let f l , . . . , f l r (resp. f l r +1 , . . . , f l m ) denote the set of polynomials which vanishat g (resp. non-vanish at g ). Consider the K -affine scheme X := Spec( K [ T ij ] ≤ i,j ≤ n / ( f l , . . . , f l r )) . Then g is a K -rational point of X ( K ) ⊆ A n × n ( K ) withdet( g ) m Y l = r +1 f l ( g ) = 0 . Therefore, by [3, Lemma 10], there exists a finite extension field L ⊆ K of K with[ L : K ] ≤ r Y l =1 deg( f l ) = d r such that X has a L -rational point g ′ in X and det( g ′ ) Q ml = r +1 f l ( g ′ ) = 0. Thus g ′ ∈ GL n ( L ) and g · v and g ′ · v have the same set of monomials with non-zerocoefficients when expanded in terms of the basis e , . . . , e m . Since e , . . . , e m is asimultaneous basis for g · λ ( t ) · g − , so µ ( g · λ ( t ) · g − , g · v ) = µ ( g · λ ( t ) · g − , g ′ · v ).Hence µ ( g ′ · λ ( t ) · g ′− , g ′ · v ) = µ ( λ ( t ) , v ) = µ ( g · λ ( t ) · g − , g · v ) = µ ( g · λ ( t ) · g − , g ′ · v ) . As g ′ · λ ( t ) · g ′− is an instability 1-PS of g ′ · v , so g · λ ( t ) · g − is also an instability1-PS of g ′ · v . It follows that g ′− ( g · λ ( t ) · g − ) g ′ is an instability 1-PS of v which isdefined over L . Thus, by Lemma 2.1, the instability parabolic P ( v ) of v is definedover L · K s . By the Galois descent argument, any instability parabolic is definedover a purely inseparable extension of K . Suppose that for some positive integer N with p N ≥ d m ≥ d r . Then P ( v ) must be defined over L ∩ K p − N . By thearbitrariness of non-semistable point v ∈ V , this theorem is followed. (cid:3) LINGGUANG LI Theorem 3.2. Let k be an algebraically closed field of characteristic p > , X a smooth projective variety over k with a fixed ample divisor H , ρ : GL n ( k ) → GL m ( k ) a GL n ( k ) -representation over k at most degree d . Let E be a rational GL n ( k ) -bundle on X such that F N ∗ X ( E ) is semistable for some integer N ≥ max 2. It correspondsto a k ( X )-rational point σ ∈ E ( GL k ( V ) /P ) ( k ( X )). Let T σ denote the torsionfree sheaf determined by the the locally free sheaf σ ∗ ( T P ) on U , where T P is thetangent bundle along the fibers of the map E ( GL k ( V ) /P ) → X . By Lemma 2.2, if σ is a semistable point for the action of E ( GL n ( k )) on E ( GL k ( V ) /P ) linearizedby line bundle E ( L ) , then deg H T σ ≥ . On the other hand, if σ is not a semistable point for the action of E ( GL n ( k )) on E ( GL k ( V ) /P ) linearized by line bundle E ( L ) , we would like to prove thatdeg H T σ ≥ . View σ as an k ( X ) s -rational point in ( GL k ( V ) /P ) ⊗ k k ( X ) s , and lift this pointto a point in V r V s , denote by σ again. Then by Theorem 3.1, we have theinstability parabolic P ( σ ) of σ is defined over k ( X ) p − l s for any integer l ≥ C rm · log p ( dr ). Then P ( σ ) is actually defined over k ( X ) p − l by the uniqueness of theinstability parabolic and Galois descent argument.Pulling back by the Frobenius morphism, the action of the generic fibre F l ∗ X ( E ( GL n ( k ))) ∼ = F l ∗ k ( X ) ( E ( GL n ( k )) )on F l ∗ X ( E ( GL k ( V ) /P )) ∼ = F l ∗ k ( X ) ( E ( GL m ( V ) /P ) )is the base change by F l ∗ k ( X ) of the E ( GL n ( k )) -action on ( GL k ( V ) /P ) . TheFrobenius F l ∗ k ( X ) factors through an isomorphism: k ( X ) ∼ = (cid:15) (cid:15) F l ∗ k ( X ) / / k ( X ) k ( X ) p − l i l : : ✉✉✉✉✉✉✉✉ where i l : Spec( k ( X ) p − l ) → Spec( k ( X )) is given by the inclusion k ( X ) ⊆ k ( X ) p − l .Therefore for this F l ∗ k ( X ) ( E ( GL n ( k )) )-action, the instability parabolic of the point F l ∗ X ( σ ) is defined over k ( X ). Since F l ∗ X ( E ) is semistable, by Lemma 2.2, we havedeg H ( T F l ∗ X ( σ ) ) = deg H F l ∗ X ( T σ ) = p l · deg H ( T σ ) ≥ . Thus deg H T σ ≥ 0. Therefore if F N ∗ X ( E ) is semistable for some integer N ≥ max The truncated symmetric powers was first introduced in [10] in order to studythe semistability of Frobenius direct images. L. Li and F. Yu [8] have studied theinstability of T l ( E ) and show that T l ( E ) is strongly semistable when E is stronglysemistable. In this section, we would like to continue the further study of thesemistability T l ( E ), and give a sufficient condition for semistability of T l ( E ).Now, we recall the construction and properties of truncated symmetric powersof vector spaces (See [10, Section 3]).Let K be an arbitrary field, V a n -dimensional K -vector space with standardrepresentation of GL n ( K ). Let l be a positive integer, S l the symmetric group of l elements with a natural action on V ⊗ l by( v ⊗ · · · ⊗ v l ) · σ = v σ (1) ⊗ · · · ⊗ v σ ( l ) for any v i ∈ V and any σ ∈ S l . Let e , · · · , e n be a basis of V . For any non-negativepartition ( k , · · · , k n ) of l (i.e. l = n P i =1 k i , k i ≥ , ≤ i ≤ n ), we define v ( k , · · · , k n ) := X σ ∈ S l ( e ⊗ k ⊗ · · · ⊗ e ⊗ k n n ) · σ. Let T l ( V ) ⊂ V ⊗ l be the linear subspace generated by all vectors { v ( k , · · · , k n ) | l = n X i =1 k i , k i ≥ , ≤ i ≤ n } . Then T l ( V ) is a GL n ( K )-subrepresentation of V ⊗ l with N ( p, n, l ) := dim k T l ( V ) = l ( p ) X q =0 ( − q · C qn · C l − pqn + l − q − , where l ( p ) is the unique integer such that 0 ≤ l − l ( p ) · p < p .If char( K ) = 0 then we have GL n ( K )-equivalent T l ( V ) ∼ = Sym l ( V ) for anyinteger l > 0. On the other hand, if char( K ) = p > 0, then we have GL n ( K )-equivalent T l ( V ) ∼ = Sym l ( V ) when 0 < l < p and T l ( V ) = 0 for l > n ( p − Proposition 4.1. For any integer l > , T l ( V ) = ( V ⊗ l ) S l .Proof. Fix a basis e , · · · , e n of V . Let ( k , · · · , k n ) be a non-negative partition of l , W k , ··· ,k n the linear subspace of V ⊗ l generated by vectors { e i ⊗ · · · ⊗ e i l | k m = ♯ { i j | i j = m, ≤ j ≤ l } , ≤ m ≤ n } . Thus W ( k , ··· ,k n ) is a S l -invariant linear subspace of V ⊗ l .By definition, it is obvious that T l ( V ) ⊆ ( V ⊗ l ) S l . It is easy to see that, anyelement in ( V ⊗ l ) S l can be expressed as the form α = X l = n P i =1 k i ,k i ≥ α k , ··· ,k n , where α k , ··· ,k n ∈ W k , ··· ,k n . Then we have α k , ··· ,k n ∈ ( V ⊗ l ) S l . In order to prove α ∈ T l ( V ), it suffices to show that α k , ··· ,k n ∈ T l ( V ). By simple observation, wehave α k , ··· ,k n = a · v ( k , · · · , k n ) for some a ∈ k . It follows that ( V ⊗ l ) S l ⊆ T l ( V ).Hence T l ( V ) = ( V ⊗ l ) S l . (cid:3) EMISTABILITY OF RATIONAL PRINCIPAL GL n -BUNDLES IN POSITIVE CHARACTERISTIC11 Proposition 4.2. Let K be a field of characteristic p > , V a n -dimensional K -vector space. Then for any integer < l ≤ n ( p − , the GL K ( V ) -representation ρ l : GL K ( V ) → GL K (T l ( V )) is a polynomial representation at most degree l .Proof. Fix a basis e , · · · , e n of V . Endowing the basis { e i ⊗ · · · ⊗ e i l | ≤ i j ≤ n } of V ⊗ l with lexicographic order. Under this basis, the GL K ( V )-representation V ⊗ l is equivalent to the homomorphism of algebraic groups ρ l : GL n ( K ) → GL n l ( K )( g ij ) ( h st ) := (( g ij ) ⊗ · · · ⊗ ( g ij )) . Therefore ρ l is given by a matrix of polynomials h st in K [ T ij ] ≤ i,j ≤ n of degree l forany integers 1 ≤ s, t ≤ n l . Then by the change of basis, under the basis e , · · · , e n ,(resp. { v ( k , · · · , k n ) | l = n X i =1 k i , ≤ k i < p, ≤ i ≤ n } )of V (resp. T l ( V )), the GL K ( V )-representation ρ l : GL K ( V ) → GL K (T l ( V ))is also give by a matrix of polynomials f ij in K [ T ij ] ≤ i,j ≤ n of degree l for anyintegers 1 ≤ i, j ≤ dim T l ( V ). Hence ρ l : GL K ( V ) → GL K (T l ( V )) is a polynomialrepresentation at most degree l . (cid:3) Corollary 4.3. Let K be a field of characteristic p > , V a n -dimensional K -vector space, l an integer with < l ≤ n ( p − . Then the instability parabolic ofany unstable K -rational point in the GL K ( V ) -representation space T l ( V ) is definedover K p − N for some integer N ≥ log p ( l ) · dim K T l ( V ) , i.e., N ≥ log p ( l ) · l ( p ) X q =0 ( − q · C qn · C l − pqn + l − q − , where l ( p ) is the unique integer such that ≤ l − l ( p ) · p < p .Proof. By Proposition 4.2, we have ρ l : GL K ( V ) → GL K (T l ( V )) a polynomialrepresentation of GL K ( V ) at most degree l . Hence by Theorem 3.1, if an integer N ≥ (dim T l ( V )) · log p ( l ), then the instability parabolic of any unstable K -rationalpoint in T l ( V ) is defined over K p − N . (cid:3) Let X be a smooth variety over an algebraically closed field k of characteristic p > E a locally free sheaf of rank n on X . Then the locally free sheaf T l ( E ) ⊂ E ⊗ l is defined to be the sheaf of sections of the associated vector bundle of the framebundle of E through the representation T l ( V ) (See [10, Definition 3.4]). Theorem 4.4. Let k be an algebraically closed field of characteristic p > , X asmooth projective variety over k with a fixed ample divisor H . Let E be a torsionfree sheaf of rank n on X such that F N ∗ X ( E ) is semistable for some integer N ≥ max ≤ r ≤ N ( p,n,l ) C rN ( p,n,l ) · log p ( lr ) , where N ( p, n, l ) = P l ( p ) q =0 ( − q · C qn · C l − pqn + l − q − , and l ( p ) is the unique integer suchthat ≤ l − l ( p ) · p < p . Then the torsion free sheaf T l ( E ) is also a semistable sheaf.Proof. Let V be a n -dimensional k -vector space. Then by Proposition 4.2, the GL K ( V )-representation on T l ( V ) is a polynomial representation of dimension N ( p, n, l )at most degree l . Hence this Proposition followed by Theorem 3.2. (cid:3) Semistability of Frobenius direct image of ρ ∗ (Ω X )Let k be an algebraically closed field of characteristic p > X a smoothprojective surface with a fixed ample divisor H such that Ω X is semistable withdeg H (Ω X ) > 0, and L ∈ Pic( X ). Y. Kitadai, H. Sumihiro [7, Theorem 3.1] showedthat the F X ∗ (Ω X ) is semistable. Moreover, X. Sun [11, Theorem 4.11] generalizedthe [7, Theorem 3.1] and showed that F X ∗ (Ω X ) is stable if Ω X is stable. In this sec-tion, we would like to study the semistability of Frobenius direct image of ρ ∗ (Ω X )on higher dimensional base space, where ρ ∗ (Ω X ) is the locally free sheaf obtainedfrom Ω X via a GL n ( k )-representation ρ . Lemma 5.1. [10, Theorem 4.8] Let k be an algebraically closed field of character-istic p > , X a smooth projective variety over k of dimension n with a fixed ampledivisor H such that deg H (Ω X ) ≥ . Let E be a torsion free sheaf on X such that E ⊗ T l (Ω X ) , ≤ l ≤ n ( p − , are semistable sheaves. Then the Frobenius directimage F X ∗ ( E ) is a semistable sheaf. Lemma 5.2. Let K be a field, V , W and W are k -vector spaces. Let ρ l : GL K ( V ) → GL K ( W l ) be GL K ( V ) -representation at most degree d l , l = 1 , . Then the GL K ( V ) -representation ρ ⊗ ρ : GL K ( V ) → GL K ( W ⊗ W ) at most degree d + d .Proof. The GL K ( V )-representation ρ l is given by a matrix M l of regular functions f ( l ) ij det( T ij ) a l ∈ K [ T ij , det( T ij ) − ] ≤ i,j ≤ dim K V , where a l ∈ N and f ( l ) ij ∈ K [ T ij ] ≤ i,j ≤ dim K V with deg( f ( l ) ij ) ≤ d l , l = 1 , 2. Then GL K ( V )-representation ρ ⊗ ρ : GL K ( V ) → GL K ( W ⊗ W )is determined by the matrix M ⊗ M of regular functions on GL K ( V ). Hence it isat most degree d + d . (cid:3) Proposition 5.3. Let k be an algebraically closed field of characteristic p > , X a smooth projective variety over k with a fixed ample divisor H , and ρ : GL n ( k ) → GL m ( k ) a GL n ( k ) -representation at most degree d . Let E be a torsion free sheafof rank n on X , and F N ∗ X ( E ) is semistable for some integer N ≥ max Let k be an algebraically closed field of characteristic p > , X a smooth projective variety over k of dimension n with a fixed ample divisor H such that deg H (Ω X ) ≥ , ρ : GL n ( k ) → GL m ( k ) a GL n ( k ) -representation at mostdegree d . Suppose that F N ∗ X (Ω X ) is semistable for some integer N ≥ max ≤ l ≤ n ( p − Let k be an algebraically closed field of characteristic p > , X asmooth projective variety over k of dimension n with a fixed ample divisor H suchthat deg H (Ω X ) ≥ , d and m ∈ N + . Suppose that F N ∗ X (Ω X ) is semistable for someinteger N ≥ max ≤ l ≤ n ( p − 0, the natural GL k ( V )-action on V ⊗ d , Sym d ( V ) and V d ( V ) are all polynomial representation of degree d . Sincerk((Ω X ) ⊗ d ) = n d , rk(Sym d (Ω X )) = C r − n + r − and rk( V d (Ω X )) = C dn , this corollaryfollows from Theorem 5.4. 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