On the Stability of Tangent Bundle on Double Coverings

Acta Mathematica Sinica, English Series Aug., 2017, Vol. 33, No. 8, pp. 1039 1047 Published online: March 30, 2017 DOI: 10.1007/s10114-017-6190-7 Http://www.ActaMath.com Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2017 On the Stability of Tangent Bundle on Double Coverings Yongming ZHANG School of Mathematical Sciences, Fudan University, Shanghai 200433, P. R. China E-mail : zhangyongming@amss.ac.cn Abstract Let Y be a smooth projective surface defined over an algebraically closed field k with char k 2,andletπ : X Y be a double covering branched along a smooth divisor. We show that T X is stable with respect to π H if the tangent bundle T Y is semi-stable with respect to some ample line bundle H on Y. Keywords Stability, tangent bundle, double covering, Frobenius morphism MR(2010) Subject Classification 14J60 1 Introduction Stability of tangent bundles plays an important role in algebraic geometry. It is related to whether the tangent bundles admit Hermitian Einstein metrics in differential geometry. In [8], the authors study that the strong stability of tangent bundle of higher dimensional variety which is a cyclic cover over a variety with Picard number 1 in positive characteristic. We study the stability of tangent bundle of a surface which is a double cover of another surface (in fact, the proof can be promoted to a cyclic cover). In this paper, let k denote an algebraically closed field with char k 2 if it is not specified. Let π : X Y be a finite covering defined over k. We know that the pull back of semi-stable sheaves under π are semi-stable in characteristic zero (see [3, Lemma 1.1]), but it does not generally hold that the pull back of stable sheaves are stable. In some special cases, for example when π is a double cover, the pull back of stable sheaves are stable with char k 2 (see Lemma 3.2). Then, using this result we prove: Theorem 1.1 ([Theorem 3.7]) Let Y be a smooth projective surface over an algebraically closed field k with char k 2and let π : X Y be a double covering branched along a smooth curve B Γ(Y,L 2 ) for some line bundle L Pic (Y ). If tangent bundle T Y is semi-stable with respect to some ample line bundle H Pic (Y ), then the tangent bundle T X is stable with respect to π H. So we conclude that when X is a double plane T X is stable since T P 2 is stable. And in this case, as an application, when char k>2we obtain that the Frobenius push forward F L is semi-stable for a torsion free sheaf L of rank 1 on X when the degree of branched curve d 6. In addition, we already know that the tangent bundle of a K3 surface of characteristic zero is stable (see Chapter 9 of [4]). Note that when char k 2 the general K3 surface of degree 2 can be realized as a double covering of P 2 branched along a smooth curve of degree 6 (when the Received April 8, 2016, revised September 28, 2016, accepted January 17, 2017

1040 Zhang Y. M. polarizing line bundle on a K3 surface is base point free). So we obtain a partial result about the stability of the tangent bundle of K3 surfaces in positive characteristic: the tangent bundle of a general K3 surface of degree 2 is stable when char k>2. 2 Preliminaries Let k denote an algebraically closed field in this section. A variety over k means a separated, integral scheme of finite type over k. 2.1 Double Coverings and K3 Surfaces First we recall the definition of double covering. Let Y be a smooth variety over k and let B be a effective divisor on Y. Suppose that O Y (B) is divisible by 2 in Pic(Y ), i.e., there is a line bundle L Pic(Y ) such that L 2 = O Y (B). Note that the divisor B gives a morphism L 2 O Y, and this morphism gives E = O Y L 1 a ring structure. Let X =Spec(E ). Then with the canonical projection we get a double covering over Y : π : X Y. Note that the double cover X over Y branched along B associated to L is a subvariety of the total space of the bundle L. When char k 2,X is normal if and only if B is reduced and X is smooth if and only if B is smooth. It is a Galois covering since the degree of π is 2. Conversely, let π : X Y be a finite surjective morphism of degree 2 between two smooth varieties and ι : X X be the sheet interchange involution over Y.ThenB is the image under π of the fixed set of ι and π O X = O Y L 1, where the direct sum decomposition corresponds to taking the +1 and 1 eigenspaces of ι. Let B 1 denote the reduced divisor of the inverse image π 1 (B). Then we have the following lemma: Lemma 2.1 ([1, Chapter 1.17]) (i) π L = O X (B 1 ); (ii) K X = π (K Y L); (iii) π O X = O Y L 1. Keep the above notation. Then we have Next, we recall the definition of K3 surfaces: Definition 2.2 An algebraic K3 surface over k is a projective smooth variety X of dimension 2 such that the canonical bundle K X = O X and H 1 (X, O X )=0. By (ii), (iii) of Lemma 2.1, when Y is P 2 and B is a smooth curve of degree 6 on Y, X is a K3 surface of degree 2. 2.2 Stability We collect here some well known facts about stability of coherent sheaves. Let X be a smooth projective variety of dimension n over an algebraically closed field k and H an ample line bundle on X. LetE be a torsion free sheaf on X. The rank of E, denoted by rk(e), is the rank of E at the generic point of X. The first Chern class of E is denoted by c 1 (E), then the degree of E denoted by deg(e) is the intersection number c 1 (E) H n 1. We define μ(e) := deg(e) rk(e), which depends on the choice of the ample line bundle H on X.

On the Stability of Tangent Bundle on Double Coverings 1041 Definition 2.3 we have E is stable (resp. semi-stable) if for any sub-sheaf F with 0 < rk(f ) < rk(e), μ(f ) <μ(e) (resp. μ(f ) μ(e)). Actually, in the above definition we only need to consider the saturated sub-sheaves of E. If E is a torsion free sheaf on X, then there is a unique filtration (the Harder Narasimhan filtration) 0=E 0 E 1 E l = E such that the factors gr HN i := E i /E i 1 for i =1,...,l are semi-stable sheaves and μ max (E) :=μ(gr1 HN ) > >μ(grl HN )=:μ min (E). The instability of E is defined as I(E) :=μ max (E) μ min (E), and E is semi-stable if and only if I(E) =0. If E is a semi-stable torsion free sheaf on X, then there is a filtration (the Jordan Hölder filtration) 0=E 0 E 1 E l = E such that the factors gr i = E i /E i 1 for i =1,...,l are stable sheaves and μ(gr 1 )= = μ(gr l )=μ(e). Note that the Jordan Hölder filtration is not unique but the factors gr i := E i /E i 1 for i = 1,...,l are unique determined by E. 3 The Main Results and Proof For a Galois covering π : X Y with Galois group G and coherent sheaf E on Y, then there is an isomorphism σ π E π E. For any sub-sheaf F π E, we will denote the image of sub-sheaf σ F σ π E under this isomorphism by F σ for σ G. Next we relate the following fact in Galois decent theory: Theorem 3.1 ([2, Chapte 9, Theorem 28]) Let p : X Y be a finite flat Galois morphism with Galois group G, and suppose that X and Y are both integral. Let F be a locally free sheaf on Y and suppose that G is a coherent sub-sheaf of p F such that For all σ G, (G ) σ = G as a sub-sheaf of p F; p F/G is torsion free. Then there exists a coherent sub-sheaf G of F such that G = p G as a sub-sheaf of p F. Immediately, we have the following observation: Lemma 3.2 Let Y be a smooth projective variety over k with char k 2and let π : X Y be a double covering branched along a smooth effective divisor B Γ(Y,L 2 ) for some line bundle L Pic (Y ). If F is locally free sheaf on Y and stable with respect to an ample line bundle H on Y,thenπ F is stable with respect to the ample line bundle π H. Proof Since degree π =2andX is smooth, π is a flat Galois covering. Let G = {1,σ} be the Galois group acting on X. We first prove π F is semi-stable. Suppose π F is not semi-stable. Then there is a unique maximal destabilizing sub-sheaf M π F with μ(m) >μ(π F).

1042 Zhang Y. M. Note that M is a saturated sub-sheaf and M σ = M. By Theorem 3.1, M = π G for some sub-sheaf G F.Sinceμ(π G)=deg(π) μ(g) andμ(π F)=deg(π) μ(f) wegetμ(g) >μ(f) which is a contradiction. Second, we suppose π F is not stable. Then there is a filtration (the Jordan Hölder filtration) 0=F 0 F 1 F l = π F such that the factors gr i = F i /F i 1 for i =1,...,l (l 2) are stable sheaves and μ(gr 1 )= = μ(gr l )=μ(π F). There is no saturated sub-sheaf E π F with μ(e) μ(π F)andE σ = E (if not then E = π E for some sub-sheaf E Fand by the argument as before we will get a contradiction), so Fi σ F i for i<l. Using the fact that the only nontrivial map between two stable sheaves with the same slope is isomorphism, we have l =2, π F = gr 1 gr 2 and gr 1 gr 2. In fact, after the action of σ we get another Jordan Hölder filtration: 0=F σ 0 F σ 1 F σ l = π F such that the factors gr i = F i σ/f i 1 σ for i =1,...,l are stable sheaves and μ(gr 1)= = μ(gr l)=μ(π F). Let l 0 be the smallest number satisfies F σ 1 F l0. Then we get a nontrivial morphism between stable sheaves with the same slope: F σ 1 gr l0 = F l0 /F l0 1, so it is an isomorphism, and F σ 1 F 1 = 0. It follows that F σ 1 F 1 π F,sowehaveF σ 1 F 1 = π F. On the other hand, after pushing forward we obtain π π (F) =F F(L 1 )=π gr 1 π gr 2. Note that σ acts on F as Id, F(L 1 ) by multiplying 1, and π gr 1 onto another component. So π gr 1 F = 0, hence we can get an injection π gr 1 F(L 1 ). Note that F(L 1 ) is stable and μ(f(l 1 )) <μ(f), so we will get μ(π gr 1 ) <μ(f(l 1 )) <μ(f F(L 1 )=μ(π π (F)). But μ(π π (F)) = μ(π gr 1 ) which leads to a contradiction. Consequently, π F is stable and we complete the proof. Remark 3.3 In general, the pull back π preserves semi-stability of vector bundles under separated finite morphism π (see [3, Lemma 1.1]) in characteristic zero. Here it holds for stable bundles just because the finite morphism π is a Galois cyclic covering which can be written exactly. Next, we relate a well-known fact and present a proof. Lemma 3.4 With the same assumption in Lemma 3.2, ifm is an invertible sheaf on X such that f : σ M M,thenM π M for some M Pic (Y ).

On the Stability of Tangent Bundle on Double Coverings 1043 Proof First, we note the fact that if D is an effective divisor and D = σ D,thenD = π (D ) for some effective divisor D on Y. Suppose that M is effective and G acts on P(H 0 (X, M)) by s f(σ s), s P(H 0 (X, M)). Then there exists a fixed point D P(H 0 (X, M)). So D = π (D ) for some effective divisor D on Y, i.e., M π O Y (D ). If M is not effective, we could take an ample line bundle H Pic (Y ). Then M 1 := M π H n for some n Z is effective and σ (M 1 ) M 1. So by the same argument as before, we have M 1 π O Y (D ) for some divisor D on Y and then M π O Y (D ) π H n. Lemma 3.5 With the same assumption in Lemma 3.2, lete and F be two torsion free sheaf on Y. If there is a morphism f : π E π F commuting with the nature isomorphism σ π E π E, i.e., π E f π F σ π E σ (f) σ π F commutes, then f = π (f ) for some morphism f : E F. Proof Note that π f : E (E L 1 ) F (F L 1 ) commutes with the action induced by the involution σ which acts on the first factor as Id and the second factor by multiplying 1. So let f = π f E and it easy to see that f = π (f ). Lemma 3.6 Let Y be a smooth projective surface over an algebraically closed field k with char k 2and let π : X Y be a double covering branched along a smooth curve B Γ(Y,L 2 ) for some line bundle L Pic (Y ). If the cotangent bundle Ω X contains a saturated sub-line bundle M 1 which is stable under the action of σ, then it is a sub-line bundle contained in π Ω Y Ω X ; i.e., if sub-line bundle M 1 Ω X satisfies M 1 = M σ 1 where M σ 1 is the image of σ M 1 σ Ω X under the canonical isomorphism f : σ Ω X Ω X,thenM 1 π Ω Y Ω X. Proof First we have the exact sequence 0 π Ω Y i Ω X Ω X/Y 0 and the canonical isomorphism f : σ Ω X Ω X commuting with σ (i). Suppose that M 1 is a sub-line bundle of Ω X satisfying M 1 = M σ 1 ;thenm 1 = π M 1 for some line bundle on Y by Lemma 3.4. Setting N 1 := π Ω Y M 1, we have the following diagram: 0 N 2 M 2 Q 2 0 0 π Ω Y i 0 i N1 N 1 Ω X M 1 Ω X/Y Q 1 0 0 Since N 2 M 2 is torsion free N 1 is saturated in π Ω Y. So it is a line bundle and N 1 = π N 1 for some line bundle on Y by Lemma 3.4. And i N1 = π (i ) for some morphism i : N 1 M 1 by Lemma 3.5. Now the third line in the above diagram is in fact the pull back of the following

1044 Zhang Y. M. sequence: 0 N 1 i M 1 Q 1 0 under π. Now suppose that Q 1 0. We may assume B is connected (if not, we can restrict the following argument to an irreducible component). Let B 1 denote the reduced divisor of the inverse image π 1 (B). It is easy to find that Supp Ω X/Y is contained in B 1,andbylocal calculation we find that Ω X/Y is in fact a locally free O B1 -module of rank 1. On the other hand, we have (M 1) 1 Q 1 O nb for some multiplicity n Z +. Note that 2B 1 = π 1 (B) andπ O nb = O 2nB1. So π (M 1 O nb ) Ω X/Y is a locally free O 2nB1 -module of rank 1. But Ω X/Y is a locally free O B1 -module of rank 1. So we get a contradiction. Theorem 3.7 Let Y be a smooth projective surface over an algebraically closed field k with char k 2and let π : X Y be a double covering branched along a smooth curve B Γ(Y,L 2 ) for some line bundle L Pic(Y ). If tangent bundle T Y is semi-stable with respect to some ample line bundle H Pic(Y ), then the tangent bundle T X is stable with respect to π H. Proof We may consider cotangent bundle instead of tangent bundle. Suppose Ω X is not semi-stable, then there is a unique filtration (H-N filtration) 0 M 1 Ω X with μ(m 1 ) >μ(ω X ). Also note that M 1 is a saturated sub-line bundle. By the uniqueness of M 1 we have M σ 1 = M 1. By Lemma 3.6 we know that M 1 π Ω Y Ω X.Soμ(M 1 ) μ(π Ω Y ) <μ(ω X ) by Lemma 3.2 and we get a contradiction. Thus Ω X is semi-stable with respect to π H. Now we suppose that Ω X is not stable with respect to π H, then there is a Jordan Hölder filtration 0 M 1 Ω X with μ(m 1 )=μ(ω X ). Now we get a saturated sub-line bundle M 1. By the same argument as above we know that M σ 1 M 1, then we get another saturated stable sub-line bundle M σ 1 with μ(m σ 1 )=μ(ω X ). Since the graded factors of Jordan Hölder filtration are unique determined, we conclude that the filtration split Ω X = M 1 M σ 1. Since π is exact we have the following diagram after push forward a diagram on X: 0 π N 2 π M σ 1 π Q 2 0 0 π π π Ω (i) Y π Ω X 0 π (i N1 ) π N 1 where N 1 = π Ω Y M 1 and N 2 = π Ω Y /N 1. π M 1 π Ω X/Y π Q 1 0 0

On the Stability of Tangent Bundle on Double Coverings 1045 In the following argument the degree and slope of coherent sheaves are calculated with respect to H or π H. Now we relate some facts which will be used later. For any coherent sheaf E on X, wehave deg(π E)= rk(π E) deg(ω Y ) rk(e) deg(ω X )+deg(e), 2 2 which can be calculated using Grothendieck Riemann Roch theorem. And since rk(π E)= 2rk(E), we have μ(π E)=μ(Ω Y ) 1 2 μ(ω X)+ 1 2 μ(e) if rk(e) > 0. By Lemma 2.1 (ii) we can get μ(ω X )=μ(π Ω Y )+deg(l) =2μ(Ω Y )+deg(l), so we have μ(π E)= 1 2 μ(e) 1 2 deg(l) if rk(e) > 0. Next we will conclude a contradiction by calculating the slope of sheaves. First by the above facts, we have μ(π N 2 )= 1 2 μ(n 2) 1 2 deg(l) and since M σ 1 is stable μ(n 2 ) <μ(m σ 1 )=μ(ω X ), we have Second we note that μ(π N 2 ) < 1 2 μ(ω X) 1 2 deg(l) =μ(ω Y ). π N 1 π N σ 1 π π (Ω Y )=Ω Y Ω Y (L 1 ), where N σ 1 = π Ω Y M σ 1 and σ acts on Ω Y as Id, Ω Y (L 1 ) by multiplying 1, and π N 1 onto π N σ 1.Soπ N 1 Ω Y = 0, hence we can get an injection Noting that Ω Y (L 1 ) is semi-stable, we have So we conclude that π N 1 Ω Y (L 1 ). μ(π N 1 ) μ(ω Y (L 1 )) = μ(ω Y ) deg(l). μ(π π (Ω Y )) = 1 2 (μ(π N 1 )+μ(π N 2 )) < 1 2 (μ(ω Y ) deg(l)+μ(ω Y )) = μ(ω Y ) 1 2 deg(l), but μ(π π (Ω Y )) = μ(ω Y Ω Y (L 1 )) = μ(ω Y ) 1 2 deg(l). So we get a contradiction and T X is stable with respect to π H. Next, we relate the following well-known result. Theorem 3.8 ([4, Chapter 1.4]) Let k be an algebraically closed field, and Ω 1 P is the cotangent N bundle on P N k.thenω1 P N is stable with respect to O P N (1). Then we get the following result using Theorem 3.8. Corollary 3.9 Let k be an algebraically closed field with char k 2and π : X P 2 be a double covering branched along a smooth curve in P 2. Then T X is stable with respect to π O P N (1).

1046 Zhang Y. M. Remark 3.10 In particular, when the branched curve in P 2 is of degree 6, X is a K3 surface of degree 2 defined over k with char k 2 and the tangent bundle T X is stable with respect to π O P N (1). 4 Application Let X be a smooth projective variety over an algebraically closed field k with char k = p>0. The absolute Frobenius F X : X X is defined by the map: f f p in O X.LetF : X X 1 := X k k denote the relative Frobenius morphism over k. We know that F preserves semi-stability of vector bundles on curves of genus g 1. When W is a vector bundle on a higher dimensional variety, the instability of F W is bounded by instability of W T l (Ω 1 X )(0 l n(p 1)) which is proved by Sun in [9] and we will use this result to get our corollary. Definition 4.1 ([6, Section 5.1]) Let V 0 := V = F (F W ), V 1 =ker(f (F W ) W ), and V l+1 := ker(v l V OX Ω 1 X (V/V l) OX Ω 1 X ) where : V V O X Ω 1 X is the canonical connection with zero p-curvature (see [7, Theorem 5.1]). Then we have the filtration 0=V n(p 1)+1 V n(p 1) V 1 V 0 = V = F (F W ). Next,werecallthedefinitionofT l (V ). Let V be an n-dimensional vector space over k with standard representation of GL(n), S l be the symmetric group with natural action on V l by (v 1 v l ) σ = v σ(1) v σ(l) for v i V and σ S l.lete 1,..., e n be a basis of V ;for k i 0withk 1 + + k n = l, define v(k 1,...,k n )= σ S l (e k 1 1 e k n n ) σ. Let T l (V ) V l be the linear subspace generated by all vectors v(k 1,...,k n ) for all k i 0 satisfying k 1 + + k n = l. Then we have the following property. Theorem 4.2 ([9, Theorem 3.7]) The filtration defined in Definition 4.1 has the following properties : (i) (V l+1 ) V l Ω 1 X for l 1, andv 0/V 1 = W. (ii) V l /V l+1 (Vl 1 /V l ) Ω 1 X are injective morphisms of vector bundles for 1 l n(p 1), which induced isomorphisms l : V l /V l+1 = W OX T l (Ω 1 X ), 0 l n(p 1). Let H be an ample line bundle on X. For any vector bundle W on X, let I(W, X) :=max{i(w T l (Ω 1 X)) 0 l n(p 1)} be the maximal value of instabilities I(W T l (Ω 1 X )). Theorem 4.3 ([9, Theorem 4.8]) When K X H n 1 0, we have, for any E F W, I(W, X) μ(f W ) μ(e). p In particular, if W T l (Ω 1 X ), 0 l n(p 1), are semi-stable, then F W is semi-stable. Moreover, if K X H n 1 > 0, the stability of the bundles W T l (Ω 1 X ), 0 l n(p 1), implies the stability of F W. When X is a surface we have a better description of T l (Ω 1 X ).

On the Stability of Tangent Bundle on Double Coverings 1047 Theorem 4.4 ([9, Theorem 3.8]) When dim(x) = 2, we have W Sym l (Ω 1 X V l /V l+1 = ), when l<p, W Sym 2(p 1) l (Ω 1 X ) ωl (p 1) X, when l p. Now using the above results, we get the following corollary. Corollary 4.5 Let π : X P 2 be a double covering over an algebraically closed field k with char k = p>2 branched along a smooth curve of degree d and H = π O(1). Then F L is semi-stable (resp. stable) if d (resp. >) 6for any torsion free sheaf L of rank 1. Proof First, by Lemma 2.1 we know that K X H=0ifd =6andK X H> 0ifd>6. And we have proved that Ω 1 X is stable in Corollary 3.9, so Syml (Ω 1 X ) is semi-stable when l<p. Then we get T l (Ω 1 X ) is semi-stable by Theorem 4.4 and I(L Tl (Ω 1 X )) = I(Tl (Ω 1 X )) = 0 since L is of rank 1. At last by using Theorem 4.3 we obtain our result. Acknowledgements The author would like to express sincere thanks to his advisor Professor Xiaotao Sun for careful reading and helpful discussions to improve the paper in both mathematics and presentations. References [1] Barth, W. P., Hulek, K., Peters, C. A. M., et al.: Complex Compact Surfaces, Springer Science & Business Media, Berlin, 2004 [2] Friedman, R.: Algebraic Surfaces and Holomorphic Vector Bundles, Springer Science & Business Media, New York, 2012 [3] Gieseker, D.: On a theorem of Bogomolov on Chern classes of stable bundles. Amer. J. Math., 101, 77 85 (1979) [4] Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, Cambridge University Press, Cambridge, 2010 [5] Huybrechts, D.: Lectures on K3 Surfaces, Cambridge University Press, Cambridge, 2016 [6] Joshi, K., Ramanan, S., Xia, E. Z., et al.: On vector bundles destabilized by Frobenius pull-back. Compos. Math., 142, 616 630 (2006) [7] Katz, N. M.: Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Publ. Math., Inst. Hautes, Études Sci., 39, 175 232 (1970) [8] Li, L., Shentu, J.: Strong stability of cotangent bundles of cyclic covers. C. R. Math. Acad. Sci. Paris, 352(7), 639 644 (2014) [9] Sun, X.: Direct images of bundles under Frobenius morphism. Invent. Math., 173, 427 447 (2008)