Ramified covering maps and stability of pulled back bundles
aa r X i v : . [ m a t h . AG ] F e b RAMIFIED COVERING MAPS AND STABILITY OF PULLED BACKBUNDLES
INDRANIL BISWAS AND A. J. PARAMESWARAN
Abstract.
Let f : C −→ D be a nonconstant separable morphism between irreduciblesmooth projective curves defined over an algebraically closed field. We say that f isgenuinely ramified if O D is the maximal semistable subbundle of f ∗ O C (equivalently,the homomorphism f ∗ : π et1 ( C ) −→ π et1 ( D ) is surjective). We prove that the pullback f ∗ E −→ C is stable for every stable vector bundle E on D if and only if f is genuinelyramified. Contents
1. Introduction 12. Genuinely ramified morphism 23. Properties of genuinely ramified morphisms 74. Pullback of stable bundles and genuinely ramified maps 145. Characterizations of genuinely ramified maps 19Acknowledgements 21References 211.
Introduction
Let k be an algebraically closed field. Let f : C −→ D be a nonconstant separablemorphism between irreducible smooth projective curves defined over k . For any semistablevector bundle E on D , the pullback f ∗ E is also semistable. However, f ∗ E need not bestable for every stable vector bundle E on D . Our aim here is to characterize all f suchthat f ∗ E remains stable for every stable vector bundle E on D . It should be mentionedthat E is stable (respectively, semistable) if f ∗ E is stable (respectively, semistable).For any f as above the following five conditions are equivalent:(1) The homomorphism between ´etale fundamental groups f ∗ : π et1 ( C ) −→ π et1 ( D )induced by f is surjective. Mathematics Subject Classification.
Key words and phrases.
Genuinely ramified map, stable bundle, maximal semistable subbundle, socle. (2) The map f does not factor through some nontrivial ´etale cover of D (in particular, f is not nontrivial ´etale).(3) The fiber product C × D C is connected.(4) dim H ( C, f ∗ f ∗ O C ) = 1.(5) The maximal semistable subbundle of the direct image f ∗ O C is O D .The map f is called genuinely ramified if any (hence all) of the above five conditions holds.Proposition 2.6 and Definition 2.5 show that the above statements (1), (2) and (5) areequivalent; in Lemma 3.1 it is shown that the statements (3), (4) and (5) are equivalent.We prove the following (see Theorem 5.3): Theorem 1.1.
Let f : C −→ D be a nonconstant separable morphism between irre-ducible smooth projective curves defined over k . The map f is genuinely ramified if andonly if f ∗ E −→ C is stable for every stable vector bundle E on D . The key technical step in the proof of Theorem 1.1 is the following (see Proposition3.5):
Proposition 1.2.
Let f : C −→ D be a genuinely ramified Galois morphism, of degree d , between irreducible smooth projective curves defined over k . Then f ∗ (( f ∗ O C ) / O D ) ⊂ d − M i =1 L i , where each L i is a line bundle on D of negative degree. When k = C , a vector bundle F on a smooth complex projective curve is stable ifand only if F admits an irreducible flat projective unitary connection [NS]. From thischaracterization of stable vector bundles it follows immediately that given a nonconstantmap f : C −→ D between irreducible smooth complex projective curves, f ∗ E is stablefor every stable vector bundle E on D if the homomorphism of topological fundamentalgroups induced by f f ∗ : π ( C, x ) −→ π ( D, f ( x ))is surjective.Theorem 4.4 was stated in [PS] (it is [PS, p. 524, Lemma 3.5(b)]) without a completeproof. In the proof of Lemma 3.5(b), which is given in three sentences in [PS, p. 524],it is claimed that the socle of a semistable bundle descends under any ramified coveringmap (the first sentence). 2. Genuinely ramified morphism
The base field k is assumed to be algebraically closed; there is no restriction on thecharacteristic of k .Let V be a vector bundle on an irreducible smooth projective curve X defined over k .If 0 = V ⊂ V ⊂ · · · ⊂ V n − ⊂ V n = V AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 3 is the Harder-Narasimhan filtration of V [HL, p. 16, Theorem 1.3.4], then define µ max ( V ) := µ ( V ) := deg( V )rk( V ) and µ min ( V ) := µ ( V /V n − ) . Furthermore, the above subbundle V ⊂ V is called the maximal semistable subbundle of V .If V and W are vector bundles on X , and β ∈ H ( X, Hom(
V, W )) \ { } , then it canbe shown that µ max ( W ) ≥ µ min ( V ) . (2.1)Indeed, we have µ min ( V ) ≤ µ ( β ( V )) ≤ µ max ( W ) . Remark 2.1.
Let f : X −→ Y be a nonconstant separable morphism between irre-ducible smooth projective curves, and let F be a semistable vector bundle on Y . Then itis known that f ∗ F is also semistable. Indeed, fixing a nonconstant separable morphism h : Z −→ X , where Z is an irreducible smooth projective curve and f ◦ h is Galois, we seethat ( f ◦ h ) ∗ F is semistable, because its maximal semistable subbundle, being Gal( f ◦ h )invariant, descends to a subbundle of F . The semistability of ( f ◦ h ) ∗ F = h ∗ f ∗ F imme-diately implies that f ∗ F is semistable. Lemma 2.2.
Let f : X −→ Y be a nonconstant separable morphism of irreduciblesmooth projective curves. Then for any semistable vector bundle E on X , µ max ( f ∗ E ) ≤ µ ( E ) / deg( f ) . More generally, for any vector bundle E on X , µ max ( f ∗ E ) ≤ µ max ( E ) / deg( f ) . Proof.
Let E be any vector bundle on X . The coherent sheaf f ∗ E on Y is locally free,because it is torsion-free. We have H ( Y, Hom(
F, f ∗ E )) ∼ = H ( X, Hom( f ∗ F, E )) (2.2)for any vector bundle F on Y ; see [Ha, p. 110]. Setting F in (2.2) to be a semistable sub-bundle V of f ∗ E we see that H ( X, Hom( f ∗ V, E )) = 0. The pullback f ∗ V is semistablebecause V is semistable and f is separable; see Remark 2.1.First take E to be semistable. Hence for any nonzero homomorphism β : f ∗ V −→ E ,deg( f ) · µ ( V ) = µ ( f ∗ V ) ≤ µ ( E ) (2.3)(see (2.1)). Now setting V in (2.3) to be the maximal semistable subbundle of f ∗ E weconclude that µ max ( f ∗ E ) ≤ µ ( E ) / deg( f ) . (2.4)To prove the general (second) statement, for any vector bundle E on X , let0 = E ⊂ E ⊂ · · · ⊂ E n − ⊂ E n = E I. BISWAS AND A. J. PARAMESWARAN be the Harder–Narasimhan filtration of E [HL, p. 16, Theorem 1.3.4]. Consider thefiltration of subbundles0 = f ∗ E ⊂ f ∗ E ⊂ · · · ⊂ f ∗ E n − ⊂ f ∗ E n = f ∗ E . (2.5)From (2.4) we know that µ max (( f ∗ E i ) / ( f ∗ E i − )) = µ max ( f ∗ ( E i /E i − )) ≤ µ ( E i /E i − ) / deg( f ) ≤ µ ( E ) / deg( f ) = µ max ( E ) / deg( f ) (2.6)for all 1 ≤ i ≤ n . Observe that using (2.1) and the filtration in (2.5) it follows that µ max ( f ∗ E ) ≤ Max { µ max (( f ∗ E i ) / ( f ∗ E i − )) } ni =1 , while from (2.6) we haveMax { µ max (( f ∗ E i ) / ( f ∗ E i − )) } ni =1 ≤ µ max ( E ) / deg( f ) . Therefore, µ max ( f ∗ E ) ≤ µ max ( E ) / deg( f ), and this completes the proof. (cid:3) The following lemma characterizes the ´etale maps among the separable morphisms.
Lemma 2.3.
Let f : X −→ Y be a nonconstant separable morphism between irreduciblesmooth projective curves. Then the following three conditions are equivalent: (1) The map f is ´etale. (2) The degree of f ∗ O X is zero. (3) The vector bundle f ∗ O X is semistable.Proof. We have µ max ( f ∗ O X ) ≥
0, because O Y ⊂ f ∗ O X (see (2.2) and (2.1)). On theother hand, from Lemma 2.2 it follows that µ max ( f ∗ O X ) ≤
0, so µ max ( f ∗ O X ) = 0 . (2.7)From (2.7) it follows immediately that f ∗ O X is semistable if and only if deg( f ∗ O X ) = 0.Therefore, statements (2) and (3) are equivalent.Let R be the ramification divisor on X for f ; define the effective divisor B := f ∗ R on Y . We know that (det f ∗ O X ) ⊗ = O Y ( − B )(see [Ha, p. 306, Ch. IV, Ex. 2.6(d)], [Se]). Therefore, f is ´etale (meaning B = 0) if andonly if deg( f ∗ O X ) = 0. So statements (1) and (2) are equivalent. (cid:3) Let f : X −→ Y be a nonconstant separable morphism between irreducible smoothprojective curves. The algebra structure of O X produces an O Y –algebra structure on thedirect image f ∗ O X . Lemma 2.4.
Let
V ⊂ f ∗ O X be the maximal semistable subbundle. Then V is a sheaf of O Y –subalgebras of f ∗ O X . AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 5
Proof.
The action of O Y on f ∗ O X is the standard one. Let m : ( f ∗ O X ) ⊗ ( f ∗ O X ) −→ f ∗ O X be the O Y –algebra structure on the direct image f ∗ O X given by the algebra structure ofthe coherent sheaf O X . We need to show that m ( V ⊗ V ) ⊂ V , (2.8)where V is the maximal semistable subbundle of f ∗ O X .Since V is semistable of degree zero (see (2.7)), and µ max (( f ∗ O X ) / V ) <
0, using (2.1)we conclude that in order to prove (2.8) it suffices to show that
V ⊗ V is semistable ofdegree zero. Indeed, there is no nonzero homomorphism from
V ⊗ V to ( f ∗ O X ) / V , if V ⊗ V is semistable of degree zero.We have deg(
V ⊗ V ) = 0, because deg( V ) = 0. So V ⊗ V is semistable if it does notcontain any coherent subsheaf of positive degree. As
V ⊗ V ⊂ ( f ∗ O X ) ⊗ ( f ∗ O X ) , if V ⊗ V contains a subsheaf of positive degree, then ( f ∗ O X ) ⊗ ( f ∗ O X ) also contains asubsheaf of positive degree.Therefore, to prove the lemma it is enough to show that ( f ∗ O X ) ⊗ ( f ∗ O X ) does notcontain any subsheaf of positive degree.The projection formula, [Ha, p. 124, Ch. II, Ex. 5.1(d)], [Se], says that( f ∗ O X ) ⊗ O Y ( f ∗ O X ) ∼ = f ∗ ( f ∗ ( f ∗ O X )) . (2.9)Since O Y ⊂ f ∗ O X , we have O Y = O Y ⊗ O Y O Y ⊂ ( f ∗ O X ) ⊗ O Y ( f ∗ O X ) , and hence µ max (( f ∗ O X ) ⊗ O Y ( f ∗ O X )) ≥
0. Now from (2.9) it follows that µ max ( f ∗ ( f ∗ ( f ∗ O X ))) ≥ . (2.10)Since f is separable, the pullback, by f , of a semistable bundle on Y is semistable (seeRemark 2.1), and consequently the Harder–Narasimhan filtration of f ∗ F is the pullback,by f , of the Harder–Narasimhan filtration of F . Therefore, from (2.7) it follows that µ max ( f ∗ ( f ∗ O X )) = 0 . Now applying the second part of Lemma 2.2,0 = µ max ( f ∗ ( f ∗ O X )) / deg( f ) ≥ µ max ( f ∗ ( f ∗ ( f ∗ O X ))) . This and (2.10) together imply that µ max ( f ∗ ( f ∗ ( f ∗ O X ))) = 0 . Therefore, using (2.9) it follows that µ max (( f ∗ O X ) ⊗ ( f ∗ O X )) = 0 . Hence ( f ∗ O X ) ⊗ ( f ∗ O X ) does not contain any subsheaf of positive degree. It was shownearlier that the lemma follows from the statement that ( f ∗ O X ) ⊗ ( f ∗ O X ) does not containany subsheaf of positive degree. (cid:3) I. BISWAS AND A. J. PARAMESWARAN
Definition 2.5.
A nonconstant separable morphism f : X −→ Y between irreduciblesmooth projective curves is called genuinely ramified if O Y is the maximal semistablesubbundle of f ∗ O X . Proposition 2.6.
Let f : X −→ Y be a nonconstant separable morphism betweenirreducible smooth projective curves. Then the following three conditions are equivalent: (1) The map f is genuinely ramified. (2) The map f does not factor through any nontrivial ´etale cover of Y (in particular, f is not nontrivial ´etale). (3) The homomorphism between ´etale fundamental groups induced by ff ∗ : π et1 ( X ) −→ π et1 ( Y ) is surjective.Proof. (1) = ⇒ (2): If f factors through a nontrivial ´etale covering g : e Y −→ Y , then g ∗ O e Y is semistable of degree zero (see Lemma 2.3) and its rank coincides with the degreeof g . Since g ∗ O e Y ⊂ f ∗ O X , this implies that f is not genuinely ramified.(2) = ⇒ (1): Lemma 2.4 says that the maximal semistable subbundle V ⊂ f ∗ O X is asubalgebra. If f is not genuinely ramified, then by taking the spectrum of V we obtain aseparable, possibly ramified, covering map g : e Y = Spec V −→ Y (2.11)whose degree coincides with the rank of V . We have g ∗ O e Y = V , and the inclusion map V ֒ → f ∗ O X defines a map h : X −→ e Y such that g ◦ h = f. (2.12)Since f is separable, from (2.12) it follows that g is also separable. It can be shown that g is ´etale. To prove this, first note that g ∗ O e Y is semistable, because g ∗ O e Y = V and V is semistable. Next, from (2.7) and the semistability of V it follows that µ ( g ∗ O e Y ) = µ max ( g ∗ O e Y ) = 0. Now Lemma 2.3 gives that the map g in (2.11) is ´etale.Since g is ´etale, and (2.12) holds, we conclude that the statement (2) fails. Hence thestatement (2) implies the statement (1).The equivalence between the statements (2) and (3) follows from the definition of the´etale fundamental group. (cid:3) Let f : X −→ Y be a nonconstant separable morphism between irreducible smoothprojective curves. Let g : e Y := Spec V −→ Y AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 7 be the ´etale covering corresponding to the maximal semistable subbundle
V ⊂ f ∗ O X (see(2.11); it was shown that the map in (2.11) is ´etale). Let h : X −→ e Y (2.13)be the morphism given by the inclusion map V ֒ → f ∗ O X . Corollary 2.7.
The map h in (2.13) is genuinely ramified.Proof. Let β : Z −→ e Y be an ´etale covering such that there is a map γ : X −→ Z satisfying the condition β ◦ γ = h . Since ( g ◦ β ) ◦ γ = f , we have g ∗ O e Y ⊂ ( g ◦ β ) ∗ O Z ⊂ f ∗ O X ; (2.14)also, we have deg(( g ◦ β ) ∗ O e Y ) = 0, because g ◦ β is ´etale (see Lemma 2.3). But V = g ∗ O e Y is the maximal semistable subsheaf of f ∗ O X . Hence from (2.14) it follows that g ∗ O e Y =( g ◦ β ) ∗ O Z . This implies that deg( β ) = 1. Therefore, from Proposition 2.6 we concludethat the map h in (2.13) is genuinely ramified. (cid:3) Properties of genuinely ramified morphisms
Lemma 3.1.
Let f : C −→ D be a nonconstant separable morphism between irreduciblesmooth projective curves. Then the following three conditions are equivalent: (1) The map f is genuinely ramified. (2) dim H ( C, f ∗ f ∗ O C ) = 1 . (3) The fiber product C × D C is connected.Proof. Let ^ C × D C be the normalization of the fiber product C × D C ; it is a smooth projec-tive curve, but it is not connected unless f is an isomorphism. We have the commutativediagram ^ C × D C e π " " ν % % ▲▲▲▲▲▲▲▲▲▲▲ e π ( ( C × D C π / / π (cid:15) (cid:15) C f (cid:15) (cid:15) C f / / D (3.1)By flat base change [Ha, p. 255, Proposition 9.3], f ∗ ( f ∗ O C ) ∼ = π ∗ ( π ∗ O C ) = π ∗ O C × D C . (3.2)(1) = ⇒ (2): Since f is separable, f ∗ F is semistable if F is so (see Remark 2.1), andhence the maximal semistable subbundle of f ∗ f ∗ O C is f ∗ V , where V ⊂ f ∗ O C is themaximal semistable subbundle. If f is genuinely ramified, then the maximal semistablesubbundle of f ∗ f ∗ O C is f ∗ O D = O C . On the other hand, H ( C, ( f ∗ f ∗ O C ) / ( f ∗ V )) = 0 , I. BISWAS AND A. J. PARAMESWARAN because µ max (( f ∗ f ∗ O C ) / ( f ∗ V )) < H ( C, f ∗ f ∗ O C ) = 1 ;to see this consider the long exact sequence of cohomologies associated to the short exactsequence 0 −→ f ∗ V −→ f ∗ f ∗ O C −→ ( f ∗ f ∗ O C ) / ( f ∗ V ) −→ . (2) ⇐⇒ (3): From (3.2) it follows that H ( C, f ∗ f ∗ O C ) = H ( C, π ∗ O C × D C ) = H ( C × D C, O C × D C ) . (3.3)Consequently, C × D C is connected if and only if dim H ( C, f ∗ f ∗ O C ) = 1.(3) = ⇒ (1): Assume that f is not genuinely ramified. We will prove that C × D C isnot connected.Let g : e D −→ D be the ´etale cover of D given by Spec W , where W ⊂ f ∗ O C is themaximal semistable subbundle (as in (2.11)). The degree of this covering g is at least two,because f is not genuinely ramified. To prove that C × D C is not connected it suffices toshow that e D × D e D is not connected.The projection γ : e D × D e D −→ e D to the first factor is evidently the base change of g : e D −→ D to e D , and hence themap γ is ´etale. The diagonal e D ֒ → e D × D e D is a connected component of e D × D e D . Thisimplies that e D × D e D is not connected, because the degree of γ is at least two. (cid:3) Definition 3.2.
A nonconstant morphism f : C −→ D between irreducible smoothprojective curves will be called a separable Galois morphism if f is separable, and thereis a reduced finite subgroup Γ ⊂ Aut( C ) such that D = C/ Γ and f is the quotient map C −→ C/ Γ. Note that a separable Galois morphism need not be ´etale. A separableGalois morphism which is genuinely ramified will be called a genuinely ramified Galoismorphism . Proposition 3.3.
Let f : C −→ D be a separable Galois morphism, of degree d , be-tween irreducible smooth projective curves. Then f ∗ (( f ∗ O C ) / O D ) is a coherent subsheafof O ⊕ ( d − C .Proof. The Galois group Gal( f ) of f will be denoted by Γ. For any point x ∈ C , letΓ x ⊂ Γbe the isotropy subgroup that fixes x for the action of Γ on C . A point ( x, y ) ∈ C × D C is singular if and only if Γ x is nontrivial. Note that for any ( x, y ) ∈ C × D C the twoisotropy subgroups Γ x and Γ y are conjugate, because y lies in the orbit Γ · x of x . For any σ ∈ Γ, let C σ ⊂ C × D C (3.4)be the irreducible component given by the image of the map β σ : C −→ C × C , x ( x, σ ( x )) ; AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 9 clearly we have β σ ( C ) ⊂ C × D C . In this way, the irreducible components of C × D C are parametrized by the elements of the Galois group Γ. Note that there is a canonicalidentification C ∼ −→ C σ (3.5)for every σ ∈ Γ.Let ^ C × D C be the normalization of C × D C . The maps β σ , σ ∈ Γ, in (3.4) togetherproduce an isomorphism C × Γ ∼ −→ ^ C × D C ; (3.6)this map sends any ( y, σ ) ∈ C × Γ to ( y, σ ( y )) if Γ y is trivial; if Γ y is trivial, then( y, σ ( y )) is a smooth point of C × D C and hence ( y, σ ( y )) gives a unique point of ^ C × D C .Consequently, we have e π ∗ O ^ C × D C = O C ⊗ k k [Γ] , (3.7)where e π is the projection in (3.1), and k [Γ] is the group ring. The natural inclusion O C × D C ֒ → ν ∗ O ^ C × D C , where ν is the map in (3.1), induces an injective homomorphism ϕ : π ∗ O C × D C ֒ → π ∗ ν ∗ O ^ C × D C = e π ∗ O ^ C × D C , (3.8)where π and e π are the maps in (3.1).Let ξ : O C −→ O C ⊗ k k [Γ] (3.9)be the composition of homomorphisms O C −→ π ∗ O C × D C ϕ −→ e π ∗ O ^ C × D C = O C ⊗ k k [Γ](see (3.8) and (3.7)). Note that the image ξ ( O C ) in (3.9) is a subbundle of O C ⊗ k k [Γ],because the section ξ (1 C ) ⊂ H ( C, e π ∗ O ^ C × D C ) = k [Γ]is nowhere vanishing, where 1 C is the constant function 1 on C . There is a trivial sub-bundle E of the trivial bundle O C ⊗ k k [Γ] O ⊕ ( d − C = E ⊂ O C ⊗ k k [Γ]such that E ⊕ ξ ( O C ) = O C ⊗ k k [Γ] . (3.10)To see this, take any point x ∈ C , and choose a subspace E x ⊂ ( O C ⊗ k k [Γ]) x = k [Γ]such that k [Γ] = E x ⊕ ξ ( O C ) x ; then take E := O C ⊗ k E x ⊂ O C ⊗ k k [Γ] . This subbundle E clearly satisfies the condition in (3.10).From the decomposition in (3.10) we conclude that ( O C ⊗ k k [Γ]) /ξ ( O C ) = E . Usingthe isomorphism in (3.7), the homomorphism ϕ in (3.8) gives a homomorphism ϕ ′ : ( π ∗ O C × D C ) / O C −→ ( O C ⊗ k k [Γ]) / ( ϕ ( O C )) = ( O C ⊗ k k [Γ]) / ( ξ ( O C )) = E . (3.11) On the other hand, the isomorphism in (3.2) produces an isomorphism( π ∗ O C × D C ) / O C ∼ = f ∗ (( f ∗ O C ) / O D ) . Combining this isomorphism with the homomorphism ϕ ′ in (3.11) we get a homomorphism f ∗ (( f ∗ O C ) / O D ) −→ E = O ⊕ ( d − C . This homomorphism is clearly an isomorphism over the nonempty open subset of C where f is ´etale. (cid:3) Note that f ∗ (( f ∗ O C ) / O D ) = ( f ∗ f ∗ O C ) / O C ; but we use f ∗ (( f ∗ O C ) / O D ) due to therelevance of ( f ∗ O C ) / O D .Let f : C −→ D be a genuinely ramified Galois morphism, of degree d , between irreducible smooth pro-jective curves; see Definition 3.2. As before, the Galois group Gal( f ) will be denoted byΓ, so we have d . Assume that d > C × D C corresponding to σ ∈ Γ will bedenoted by C σ .The following lemma formulated in the above set-up will be used in proving a variationof Proposition 3.3. Lemma 3.4.
There is an ordering of the elements of
ΓΓ = { γ , γ , · · · , γ d } and a self-map η : { , , · · · , d } −→ { , , · · · , d } such that (1) γ = e (the identity element of Γ ), (2) η (1) = 1 , (3) η ( j ) < j for all j ∈ { , · · · , d } , and (4) C γ j T C γ η ( j ) = ∅ (see (3.4) for notation).Proof. Set Γ := γ to be the identity element e ∈ Γ; also, set η (1) = 1. Set N = 1.Let Γ ⊂ Γbe the subset consisting of all γ = e such that the action of γ on C has a fixed point.Therefore, Γ consists of all γ = e such that the irreducible component C γ ⊂ C × D C intersects the component C e = C γ . We note that Γ is nonempty, because otherwise C γ would be a connected component of C × D C , while from Lemma 3.1(3) we know that C × D C is connected; recall that Γ = { e } and f is genuinely ramified. AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 11
If = N − N − N , set γ j ∈ Γ, 2 ≤ j ≤ N , to be distinct elements of Γ in an arbitrary order. Set η ( j ) = 1for all 2 ≤ j ≤ N .If Γ S Γ = Γ, let Γ ⊂ Γ \ (Γ ∪ Γ )be the subset consisting of all γ ∈ Γ \ (Γ S Γ ) such that the irreducible component C γ ⊂ C × D C intersects the component C σ for some σ ∈ Γ . Note that such a component C γ does notintersect C γ , because in that case we would have γ ∈ Γ .If = N − N , set γ j ∈ Γ, N + 1 ≤ j ≤ N , to be distinct elements of Γ in anarbitrary order. For every N + 1 ≤ j ≤ N , set η ( j ) ∈ { , · · · , N } such that the component C γ j ⊂ C × D C intersects the component C γ η ( j ) ; the above defi-nition of Γ ensures that such a η ( j ) exists. If there are more than one m ∈ { , · · · , N } such that C γ j ⊂ C × D C intersects the component C γ m , then choose η ( j ) arbitrarily fromthem.Now inductively define Γ n ⊂ Γ \ ( n − [ i =0 Γ i ) , if Γ n = ∅ , to be the subset consisting of all γ ∈ Γ \ ( S n − i =0 Γ i ) such that the irreduciblecomponent C γ ⊂ C × D C intersects the component C σ for some σ ∈ Γ n − . Note thatsuch a component C γ does not intersect S n − i =0 Γ i , because in that case γ ∈ S n − i =0 Γ i .If n = N n − P n − i =0 i = N n − N n − , set γ j ∈ Γ, N n − + 1 ≤ j ≤ N n , to bedistinct elements of Γ n in an arbitrary order. For N n − + 1 ≤ j ≤ N n , set η ( j ) ∈ [ N n − + 1 , N n − ] = { n − X i =0 i , · · · , n − X i =0 i } such that the component C γ j ⊂ C × D C intersects the component C γ η ( j ) . If C γ j intersectsmore than one such component, choose η ( j ) to be any one from them, as before.Since Γ is a finite group, we have Γ n = ∅ for all n sufficiently large. Set S = ∞ X i =0 i = Max i ≥ { N i } . Note that S [ i =1 C γ i ⊂ C × D C is the connected component of C × D C containing C γ . Hence from Lemma 3.1(3) weknow that S [ i =1 C γ i = C × D C. In other words, we have S = d := (cid:3) Proposition 3.5.
Let f : C −→ D be a genuinely ramified Galois morphism, of degree d , between irreducible smooth projective curves. Then f ∗ (( f ∗ O C ) / O D ) ⊂ d − M i =1 L i , where each L i is a line bundle on D of negative degree.Proof. As in Lemma 3.4, the Galois group Gal( f ) is denoted by Γ. The ordering in Lemma3.4 of the elements of Γ produces an isomorphism of k [Γ] with k ⊕ d . Consequently, from(3.7) we have e π ∗ O ^ C × D C = O C ⊗ k k [Γ] = O ⊕ dC . (3.12)Let Φ : e π ∗ O ^ C × D C = O ⊕ dC −→ O ⊕ dC = e π ∗ O ^ C × D C be the homomorphism defined by( f , f , · · · , f d ) ( f − f η (1) , f − f η (2) , · · · , f d − f η ( d ) ) , (3.13)where η is the map in Lemma 3.4; more precisely, the i -th component of Φ( f , f , · · · , f d )is f i − f η ( i ) . It is straightforward to check that F := Φ( O ⊕ dC ) ⊂ O ⊕ dC = e π ∗ O ^ C × D C is a trivial subbundle of rank d −
1; the first component of Φ( f , f , · · · , f d ) vanishesidentically, because η (1) = 1. More precisely, we have F = O ⊕ ( d − C ⊂ O ⊕ dC = e π ∗ O ^ C × D C , (3.14)where O ⊕ ( d − C is the subbundle of O ⊕ dC spanned by all ( f , f , · · · , f d ) such that f = 0.From (3.14) it follows immediately that e π ∗ O ^ C × D C = O ⊕ dC = F ⊕ ξ ( O C ) , (3.15)where ξ ( O C ) is the subbundle of O ⊕ dC = O C ⊗ k k [Γ] in (3.9) (see (3.12)).In (3.1) we have e π = π ◦ ν , and hence, as in (3.8), there is a natural homomorphism ϕ : π ∗ O C × D C ֒ → e π ∗ O ^ C × D C (3.16)which is an isomorphism over the open subset of C where f is ´etale. Therefore, from (3.2)and (3.15) we get an injective homomorphism of coherent sheavesΨ : f ∗ (( f ∗ O C ) / O D ) −→ F = O ⊕ ( d − C ; (3.17)it is similar to (3.11), except that now the direct summand F is chosen carefully (it was E in(3.11)). Note that since rk( f ∗ (( f ∗ O C ) / O D )) = d − O ⊕ ( d − C ), the homomorphism AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 13
Ψ in (3.17) is generically an isomorphism, because it is an injective homomorphism ofcoherent sheaves. More precisely, Ψ is an isomorphism over the open subset of C wherethe map f is ´etale.Consider the map η in Lemma 3.4. For every 1 ≤ i ≤ d −
1, choose a point z i ∈ C γ i +1 \ C γ η ( i +1) ; (3.18)this is possible because the fourth property in Lemma 3.4 says that the intersection C γ i +1 T C γ η ( i +1) is nonempty. Recall from (3.5) that C is identified with C γ i +1 . The point z i ∈ C γ i +1 in (3.18) will be considered as a point of C using this identification. Let L i := O C ( − z i )be the line bundle corresponding to the point z i ∈ C .For every 1 ≤ i ≤ d −
1, let P i : O ⊕ ( d − C −→ O C (3.19)be the natural projection to the i -th factor.Consider the composition of homomorphisms P i ◦ Ψ, where P i and Ψ are constructedin (3.19) and (3.17) respectively. It can be shown that P i ◦ Ψ vanishes when restricted tothe point z i in (3.18). To see this, for any 1 ≤ j ≤ d , let b P j : O ⊕ dC −→ O C be the natural projection to the j -th factor. Recall the homomorphism Φ constructed in(3.13). If ( f , f , · · · , f d ) in (3.13) actually lies in the image of π ∗ O C × D C by the inclusionmap ϕ in (3.16), then from (3.18) we have( b P i +1 ◦ Φ)( f , f , · · · , f d )( z i , γ i +1 ) = f i +1 ( z i , γ ) − f η ( i +1) ( z i , γ η ( i +1) ) = 0 , (3.20)where ( z i , γ i +1 ) ∈ C × Γ = ^ C × D C (see (3.6)) and the same for ( z i , γ η ( i +1) ); note thatfrom (3.18) it follows that the point in C corresponding to z i ∈ C γ η ( i +1) (see (3.18)) by theidentification C ∼ −→ C γ η ( i +1) in (3.5) coincides with the point corresponding to z i ∈ C γ i +1 (the element γ − i +1 γ η ( i +1) ∈ Γ fixes this point of C ). To clarify, there is a slight abuse ofnotation in (3.8) in the following sense: sections of e π ∗ O ^ C × D C over an open subset U ⊂ C are identified with function on e π − ( U ). So ( f , f , · · · , f d ) in (3.20) is considered as afunction on e π − ( U ); the above condition that ( f , f , · · · , f d ) in (3.20) lies in the imageof π ∗ O C × D C by the inclusion ϕ map in (3.16) means that ( f , f , · · · , f d ) coincides with b f ◦ ν for some function b f on π − ( U ), where ν is the map in (3.1). Now from (3.20) itfollows that P i ◦ Ψ vanishes when restricted to the point z i ∈ C .Since P i ◦ Ψ vanishes when restricted to the point z i , we have P i ◦ Ψ( f ∗ (( f ∗ O C ) / O D )) ⊂ L i = O C ( − z i ) ⊂ O C . (3.21)From (3.17) and (3.21) it follows immediately that f ∗ (( f ∗ O C ) / O D ) ֒ → d − M i =1 L i . Since deg( L i ) = −
1, the proof of the proposition is complete. (cid:3) Pullback of stable bundles and genuinely ramified maps
Lemma 4.1.
Let f : C −→ D be a genuinely ramified morphism between irreduciblesmooth projective curves. Let V be a semistable vector bundle on D . Then µ max ( V ⊗ (( f ∗ O C ) / O D )) < µ ( V ) . Proof.
First assume that the map f is Galois. Take the line bundles L i , 1 ≤ i ≤ d − d = deg( f ). Then from Proposition 3.5 we have µ max ( V ⊗ (( f ∗ O C ) / O D )) ≤ µ max ( V ⊗ ( d − M i =1 L i )) ≤ max { µ ( V ⊗ L i ) } d − i =1 , because V ⊗ L i is semistable. On the other hand, µ ( V ⊗ L i ) < µ ( V ) , because deg( L i ) <
0. Combining these, we have µ max ( V ⊗ (( f ∗ O C ) / O D )) < µ ( V ) , giving the statement of the proposition.If the map f is not Galois, consider the smallest Galois extension F : b C −→ D (4.1)such that there is a morphism b f : b C −→ C for which f ◦ b f = F . (4.2)Note that b C is irreducible and smooth, and F is separable. From (4.2) it follows that f ∗ O C ⊂ F ∗ O b C . (4.3)First assume that the map F in (4.1) is genuinely ramified. From (4.3) it follows that( f ∗ O C ) / O D ⊂ ( F ∗ O b C ) / O D . (4.4)Since F is Galois, from Proposition 3.5 we know that ( F ∗ O b C ) / O D is contained in a directsum of line bundles of negative degree. Hence the subsheaf ( f ∗ O C ) / O D in (4.4) is alsocontained in a direct sum of line bundles of negative degree. This implies that µ max ( V ⊗ (( f ∗ O C ) / O D )) < µ ( V ) , giving the statement of the proposition.Therefore, we now assume that F is not genuinely ramified. Let( F ∗ O b C ) ⊂ F ∗ O b C be the maximal semistable subbundle. Let g : b D −→ D (4.5)be the ´etale cover defined by the spectrum of the bundle ( F ∗ O b C ) of O D –algebras (seeLemma 2.4); that the map g in (4.5) is ´etale follows from Lemma 2.3 and (2.7), because g ∗ O b D = ( F ∗ O b C ) and ( F ∗ O b C ) is semistable. We note that the Galois group Gal( F ) for F acts naturally on F ∗ O b C , and this action of Gal( F ) preserves the subbundle ( F ∗ O b C ) ; AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 15 indeed, this follows from the uniqueness of the maximal semistable subbundle ( F ∗ O b C ) .Therefore, Gal( F ) acts on b D , and the map g in (4.5) is Gal( F )–equivariant for the trivialaction of Gal( F ) on D . Consequently, the covering g in (4.5) is Galois.Consider the following commutative diagram b C b f (cid:30) (cid:30) h ❋❋❋❋❋❋❋❋❋ b g ( ( C × D b D π / / π (cid:15) (cid:15) b D g (cid:15) (cid:15) C f / / D (4.6)The existence of the map h in (4.6) is evident. The map f being genuinely ramified, itfollows from Lemma 2.6 that the homomorphism between ´etale fundamental groups f ∗ : π et1 ( C ) −→ π et1 ( D )induced by f is surjective. This implies that the fiber product C × D b D is connected. Thediagram in (4.6) should not be confused with the one in (3.1) — in (4.6), C × D b D issmooth as g is ´etale.We will prove that the map b g in (4.6) is genuinely ramified and Galois. For this, firstrecall the earlier observation that Gal( F ) acts on b D . The map b g is evidently equivariantfor the actions of Gal( F ) on b C and b D . This immediately implies that the map b g is Galois.From Corollary 2.7 it follows that b g is genuinely ramified.We will next prove that b C × D b C is a disjoint union of curves isomorphic to b C × b D b C .For this, first note that b C × D b C maps to b D × D b D , and the curve b D × D b D is a disjointunion of copies of b D as g is ´etale Galois. The component of b C × D b C lying over any ofthese copies of b D is isomorphic to b C × b D b C , and therefore b C × D b C is a disjoint union ofcurves isomorphic to b C × b D b C .From (4.3) we have F ∗ f ∗ O C ⊂ F ∗ F ∗ O b C . (4.7)Since b C × D b C is a disjoint union of curves isomorphic to b C × b D b C , from (3.3) it followsthat F ∗ F ∗ O b C is a direct sum of copies of b g ∗ b g ∗ O b C . It was shown above that b g is genuinelyramified and Galois. So from Proposition 3.5 we know that b g ∗ (( b g ∗ O b C ) / O b D ) is containedin a direct sum of line bundles of negative degree.Since b g is genuinely ramified, we know from Lemma 3.1, Remark 2.1 and (2.7) that O b C = H ( b C, b g ∗ b g ∗ O D ) ⊗ O b C is the maximal semistable subbundle of b g ∗ b g ∗ O D . Since F ∗ F ∗ O b C is a direct sum of copiesof b g ∗ b g ∗ O b C , this implies that H ( b C, F ∗ F ∗ O b C ) ⊗ O b C ⊂ F ∗ F ∗ O b C (4.8) is the maximal semistable subbundle. On the other hand, we have F ∗ f ∗ O C = b f ∗ f ∗ f ∗ O C .So from Lemma 3.1, Remark 2.1 and (2.7) we know that O b C = H ( b C, F ∗ f ∗ O C ) ⊗ O b C ⊂ F ∗ f ∗ O C (4.9)is the maximal semistable subbundle.Consider the inclusion homomorphism in (4.7). From (4.8) and (4.9) we conclude thatusing this homomorphism, the quotient F ∗ (( f ∗ O C ) / O D ) is contained in F ∗ F ∗ O b C / ( H ( b C, F ∗ F ∗ O b C ) ⊗ O b C ) . (4.10)Since F ∗ F ∗ O b C is a direct sum of copies of b g ∗ b g ∗ O b C , the vector bundle in (4.10) is isomorphicto a direct sum of copies of b g ∗ (( b g ∗ O b C ) / O b D ).It was shown above that b g ∗ (( b g ∗ O b C ) / O b D ) is contained in a direct sum of line bundlesof negative degree. Therefore, the vector bundle in (4.10) is contained in a direct sum ofline bundles of negative degree. Consequently, the subsheaf F ∗ (( f ∗ O C ) / O D ) ⊂ F ∗ F ∗ O b C / ( H ( b C, F ∗ F ∗ O b C ) ⊗ O b C )is also contained in a direct sum of line bundles of negative degree.Since F ∗ ( f ∗ O C ) / O D ) is contained in a direct sum of line bundles of negative degree,we conclude that µ max (( F ∗ V ) ⊗ ( F ∗ (( f ∗ O C ) / O D ))) < µ ( F ∗ V ) ;note that F ∗ V is semistable by Remark 2.1 as F is separable. From this it follows that µ max ( V ⊗ (( f ∗ O C ) / O D )) = µ max ( F ∗ ( V ⊗ (( f ∗ O C ) / O D ))) / deg( F ) < µ ( F ∗ V ) / deg( F ) = µ ( V ) , because F ∗ V is semistable. This completes the proof. (cid:3) Remark 4.2.
When the characteristic of the base field k is zero, the tensor product oftwo semistable bundles remains semistable [RR, p. 285, Theorem 3.18]. We note thatLemma 4.1 is a straight-forward consequence of it, provided the characteristic of k is zero. Lemma 4.3.
Let f : C −→ D be a genuinely ramified morphism between irreduciblesmooth projective curves. Let V and W be two semistable vector bundles on D with µ ( V ) = µ ( W ) . Then H ( D, Hom(
V, W )) = H ( C, Hom( f ∗ V, f ∗ W )) . Proof.
Using the projection formula, and the fact that f is a finite map, we have H ( C, Hom( f ∗ V, f ∗ W )) ∼ = H ( D, f ∗ Hom( f ∗ V, f ∗ W )) ∼ = H ( D, f ∗ f ∗ Hom(
V, W )) ∼ = H ( D, Hom(
V, W ) ⊗ f ∗ O C ) ∼ = H ( D, Hom(
V, W ⊗ f ∗ O C )) . (4.11)Let 0 = B ⊂ B ⊂ · · · ⊂ B m − ⊂ B m = W ⊗ (( f ∗ O C ) / O D ) AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 17 be the Harder–Narasimhan filtration of W ⊗ (( f ∗ O C ) / O D ) [HL, p. 16, Theorem 1.3.4].Since W is semistable, and f is genuinely ramified, from Lemma 4.1 we know that µ ( B i /B i − ) ≤ µ ( B ) = µ max ( W ⊗ (( f ∗ O C ) / O D )) < µ ( W )for all 1 ≤ i ≤ m . In view this and the given condition that µ ( V ) = µ ( W ), from (2.1)we conclude that H ( D, Hom(
V, B i /B i − )) = 0for all 1 ≤ i ≤ m ; note that both V and B i /B i − are semistable. This implies that H ( D, Hom(
V, W ⊗ (( f ∗ O C ) / O D ))) = 0 . Consequently, we have H ( D, Hom(
V, W ⊗ f ∗ O C )) = H ( D, Hom(
V, W ))by examining the exact sequence0 −→ Hom(
V, W ) −→ Hom(
V, W ⊗ f ∗ O C ) −→ Hom(
V, W ⊗ (( f ∗ O C ) / O D )) −→ . From this and (4.11) it follows that H ( C, Hom( f ∗ V, f ∗ W )) = H ( D, Hom(
V, W )) . This completes the proof. (cid:3)
Theorem 4.4.
Let f : C −→ D be a genuinely ramified morphism between irreduciblesmooth projective curves. Let V be a stable vector bundle on D . Then the pulled backvector bundle f ∗ V is also stable.Proof. Consider the Galois extension F : b C −→ D and the diagram in (4.6). Since V isstable, from Lemma 4.3 it follows that f ∗ V is simple. As V is semistable, it follows that g ∗ V is also semistable, where g is the map in (4.6). Let E ⊂ g ∗ V be the unique maximal polystable subbundle with µ ( E ) = µ ( g ∗ V ) [HL, p. 23, Lemma1.5.5]; this subbundle E is called the socle of g ∗ V . Since g ∗ V is preserved by the actionof the Galois group Gal( g ) on g ∗ V , there is a unique subbundle E ′ ⊂ V such that E = g ∗ E ′ ⊂ g ∗ V . As V is stable, we conclude that E ′ = V , and hence g ∗ V is polystable. So we have adirect sum decomposition g ∗ V = m M j =1 V j , (4.12)where each V j is stable with µ ( V j ) = µ ( g ∗ V ).Take any 1 ≤ j ≤ m , where m is the integer in (4.12). Since V j is stable, and b g in(4.6) is Galois (this was shown in the proof of Lemma 4.1), repeating the above argumentinvolving the socle we conclude that b g ∗ V j is also polystable. On the other hand, as b g isgenuinely ramified (see the proof of Lemma 4.1), from Lemma 4.3 it follows that H ( b C, End( b g ∗ V j )) = H ( b D, End( V j )) . (4.13) But H ( b D, End( V j )) = k , because V j is stable. Hence from (4.13) we know that H ( b C, End( b g ∗ V j )) = k . This implies that b g ∗ V j is stable, because it is polystable.Since b g ∗ V j is stable, and π ◦ h = b g (see (4.6)), we conclude that π ∗ V j is also stablewith µ ( π ∗ V j ) = µ ( π ∗ g ∗ V )for all 1 ≤ j ≤ m . This implies that π ∗ f ∗ V = π ∗ g ∗ V = m M j =1 π ∗ V j (4.14)is polystable.The map π is genuinely ramified because b g is genuinely ramified (see the proof ofLemma 4.1) and b g = h ◦ π . Indeed, if π factors through an ´etale covering of b D , thenthe genuinely ramified map b g factors through that ´etale covering of b D , and hence fromProposition 2.6 it follows that π is genuinely ramified.Since π is genuinely ramified, and each V j in (4.14) is stable, from Lemma 4.3 it followsthat H ( C × D b D, Hom( π ∗ V i , π ∗ V j )) = H ( b D, Hom( V i , V j )) (4.15)for all 1 ≤ i, j ≤ m . We know that V i and π ∗ V i are stable. So from (4.15) we concludethat V i is isomorphic to V j if and only if π ∗ V i is isomorphic to π ∗ V j . From (4.15) it alsofollows that H ( C × D b D, End( π ∗ g ∗ V )) = H ( b D, End( g ∗ V )) ; (4.16)we note that this also follows from Lemma 4.3.The vector bundle f ∗ V on C is semistable, because V is semistable and f is separable.Let 0 = S ⊂ f ∗ V (4.17)be a stable subbundle with µ ( S ) = µ ( f ∗ V ) . (4.18)Since S is stable with µ ( S ) = µ ( f ∗ V ), and the map π is Galois, using the earlierargument involving the socle we conclude that e S := π ∗ S ⊂ π ∗ f ∗ V = π ∗ g ∗ V = m M j =1 π ∗ V j =: e V (4.19)is a polystable subbundle with µ ( e S ) = µ ( e V ).Consider the associative algebra H ( C × D b D, End( e V )), where e V is the vector bundlein (4.19). Define the right idealΘ := { γ ∈ H ( C × D b D, End( e V )) | γ ( e V ) ⊂ e S } ⊂ H ( C × D b D, End( e V )) , (4.20)where e S is the subbundle in (4.19). The subbundle e S ⊂ e V is a direct summand, because e V is polystable, and µ ( e S ) = µ ( e V ). Consequently, e S coincides with the subbundle generatedby the images of endomorphisms lying in the right ideal Θ. Since e V is semistable, theimage of any endomorphism of it is a subbundle. AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 19
Consider e V in (4.19). The identification H ( C × D b D, End( e V )) = H ( b D, End( g ∗ V ))in (4.16) preserves the associative algebra structures of H ( C × D b D, End( e V )) and H ( b D, End( g ∗ V )) , because it sends any γ ∈ H ( b D, End( g ∗ V )) to π ∗ γ . Let e Θ ⊂ H ( b D, End( g ∗ V )) (4.21)be the right ideal that corresponds to Θ in (4.20) by the identification in (4.16). Let S ⊂ g ∗ V (4.22)be the subbundle generated by the images of endomorphisms lying in the right ideal e Θin (4.21). Since g ∗ V is semistable, the image of any endomorphism of it is a subbundle.From the above construction of S it follows that e S = π ∗ S , where e S is the subbundle in (4.20).The isomorphism in (4.16) is equivariant for the actions of the Galois group Gal( π ) =Gal( g ) on H ( C × D b D, End( e V )) = H ( C × D b D, End( π ∗ f ∗ V ))and H ( b D, End( g ∗ V )), because the isomorphism sends any γ ∈ H ( b D, End( g ∗ V )) to π ∗ γ . Since e S = π ∗ S in (4.19) is preserved under the action of Gal( π ) on π ∗ f ∗ V , itfollows that the action of Gal( π ) on H ( C × D b D, End( π ∗ f ∗ V )) preserves the right idealΘ in (4.20). These together imply that the action of Gal( g ) on H ( b D, End( g ∗ V )) preservesthe right ideal e Θ in (4.21). Consequently, the subbundle
S ⊂ g ∗ V in (4.22) is preserved under the action of Gal( g ) on g ∗ V .Since S is preserved under the action of Gal( g ) on g ∗ V , there is a unique subbundle S ⊂ V such that S = g ∗ S ⊂ g ∗ V . Given that V is stable, and µ ( S ) = µ ( V ) (this followsfrom (4.18)), we now conclude that S = V . Hence the subbundle S in (4.17) coincideswith f ∗ V . Therefore, we conclude that f ∗ V is stable. (cid:3) Characterizations of genuinely ramified maps
Let D be an irreducible smooth projective curve, and let φ : X −→ D be a nontrivial ´etale covering with X irreducible. Let L be a line bundle on X of degreeone. Proposition 5.1. (1)
The direct image φ ∗ L is a stable vector bundle on D . (2) The pulled back bundle φ ∗ φ ∗ L is not stable.Proof. Let δ be the degree of the map φ ; note that δ >
1, because φ is nontrivial.We have deg( φ ∗ L ) = deg( L ) = 1 [Ha, p. 306, Ch. IV, Ex. 2.6(a) and 2.6(d)]. Thisimplies that deg( φ ∗ φ ∗ L ) = δ · deg( L ) = δ . We have a natural homomorphism H : φ ∗ φ ∗ L −→ L . (5.1)This H has the following property: For any coherent subsheaf W ⊂ φ ∗ L , the restrictionof H to φ ∗ W ⊂ φ ∗ φ ∗ L H W := H | φ ∗ W : φ ∗ W −→ L (5.2)is a nonzero homomorphism. Note that for any point y ∈ D , the fiber ( φ ∗ φ ∗ L ) y is H ( φ − ( y ) , L | φ − ( y ) ), and hence a nonzero element of ( φ ∗ φ ∗ L ) y must be nonzero at somepoint of φ − ( y ).We will first show that φ ∗ L is semistable. To prove this by contradiction, let V ⊂ φ ∗ L be a semistable subbundle with µ ( V ) > µ ( φ ∗ L ) = 1 δ . (5.3)Consider the nonzero homomorphism H V := H | φ ∗ V : φ ∗ V −→ L in (5.2). We have µ ( φ ∗ V ) = δ · µ ( V ) > φ ∗ V is semistable because V is so. Consequently, H V contradicts (2.1). As φ ∗ L does not contain any subbundle V satisfying (5.3), we conclude that φ ∗ L is semistable.Since rk( φ ∗ L ) is coprime to deg( φ ∗ L ), the semistable vector bundle φ ∗ L is also stable.This proves statement (1).The vector bundle φ ∗ φ ∗ L is not stable, because the homomorphism H in (5.1) is nonzeroand µ ( φ ∗ φ ∗ L ) = µ ( L ). (cid:3) Proposition 5.2.
Let f : C −→ D be a nonconstant separable morphism betweenirreducible smooth projective curves such that f is not genuinely ramified. Then there isa stable vector bundle E on D such that f ∗ E is not stable.Proof. Since f is not genuinely ramified, from Proposition 2.6 we know that there is anontrivial ´etale covering φ : X −→ D and a map β : C −→ X such that φ ◦ β = f . As in Proposition 5.1, take a line bundle L on X of degree one. The vector bundle φ ∗ L is stable by Proposition 5.1(1).The vector bundle φ ∗ φ ∗ L is not stable by Proposition 5.1(2). Therefore, f ∗ ( φ ∗ L ) = β ∗ φ ∗ ( φ ∗ L ) = β ∗ ( φ ∗ φ ∗ L )is not stable. (cid:3) AMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 21
Theorem 4.4 and Proposition 5.2 together give the following:
Theorem 5.3.
Let f : C −→ D be a nonconstant separable morphism between irre-ducible smooth projective curves. The map f is genuinely ramified if and only if f ∗ E isstable for every stable vector bundle E on D . Acknowledgements
We thank both the referees for going through the paper very carefully and makingnumerous suggestions. The first-named author is partially supported by a J. C. Bose Fel-lowship. Both authors were supported by the Department of Atomic Energy, Governmentof India, under project no.12-R&D-TFR-5.01-0500.
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