Ramified Covering Maps and Stability of Pulled Back Bundles

Home , Inclusion map

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(2) The map f does not factor through some nontrivial étale cover of D (in particular, f is not nontrivial étale). (3) The fiber product C ×D C is connected. 0 ∗ (4) dim H (C, f f∗OC ) = 1. (5) The maximal semistable subbundle of the direct image f∗OC is OD. 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) are equivalent; 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 irreducible smooth projective curves defined over k. The map f is genuinely ramified if and only 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 Proposition 3.5): 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 d−1 ∗ f ((f∗OC )/OD) ⊂ M Li , i=1 where each Li is a line bundle on D of negative degree.

When k = C, a vector bundle F on a smooth complex projective curve is stable if and only if F admits an irreducible ﬂat projective unitary connection [NS]. From this characterization of stable vector bundles it follows immediately that given a nonconstant map f : C −→ D between irreducible smooth complex projective curves, f ∗E is stable for every stable vector bundle E on D if the homomorphism of topological fundamental groups induced by f

f∗ : π1(C, x0) −→ π1(D, f(x0)) is surjective. Theorem 4.4 was stated in [PS] (it is [PS, p. 524, Lemma 3.5(b)]) without a complete proof. 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 covering map (the ﬁrst sentence).

2. Genuinely ramified morphism

The base ﬁeld k is assumed to be algebraically closed; there is no restriction on the characteristic of k. Let V be a vector bundle on an irreducible smooth projective curve X deﬁned over k. If

0 = V0 ⊂ V1 ⊂ ··· ⊂ Vn−1 ⊂ Vn = V RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 3 is the Harder-Narasimhan ﬁltration of V [HL, p. 16, Theorem 1.3.4], then deﬁne

deg(V1) µmax(V ) := µ(V1) := and µmin(V ) := µ(V/Vn−1) . rk(V1)

Furthermore, the above subbundle V1 ⊂ V is called the maximal semistable subbundle of V . If V and W are vector bundles on X, and β ∈ H0(X, Hom(V, W )) \{0}, then it can be shown that

µmax(W ) ≥ µmin(V ) . (2.1) Indeed, we have

µmin(V ) ≤ µ(β(V )) ≤ µmax(W ) . Remark 2.1. Let f : X −→ Y be a nonconstant separable morphism between irreducible smooth projective curves, and let F be a semistable vector bundle on Y . Then it is 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 see that (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 immediately implies that f ∗F is semistable.

Lemma 2.2. Let f : X −→ Y be a nonconstant separable morphism of irreducible smooth projective curves. Then for any semistable vector bundle E on X,

µmax(f∗E) ≤ µ(E)/deg(f) .

More generally, for any vector bundle E on X,

µmax(f∗E) ≤ µmax(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

0 0 ∗ 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- 0 ∗ ∗ bundle V of f∗E we see that H (X, Hom(f V, E)) =6 0. The pullback f V is semistable because 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 we conclude that

µmax(f∗E) ≤ µ(E)/deg(f). (2.4) To prove the general (second) statement, for any vector bundle E on X, let

0 = E0 ⊂ E1 ⊂ ··· ⊂ En−1 ⊂ En = E 4 I.BISWASANDA.J.PARAMESWARAN be the Harder–Narasimhan ﬁltration of E [HL, p. 16, Theorem 1.3.4]. Consider the ﬁltration of subbundles

0 = f∗E0 ⊂ f∗E1 ⊂ ··· ⊂ f∗En−1 ⊂ f∗En = f∗E. (2.5) From (2.4) we know that

µmax((f∗Ei)/(f∗Ei−1)) = µmax(f∗(Ei/Ei−1)) ≤ µ(Ei/Ei−1)/deg(f)

≤ µ(E1)/deg(f) = µmax(E)/deg(f) (2.6) for all 1 ≤ i ≤ n. Observe that using (2.1) and the ﬁltration in (2.5) it follows that

n µmax(f∗E) ≤ Max{µmax((f∗Ei)/(f∗Ei−1))}i=1, while from (2.6) we have

n Max{µmax((f∗Ei)/(f∗Ei−1))}i=1 ≤ µmax(E)/deg(f).

Therefore, µmax(f∗E) ≤ µmax(E)/deg(f), and this completes the proof.

The following lemma characterizes the ´etale maps among the separable morphisms.

Lemma 2.3. Let f : X −→ Y be a nonconstant separable morphism between irreducible smooth projective curves. Then the following three conditions are equivalent: (1) The map f is ´etale. (2) The degree of f∗OX is zero. (3) The vector bundle f∗OX is semistable.

Proof. We have µmax(f∗OX ) ≥ 0, because OY ⊂ f∗OX (see (2.2) and (2.1)). On the other hand, from Lemma 2.2 it follows that µmax(f∗OX ) ≤ 0, so

µmax(f∗OX )=0 . (2.7)

From (2.7) it follows immediately that f∗OX is semistable if and only if deg(f∗OX ) = 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 ⊗2 (det f∗OX ) = OY (−B) (see [Ha, p. 306, Ch. IV, Ex. 2.6(d)], [Se]). Therefore, f is étale (meaning B = 0) if and only if deg(f∗OX ) = 0. So statements (1) and (2) are equivalent.

Let f : X −→ Y be a nonconstant separable morphism between irreducible smooth projective curves. The algebra structure of OX produces an OY –algebra structure on the direct image f∗OX .

Lemma 2.4. Let V ⊂ f∗OX be the maximal semistable subbundle. Then V is a sheaf of OY –subalgebras of f∗OX . RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 5

Proof. The action of OY on f∗OX is the standard one. Let

m : (f∗OX ) ⊗ (f∗OX ) −→ f∗OX be the OY –algebra structure on the direct image f∗OX given by the algebra structure of the coherent sheaf OX . We need to show that m(V⊗V) ⊂ V , (2.8) where V is the maximal semistable subbundle of f∗OX .

Since V is semistable of degree zero (see (2.7)), and µmax((f∗OX )/V) < 0, using (2.1) we conclude that in order to prove (2.8) it suﬃces to show that V⊗V is semistable of degree zero. Indeed, there is no nonzero homomorphism from V⊗V to (f∗OX )/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 not contain any coherent subsheaf of positive degree. As

V⊗V ⊂ (f∗OX ) ⊗ (f∗OX ) , if V⊗V contains a subsheaf of positive degree, then (f∗OX ) ⊗ (f∗OX ) also contains a subsheaf of positive degree.

Therefore, to prove the lemma it is enough to show that (f∗OX ) ⊗ (f∗OX ) does not contain any subsheaf of positive degree. The projection formula, [Ha, p. 124, Ch. II, Ex. 5.1(d)], [Se], says that ∼ ∗ (f∗OX ) ⊗OY (f∗OX ) = f∗(f (f∗OX )) . (2.9)

Since OY ⊂ f∗OX , we have

OY = OY ⊗OY OY ⊂ (f∗OX ) ⊗OY (f∗OX ) , and hence µmax((f∗OX ) ⊗OY (f∗OX )) ≥ 0. Now from (2.9) it follows that ∗ µmax(f∗(f (f∗OX ))) ≥ 0 . (2.10) Since f is separable, the pullback, by f, of a semistable bundle on Y is semistable (see Remark 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 ∗ µmax(f (f∗OX )) = 0 . Now applying the second part of Lemma 2.2, ∗ ∗ 0 = µmax(f (f∗OX ))/deg(f) ≥ µmax(f∗(f (f∗OX ))) . This and (2.10) together imply that ∗ µmax(f∗(f (f∗OX ))) = 0 . Therefore, using (2.9) it follows that

µmax((f∗OX ) ⊗ (f∗OX )) = 0 .

Hence (f∗OX ) ⊗ (f∗OX ) does not contain any subsheaf of positive degree. It was shown earlier that the lemma follows from the statement that (f∗OX ) ⊗ (f∗OX ) does not contain any subsheaf of positive degree. 6 I.BISWASANDA.J.PARAMESWARAN

Definition 2.5. A nonconstant separable morphism f : X −→ Y between irreducible smooth projective curves is called genuinely ramified if OY is the maximal semistable subbundle of f∗OX . Proposition 2.6. Let f : X −→ Y be a nonconstant separable morphism between irreducible 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 étale cover of Y (in particular, f is not nontrivial étale). (3) The homomorphism between étale fundamental groups induced by f

et et f∗ : π1 (X) −→ π1 (Y ) is surjective.

Proof. (1) =⇒ (2): If f factors through a nontrivial ´etale covering g : Y −→ Y , then e g∗OYe is semistable of degree zero (see Lemma 2.3) and its rank coincides with the degree of g. Since

g∗OYe ⊂ f∗OX , this implies that f is not genuinely ramiﬁed.

(2) =⇒ (1): Lemma 2.4 says that the maximal semistable subbundle V ⊂ f∗OX is a subalgebra. If f is not genuinely ramified, then by taking the spectrum of V we obtain a separable, possibly ramified, covering map g : Y = Spec V −→ Y (2.11) e whose degree coincides with the rank of V. We have g∗OYe = V, and the inclusion map V ֒→ f∗OX defines a map h : X −→ Y e 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 étale. To prove this, first note that g∗OYe is semistable, because g∗OYe = V and V is semistable. Next, from (2.7) and the semistability of V it follows that µ(g∗OYe ) = µmax(g∗OYe ) = 0. Now Lemma 2.3 gives that the map g in (2.11) is étale. Since g is étale, and (2.12) holds, we conclude that the statement (2) fails. Hence the statement (2) implies the statement (1). The equivalence between the statements (2) and (3) follows from the definition of the étale fundamental group.

Let f : X −→ Y be a nonconstant separable morphism between irreducible smooth projective curves. Let g : Y := Spec V −→ Y e RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 7 be the étale covering corresponding to the maximal semistable subbundle V ⊂ f∗OX (see (2.11); it was shown that the map in (2.11) is étale). Let h : X −→ Y (2.13) e . be the morphism given by the inclusion map V ֒→ f∗OX Corollary 2.7. The map h in (2.13) is genuinely ramified.

Proof. Let β : Z −→ Y be an ´etale covering such that there is a map e γ : X −→ Z satisfying the condition β ◦ γ = h. Since (g ◦ β) ◦ γ = f, we have

g∗OYe ⊂ (g ◦ β)∗OZ ⊂ f∗OX ; (2.14) also, we have deg((g◦β)∗OYe ) = 0, because g◦β is ´etale (see Lemma 2.3). But V = g∗OYe is the maximal semistable subsheaf of f∗OX . Hence from (2.14) it follows that g∗OYe = (g ◦ β)∗OZ . This implies that deg(β) = 1. Therefore, from Proposition 2.6 we conclude that the map h in (2.13) is genuinely ramiﬁed.

3. Properties of genuinely ramified morphisms

Lemma 3.1. Let f : C −→ D be a nonconstant separable morphism between irreducible smooth projective curves. Then the following three conditions are equivalent: (1) The map f is genuinely ramified. 0 ∗ (2) dim H (C, f f∗OC )=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 projective curve, but it is not connected unless f is an isomorphism. We have the commutative diagram C^× C (3.1) D ▲ ▲▲ πe2 ▲▲ ν ▲▲▲ ▲▲ % ▲% ( π2 (/ C ×D C C πe1 π1 f " f C / D By flat base change [Ha, p. 255, Proposition 9.3], ∗ ∼ ∗ f (f∗OC ) = π1∗(π2 OC ) = π1∗OC×DC . (3.2) (1) =⇒ (2): Since f is separable, f ∗F is semistable if F is so (see Remark 2.1), and ∗ ∗ hence the maximal semistable subbundle of f f∗OC is f V, where V ⊂ f∗OC is the maximal semistable subbundle. If f is genuinely ramified, then the maximal semistable ∗ ∗ subbundle of f f∗OC is f OD = OC . On the other hand, 0 ∗ ∗ H (C, (f f∗OC )/(f V)) = 0 , 8 I.BISWASANDA.J.PARAMESWARAN

∗ ∗ because µmax((f f∗OC )/(f V)) < 0 (see (2.1)). These together imply that 0 ∗ dim H (C, f f∗OC ) = 1 ; to see this consider the long exact sequence of cohomologies associated to the short exact sequence ∗ ∗ ∗ ∗ 0 −→ f V −→ f f∗OC −→ (f f∗OC )/(f V) −→ 0. (2) ⇐⇒ (3): From (3.2) it follows that 0 ∗ 0 0 H (C, f f∗OC ) = H (C, π1∗OC×DC ) = H (C ×D C, OC×DC ) . (3.3) 0 ∗ Consequently, C ×D C is connected if and only if dim H (C, f f∗OC ) = 1.

(3) =⇒ (1): Assume that f is not genuinely ramiﬁed. We will prove that C ×D C is not connected.

Let g : D −→ D be the étale cover of D given by Spec W, where W ⊂ f∗OC is the maximal semistablee 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 to show that D ×D D is not connected. e e The projection

γ : D ×D D −→ D e e e to the first factor is evidently the base change of g : D −→ D to D, and hence the e e map γ is étale. The diagonal D ֒→ D ×D D is a connected component of D ×D D. This e e e e e implies that D ×D D is not connected, because the degree of γ is at least two. e e Definition 3.2. A nonconstant morphism f : C −→ D between irreducible smooth projective curves will be called a separable Galois morphism if f is separable, and there is 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 étale. A separable Galois morphism which is genuinely ramified will be called a genuinely ramified Galois morphism. Proposition 3.3. Let f : C −→ D be a separable Galois morphism, of degree d, be- ∗ tween irreducible smooth projective curves. Then f ((f∗OC )/OD) is a coherent subsheaf ⊕(d−1) of OC .

Proof. The Galois group Gal(f) of f will be denoted by Γ. For any point x ∈ C, let

Γx ⊂ Γ be 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 Γx is nontrivial. Note that for any (x, y) ∈ C ×D C the two isotropy subgroups Γx and Γy are conjugate, because y lies in the orbit Γ · x of x. For any σ ∈ Γ, let

Cσ ⊂ C ×D C (3.4) be the irreducible component given by the image of the map

βσ : C −→ C × C , x 7−→ (x, σ(x)) ; RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 9 clearly we have βσ(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 canonical identiﬁcation ∼ C −→ Cσ (3.5) for every σ ∈ Γ. ^ Let C ×D C be the normalization of C ×D C. The maps βσ, σ ∈ Γ, in (3.4) together produce an isomorphism ∼ ^ C × Γ −→ C ×D C ; (3.6) this map sends any (y, σ) ∈ C × Γ to (y, σ(y)) if Γy is trivial; if Γy 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

π1∗O ^ = OC ⊗k k[Γ] , (3.7) C×DC e where π1 is the projection in (3.1), and k[Γ] is the group ring. The natural inclusion

OC×DCe֒→ ν∗O ^ , where ν is the map in (3.1), induces an injective homomorphism C×DC

(ϕ : π1∗OC×DC ֒→ π1∗ν∗O ^ = π1∗O ^ , (3.8 C×DC C×DC e where π1 and π1 are the maps in (3.1). Let e

ξ : OC −→ OC ⊗k k[Γ] (3.9) be the composition of homomorphisms ϕ OC −→ π1∗OC×DC −→ π1∗O ^ = OC ⊗k k[Γ] C×DC e (see (3.8) and (3.7)). Note that the image ξ(OC) in (3.9) is a subbundle of OC ⊗k k[Γ], because the section 0 ξ(1C ) ⊂ H (C, π1∗O ^ ) = k[Γ] C×DC e is nowhere vanishing, where 1C is the constant function 1 on C. There is a trivial subbundle E of the trivial bundle OC ⊗k k[Γ] ⊕(d−1) OC = E ⊂ OC ⊗k k[Γ] such that

E ⊕ ξ(OC ) = OC ⊗k k[Γ] . (3.10) To see this, take any point x ∈ C, and choose a subspace

Ex ⊂ (OC ⊗k k[Γ])x = k[Γ] such that k[Γ] = Ex ⊕ ξ(OC )x; then take

E := OC ⊗k Ex ⊂ OC ⊗k k[Γ] . This subbundle E clearly satisﬁes the condition in (3.10).

From the decomposition in (3.10) we conclude that (OC ⊗k k[Γ])/ξ(OC ) = E. Using the isomorphism in (3.7), the homomorphism ϕ in (3.8) gives a homomorphism ′ ϕ : (π1∗OC×DC )/OC −→ (OC ⊗k k[Γ])/(ϕ(OC )) = (OC ⊗k k[Γ])/(ξ(OC)) = E. (3.11) 10 I.BISWASANDA.J.PARAMESWARAN

On the other hand, the isomorphism in (3.2) produces an isomorphism ∼ ∗ (π1∗OC×DC )/OC = f ((f∗OC )/OD) . Combining this isomorphism with the homomorphism ϕ′ in (3.11) we get a homomorphism

∗ ⊕(d−1) f ((f∗OC )/OD) −→ E = OC . This homomorphism is clearly an isomorphism over the nonempty open subset of C where f is ´etale.

∗ ∗ ∗ Note that f ((f∗OC )/OD) = (f f∗OC )/OC ; but we use f ((f∗OC )/OD) due to the relevance of (f∗OC )/OD. Let f : C −→ D be a genuinely ramiﬁed Galois morphism, of degree d, between irreducible smooth projective curves; see Deﬁnition 3.2. As before, the Galois group Gal(f) will be denoted by Γ, so we have #Γ = d . Assume that d > 1.

As in (3.4), the irreducible component of C ×D C corresponding to σ ∈ Γ will be denoted by Cσ. The following lemma formulated in the above set-up will be used in proving a variation of Proposition 3.3. Lemma 3.4. There is an ordering of the elements of Γ

Γ = {γ1, γ2, ··· , γd} and a self-map η : {1, 2, ··· , d} −→ {1, 2, ··· , d} such that

(1) γ1 = e (the identity element of Γ), (2) η(1) = 1, (3) η(j) < j for all j ∈ {2, ··· , d}, and (4) C C =6 ∅ (see (3.4) for notation). γj T γη(j)

Proof. Set Γ0 := γ1 to be the identity element e ∈ Γ; also, set η(1) = 1. Set N0 = 1. Let

Γ1 ⊂ Γ be the subset consisting of all γ =6 e such that the action of γ on C has a ﬁxed point. Therefore, Γ1 consists of all γ =6 e such that the irreducible component Cγ ⊂ C ×D C intersects the component Ce = Cγ1 . We note that Γ1 is nonempty, because otherwise

Cγ1 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 =6 {e} and f is genuinely ramiﬁed. RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 11

If #Γ1 = N1 − 1 = N1 − N0, set γj ∈ Γ, 2 ≤ j ≤ N1, to be distinct elements of Γ1 in an arbitrary order. Set η(j)=1 for all 2 ≤ j ≤ N1. If Γ1 S Γ0 =6 Γ, let Γ2 ⊂ Γ \ (Γ1 ∪ Γ0) be the subset consisting of all γ ∈ Γ \ (Γ1 S Γ0) such that the irreducible component

Cγ ⊂ C ×D C intersects the component Cσ for some σ ∈ Γ1. Note that such a component Cγ does not intersect Cγ1 , because in that case we would have γ ∈ Γ1.

If #Γ2 = N2 − N1, set γj ∈ Γ, N1 +1 ≤ j ≤ N2, to be distinct elements of Γ2 in an arbitrary order. For every N1 +1 ≤ j ≤ N2, set

η(j) ∈ {2, ··· , N1}

such that the component Cγj ⊂ C ×D C intersects the component Cγη(j) ; the above deﬁ- nition of Γ2 ensures that such a η(j) exists. If there are more than one m ∈ {2, ··· , N1} such that Cγj ⊂ C ×D C intersects the component Cγm , then choose η(j) arbitrarily from them. Now inductively deﬁne n−1 Γn ⊂ Γ \ ([ Γi) , i=0 n−1 if Γn =6 ∅, to be the subset consisting of all γ ∈ Γ \ (Si=0 Γi) such that the irreducible component Cγ ⊂ C ×D C intersects the component Cσ for some σ ∈ Γn−1. Note that n−2 n−1 such a component Cγ does not intersect Si=0 Γi, because in that case γ ∈ Si=0 Γi. n−1 If #Γn = Nn − Pi=0 #Γi = Nn − Nn−1, set γj ∈ Γ, Nn−1 +1 ≤ j ≤ Nn, to be distinct elements of Γn in an arbitrary order. For Nn−1 +1 ≤ j ≤ Nn, set

n−2 n−1 η(j) ∈ [Nn−2 +1, Nn−1] = {1+ X #Γi, ··· , X #Γi} i=0 i=0 such that the component Cγj ⊂ C ×D C intersects the component Cγη(j) . If Cγj intersects more than one such component, choose η(j) to be any one from them, as before.

Since Γ is a ﬁnite group, we have Γn = ∅ for all n suﬃciently large. Set ∞ S = X #Γi = Maxi≥0{Ni} . i=0 Note that S

[ Cγi ⊂ C ×D C i=1 12 I.BISWASANDA.J.PARAMESWARAN

is the connected component of C ×D C containing Cγ1 . Hence from Lemma 3.1(3) we know that S

[ Cγi = C ×D C. i=1 In other words, we have S = d := #Γ. This completes the proof of the lemma. Proposition 3.5. Let f : C −→ D be a genuinely ramiﬁed Galois morphism, of degree d, between irreducible smooth projective curves. Then d−1 ∗ f ((f∗OC )/OD) ⊂ M Li , i=1 where each Li 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 Lemma 3.4 of the elements of Γ produces an isomorphism of k[Γ] with k⊕d. Consequently, from (3.7) we have ⊕d π1∗O ^ = OC ⊗k k[Γ] = O . (3.12) C×DC C e Let ⊕d ⊕d Φ: π1∗O ^ = O −→ O = π1∗O ^ C×DC C C C×DC be the homomorphism deﬁnede by e

(f1, f2, ··· , fd) 7−→ (f1 − fη(1), f2 − fη(2), ··· , fd − fη(d)) , (3.13) where η is the map in Lemma 3.4; more precisely, the i-th component of Φ(f1, f2, ··· , fd) is fi − fη(i). It is straightforward to check that ⊕d ⊕d F := Φ(O ) ⊂ O = π1∗O ^ C C C×DC e is a trivial subbundle of rank d − 1; the ﬁrst component of Φ(f1, f2, ··· , fd) vanishes identically, because η(1) = 1. More precisely, we have ⊕(d−1) ⊕d F = O ⊂ O = π1∗O ^ , (3.14) C C C×DC e ⊕(d−1) ⊕d where OC is the subbundle of OC spanned by all (f1, f2, ··· , fd) such that f1 = 0. From (3.14) it follows immediately that ⊕d π1∗O ^ = O = F ⊕ ξ(OC) , (3.15) C×DC C e ⊕d where ξ(OC ) is the subbundle of OC = OC ⊗k k[Γ] in (3.9) (see (3.12)).

In (3.1) we have π1 = π1 ◦ ν, and hence, as in (3.8), there is a natural homomorphism e (ϕ : π1∗OC×DC ֒→ π1∗O ^ (3.16 C×DC e 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 sheaves ∗ ⊕(d−1) Ψ: f ((f∗OC )/OD) −→ F = OC ; (3.17) it is similar to (3.11), except that now the direct summand F is chosen carefully (it was E in ∗ ⊕(d−1) (3.11)). Note that since rk(f ((f∗OC )/OD)) = d − 1 = rk(OC ), the homomorphism RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 13

Ψ in (3.17) is generically an isomorphism, because it is an injective homomorphism of coherent sheaves. More precisely, Ψ is an isomorphism over the open subset of C where the map f is ´etale. Consider the map η in Lemma 3.4. For every 1 ≤ i ≤ d − 1, choose a point

zi ∈ Cγi+1 \ Cγη(i+1) ; (3.18) this is possible because the fourth property in Lemma 3.4 says that the intersection C C is nonempty. Recall from (3.5) that C is identiﬁed with C . The point γi+1 T γη(i+1) γi+1 zi ∈ Cγi+1 in (3.18) will be considered as a point of C using this identiﬁcation. Let

Li := OC (−zi) be the line bundle corresponding to the point zi ∈ C. For every 1 ≤ i ≤ d − 1, let ⊕(d−1) Pi : OC −→ OC (3.19) be the natural projection to the i-th factor.

Consider the composition of homomorphisms Pi ◦ Ψ, where Pi and Ψ are constructed in (3.19) and (3.17) respectively. It can be shown that Pi ◦ Ψ vanishes when restricted to the point zi in (3.18). To see this, for any 1 ≤ j ≤ d, let ⊕d Pj : OC −→ OC b be the natural projection to the j-th factor. Recall the homomorphism Φ constructed in

(3.13). If (f1, f2, ··· , fd) in (3.13) actually lies in the image of π1∗OC×DC by the inclusion map ϕ in (3.16), then from (3.18) we have

(Pi+1 ◦ Φ)(f1, f2, ··· , fd)(zi, γi+1) = fi+1(zi, γ) − fη(i+1)(zi, γη(i+1))=0 , (3.20) b ^ where (zi, γi+1) ∈ C × Γ = C ×D C (see (3.6)) and the same for (zi, γη(i+1)); note that from (3.18) it follows that the point in C corresponding to zi ∈ Cγη(i+1) (see (3.18)) by the ∼ identification C −→ Cγη(i+1) in (3.5) coincides with the point corresponding to zi ∈ Cγi+1 −1 (the element γi+1γη(i+1) ∈ Γ fixes this point of C). To clarify, there is a slight abuse of notation in (3.8) in the following sense: sections of π1∗O ^ over an open subset U ⊂ C C×DC −1 are identified with function on π (U). So (f1, f2e, ··· , fd) in (3.20) is considered as a −1 function on π (U); the above conditione that (f1, f2, ··· , fd) in (3.20) lies in the image of π1∗OC×DCeby the inclusion ϕ map in (3.16) means that (f1, f2, ··· , fd) coincides with −1 f ◦ ν for some function f on π1 (U), where ν is the map in (3.1). Now from (3.20) it b b follows that Pi ◦ Ψ vanishes when restricted to the point zi ∈ C.

Since Pi ◦ Ψ vanishes when restricted to the point zi, we have ∗ Pi ◦ Ψ(f ((f∗OC )/OD)) ⊂ Li = OC (−zi) ⊂ OC . (3.21) From (3.17) and (3.21) it follows immediately that d−1 ∗ . f ((f∗OC )/OD) ֒→ M Li i=1

Since deg(Li) = −1, the proof of the proposition is complete. 14 I.BISWASANDA.J.PARAMESWARAN

4. Pullback of stable bundles and genuinely ramified maps

Lemma 4.1. Let f : C −→ D be a genuinely ramiﬁed morphism between irreducible smooth projective curves. Let V be a semistable vector bundle on D. Then

µmax(V ⊗ ((f∗OC )/OD)) < µ(V ) .

Proof. First assume that the map f is Galois. Take the line bundles Li, 1 ≤ i ≤ d − 1, in Proposition 3.5, where d = deg(f). Then from Proposition 3.5 we have d−1 d−1 µmax(V ⊗ ((f∗OC )/OD)) ≤ µmax(V ⊗ (M Li)) ≤ max{µ(V ⊗Li)}i=1 , i=1 because V ⊗Li is semistable. On the other hand,

µ(V ⊗Li) < µ(V ) , because deg(Li) < 0. Combining these, we have

µmax(V ⊗ ((f∗OC )/OD)) < µ(V ) , giving the statement of the proposition. If the map f is not Galois, consider the smallest Galois extension F : C −→ D (4.1) b such that there is a morphism f : C −→ C for which b b f ◦ f = F. (4.2) b Note that C is irreducible and smooth, and F is separable. From (4.2) it follows that b f∗OC ⊂ F∗OCb . (4.3) First assume that the map F in (4.1) is genuinely ramiﬁed. From (4.3) it follows that

(f∗OC )/OD ⊂ (F∗OCb)/OD . (4.4)

Since F is Galois, from Proposition 3.5 we know that (F∗OCb)/OD is contained in a direct sum of line bundles of negative degree. Hence the subsheaf (f∗OC )/OD in (4.4) is also contained in a direct sum of line bundles of negative degree. This implies that

µmax(V ⊗ ((f∗OC )/OD)) < µ(V ) , giving the statement of the proposition. Therefore, we now assume that F is not genuinely ramiﬁed. Let

(F∗OCb)1 ⊂ F∗OCb be the maximal semistable subbundle. Let g : D −→ D (4.5) b be the étale cover defined by the spectrum of the bundle (F∗OCb)1 of OD–algebras (see Lemma 2.4); that the map g in (4.5) is étale follows from Lemma 2.3 and (2.7), because g∗ODb = (F∗OCb)1 and (F∗OCb)1 is semistable. We note that the Galois group Gal(F ) for F acts naturally on F∗OCb, and this action of Gal(F ) preserves the subbundle (F∗OCb)1; RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 15 indeed, this follows from the uniqueness of the maximal semistable subbundle (F∗OCb)1. Therefore, Gal(F ) acts on D, and the map g in (4.5) is Gal(F )–equivariant for the trivial action of Gal(F ) on D. Consequently,b the covering g in (4.5) is Galois. Consider the following commutative diagram

C ❋ (4.6) ❋❋ gb b ❋❋h ❋❋ ❋❋# ( π2 /( b C ×D D D f b b π1 g f C / D The existence of the map h in (4.6) is evident. The map f being genuinely ramiﬁed, it follows from Lemma 2.6 that the homomorphism between ´etale fundamental groups

et et f∗ : π1 (C) −→ π1 (D) induced by f is surjective. This implies that the fiber product C ×D D is connected. The b diagram in (4.6) should not be confused with the one in (3.1) — in (4.6), C ×D D is smooth as g is étale. b We will prove that the map g in (4.6) is genuinely ramified and Galois. For this, first recall the earlier observation thatb Gal(F ) acts on D. The map g is evidently equivariant for the actions of Gal(F ) on C and D. This immediatelyb impliesb that the map g is Galois. From Corollary 2.7 it followsb that gbis genuinely ramified. b b We will next prove that C ×D C is a disjoint union of curves isomorphic to C ×Db C. b b b b For this, first note that C ×D C maps to D ×D D, and the curve D ×D D is a disjoint b b b b b b union of copies of D as g is étale Galois. The component of C ×D C lying over any of b b b these copies of D is isomorphic to C ×Db C, and therefore C ×D C is a disjoint union of b b b b b curves isomorphic to C ×Db C. b b From (4.3) we have ∗ ∗ F f∗OC ⊂ F F∗OCb . (4.7)

Since C ×D C is a disjoint union of curves isomorphic to C ×Db C, from (3.3) it follows b∗ b ∗ b b that F F∗OCb is a direct sum of copies of g g∗OCb. It was shown above that g is genuinely ∗ ramiﬁed and Galois. So from Propositionb 3.5b we know that g ((g∗OCb)/ODb )b is contained in a direct sum of line bundles of negative degree. b b Since g is genuinely ramiﬁed, we know from Lemma 3.1, Remark 2.1 and (2.7) that b 0 ∗ OCb = H (C, g g∗OD) ⊗ OCb b b b ∗ ∗ is the maximal semistable subbundle of g g∗OD. Since F F∗OCb is a direct sum of copies ∗ of g g∗OCb, this implies that b b b b 0 ∗ ∗ H (C, F F∗OCb) ⊗ OCb ⊂ F F∗OCb (4.8) b 16 I.BISWASANDA.J.PARAMESWARAN

∗ ∗ ∗ is the maximal semistable subbundle. On the other hand, we have F f∗OC = f f f∗OC . So from Lemma 3.1, Remark 2.1 and (2.7) we know that b

0 ∗ ∗ OCb = H (C, F f∗OC ) ⊗ OCb ⊂ F f∗OC (4.9) b is the maximal semistable subbundle. Consider the inclusion homomorphism in (4.7). From (4.8) and (4.9) we conclude that ∗ using this homomorphism, the quotient F ((f∗OC )/OD) is contained in ∗ 0 ∗ F F∗OCb/(H (C, F F∗OCb) ⊗ OCb) . (4.10) b ∗ ∗ Since F F∗OCb is a direct sum of copies of g g∗OCb, the vector bundle in (4.10) is isomorphic ∗ to a direct sum of copies of g ((g∗OCb)/ObDb ).b b∗ b It was shown above that g ((g∗OCb)/ODb ) is contained in a direct sum of line bundles of negative degree. Therefore,b theb vector bundle in (4.10) is contained in a direct sum of line bundles of negative degree. Consequently, the subsheaf ∗ ∗ 0 ∗ F ((f∗OC )/OD) ⊂ F F∗OCb/(H (C, F F∗OCb) ⊗ OCb) b is also contained in a direct sum of line bundles of negative degree. ∗ Since F (f∗OC )/OD) is contained in a direct sum of line bundles of negative degree, we conclude that ∗ ∗ ∗ µmax((F V ) ⊗ (F ((f∗OC )/OD))) < µ(F V ); note that F ∗V is semistable by Remark 2.1 as F is separable. From this it follows that ∗ µmax(V ⊗ ((f∗OC )/OD)) = µmax(F (V ⊗ ((f∗OC )/OD)))/deg(F ) < µ(F ∗V )/deg(F ) = µ(V ) , because F ∗V is semistable. This completes the proof. Remark 4.2. When the characteristic of the base ﬁeld k is zero, the tensor product of two semistable bundles remains semistable [RR, p. 285, Theorem 3.18]. We note that Lemma 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 irreducible smooth projective curves. Let V and W be two semistable vector bundles on D with µ(V ) = µ(W ) . Then H0(D, Hom(V, W )) = H0(C, Hom(f ∗V, f ∗W )) .

Proof. Using the projection formula, and the fact that f is a ﬁnite map, we have 0 ∗ ∗ 0 ∗ ∗ 0 ∗ H (C, Hom(f V, f W )) =∼ H (D, f∗Hom(f V, f W )) =∼ H (D, f∗f Hom(V, W ))

0 0 ∼= H (D, Hom(V, W ) ⊗ f∗OC ) ∼= H (D, Hom(V, W ⊗ f∗OC )) . (4.11) Let

0 = B0 ⊂ B1 ⊂ ··· ⊂ Bm−1 ⊂ Bm = W ⊗ ((f∗OC )/OD) RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 17 be the Harder–Narasimhan ﬁltration of W ⊗ ((f∗OC )/OD) [HL, p. 16, Theorem 1.3.4]. Since W is semistable, and f is genuinely ramiﬁed, from Lemma 4.1 we know that

µ(Bi/Bi−1) ≤ µ(B1) = µmax(W ⊗ ((f∗OC )/OD)) < 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 0 H (D, Hom(V, Bi/Bi−1)) = 0 for all 1 ≤ i ≤ m; note that both V and Bi/Bi−1 are semistable. This implies that 0 H (D, Hom(V, W ⊗ ((f∗OC )/OD))) = 0 . Consequently, we have 0 0 H (D, Hom(V, W ⊗ f∗OC )) = H (D, Hom(V, W )) by examining the exact sequence

0 −→ Hom(V, W ) −→ Hom(V, W ⊗ f∗OC ) −→ Hom(V, W ⊗ ((f∗OC )/OD)) −→ 0 . From this and (4.11) it follows that H0(C, Hom(f ∗V, f ∗W )) = H0(D, Hom(V, W )) . This completes the proof. Theorem 4.4. Let f : C −→ D be a genuinely ramiﬁed morphism between irreducible smooth projective curves. Let V be a stable vector bundle on D. Then the pulled back vector bundle f ∗V is also stable.

Proof. Consider the Galois extension F : C −→ D and the diagram in (4.6). Since V is stable, from Lemma 4.3 it follows that f ∗Vb 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, Lemma 1.5.5]; this subbundle E is called the socle of g∗V . Since g∗V is preserved by the action of 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 a direct sum decomposition m ∗ g V = M Vj , (4.12) j=1 ∗ where each Vj is stable with µ(Vj) = µ(g V ).

Take any 1 ≤ j ≤ m, where m is the integer in (4.12). Since Vj is stable, and g in (4.6) is Galois (this was shown in the proof of Lemma 4.1), repeating the above argumentb ∗ involving the socle we conclude that g Vj is also polystable. On the other hand, as g is genuinely ramiﬁed (see the proof of Lemmab 4.1), from Lemma 4.3 it follows that b 0 ∗ 0 H (C, End(g Vj)) = H (D, End(Vj)) . (4.13) b b b 18 I.BISWASANDA.J.PARAMESWARAN

0 But H (D, End(Vj)) = k, because Vj is stable. Hence from (4.13) we know that 0 b ∗ ∗ H (C, End(g Vj)) = k. This implies that g Vj is stable, because it is polystable. b ∗ b b ∗ Since g Vj is stable, and π2 ◦ h = g (see (4.6)), we conclude that π2Vj is also stable with b b ∗ ∗ ∗ µ(π2Vj) = µ(π2g V ) for all 1 ≤ j ≤ m. This implies that m ∗ ∗ ∗ ∗ ∗ π1f V = π2 g V = M π2Vj (4.14) j=1 is polystable.

The map π2 is genuinely ramified because g is genuinely ramified (see the proof of Lemma 4.1) and g = h ◦ π2. Indeed, if π2 factorsb through an étale covering of D, then the genuinely ramifiedb map g factors through that étale covering of D, and henceb from b Proposition 2.6 it follows thatb π2 is genuinely ramified.

Since π2 is genuinely ramiﬁed, and each Vj in (4.14) is stable, from Lemma 4.3 it follows that 0 ∗ ∗ 0 H (C ×D D, Hom(π2Vi, π2Vj)) = H (D, Hom(Vi, Vj)) (4.15) b ∗ b for all 1 ≤ i, j ≤ m. We know that Vi and π2 Vi are stable. So from (4.15) we conclude ∗ ∗ that Vi is isomorphic to Vj if and only if π2Vi is isomorphic to π2Vj. From (4.15) it also follows that 0 ∗ ∗ 0 ∗ H (C ×D D, End(π2g V )) = H (D, End(g V )) ; (4.16) b b 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 =6 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 π1 is Galois, using the earlier argument involving the socle we conclude that m S ∗ ∗ ∗ ∗ ∗ ∗ := π1 S ⊂ π1 f V = π2g V = M π2Vj =: V (4.19) e j=1 e is a polystable subbundle with µ(S) = µ(V ). e 0 e Consider the associative algebra H (C ×D D, End(V )), where V is the vector bundle in (4.19). Deﬁne the right ideal b e e 0 0 Θ := {γ ∈ H (C ×D D, End(V )) | γ(V ) ⊂ S} ⊂ H (C ×D D, End(V )) , (4.20) b e e e b e where S is the subbundle in (4.19). The subbundle S ⊂ V is a direct summand, because V is polystable,e and µ(S) = µ(V ). Consequently, S coincidese e with the subbundle generatede by the images of endomorphismse e lying in the righte ideal Θ. Since V is semistable, the image of any endomorphism of it is a subbundle. e RAMIFIED MAPS AND STABILITY OF PULLED BACK BUNDLES 19

Consider V in (4.19). The identiﬁcation e 0 0 ∗ H (C ×D D, End(V )) = H (D, End(g V )) b e b in (4.16) preserves the associative algebra structures of 0 0 ∗ H (C ×D D, End(V )) and H (D, End(g V )) , b e b 0 ∗ ∗ because it sends any γ ∈ H (D, End(g V )) to π2 γ. Let b Θ ⊂ H0(D, End(g∗V )) (4.21) e b 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 Θ in (4.21). Since g∗V is semistable, the image of any endomorphism of it is a subbundle.e From the above construction of S it follows that S ∗ = π2S , e where S is the subbundle in (4.20). e The isomorphism in (4.16) is equivariant for the actions of the Galois group Gal(π1) = Gal(g) on 0 0 ∗ ∗ H (C ×D D, End(V )) = H (C ×D D, End(π1f V )) b e b and H0(D, End(g∗V )), because the isomorphism sends any γ ∈ H0(D, End(g∗V )) to ∗ b S ∗ b ∗ ∗ π2γ. Since = π1S in (4.19) is preserved under the action of Gal(π1) on π1f V , it e 0 ∗ ∗ follows that the action of Gal(π1) on H (C ×D D, End(π1f V )) preserves the right ideal Θ in (4.20). These together imply that the actionb of Gal(g) on H0(D, End(g∗V )) preserves the right ideal Θ in (4.21). Consequently, the subbundle b e 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

S0 ⊂ V ∗ ∗ such that S = g S0 ⊂ g V . Given that V is stable, and µ(S0) = µ(V ) (this follows from (4.18)), we now conclude that S0 = V . Hence the subbundle S in (4.17) coincides with f ∗V . Therefore, we conclude that f ∗V is stable.

5. 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 degree one. Proposition 5.1. 20 I.BISWASANDA.J.PARAMESWARAN

(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)]. This implies 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 restriction ∗ ∗ of H to φ W ⊂ φ φ∗L ∗ HW := 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 0 −1 ∗ H (φ (y), L|φ−1(y)), and hence a nonzero element of (φ φ∗L)y must be nonzero at some point of φ−1(y).

We will ﬁrst show that φ∗L is semistable. To prove this by contradiction, let V ⊂ φ∗L be a semistable subbundle with 1 µ(V ) > µ(φ∗L) = . (5.3) δ Consider the nonzero homomorphism ∗ HV := H|φ∗V : φ V −→ L in (5.2). We have µ(φ∗V ) = δ · µ(V ) > 1 (see (5.3)), and also φ∗V is semistable because V is so. Consequently, HV 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 nonzero ∗ and µ(φ φ∗L) = µ(L). Proposition 5.2. Let f : C −→ D be a nonconstant separable morphism between irreducible smooth projective curves such that f is not genuinely ramiﬁed. Then there is a 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 a nontrivial ´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. RAMIFIED 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 irreducible smooth projective curves. The map f is genuinely ramiﬁed if and only if f ∗E is stable for every stable vector bundle E on D.

Acknowledgements

We thank both the referees for going through the paper very carefully and making numerous 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, Government of India, under project no.12-R&D-TFR-5.01-0500.

References [Ha] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. [HL] D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. [NS] M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540–567. [PS] A. J. Parameswaran and S. Subramanian, On the spectrum of asymptotic slopes, Teichmüller theory and moduli problem, 519–528, Ramanujan Math. Soc. Lect. Notes Ser., 10, Ramanujan Math. Soc., Mysore, 2010. [RR] S. Ramanan and A. Ramanathan, Some remarks on the instability flag, Tohoku Math. Jour. 36 (1984), 269–291. [Se] J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6 (1956), 1–42.

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India Email address: [email protected]