Contents 1 Stable sheaves 1 1 What is the moduli of vector bundles? .............................. 2 2 Definition of stable sheaves ......................................... 5 3 Generalities on boundedness ........................................ 13 4 Openness of stability ............................................... 21 2 Restriction theorems and boundedness 24 1 Restriction of Harder-Narasimhan filtration ......................... 24 2 Theorem of Grauert-Mu¨lich-Spindler ................................ 31 3 Statements of boundedness ......................................... 37 4 Boundedness of a family of torsion free sheaves ..................... 42 5 Tensor product of semi-stable sheaves ............................... 49 6 Theorem of Mumford-Mehta-Ramanathan .......................... 61 7 Simpson’s stability .................................................. 72 3 Construction of Moduli Spaces 81 1 Construction of Quot-schemes ...................................... 81 2 Geometric invariant theory ......................................... 88 3 S-equivalence and e-semi-stability ................................... 97 4 General setting and a fundamental lemma .......................... 103 5 Moduli spaces of stable sheaves ..................................... 116 6 Moduli spaces of semi-stable sheaves : char 0 ....................... 120 7 Closed orbits of a Grassmann variety ............................... 129 8 Langton’s theorem .................................................. 137 9 Moduli spaces of semi-stable sheaves – general case ................. 142 Appendix A On Langer’s work 147 1 Boundedness of semi-stable sheaves ................................. 147 2 Dimension estimate of global sections of torsion free sheaves ........ 148 Appendix B Some properties of the moduli 149 Bibliography 153 Glossary of Notations 154 xi Chapter 1 Stable sheaves First we shall show that if we collect all the vector bundles on a projective variety and require a weak universal property of a moduli space, then there does not exist the moduli space. This motivates us to introduce the notion of stability and semi-stability. The idea of Harder-Narasimhan filtration plays a crucial role sometimesbehindstrongresultsandsometimesveryexplicitly. Twoofbasicresults on boundedness are proved in the section 3. The formulation of the first is due to L.S.Kleiman[K2]andthesecondisatheoremofGrothendieck[G]. Weshallshow a beautiful application of the second result in the proof of the openness of stability. Definition of quot-schemes We define quot-schemes. The construction will be given in Chapter III. Let f : X →S bea morphismof noetherian schemes andF acoherentsheaf onX. For an S-scheme T, we set Quot (T)={F →E →0|E is flat over T}/∼, F/X/S T where F =F⊗ O is the pull-back of F to X×T and where two members are T OS T equivalent under the relation ∼ if they are isomorphic as quotient sheaves of F . T This defines a contravariant functor Quot of the category (Sch/S) of locally F/X/S noetherian S-schemes to that of sets. Assuming f to be projective, we shall fix an f-ample invertible sheaf O (1). X If F → E is a member of Quot (T), then the flatness of E over T and the T F/X/S invariance of Hilbert polynomials of a flat family tell us that T is the direct sum of subschemes TP such that the Hilbert polynomial of E(t) is P(m) on every fiber of X overTP. EspeciallyifthefunctorQuot isrepresentedbyapair(Q,G)of T F/X/S a locally noetherian S-scheme Q and the universal quotient sheaf F →G, then Q Q is the direct sum of QP. For a numerical polynomial P, we define QuotP (T) F/X/S tobethesubsetofQuot (T)consistingofquotientsheavesofF withHilbert F/X/S T polynomial P on each fiber of X over T. Then we have a subfunctor QuotP T F/X/S of Quot . What we have seen in the above is that Quot is representable F/X/S F/X/S if and only if so is QuotP is representable for all P. Moreover, in this case the F/X/S pair Q = (cid:96) QP and G = (cid:96) GP represents Quot , where (QP,GP) does P P F/X/S QuotP . We shall prove in Theorem III 1.5 that if f : X → S is a projective F/X/S morphism of noetherian schemes and F is a coherent sheaf on X, then QuotP F/X/S is representable by a projective S-scheme, which we denote by QuotP . F/X/S 1 2 Chapter1 STABLESHEAVES By studying infinitesimal deformations of a quotient sheaf, we can obtain a de- scriptionoftheZariskitangentspaceofaquot-scheme. Weheregivethedescription withouttheproof. LetX beaprojectiveschemeoverS =Speck withk afield,and let F a coherent sheaf on X. Consider a point x ∈ Q = Quot and let K be F/X/S the residue field of O . The point x corresponds to a surjective homomorphism Q,x α:F →E ofcoherentsheavesonX . PutK =ker(α). ThentheZariskitangent K K space Hom (m /m2,K) of Q at x is isomorphic to Hom (K,E). K x x OXK 1 What is the moduli of vector bundles? Let X be a non-singular projective variety over an algebraically closed field k and VB(X) be the set of isomorphism classes of algebraic vector bundles on X. The problem of moduli of algebraic vector bundles is intuitively to endow VB(X) with a natural structure VB of scheme. What is the meaning of a natural structure X then? We require at least VB to have the following property: X (1.1) Let S be an algebraic k-scheme and E a vector bundle on X× S. Then the k map S(k) (cid:51) s (cid:55)→ [E(s)] = [E⊗ k(s)] ∈ VB(X) is induced by a morphism OS of S to VB , where [ ] denotes the isomorphism class. X Fix an ample line bundle O (1) on X. Let H be a numerical polynomial of X degree = dimX. Pick two vector bundles E and E on X with χ(E (m)) = 1 2 1 χ(E (m))=H(m). Ifmisasufficientlylargeinteger,thenE (m)hasthefollowing 2 i property: (1.2) E (m)isgeneratedbyitsglobalsectionsandHq(X,E (m))=0forallq ≥1. i i By replacing E by E (m), we may assume that E itself has the property. Let i i i us take a look at the quot-scheme Q=QuotH , where V is a vector space V⊗kOX/X/k of dimension N = H(0) = dimH0(X,E ). If θ : V ⊗ O → F is the universal i k X×Q quotient sheaf on X × Q, then there exists an open set U of Q such that U(k) is k exactly the set {y ∈Q(k)|F(y)=F ⊗ k(y) has the following properties (a), (b) OQ and (c) }: (a) F(y) is locally free, (b) θ(y):V⊗ O →F(y)inducesanisomorphismΓ(θ(y)):V =H0(X,V⊗ k X k O )→H0(X,F(y)), X (c) Hq(X,F(y))=0 for all q ≥1. By abuse of notation we denote the restriction of the universal quotient sheaf to X × U by θ : V ⊗ O → F. Through the natural action on V, G = GL(V) k k X×U acts on Q and U is a G-invariant open subscheme. It is easy to see that the center G ofGactstriviallyonQandhenceG=G/G actsonU. Forapointy ∈U(k), m m an automorphism σ of F(y) induces an element σ(cid:48) of G by the property (b) : 1 WHATISTHEMODULIOFVECTORBUNDLES? 3 V −−Γ(−θ−(y−→)) H0(X,F(y))   σ(cid:48)(cid:121) (cid:121)Γ(σ) V −−Γ(−θ−(y−→)) H0(X,F(y)) Obviously, σ(cid:48) is an element of the stabilizer group St (y) of G at y. Conversely, τ G is an element of St (y) if and only if there is an automorphism τ(cid:48) of F(y) which G makes the following diagram commutative θ(y) V ⊗ O −−−−→ F(y) k X   τ(cid:121) (cid:121)τ(cid:48) θ(y) V ⊗ O −−−−→ F(y) k X Thus τ gives rise to an automorphism of F(y). Thanks to (b) again, τ(cid:48) is not ∼ the identity unless τ = id. We see therefore that St (y) → Aut(F(y)) for every G y ∈U(k). TheimageofG bythisisomorphismisthemultiplicationsbyelements m of k× on F(y), that is, ∼ (1.3) St (y)−→Aut(F(y))/k×. G An observation similar to the above shows that (1.4) for y ,y ∈ U(k), both y and y belong to the same orbit of G if and only 1 2 1 2 if F(y )∼=F(y ). 1 2 Now, by (1.2) for E with m=0, we have a surjective η :V ⊗ O →E such i i k X i that Γ(η ) : V → H0(X,E ) is bijective. The universal property of Q provides us i i with points z , z of Q(k) such that F(z )∼=E and θ(z )∼=η . Both z and z are 1 2 i i i i 1 2 in U(k) because of (1.2). Assume that VB exists. Then, by the property (1.1) for VB , we obtain X X a morphism f : U → VB . For each k-rational point y of U, f−1f(y) is the X G-orbit of y by (1.4). On one hand, dim f−1f(y) is upper semi-continuous on y U by a theorem of Chevalley. On the other hand, dim f−1f(y) = dimo(y) = y dimG−dimSt (y)=N2−1−dimSt (y)islowersemi-continuous. ThusdimSt (y) G G G is constant over each connected component of U. By this and (1.3) we see that dim End (F(y)) = dimAut(F(y)) depends only on the connected component k OX containing y. We have therefore (1.5) if dim End (E ) (cid:54)= dim End (E ), then z and z belong to different k OX 1 k OX 2 1 2 connected components of U. Letusgiveanexampleofafamilyofvectorbundleswhichareparameterizedby an irreducible curve but whose spaces of endomorphisms jump at a special point. 4 Chapter1 STABLESHEAVES Example 1.6. Assume that dimX ≥ 2. Let T be a non-singular subvariety of codimension 1 in X and D a very ample divisor on T. Assume (1.6.1) H0(T,O (T)⊗ O (−D))=0, X OX T (1.6.2) dimH0(T,O (D))≥dimX+1=n+1. T Choose an (n+1)-ple (s ,...,s ) of elements of H0(T,O (D)) so generally that 0 n T s ,...,s are linearly independent over k and (cid:84)n−1(s ) = ∅. Set Z = X × A1 0 n i=0 i 0 k and S = T × A1. On S we have a homomorphism α : On+1 → p∗(O (D)) such k S T thatonS =T×u, u∈A1,α(u)isdefinedbyΓ(α(u))(e )=s (0≤i≤n−1)and u i i Γ(α(u))(e ) = us , where p : S → T is the projection and {e ,...,e } is a fixed n n 0 n freebasisofOn+1. Bythefactthat(cid:84)n−1(s ) =∅weseethatα(u)issurjectivefor S i=0 i 0 all u and hence α is surjective by Nakayama’s lemma. Put K =ker(α). Then K is a vector bundle of rank n on S. Dualizing K (cid:44)→ On+1 and composing it with the S natural On+1 → On+1, we have a surjective homomorphism β : On+1 → K∨. Put Z S Z E =ker(β)⊗ p(cid:48)∗O (T),socalled,anelementarytransformofOn+1alongK∨and OZ X Z itiswell-knownthatE isavectorbundleofrankn+1onZ,wherep(cid:48) :Z →X isthe projection. At the origin 0∈A1, E(0)∼=E(cid:48)⊕O , whence dimEnd (E(0))≥2. X OX Let us show that if u ∈ A1 is not zero, then End (E(u)) ∼= k. By a property of OX elementary transform, there is a short exact sequence 0→On+1 −→ϕ E(u)→O (T)⊗ O (−D)→0. X X OX T By (1.6.1), the map Γ(ϕ) : kn+1 → H0(X,E(u)) is bijective. It follows that if γ ∈End (E(u)), then we have a commutative diagram OX On+1 −−−ϕ−→ E(u) X   Γ(γ)(cid:121) (cid:121)γ On+1 −−−ϕ−→ E(u). X From this we obtain a commutative diagram O (−D) −−−−→ On+1 T T     (cid:121) Γ(γ)(cid:121) O (−D) −−−−→ On+1, T T wheretheleftverticalarrowisthemultiplicationofsomea∈k. Sinces ,...,s , 0 n−1 us are linearly independent, Γ(γ) is also the multiplication of a. This implies that n γ is the multiplication of a. Let us consider the E in the above example. If m is sufficiently large, then for E(m) = E ⊗ O (m) and for all u in A1, E(m)(u) is generated by its global OX X sections and Hi(X,E(m)(u)) = 0 for all positive i. By virtue of the base change theorem of cohomologies, q (E(m)) is locally free, the natural map ∗ ψ :q∗q (E(m))→E(m) ∗ 2 DEFINITIONOFSTABLESHEAVES 5 is surjective and k⊕N ∼=H0(X,ON)∼=H0(X,q∗q (E(m))(u))→H0(X,E(m)(u)) X ∗ is bijective, where q is the projection of Z to A1. Since every vector bundle on A1 is trivial, q (E(m))∼=V ⊗ O with V a k-vector space. We obtain therefore ∗ k A1 a surjective ψ : V ⊗ O → E(m). What we have shown implies that there is k Z a morphism g : A1 → QuotH such that (1 ×g)∗(θ) is isomorphic to V⊗kOX/X/k X ψ :V ⊗ O →E(m), where H is the Hilbert polynomial of an E(m)(u). g(A1) is k Z a subset of the open set U for QuotH . Set E =E and E =E(u) with V⊗kOX/X/k 2 0 1 u (cid:54)= 0. Then z = g(u) and z = g(0) belong to the same connected component 1 2 of U. This violates (1.5) because dim End (E ) = 1 < 2 ≤ dim EndO (E ). k OX 1 k X 2 Foreverypair(X,T),thereareD’swhichsatisfytheconditions(1.6.1)and(1.6.2). Moreover, when dimX =1, the construction of the family of vector bundles on X as in Example 1.6 is much easier. We see therefore the following. Theorem 1.7. If X is a non-singular projective variety with positive dimension, then there does not exist VB with the property (1.1). X The above theorem tells us that to get a moduli space of vector bundles on a projective variety X we have to restrict ourselves to a good subfamily of VB(X). What is the good family then? 2 Definition of stable sheaves Throughout this section we shall work on a field k which may not be algebraically closed. We fix an algebraic closure k¯ of k. An algebraic scheme X over k is said to be geometrically integral if X ⊗ k¯ is integral. It seems more natural to study our k moduli problem in a category wider than that of vector bundles. Definition 2.1. Let X be an integral algebraic scheme over k and E a coherent sheaf on X. (1)Thereisanon-emptyopensetU ofX overwhichE islocallyfree. Therank of E| is said to be the rank of E and denoted by r(E). U (2) E is said to be torsion free if the natural map of E to E ⊗ K(X) is OX injective, where K(X) is the sheaf of function field of X. IfX isquasi-projective,thenthedefinitionoftorsionfreesheavesissimplerand clearer. Lemma 2.2. Assume that X is a quasi-projective, integral scheme. A coherent sheaf E on X is torsion free if and only if E is a subsheaf of a coherent locally free sheaf F (that is, a vector bundle) on X. In this case F can be chosen to be of rank r(E). Proof. AssumethatthereisavectorbundleF onX andaninjectionαofE toF. SincethesheafK(X)containsO ,F isasubsheafofF⊗ K(X). Thecomposition X OX α of E → E⊗ K(X) with α⊗ K(X) is the same as E −→ F → F ⊗ K(X). OX OX OX 6 Chapter1 STABLESHEAVES Since the latter is injective, we see that the natural map E → E ⊗ K(X) is OX injective and hence E is torsion free. Conversely, if E is torsion free, then the natural map of E to its double dual (E∨)∨ is injective because E has no local sections supported by proper closed sets. Taking an ample line bundle O (1) on X X, E∨(m) = E∨⊗ O (m) is generated by its global sections. Then, if we pick OX X sufficiently general members s ,...,s of H0(X,E∨(m)) with r = r(E) = r(E∨), 1 r then they define a generically surjective map β : O⊕r → E∨(m). Dualizing β and V tensoring O (m), we have an injection of (E∨)∨ to O (m)⊕r because (E∨)∨ is X X torsion free. As a simple application of the above lemma we see that the restriction of a torsion free coherent sheaf to a general hyperplane section is again torsion free. Indeed, we have the following result. Corollary 2.3. LetE beatorsionfreecoherentsheafonaquasi-projectiveintegral scheme X over an infinite field k, L a line bundle on X and let W be a k-vector subspace of H0(X,L) which generates L. Assume that for a general member s of W, the zero scheme Z(s) of s is integral. Then, for a sufficiently general member s of W, the restriction E| =E⊗O of E to Z(s) is torsion free. Z(s) Z(s) Proof. Since X is quasi-projective and E is torsion free, E is a subsheaf of a vector bundle F by the above lemma. Let C be the quotient sheaf F/E. If s is a sufficiently general member of W, then Z(s) is integral and does not contain any point of Ass(C) because W generates L and k is infinite. Using the section s, we have the following exact commutative diagram 0 0     (cid:121) (cid:121) 0 −−−−→ E⊗L∨ −−×−−s→ E −−−−→ E| −−−−→ 0 Z(s)      γ (cid:121) (cid:121) (cid:121) 0 −−−−→ F ⊗L∨ −−×−−s→ F −−−−→ F| −−−−→ 0 Z(s)       (cid:121) (cid:121) (cid:121) 0 −−−−→ D −−−−→ C⊗L∨ −−×−−s→ C −−−−→ C| −−−−→ 0 Z(s)       (cid:121) (cid:121) (cid:121) 0 0 0 Thankstothewayofchoiceofs,D mustbezeroandthenthesnakelemmaimplies that γ is injective. Since Z(s) is integral, Lemma 2.2 implies that E| is torsion Z(s) free. Let us fix a pair (X,O (1)) of a geometrically integral projective scheme X X over k and an ample line bundle O (1) on X. Assume that X is of dimension n. X 2 DEFINITIONOFSTABLESHEAVES 7 For a coherent sheaf E of rank r on X, there are integers a (E),a (E),...,a (E) 1 2 n such that the Hilbert polynomial χ(E(m)) = (cid:80)n (−1)idimHi(X,E(m)) can be i=0 written in the form (cid:18) (cid:19) n−1 (cid:18) (cid:19) m+n (cid:88) m+i χ(E(m))=rd + a (E) (2.4) n n−i i i=0 where d is the degree of X with respect to O (1). We introduce the following X notation which plays a key role in the sequel. Definition 2.5. Let E be a coherent sheaf on a geometrically integral projective scheme X over k and let O (1) be an ample line bundle on X. Assume that X r =r(E) is positive. (1) For the integer a (E) in (2.4), we define µ (E) to be the rational number 1 0 a (E)/r. 1 (2) We denote the polynomial χ(E(m))/r by P (m). E By Riemann-Roch theorem we have Lemma 2.6. If X is non-singular, then we have rd(K ,O (1)) rd(n+1) a (E)=d(E,O (1))− X X − , 1 X 2 2 where for a coherent sheaf F on X, d(F,O (1)) is the degree of the first Chern X class of F with respect to O (1) and where K is the canonical sheaf of X. X X Now a usual definition of µ is as follows. Definition 2.7. Let(X,O (1))andE beasinDefinition2.5. IfX is,inaddition, X smooth, then we define a rational number µ(E) to be d(E,O (1)) µ(E)= X . r The above lemma shows that the difference µ(E)−µ (E) is d(K ,O (1))/2+ 0 X X d(n+1)/2, which depends only on the pair (X,O (1)). X Now we are ready to introduce the notion of stability of coherent sheaves. Definition 2.8. Let (X,O (1)) be a pair of a geometrically integral projective X scheme X over k and an ample line bundle O (1) on X and let E be a coherent X O -module. X (1) E is said to be µ-stable (or, µ-semi-stable) if E is torsion free and for every coherent subsheaf F of E ⊗ k¯ with 0 < r(F) < r(E), we have the inequality k µ (F)<µ (E) (or, µ (F)≤µ (E), resp.). 0 0 0 0 (2) E is said to be stable (or, semi-stable) if E is torsion free and for every coherent subsheaf F of E⊗ k¯ with 0 < r(F) < r(E) and for all sufficiently large k integers m, we have the inequalities P (m)<P (m) (or, P (m)≤P (m), resp.). F E F E 8 Chapter1 STABLESHEAVES Remark 2.9. (1) In Definition 2.8 we may take only F with E⊗ k¯/F is torsion k free. Indeed, if T is the torsion part of E⊗ k¯/F, then for the inverse image F(cid:48) of k T to E⊗ k¯, we obtain χ(F(cid:48)(m))=χ(F(m))+χ(T(m)), degχ(T(m))<n and the k leading coefficient of χ(T(m)) is positive. Thus the inequality for F(cid:48) implies that of F. (2) By Lemma 2.6 we see that if X is smooth, then the above definition of the µ-stability and µ-semi-stability is the same as the usual, that is, we can use µ instead of µ to define the µ-(semi-)stability. 0 (3) Directly from our definition we have the following implications: µ-stability =⇒ stability (cid:119) (cid:119) (cid:119) (cid:119) (cid:127) (cid:127) µ-semi-stability ⇐= semi-stability Oneofthemainaimsofthisbookistoconstructmodulispacesofstablesheaves and then, from the viewpoint of (1.5), (5) in the following is an evidence that the stable sheaves must form a good family to construct their moduli space. Proposition 2.10. Let (X,O (1)) be as in Definition 2.8 and let E and F be X coherent O -modules. X (1) If both E and F are µ-semi-stable and if µ (E)>µ (F), then every homo- 0 0 morphism of E to F is 0. (2) If E is µ-stable, F is µ-semi-stable and if µ (E) = µ (F), then every non- 0 0 zero homomorphism of E to F is injective. (3) If both E and F are semi-stable and if P (m) > P (m) for all sufficiently E F large m, then every homomorphism of E to F is 0. (4) If E is stable, F is semi-stable and if P (m)=P (m), then every non-zero E F homomorphism of E to F is injective and F/E is semi-stable. (5) If E is stable, then E is simple, that is, End (E) is isomorphic to k which OX is naturally embedded in End (E) as the multiplication by constants. OX (6) Assume that E is an extension of torsion free coherent sheaves 0−→E −→E −→E −→0 1 2 such that P (m)=P (m)=P (m) (or, µ (E)=µ (E )=µ (E )). Then E is E E1 E2 0 0 1 0 2 semi-stable (or, µ-semi-stable, resp.) if and only if so are E and E . 1 2 Proof. Weshallproveonly(3),(4)and(5). Theproofsof(1)and(2)areobtained from those of (3) and (4) by replacing the polynomials P (m), P (m) with µ (E), E F 0 µ (F). (6) is obvious by the definition. 0 (3) Let α be a non-zero homomorphism of E to F and G the image of α. The semi-stability of E implies that for K = ker(α) and all sufficiently large m, P (m)≤P (m) and hence K E χ(E(m))−χ(K(m)) r(E)P (m)−r(K)P (m) P (m)= ≥ E E =P (m) G r(G) r(G) E 2 DEFINITIONOFSTABLESHEAVES 9 for all sufficiently large m. On the other hand, we have that P (m) ≤ P (m) for G F all sufficiently large m because G is a subsheaf of F. Then, by our assumption, P (m) < P (m) for all sufficiently large m. This is a contradiction. Thus α must G E be 0. (4) Let α, G and K be as in the proof of (3). By the semi-stability of F and our assumption we get P (m)≤P (m). Then a computation similar to the above G F showsthatP (m)≥P (m). ThenthestabilityofE impliesthatK is0andhence K E α is injective. If F/E is not torsion free, then we obtain a coherent subsheaf E(cid:48) of F with P (m) > P (m) as in Remark 2.9. Thus F/E must be torsion free and E(cid:48) F then F/E is semi-stable by (6). (5) Since End (E ⊗ k¯) ∼= End (E)⊗ k¯ and since we have only to prove OX k OX k dim End (E)=1,wemayassumethatk isalgebraicallyclosed. Pickanon-zero k OX elementαofEnd (E). Then,byvirtueof(4)αisinjective. Sinceχ(coker(α)(m)) OX =0,weseethatcoker(α)=0orαissurjective. Thuseverynon-zeroendomorphism of E is an isomorphism. Take a k-rational point x of X. α induces a linear trans- formation α(x) of the vector space E(x). Let λ be an eigen value of α(x). Then β = α−λid cannot be an isomorphism because β(x) is not surjective. Therefore, β must be 0, which implies that α is the multiplication by λ. Harder-Narasimhanfiltrationaffordsusatooltomeasurehowfaratorsionfree coherent sheaf is from the semi-stability. Proposition 2.11. Let (X,O (1)) be as in Definition 2.8 and let E be a torsion X free coherent O -module. X (1) There is a unique filtration 0 = E ⊂ E ⊂ ··· ⊂ E = E by coher- 0 1 α ent subsheaves such that (a) E /E is µ-semi-stable for 0 < i ≤ α and (b) i i−1 µ (E /E )>µ (E /E ) for 0<i<α. 0 i i−1 0 i+1 i (2) There is a unique filtration 0 = E(cid:48) ⊂ E(cid:48) ⊂ ··· ⊂ E(cid:48) = E by coherent sub- 0 1 β sheaves such that (c) E(cid:48)/E(cid:48) is semi-stable for 0<i≤α and (d) P (m)> i i−1 E(cid:48)/E(cid:48) i i−1 P (m) for all sufficiently large integer m and for 0<i<β. E(cid:48) /E(cid:48) i+1 i Definition 2.12. The filtration in (1) (or, (2)) of Proposition 2.11 is called the Harder-Narasimhan filtration by µ-semi-stability (or, semi-stability, resp.). Proof of Proposition 2.11. We shall prove only (1) because the idea of the proofs of (1) and (2) is common. Let us prove the uniqueness by induction on r(E). If E is µ-semi-stable, especially if r(E) = 1, then there is nothing to prove. Assume that E is not µ-semi-stable and hence we have a filtration 0=E ⊂E ⊂ 0 1 ··· ⊂ E = E with the properties (a) and (b) and with α > 1. Suppose that we α have another filtration 0 = F ⊂ F ⊂ ··· ⊂ F = E with the properties (a) and 0 1 β (b). We may assume, without losing any generality, that µ (F ) ≥ µ (E ). Let 0 1 0 1 i be the smallest integer such that E contains F . If i > 1, then the composi- i 1 tion F (cid:44)→ E → E /E is a non-zero homomorphism. On the other hand, since 1 i i i−1 µ (F )≥µ (E )>µ (E /E ),Proposition2.10,(1)impliesthatthecomposition 0 1 0 1 0 i i−1