The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Shrödinger International Boltzmanngasse 9 Institute for Mathematial Physis A-1090 Wien, Austria Moduli Spaes of Holomorphi Triples over Compat Riemann Surfaes Steven B. Bradlow Osar Garia-Prada Peter B. Gothen Vienna, Preprint ESI 167 (003) January 13, 003 Supported by the Austrian Federal Ministry of Eduation, Siene and Culture Available via anonymous ftp from FTP.ESI.AC.AT or via WWW, URL:

2 Moduli spaes of holomorphi triples over ompat Riemann surfaes Steven B. Bradlow 1, Department of Mathematis, University of Illinois, Urbana, IL 61801, USA Osar Garía Prada 1,3,5,6 Instituto de Matemátias y Físia Fundamental, Consejo Superior de Investigaiones Científias, Serrano 113 bis, 8006 Madrid, Spain osar.garia-prada@uam.es Peter B. Gothen 1,4,5 Departamento de Matemátia Pura, Fauldade de Ciênias, Universidade do Porto, Rua do Campo Alegre 687, Porto, Portugal pbgothen@f.up.pt November 7, 00 Abstrat. A holomorphi triple over a ompat Riemann surfae onsists of two holomorphi vetor bundles and a holomorphi map between them. After fixing the topologial types of the bundles and a real parameter, there exist moduli spaes of stable holomorphi triples. In this paper we study non-emptiness, irreduibility, smoothness, and birational desriptions of these moduli spaes for a ertain range of the parameter. Our results have important appliations to the study of the moduli spae of representations of the fundamental group of the surfae into unitary Lie groups of indefinite signature ([5, 7]). Another appliation, that we study in this paper, is to the existene of stable bundles on the produt of the surfae by the omplex projetive line. 1 Members of VBAC (Vetor Bundles on Algebrai Curves), whih is partially supported by EAGER (EC FP5 Contrat no. HPRN-CT ) and by EDGE (EC FP5 Contrat no. HPRN-CT ). Partially supported by the National Siene Foundation under grant DMS Partially supported by the Ministerio de Cienia y Tenología (Spain) under grant BFM Partially supported by the Fundação para a Ciênia e a Tenologia (Portugal) through the Centro de Matemátia da Universidade do Porto and through grant no. SFRH/BPD/1606/ Partially supported by the Portugal/Spain bilateral Programme Aiones Integradas, grant nos. HP and AI-01/4 6 Partially supported by a British EPSRC grant (Otober-Deember 001)

3 1. Introdution Let X be a losed Riemann surfae of genus g. The theory of holomorphi triples has its origins [13, 4] in the searh for solutions to ertain gauge theoreti equations on X, obtained by dimensional redution of the Hermitian Einstein equation in 4 dimensions. More preisely, solutions to the Hermitian Einstein equation on X P 1 whih are invariant under the standard ation of SU() on P 1 orrespond to solutions to the so-alled vortex equations on X. The Hithin Kobayashi orrespondene states that a solution to the Hermitian Einstein equation on X P 1 gives rise to a stable holomorphi bundle and that, onversely, any stable holomorphi bundle admits a Hermitian Einstein metri. The ounterpart on X states that there is a Hithin Kobayashi orrespondene between solutions to the oupled vortex equations and stable holomorphi triples. A holomorphi triple onsists of a pair of holomorphi vetor bundles, E 1 and E, over X and a holomorphi map φ: E E 1 between them. An important feature of the stability ondition for triples is that it depends on a real parameter α, orresponding to the fat that there is a real parameter in the vortex equations; thus one is led to the onept of α-stability of a holomorphi triple. This parallels the fat that when studying Hermitian Einstein metris and stable bundles on X P 1 it is neessary to hoose a polarization on this omplex surfae. We note that, as usual, there are orresponding onepts of α-polystable and α-semistable triples (see Setion below for preise definitions). It was shown in [4] (see also [13]) that projetive moduli spaes for holomorphi triples exist. (Later a diret onstrution was given by Shmitt using geometri invariant theory [6].) Sine the stability ondition depends on the real parameter α, so do the moduli spaes. Fixing the topologial invariants n i = rk(e i ) and d i = deg(e i ), we denote the moduli spae of α-polystable triples with the given invariants by N α = N α (n 1, n, d 1, d ), and the moduli spae of α-stable triples by Nα s N α. In this paper we address the questions of smoothness, non-emptiness and irreduibility of these moduli spaes. Before desribing our results in more detail, we explain our motivation, whih omes from the problem of determining the onneted omponents of the moduli spae of representations of the fundamental group of X in PU(p, q). A detailed study of this moduli spae appears in a ompanion paper [7]; in the following we briefly outline the main ideas. The first point to notie is that we may as well study the onneted omponents of the moduli spae of projetively flat U(p, q) bundles on X. This moduli spae an be divided into disjoint losed subspaes M(a, b) indexed by a pair of integers (a, b), the Chern lasses obtained from a redution of struture group to the maximal ompat subgroup U(p) U(q). The values of (a, b) are bounded by the Milnor Wood type inequality aq bp p q min{p, q}(g 1). For eah allowed value of (a, b) one expets the spae M(a, b) to be non-empty and onneted, thus forming a onneted omponent of the moduli spae.

4 By the work of Hithin [18, 19], Donaldson [1], Simpson [7, 8, 9, 30] and Corlette [9], the moduli spaes M(a, b) are homeomorphi to moduli spaes of so-alled U(p, q)- Higgs bundles on X: these are pairs (E, Φ), where E is a holomorphi vetor bundle whih deomposes as a diret sum E = V W and the Higgs field Φ: E E K is of the form ( ) 0 β φ = γ 0 with respet to the diret sum deomposition of E. Here K is the anonial line bundle of X and the invariants a and b appear as the degrees of V and W respetively. The L -norm of the Higgs field gives us a Bott-Morse funtion on the moduli spae (f. Hithin [18, 19]). Thus, onnetedness of the spaes M(a, b) will be a onsequene of onnetedness of the orresponding subspaes of loal minima. In the ase of flat U(, )-bundles, it was shown in [15] that the loal minima are represented by Higgs bundles for whih either β or γ vanishes. One of the main results in [7] is that this is true in general. The ruial observation is now that there is a bijetive orrespondene between U(p, q)-higgs bundles (E, Φ) with β = 0 or γ = 0 and holomorphi triples: if, say, γ = 0, we obtain a holomorphi triple T = (E 1, E, φ) by setting E 1 = V K, E = W and φ = β. It turns out that (E, Φ) is (poly)stable as a U(p, q)-higgs bundle if and only if the orresponding holomorphi triple T is α-(poly)stable for α = g. It follows that the subspae of loal minima on M(a, b) is isomorphi to a moduli spae of (g )-polystable holomorphi triples. Thus the results of the present paper imply results on non-emptiness and onnetedness of the moduli spaes M(a, b). We refer the reader to [7] for the preise statements. We now return to our main subjet of study, the holomorphi triples. In order for N α to be non-empty, one must have α α m with α m = d 1 /n 1 d /n 0. In the ase n 1 n there is also a finite upper bound α M. When the parameter α varies, the nature of the α-stability ondition only hanges for a disrete number of so-alled ritial values of α (see Setion for the preise statements). We an now state our main results. Theorem A. (1) A triple T = (E 1, E, φ) of type (n 1, n, d 1, d ) is α m -polystable if and only if φ = 0 and E 1 and E are polystable. We thus have N αm (n 1, n, d 1, d ) = M(n 1, d 1 ) M(n, d ). where M(n, d) denotes the moduli spae of polystable bundles of rank n and degree d. In partiular, N αm (n 1, n, d 1, d ) is non-empty and irreduible. () If α > α m is any value suh that g α (and α < α M if n 1 n ) then the moduli spae N s α(n 1, n, d 1, d ) is non-empty, irreduible, and smooth of dimension (g 1)(n 1 n n 1 n ) n 1 d n d 1 1. Moreover: If n 1 = n = n then the moduli spae N s α(n, n, d 1, d ) is birationally equivalent to a P N -fibration over M s (n, d ) Sym d 1 d (X), where M s (n, d ) denotes the subspae of stable bundles of type (n, d ), Sym d 1 d (X) is the symmetri produt, and the fiber dimension is N = n(d 1 d ) 1. If n 1 > n then the moduli spae N s α(n 1, n, d 1, d ) is birationally equivalent to a P N -fibration over M s (n 1 n, d 1 d ) M s (n, d ), where the fiber dimension is N = n d 1 n 1 d n (n 1 n )(g 1) 1. 3

5 4 If n 1 < n then the moduli spae Nα(n s 1, n, d 1, d ) is birationally equivalent to a P N -fibration over M s (n n 1, d d 1 ) M s (n 1, d 1 ), where the fiber dimension is N = n d 1 n 1 d n 1 (n n 1 )(g 1) 1. (3) If n 1 n then the moduli spae N αm (n 1, n, d 1, d ) is non-empty and irreduible. Moreover { N αm (n 1, n, d 1, d ) M(n, d ) M(n 1 n, d 1 d ) if n 1 > n = M(n 1, d 1 ) M(n n 1, d d 1 ) if n 1 < n. Our strategy for studying the moduli spaes is similar in spirit to the one used by Thaddeus [3]: basially it onsists in obtaining a good understanding of the moduli spae for a partiular (large) value of α and then keeping trak of how the moduli spae hanges as α varies. In the following we explain this in more detail and outline the ontents of the paper. After realling the basi fats about holomorphi triples in Setion, we go on to study extensions and deformations of triples in Setion 3. Here we show that the quasiprojetive variety Nα s N α orresponding to α-stable triples is smooth for all values of α greater than or equal to g (Theorem 3.8). In Setions 4 and 5 we examine how the moduli spaes differ for values of α on opposite sides of a ritial value. If N α denote the moduli spaes for values of α above and below a ritial value α, we denote the loi along whih they differ by S α respetively. Our main result (Theorem 5.1) is that for all α g the odimension of S α is stritly positive. It follows that the number of irreduible omponents of the spaes Nα s are the same for all α satisfying α g and α m < α < α M. In order to estimate the odimension of the S α we need to estimate the dimension of ertain spaes of extensions of triples. It is notable that this requires us to onsider objets more general than triples, namely the holomorphi hains studied in [1]. The rather tehnial details are in Setion 4: the main result is Proposition 4.3 whih is then used to dedue the key Proposition 4.7. Next we turn to the question of understanding the moduli spaes N α for large values of the parameter α. After obtaining some preliminary results in Setion 6, we onsider the ase of triples with n 1 n in Setion 7. Let N L denote the moduli spae of α-polystable triples for α between α M and the largest ritial value smaller than α M. We show that this large α moduli spae is birationally equivalent to a P N -fibration over a produt of moduli spaes of stable bundles (Theorem 7.7). Combining this fat with our odimension estimates we obtain our main results on non-emptiness and irreduibility of the moduli spaes N α and Nα; s these appear as Theorem 7.9 and Corollary In Setion 8 we obtain analogous results in the ase when n 1 = n. Even though there is no upper limit to α in this ase, the moduli spaes do stabilize for α suffiiently large (Theorem 8.6) and hene it makes sense to onsider the large α moduli spae N L also in this ase. The birational desription of N L is given in Theorem 8.15, while the main results on non-emptiness and irreduibility are in Theorem Finally, in Setion 9, we go bak to the origins of the theory of holomorphi triples and apply our results on moduli of triples to dedue the existene of SU()-invariant Hermitian Einstein metris on omplex vetor bundles on X P 1 ; equivalently, our results imply the existene of stable vetor bundles on X P 1.

6 This paper and its ompanion [7] form a substantially revised version of the preprint [6]. The main results proved in this paper were announed in the note [5]. In that note we laim (without proof) that for α g, the moduli spaes N α are irreduible without imposing the onditions in () or (3) of Theorem A. This is a reasonable onjeture, whih we hope to ome bak to in a future publiation. Aknowledgements. We thank the Mathematis Departments of the University of Illinois at Urbana-Champaign, the University Autónoma of Madrid and the University of Aarhus, the Department of Pure Mathematis of the University of Porto, the Mathematial Sienes Researh Institute of Berkeley, the Mathematial Institute of the University of Oxford, and the Erwin Shrödinger International Institute for Mathematial Physis in Vienna for their hospitality during various stages of this researh. We thank Ron Donagi, Tomás Gómez, Rafael Hernández, Nigel Hithin, Alastair King, Viente Muñoz, Peter Newstead, and S. Ramanan for many insights and patient explanations.. Definitions and basi fats.1. Holomorphi triples and their moduli spaes. Let X be a ompat Riemann surfae (some of what follows is also true also for a ompat Kähler manifold [13, 1]). Reall ([4] and [13]) that a holomorphi triple T = (E 1, E, φ) on X onsists of two holomorphi vetor bundles E 1 and E on X and a holomorphi map φ: E E 1. A homomorphism from T = (E 1, E, φ ) to T = (E 1, E, φ) is a ommutative diagram E E φ E 1 φ E 1, where the vertial arrows are holomorphi maps. A triple T = (E 1, E, φ ) is a subtriple of T = (E 1, E, φ) if the sheaf homomorphims E 1 E 1 and E E are injetive. A subtriple T T is alled proper if T 0 and T T. Definition.1. For any α R the α-degree and α-slope of T are defined to be deg α (T ) = deg(e 1 ) deg(e ) α rk(e ), deg µ α (T ) = α (T ) rk(e 1 ) rk(e ) rk(e ) = µ(e 1 E ) α rk(e 1 ) rk(e ), where deg(e), rk(e) and µ(e) = deg(e)/ rk(e) are the degree, rank and slope of E, respetively. We say T = (E 1, E, φ) is α-stable if µ α (T ) < µ α (T ) for any proper subtriple T = (E 1, E, φ ). Sometimes it is onvenient to use α (T ) = µ α (T ) µ α (T ), (.1) 5

7 6 in terms of whih the α-stability of T is equivalent to α (T ) < 0 for any proper subtriple T. We define α-semistability by replaing the above strit inequality with a weak inequality. A triple is alled α-polystable if it is the diret sum of α-stable triples of the same α-slope. Write n = (n 1, n ) and d = (d 1, d ). We denote by N α = N α (n, d) = N α (n 1, n, d 1, d ) the moduli spae of α-polystable triples T = (E 1, E, φ) whih have rk(e i ) = n i and deg(e i ) = d i for i = 1,. The subspae of α-stable triples is denoted by N s α. We refer to (n, d) = (n 1, n, d 1, d ) as the type of the triple. There are ertain neessary onditions in order for α-semistable triples to exist. Let µ i = d i /n i for i = 1,. We define α m =µ 1 µ, (.) α M =(1 n 1 n n 1 n )(µ 1 µ ), n 1 n. (.3) Proposition.. [4, Theorem 6.1] The moduli spae N α (n 1, n, d 1, d ) is a omplex analyti variety, whih is projetive when α is rational. A neessary ondition for N α (n 1, n, d 1, d ) to be non-empty is 0 α m α α M if n 1 n, 0 α m α if n 1 = n. Remark.3. If α m = 0 and n 1 n then α m = α M = 0 and the moduli spae of α stable triples is empty unless α = 0. A diret onstrution of these moduli spaes has been given by Shmitt [6] using geometri invariant theory. Given a triple T = (E 1, E, φ) one has the dual triple T = (E, E 1, φ ), where E i is the dual of E i and φ is the transpose of φ. The following is not diffiult to prove ([4, Proposition 3.16]). Proposition.4. The α-(semi)stability of T is equivalent to the α-(semi)stability of T. The map T T defines a bijetion whih is moreover an isomorphism. N α (n 1, n, d 1, d ) = N α (n, n 1, d, d 1 ), This an be used to restrit our study to n 1 n and appeal to duality to deal with the ase n 1 < n... Critial values. A holomorphi triple T = (E 1, E, φ) of type (n 1, n, d 1, d ) is stritly α-semistable if and only if it has a proper subtriple T = (E 1, E, φ ) suh that µ α (T ) = µ α (T ), i.e. µ(e 1 E ) α n n 1 n = µ(e 1 E ) α n n 1 n. (.4)

8 There are two ways in whih this an happen: The first one is if there exists a subtriple T suh that n = n, and n 1 n n 1 n µ(e 1 E ) = µ(e 1 E ). In this ase the terms ontaining α drop from (.4) and T is stritly α-semistable for all values of α. We refer to this phenomenon as α-independent semistability. This annot happen if GCD(n, n 1 n, d 1 d ) = 1. The other way in whih strit α-semistability an happen is if equality holds in (.4) but n n. (.5) n 1 n n 1 n The values of α for whih this happens are alled ritial values. Definition.5. We say that α [α m, ) is a ritial value if there exist integers n 1, n, d 1 and d suh that that is, d 1 d α n n 1 n n 1 n = d 1 d n 1 n α n n 1 n, α = (n 1 n )(d 1 d ) (n 1 n )(d 1 d ), n 1n n 1 n with 0 n i n i, (n 1, n, d 1, d ) (n 1, n, d 1, d ), (n 1, n ) (0, 0) and n 1n n 1 n. We say that α is generi if it is not ritial. Proposition.6. [4] Fix (n 1, n, d 1, d ). (1) The ritial values of α form a disrete subset of α [α m, ), where α m is as in (.). () If n 1 n the number of ritial values is finite and lies in the interval [α m, α M ], where α M is as in (.3). (3) The stability riteria for two values of α lying between two onseutive ritial values are equivalent; thus the orresponding moduli spaes are isomorphi. (4) If α is generi and GCD(n, n 1 n, d 1 d ) = 1, then α-semistability is equivalent to α-stability. For the appliation of triples to U(p, q)-higgs bundles ([7]; see also the Introdution), it is important to have riteria to rule out strit α-semistability when α = g, where g is the genus of the surfae. One suh riterion, dealing atually with any integral values of α, is given by the following. Lemma.7. Let m be an integer suh that GCD(n 1 n, d 1 d mn 1 ) = 1. Then (1) α = m is not a ritial value, () there are no α-independent semistable triples. Proof. To prove (1), suppose that α = m is a ritial value. There exist then a triple T and a proper subtriple T so that (d 1 d mn )(n 1 n ) = (d 1 d mn )(n 1 n ). 7

9 8 Thus n 1 n divides (d 1 d mn )(n 1 n ). But n 1 n > n 1 n, so we get that GCD(n 1 n, d 1 d mn ) > 1. Writing d 1 d mn = d 1 d mn 1 m(n 1 n ), we see that GCD(n 1 n, d 1 d mn 1 ) > 1, in ontradition with the hypothesis. To prove (), we show that GCD(n, n 1 n, d 1 d ) = 1, from whih the result follows by (4) in Proposition.6. Suppose that GCD(n, n 1 n, d 1 d ) 1. Then there is (n 1, n, d 1, d ) suh that n n 1 n = n n 1 n and d 1 d n 1 n = d 1d n 1 n. It follows that d 1 d mn 1 n 1 n = d 1 d mn 1 n 1 n, and hene GCD(n 1 n, d 1 d mn 1 ) 1, in ontradition with the hypothesis..3. Vortex equations. There is a orrespondene between stability and the existene of solutions to ertain gauge-theoreti equations on a triple T = (E 1, E, φ), known as the vortex equations ([4] and [13]). The vortex equations 1ΛF (E1 ) φφ = τ 1 Id E1, (.6) 1ΛF (E ) φ φ = τ Id E, are equations for Hermitian metris on E 1 and E. Here Λ is ontration by the Kähler form of a metri on X (normalized so that vol(x) = π), F (E i ) is the urvature of the unique onnetion on E i ompatible with the Hermitian metri and the holomorphi struture of E i, and τ 1 and τ are real parameters satisfying d 1 d = n 1 τ 1 n τ. Here φ is the adjoint of φ with respet to the Hermitian metris. One has the following. Theorem.8. [4, Theorem 5.1] A solution to (.6) exists if and only if T is α- polystable for α = τ 1 τ. Using the vortex interpretation of the moduli spae of triples one an easily identify the moduli spae of triples for α = α m. Proposition.9. A triple T = (E 1, E, φ) is α m -polystable if and only if φ = 0 and E 1 and E are polystable. We thus have N αm (n 1, n, d 1, d ) = M(n 1, d 1 ) M(n, d ), where M(n i, d i ) is the moduli spae of semistable bundles of rank n i and degree d i. Proof. Consider equations (.6) on T. If α = α m then τ 1 = µ 1 and τ = µ and hene in order to have solutions of (.6) we must have φ = 0. In this ase, (.6) say that the Hermitian metris on E 1 and E have onstant entral urvature. But this is equivalent to the polystability of E 1 and E by the theorem of Narasimhan and Seshadri [5]. 3. Extensions and deformations of triples In order to analyse the differenes between the moduli spaes N α as α hanges, as well as the smoothness properties of the moduli spae for a given value of α, we need to study the homologial algebra of triples. This is done by onsidering the hyperohomology of a ertain omplex of sheaves, in a similar way to what is done in the study of infinitesimal deformations by Biswas and Ramanan [3]. In fat, it is a speial ase of the more general situation onsidered in [16].

10 3.1. Extensions. Let T = (E 1, E, φ ) and T = (E 1, E, φ ) be two triples and, as usual, let (n, d ) = (n 1, n, d 1, d ), (n, d ) = (n 1, n, d 1, d ), where n i = rk(e i), n i = rk(e i ), d i = deg(e i) and d i = deg(e i ). Let Hom(T, T ) denote the linear spae of homomorphisms from T to T, and let Ext 1 (T, T ) denote the linear spae of equivalene lasses of extensions of the form 0 T T T 0, where by this we mean a ommutative diagram 0 E 1 E 1 E 1 0 φ φ φ 0 E E E 0. Hene, to analyse Ext 1 (T, T ) one onsiders the omplex of sheaves where the map is defined by C (T, T ): E 1 E 1 E E E E 1, (3.1) (ψ 1, ψ ) = φ ψ ψ 1 φ. Proposition 3.1. There are natural isomorphisms Hom(T, T ) = H 0 (C (T, T )), Ext 1 (T, T ) = H 1 (C (T, T )), and a long exat sequene assoiated to the omplex C (T, T ): 0 H 0 (C (T, T )) H 0 (E 1 E 1 E E ) H 0 (E E 1) H 1 (C (T, T )) H 1 (E 1 E 1 E E ) H 1 (E E 1) H (C (T, T )) 0. (3.) Proof. The proof is omitted sine it is very similar to that given in [3] in the study of deformations, and it is a speial ase of a muh more general result proved in [16]. We introdue the following notation: h i (T, T ) = dim H i (C (T, T )), χ(t, T ) = h 0 (T, T ) h 1 (T, T ) h (T, T ). (3.3) Proposition 3.. For any holomorphi triples T and T we have χ(t, T ) = χ(e 1 E 1) χ(e E ) χ(e E 1) = (1 g)(n 1n 1 n n n n 1) n 1d 1 n 1d 1 n d n d n d 1 n 1d, where χ(e) = dim H 0 (E) dim H 1 (E) is the Euler harateristi of E. 9

11 10 Proof. Immediate from the long exat sequene (3.) and the Riemann Roh formula. Corollary 3.3. For any extension 0 T T T 0 of triples, χ(t, T ) = χ(t, T ) χ(t, T ) χ(t, T ) χ(t, T ). Remark 3.4. Proposition 3. shows that χ(t, T ) depends only on the topologial invariants (n, d ) and (n, d ) of T and T. Whenever onvenient we shall therefore use the notation χ(n, d, n, d ) = χ(t, T ). 3.. Vanishing of H 0 and H. The following vanishing results play a entral role in our study. Proposition 3.5. Suppose that T and T are α-semistable. (1) If µ α (T ) < µ α (T ) then H 0 (C (T, T )) = 0. () If µ α (T ) = µ α (T ) and T is α-stable, then { H 0 (C (T, T )) C if T = T = 0 if T = T. Proof. By Proposition 3.1 we an identify H 0 (C (T, T )) with Hom(T, T ). The statements (1) and () are thus the diret analogs for triples of the same results for semistable bundles. The proof is idential. Suppose that h : T T is a non-trivial homomorphism of triples. If T = (E 1, E, Φ ) and T = (E 1, E, Φ ) then h is given by a pair of holomorphi maps u i : E i E i for i = 1, suh that Φ u = u 1 Φ. We an thus define subtriples of T and T respetively by T N = (ker(u 1 ), ker(u ), Φ ) and T I = (im(u 1 ), im(u ), Φ ), where in T I, it is in general neessary to take the saturations of the image im(u 1 ) and im(u ). By the semistability onditions, we get The onlusions follow diretly from this. µ α (T N ) µ α (T ) µ α (T I ) µ α (T ). Proposition 3.6. Suppose that the triples T and T are α-semistable and satisfy µ α (T ) = µ α (T ). Then (1) H (C (T, T )) = 0 whenever α > g. () If one of T, T is α ɛ-stable for some ɛ 0, then H (C (T, T )) = 0 whenever α g. Proof. From (3.) it is lear that the vanishing of H (C (T, T )) is equivalent to the surjetivity of the map H 1 (E 1 E 1 E E ) H 1 (E E 1). By Serre duality this is equivalent to the injetivity of the map H 0 (E 1 E K) P E 1 K) H 0 (E E K) H 0 (E 1 ψ ((φ Id) ψ, ψ φ ). (3.4) Proof of (1): Suppose that P is not injetive. Then there is a non-trivial homomorphism ψ : E 1 E K in ker P. Let I = im ψ and N = ker ψ. Sine (φ Id K ) ψ = 0,

12 I ker φ and hene T I = (0, I K, 0) is a proper subtriple of T. Similarly, the fat that ψ φ = 0 implies that im φ N and thus T N = (ker ψ, E, φ ) is a proper subtriple of T. Let k = rk(n) and l = deg(n). Then, from the exat sequene 0 N E 1 I 0 we see that rk(i) = n 1 k and deg(i) = d 1 l. Hene µ α (T N) = l d α n, k n k n µ α (T I ) = d 1 l g α. n 1 k Adding these two expressions, and learing denominators we see that d 1 d (n 1 k)( g) α(n 1 n k) = (k n )µ α (T N) (n 1 k)µ α (T I ). But µ α (T N ) µ α(t ), µ α (T I ) µ α(t ) and µ α (T ) = µ α (T ). From this we obtain that d 1 d (n 1 k)( g) α(n 1 n k) d 1 d αn, (3.5) and hene α(n 1 k) (n 1 k)(g ). Sine n 1 k > 0 we get that α g. Hene P must be injetive if the hypotheses of the part (1) of the proposition are satisfied. Proof of (): Suppose that T is αɛ-stable for some ɛ 0. It follows that µ αɛ (T I ) < µ αɛ (T ), i.e µ α (T I ) µ α (T ) < ɛ( n n 1 n 1) 0. Thus, following exatly the same argument as in the proof of (1), we get a strit inequality in (3.5). We onlude that that if P is not injetive then α < g, i.e. if α g then P must be injetive. If T is α ɛ-stable for some ɛ 0 then we get that µ α (T N) µ α (T ) < ɛ( n ) 0. n 1 n k n The rest of the argument is the same as in the ase that T is α ɛ-stable. Corollary 3.7. Let T and T be α-semistable triples with µ α (T ) = µ α (T ), and α > g. Then n dim Ext 1 (T, T ) = h 0 (T, T ) χ(t, T ). The same holds for α g if in addition T or T is α ɛ-stable for some ɛ 0. Proof. It follows from Proposition 3.1 and (3.3) that dim Ext 1 (T, T ) = h 0 (T, T ) h (T, T ) χ(t, T ). (3.6) The result follows immediately from this and the vanishing of h (T, T ) given by Proposition

13 Deformation theory for triples. Sine the spae of infinitesimal deformations of T is isomorphi to H 1 (C (T, T )), the onsiderations of the previous setions also apply to studying deformations of a holomorphi triple T. To be preise, one has the following. Theorem 3.8. Let T = (E 1, E, φ) be an α-stable triple of type (n 1, n, d 1, d ). (1) The Zariski tangent spae at the point defined by T in the moduli spae of stable triples is isomorphi to H 1 (C (T, T )). () If H (C (T, T )) = 0, then the moduli spae of α-stable triples is smooth in a neighbourhood of the point defined by T. (3) H (C (T, T )) = 0 if and only if the homomorphism H 1 (E 1 E 1 E E ) H 1 (E E 1 ) in the orresponding long exat sequene is surjetive. (4) At a smooth point T N s α(n 1, n, d 1, d ) the dimension of the moduli spae of α-stable triples is dim N s α(n 1, n, d 1, d ) = h 1 (T, T ) = 1 χ(t, T ) = (g 1)(n 1 n n 1 n ) n 1 d n d 1 1. (3.7) (5) If φ is injetive or surjetive then T = (E 1, E, φ) defines a smooth point in the moduli spae. (6) If α g, then T defines a smooth point in the moduli spae, and hene N s α(n 1, n, d 1, d ) is smooth. Proof. Statements (1) and () follow from Theorems.3 and 3.1 in [3], respetively. An indiret proof of (1) and (), exploiting the orrespondene between triples on X and stable bundles on X P 1 (see Setion 9) also follows from [4]. Statement (3) follows from the long exat sequene (3.) with T = T = T. (4) follows from (1), () and Propositions 3. and 3.7. (5) is proved in [4, Proposition 6.3]. (6) is a onsequene of Proposition Bounds for χ In our approah to the study of how the moduli spaes of triples vary with the parameter, it is of ruial importane to be able to estimate the Euler harateristis χ(t, T ) = χ(n, d, n, d ) when T and T are polystable triples with the same α- slope. The basi idea is to identify χ(t, T ) as a hyperohomology Euler harateristi for the omplex C (T, T ) defined in (3.1) and to notie that the omplex is itself a holomorphi triple. As suh it ought to satisfy a stability ondition indued from the stability ondition of T and T. In priniple, a way to obtain the stability ondition for C (T, T ) should be provided by the orrespondene between the stability of the holomorphi triples and the existene of solutions to the vortex equations given by Theorem.8. However, there seem to be no simple way to onstrut a solution to the vortex equations for C (T, T ) from solutions on T and T. Instead we onsider slightly more general objets than triples, known as holomorphi hains. These are studied in [1].

14 Holomorphi hains. A holomorphi hain is a diagram C : E m φ m φ m 1 φ 1 Em 1 E0, where eah E i is a holomorphi vetor bundle and φ i : E i E i 1 is a holomorphi map. Let µ(c) = µ(e 0 E m ), λ i (C) = rk(e i ) m i=0 rk(e, i = 0,..., m. i) For α = (α 1,..., α m ) R m, the α-slope of C is defined to be m µ α (C) = µ(c) α i λ i (C). The notion of α-stability is defined via the standard α-slope ondition on subhains, that is, for any holomorphi subhain C C we must have µ α (C ) < µ α (C). Semistability and polystability are defined as usual. A holomorphi triple is a holomorphi hain of length, and the stability notions oinide, taking α = (α). As for triples, there are natural gauge-theoreti equations for holomorphi hains, whih we now desribe. Define τ = (τ 0,..., τ m ) R m1 by i=1 τ i = µ α (C) α i, i = 0,..., m, (4.1) where we make the onvention α 0 = 0. Then α an be reovered from τ by α i = τ 0 τ i, i = 0,..., m. (4.) The τ -vortex equations 1ΛF (Ei ) φ i1 φ i1 φ i φ i = τ i Id Ei, i = 0,..., m, are equations for Hermitian metris on E 0,..., E m. Here, as in (.6), F (E i ) is the urvature of the Hermitian onnetion on E i, Λ is ontration with the Kähler form and vol(x) = π. By onvention φ 0 = φ m1 = 0. One has the generalization of Theorem.8 to the ase of holomorphi hains. Theorem 4.1. [1, Theorem 3.4] A holomorphi hain C is α-polystable if and only if the τ -vortex equations have a solution, where α and τ are related by (4.1). 4.. A length 3 holomorphi hain. Let T = (E 1, E, φ ) and T = (E 1, E, φ ) be two triples. Let us onsider the length 3 holomorphi hain where C (T, T ): E 1 E a E 1 E 1 E E a 1 E E 1, (4.3) a (ψ) = (φ ψ, ψφ ), a 1 (ψ 1, ψ ) = φ ψ ψ 1 φ. We shall sometimes write this hain briefly as C (T, T ): C a C1 a 1 C0. Note that the last two terms of C (T, T ) oinide with the omplex C (T, T ). Note also that C (T, T ) is not in general a omplex. Our goal in this setion is to prove,

15 14 using Theorem 4.1, that if T and T are α-polystable then C (T, T ) is α-polystable for a suitable hoie of α. Lemma 4.. Let T and T be holomorphi triples and suppose we have solutions to the (τ 1, τ )-vortex equations on T and the (τ 1, τ )-vortex equations on T, suh that τ 1 τ 1 = τ τ. Then the indued Hermitian metri on C (T, T ) satisfies the hain vortex equations 1ΛF (C0 ) a 1 a 1 = τ 0 Id C0, (4.4) 1ΛF (C1 ) a a a 1 a 1 = τ 1 Id C1, (4.5) 1ΛF (C ) a a = τ Id C, (4.6) for τ = ( τ 0, τ 1, τ ) given by τ 0 = τ 1 τ, τ 1 = τ 1 τ 1 = τ τ, τ = τ τ 1. Proof. We shall only show that the indued Hermitian metri satisfies (4.5), sine the proofs that it satisfies the two remaining equations are similar (but simpler). The vortex equations for T and T are 1ΛF (E 1 ) φ φ = τ 1 Id E 1, 1ΛF (E 1 ) φ φ = τ 1 Id E 1ΛF (E ) φ φ = τ Id E, 1ΛF (E ) φ φ = τ Id E. 1, We shall write the left hand side of (4.5) in terms of these known data of the triples T and T. First, we note that and similarly for F (E ). Hene i F (E i ) = F (E i) t, i = 1,, F (C 1 ) = F (E 1 E 1 E E ) = ( F (E 1 ) Id Id F (E 1), F (E ) Id Id F (E ) ) = ( F (E 1 ) t Id Id F (E 1), F (E ) t Id Id F (E ) ). (4.7) Next we alulate a 1: note that for ξ x C 0 and (η 1 y 1, η y ) C 1 we have a 1 (ξ x), (η 1 y 1, η y ) C 1 = ξ x, a 1 (η 1 y 1, η y ) C 0 = ξ x, η 1 φ y 1 η φ (y ) C 0 = ξ x, φ t (η 1 ) y 1 η φ (y ) C 0 = ξ, φ t (η 1 ) E x, y1 ξ, η E 1 E = φ t (ξ), η 1 E 1 x, y1 E 1 ξ, η E x, φ (y ) E 1 φ (x), y = ( φ t (ξ) x, ξ φ (x)), (η 1 y 1, η y ) C 1. E

16 Hene, a 1(ξ x) = ( φ t (ξ) x, ξ φ (x) ). (4.8) Similarly, to alulate a onsider ξ x C and (η 1 y 1, η y ) C 1. Then a (η 1 y 1, η y ) = η 1 φ (y 1 ) φ t (η ) y. (4.9) Using (4.9) and (4.8) we an now alulate for (η 1 y 1, η y ) C 1 : a a (η 1 y 1, η y ) = ( η 1 φ φ (y 1 ) φ t (η ) φ (y ), φ t (η 1 ) φ (y 1 ) φ t φ t (η ) y ), (4.10) and a 1a 1 (η 1 y 1, η y ) = ( φ t φ t (η 1 ) y 1 φ t (η ) φ (y ), φ t (η 1 ) φ (y 1 ) η φ φ (y ) ). (4.11) Putting together (4.7), (4.10) and (4.11) we finally obtain ( ) 1ΛF (C1 ) a a a 1 a 1 (η1 y 1, η y ) ( = η 1 ( 1ΛF (E 1) φ φ ) (y 1 ) ( 1ΛF (E 1 ) t φ t φ t ) (η 1 ) y 1, η ( 1ΛF (E ) φ φ ) (y ) ( 1ΛF (E ) t φ t φ t ) (η ) y ). (4.1) Notie that the unpleasant mixed term ( φ t (η ) φ (y ), φ t (η 1 ) φ (y 1 ) ) appears both in a 1a 1 and a a and therefore anels. This would not have been the ase if we had onsidered the vortex equations on the triple C (T, T ) and is the reason why we must onsider the hain C (T, T ). Combining (4.1) with the vortex equations (or their transposes) for the triples T and T we get ( 1ΛF (C1 ) a a a 1 a 1 ) (η1 y 1, η y ) 15 = ( (τ 1 τ 1 )η 1 y 1, (τ τ )η y ). (4.13) Sine τ 1 τ 1 = τ τ this onludes the proof. Proposition 4.3. Let T and T be α-polystable triples. Then the holomorphi hain C (T, T ) is α-polystable for α = (α 1, α ) = (α, α). Proof. Sine the triples T and T are α-polystable, it follows from Theorem.8 that they support solutions to the (τ 1, τ )- and (τ 1, τ )-vortex equations, respetively, where α = τ 1 τ = τ 1 τ. Notie that τ 1 τ 1 = τ τ. Thus it follows from Lemma 4. that the holomorphi hain C (T, T ) supports a solutions to the hain vortex equations for τ = (τ 1 τ, τ 1 τ 1, τ τ 1 ). Now Theorem 4.1 and (4.) imply that C (T, T ) is α-polystable for α 1 = τ 1 τ τ τ = α, α = τ 1 τ τ τ 1 = α.

17 Bounds for χ(t, T ). We start with some tehnial lemmas needed to estimate the Euler harateristi χ(t, T ). Lemma 4.4. Let T = (E 1, E, φ ) and T = (E 1, E, φ ) be triples for whih the hain C (T, T ) is α = (α, α)-polystable. Let C 1 = E 1 E 1 E E, C 0 = E E 1, and a 1 : C 1 C 0 be defined as in (4.3). Then the following inequalities hold. deg(ker(a 1 )) rk(ker(a 1 ))(µ α (T ) µ α (T )), (4.14) deg(im(a 1 )) ( rk(c 0 ) rk(im(a 1 )) ) (µ α (T ) µ α (T ) α) deg(c 0 ). (4.15) Proof. If rk(ker(a 1 )) = 0 then (4.14) is obvious. Assume therefore that rk(ker(a 1 )) > 0. Using ker(a 1 ), we an then define a quotient of the hain C (T, T ) by K : 0 ker(a 1 ) 0. Thus, sine µ α (K) = µ(ker(a 1 )) α, it follows from the definition of α-polystability that We therefore have µ(ker(a 1 )) α µ α ( C (T, T )) = µ α (T ) µ α (T ) α. µ(ker(a 1 )) µ α (T ) µ α (T ), whih is equivalent to (4.14). The seond inequality, i.e. (4.15), is obvious when rk(im(a 1 )) = rk(c 0 ). We thus assume rk(im(a 1 ) < rk(c 0 ). Using the okernel oker(a 1 ) (or its saturation if it is not torsion free), we an define a subhain of the hain C (T, T ) by Q: 0 0 oker(a 1 ). By the α-polystability of C (T, T ) we have µ α (Q) µ α ( C (T, T )). This, together with the fat that µ(oker(a 1 )) deg(c 0) deg(im(a 1 )), rk(c 0 ) rk(im(a 1 )) leads diretly to Lemma 4.5. Let : V V 1 and : V V 1 be linear maps between finite dimensional vetor spaes. Assume that V 1 V 0 and V 1 V 0. Define f : Hom(V 1, V 1) Hom(V, V ) Hom(V, V 1) (ψ 1, ψ ) ψ ψ 1. If f is an isomorphism, then exatly one of the following alternatives must our: (1) V 1 = V = 0 and = = 0. () V 1 = 0, V 1, V, V 0 and : V = V 1. (3) V = 0, V 1, V 1, V 0 and : V = V 1. In partiular, if V 1, V, V 1 and V are all non-zero then f annot be an isomorphism.

18 Proof. If (, ) = (0, 0) then f = 0 and therefore Hom(V, V 1) = Hom(V 1, V 1) = Hom(V, V ) = 0. If V 1 0 then V 1 = V = 0 i.e. V 1 V = 0. Hene V 1 = 0. Similarly one sees that V = 0 and thus alternative (1) ours. Heneforth assume that (, ) (0, 0). Let r i = dim V i and r i = dim V i for i = 1,. If f is an isomorphism then r 1r 1r r = r r 1 from whih it follows that r (r 1 r ) = r 1r 1 and r 1(r r 1) = r r. Hene r 1 r, (4.16) 17 r r 1. (4.17) Assume that we have strit inequality in (4.16) and (4.17). Then, in partiular, oker( ) and ker( ) must both be non-zero. Choose a omplement to im( ) in V 1 so that V 1 = im( ) im( ). We then have an inlusion Hom(ker( ), im( ) ) Hom(V, V 1). Let ψ = (ψ 1, ψ ) Hom(V 1, V 1) Hom(V, V ) and x ker( ), then f(ψ)(x) = ψ (x) ψ 1 (x) = ψ (x), whih belongs to im( ). Hene im(f) and Hom(ker( ), im( ) ) have trivial intersetion and, therefore, f annot be an isomorphism, whih is absurd. It follows that equality must hold in at least one of the inequalities (4.16) and (4.17). Suppose that equality holds in (4.16), i.e. r 1 = r = 0. Then r 1r 1 = 0, i.e. r 1 = 0 or r 1 = 0. Suppose first that r 1 = 0, then r = r 1 = 0, whih ontradits our assumption that V 1 V 0. Thus we must have r 1 = 0 and r 1 0. We thus also have V 0 (sine r = r 1) and V 0 (sine r r 1 0). Furthermore, sine (, ) (0, 0) we an assume that 0. In this ase f(ψ 1, ψ ) = f(ψ 1, 0) = ψ 1. In partiular, if f is an isomorphism then so is. Thus alternative () ours. In a similar manner one sees that if equality holds in (4.17) then alternative (3) ours. Obviously the three alternatives are mutually exlusive. Lemma 4.6. Suppose that T and T are non-zero triples of types (n 1, n, d 1, d ) and (n 1, n, d 1, d ) respetively. Let n 1 = n 1 n 1, n = n n, d 1 = d 1 d 1, d = d d, µ 1 = d 1 /n 1, and µ = d /n. Let α m and α M be the extreme α values for the triples of type (n 1, n, d 1, d ), as defined in (.) and (.3), with the onvention that α M = if n 1 = n. Let α m < α < α M and suppose that µ α (T ) = µ α (T ), then the map annot be an isomorphism. a 1 : E 1 E 1 E E E E 1 Proof. Let us onsider the triple T = T T. It is lear that µ α (T ) = µ α (T ) = µ α (T ). If a 1 is an isomorphism then, applying Lemma 4.5 fibrewise, it follows that one of the following alternatives must our: (a) E 1 = E = 0 and φ = φ = 0. (b) E 1 = 0, E 1, E, E 0 and φ : E () E = 0, E 1, E 1, E 0 and φ : E = E 1. = E 1.

19 18 We shall onsider eah ase in turn. Case (a). In this ase we have T = (0, E, 0), T = (E 1, 0, 0) and T = (E 1, E, 0). It follows from µ α (T ) = µ α (T ) that α = µ(e 1 ) µ(e ) = α m. Case (b). In this ase we have n 1 = n 1 and n = n n = n 1n. Hene n > n 1. Furthermore, from µ α (T ) = µ α (T ) we get µ(e 1 ) α = µ(e 1 E ) n n 1 n, i.e. α = n n n 1 α m = α M. Case (). In this ase we have n = n and n 1 = n 1n 1 = n 1n. Hene n 1 > n. Furthermore, from µ α (T ) = µ α (T ) we get α = n 1 n 1 n α m = α M. If n 1 = n then ase (a) is the only possibility, so α = α m. If n 1 n, then (a) or exatly one of (b) and () are the only possibilities, depending on whether n 1 < n or n 1 > n. In both ases we see that α = α m or α = α M. Proposition 4.7. Suppose that T and T are non-zero triples of types (n 1, n, d 1, d ) and (n 1, n, d 1, d ) respetively. Let n 1 = n 1 n 1, n = n n, d 1 = d 1 d 1, d = d d, µ 1 = d 1 /n 1, and µ = d /n. Let α m and α M be the extreme α values for the triples of type (n 1, n, d 1, d ), as defined in (.) and (.3), with the onvention that α M = if n 1 = n. Let α m < α < α M. Suppose that µ α (T ) = µ α (T ) and that the hain C (T, T ), as defined in (4.3), is (α, α)-stable. Then χ(t, T ) 1 g if α g. In partiular, if g then χ(t, T ) 0. Proof. From the long exat sequene (3.) and the Riemann-Roh formula we obtain χ(t, T ) = (1 g) ( rk(c 1 ) rk(c 0 ) ) deg(c 1 ) deg(c 0 ), (4.18) where C 1 and C 0 are as in (4.3). We an apply Lemma 4.4, and then use the estimates (4.14) and (4.15). Together with these yield deg(c 1 ) = deg(ker(a 1 )) deg(im(a 1 )), (4.19) rk(c 1 ) = rk(ker(a 1 )) rk(im(a 1 )), (4.0) deg(c 1 ) (µ α (T ) µ α (T )) ( rk(c 1 ) rk(c 0 ) ) Using that µ α (T ) = µ α (T ), we an then dedue that Combining this with (4.18) we get α ( rk(c 0 ) rk(im(a 1 )) ) deg(c 0 ). deg(c 1 ) deg(c 0 ) α ( rk(c 0 ) rk(im(a 1 )) ). χ(t, T ) (1 g) ( rk(c 1 ) rk(c 0 ) ) α ( rk(c 0 ) rk(im(a 1 )) ). (4.1) If α g then we get χ(t, T ) (1 g) ( rk(c 0 ) rk(c 1 ) rk(im(a 1 )) ), with equality if and only if α = g. Furthermore rk(im(a 1 )) rk(c 0 ) and rk(a 1 ) rk(c 1 ), with equality in both if and only if a 1 is an isomorphism. Thus in all ases we get χ(t, T ) 0, with equality if and only if α = g and a 1 is an isomorphism. But by Lemma 4.6, sine α m < α < α M, then a 1 annot be an isomorphism. Thus in all ases we get rk(c 0 ) rk(c 1 ) rk(im(a 1 )) 1 and hene χ(t, T ) 1 g.

20 Remark 4.8. Sine the roles of T and T in Proposition 4.7 are symmetri, we obtain the same bound for χ(t, T ). 5. Crossing ritial values In this setion we study the differenes between the stable loi N s α(n, d) in the moduli spaes N α (n, d), for fixed values of n = (n 1, n ) and d = (d 1, d ) but different values of α. Sine in this setion n and d are fixed, we use the abbreviated notation N s α = N s α(n, d) and N α = N α (n, d). Our main result is that for all α g any differenes between the N s α are onfined to subvarieties of positive odimension. In partiular, the number of irreduible omponents of the spaes N s α are the same for all α satisfying α g and α m < α < α M 1. If the oprimality ondition GCD(n, n 1 n, d 1 d ) = 1 is satisfied, then N s α = N α at all non-ritial vales of α, so the results apply to N α for all non-ritial α g. We begin with a set theoreti desription of the differenes between two spaes N s α 1 and N s α when α 1 and α are separated by a ritial value (as defined in setion.). For the rest of this setion we adopt the following notation: Let α be a ritial value suh that α m < α < α M. (5.1) Set α = α ɛ, α = α ɛ, (5.) where ɛ > 0 is small enough so that α is the only ritial value in the interval (α, α ) Flip Loi. Definition 5.1. Let α (α m, α M ) be a ritial value for triples of type (n, d). We define flip loi S α N s by the onditions that the points in S α α represent triples whih are α -stable but α -unstable, while the points in S α represent triples whih are α -stable but α -unstable. Remark 5.. The definition of S α an be extended to the extreme ase α = α m. However, sine all α m-stable triples must be αm-unstable, we see that S α m = N s α m. Similarly, when n 1 n we get S α = N s. The only interesting ases are thus those M α M those for whih α m < α < α M. Lemma 5.3. In the above notation: N s α S α = N s α = N s α S α. (5.3) Proof. By definition we an identify N s S α α = N s S α α. Suppose now that t is a point in N s S α α = N s S α α, but that t is not in Nα s. Let T be a triple representing t. Then T has a subtriple T T for whih µ α (T ) µ α (T ), and also µ α (T ) < µ α (T ). This is not possible, and hene t Nα s. Finally, suppose that t Nα s and let T be a triple representing t. Then µ α (T ) < µ α (T ) for all subtriples T T. But sine the set of possible values for µ α (T ) is a disrete subset of R, we an find a δ > 0 suh that µ α (T ) µ α (T ) δ for all 1 When n 1 n the bounds α m and α M are as in (.) and (.3). When n 1 = n we adopt the onvention that α M = 19

21 0 subtriples T T. Thus µ α (T ) µ α (T ) < 0. That is, t is in N s α, and hene Nα s N s α S α. Our goal is to show that the flip loi S α odimension in N s respetively. α are ontained in subvarieties of positive Proposition 5.4. Let α (α m, α M ) be a ritial value for triples of type (n, d) = (n 1, n, d 1, d ). Let T = (E 1, E, φ) be a triple of this type. (1) Suppose that T represents a point in S α, i.e. suppose that T is α -stable but α -unstable. Then T has a desription as the middle term in an extension 0 T T T 0 (5.4) in whih (a) T and T are both α -stable, with µ α (T ) < µ α (T ), (b) T and T are both α -semistable with µ α (T ) = µ α (T ). () Similarly, if T represents a point in S α, i.e. if T is α -stable but α -unstable, then T has a desription as the middle term in an extension (5.4) in whih (a) T and T are both α -stable with µ α (T ) < µ α (T ), (b) T and T are both α -semistable with µ α (T ) = µ α (T ). Proof. In both ases (i.e. (1) and ()), sine its stability property hanges at α, the triple T must be stritly α -semistable, i.e. it must have a proper subtriple T with µ α (T ) = µ α (T ). We an thus onsider the (non-empty) set F 1 = {T T µ α (T ) = µ α (T ) }. Proof of (1) Suppose first that T is α -stable but α -unstable. We observe that if T n F 1, then < n n 1 n n 1 n, sine otherwise T ould not be α -stable. But the n allowed values for are limited by the onstraints 0 n n 1 n 1 n 1, 0 n n and n 1 n 0. We an thus define { } n λ 0 = max n 1 n T F 1 and set { } n F = T 1 F 1 = λ n 1 n 0. Now let T be any triple in F. Sine T has maximal α -slope, we an assume that T = T/T is a loally free triple, i.e. if T = (E, E 1, Φ) then E and E 1 are both loally free. Furthermore, sine T is α -semistable and µ α (T ) = µ α (T ) = µ α (T ), it follows that both T and T are α -semistable and of the same α -slope. We now show that T is α -stable. Suppose not. Then there is a proper subtriple T T with µ α ( T ) µ α (T ). However, sine we an assume that α is not a ritial value for triples of type ( T ), we must have µ α ( T ) > µ α (T ). Thus, sine (T ) is α -semistable, we must have µ α ( T ) µ α (T ) and also ñ ñ 1 ñ > n. n 1 n

22 If µ α ( T ) < µ α (T ), say µ α ( T ) = µ α (T ) δ, then in order to have µ α ( T ) > µ α (T ) we must have ñ ñ 1 ñ > n n 1 n ñ ñ 1 ñ δ ɛ. Letting ɛ approah zero, we see that must be arbitrarily large. This annot be if 0 ñ 1 n 1 and 0 ñ n (and ñ 1 ñ > 0). We may thus assume that µ α ( T ) = µ α (T ). Consider now the subtriple T T defined by the pull-bak diagram 0 T T T 0. This has µ α ( T ) = µ α (T ) = µ α (T ) and thus ñ λ ñ 1 ñ 0 = n. n 1 n It follows from this and the above extension that ñ λ ñ 1 ñ 0 = n. n 1 n However, sine µ α (T ) = µ α (T ) but µ α (T ) < µ α (T ), we have that < n. n 1 n n 1 n Combining the previous two inequalities we get n ñ ñ 1 ñ < n n 1 n whih is a ontradition. Now take T F with minimum rank (i.e. minimum n 1n ) in F. We laim that T is α -stable. If not, then as before it has a proper subtriple T with µ α ( T ) µ α (T ) and ñ ñ 1 ñ > n n 1 n. Then ñ 1ñ < n 1n, whih ontradits the minimality of n 1n. Thus T is α -stable. Moreover, sine T is α -stable it follows that µ α (T ) < µ α (T ). Thus taking T F with minimum rank, and T = T/T, we get a desription of T as an extension in whih (a)-(b) are satisfied. n n 1 n Proof of (). If T is α -stable but α -unstable, then The proof of (a) must thus be modified as follows. With { } n λ 0 = min n 1 n T F 1 1 > n n 1 n for all T F 1. we an define { } F = T n F 1 = λ n 1 n 0 and selet T F suh that T has minimal rank in F. It follows in a similar fashion to that above that T has a desription as 0 T T T 0 in whih all the requirements of the proposition are satisfied.

23 Remark 5.5. Unlike for Jordan-Hölder filtrations for semistable objets, the filtrations produed by the above proposition are always of length two, i.e. always yield a desription of the semistable objet as an extension of stable objets. This is ahieved by exploiting the extra degree of freedom provided by the parameter α. The true advantage of never having to onsider extensions of length greater than two is that it removes the need for indutive proedures in the analysis of the flip loi. Definition 5.6. Let α (α m, α M ) be a ritial value for triples of type (n, d). Let (n, d ) = (n 1, n, d 1, d ) and (n, d ) = (n 1, n, d 1, d ) be suh that (n, d) = (n, d ) (n, d ), (5.5) (i.e. n 1 = n 1 n 1, n = n n, d 1 = d 1 d 1, and d = d d ), and also d 1 d n α n 1 n n 1 n = d 1 d n 1 n α n n. (5.6) 1 n (1) Define S α (n, d, n, d ) to be the set of all isomorphism lasses of extensions 0 T T T 0, where T and T are α -stable triples with topologial invariants (n, d ) and (n, d ) respetively, and the isomorphism is on the triple T. () Define S 0 (n, d, n, d ) S α α (n, d, n, d ) to be the set of all extensions for whih moreover T is α -stable. In an analogous manner, define S α (n, d, n, d ) and S 0 (n, d, n, d ) S α α (n, d, n, d ). (3) Define S α = Sα (n, d, n, d ), S0 α = S0 α (n, d, n, d ) where the union is over all (n 1, n, d 1, d ) and (n 1, n, d 1, d ) suh that the above n onditions apply, and also < n n. 1 n n 1 n (4) Similarly, define S α = Sα (n, d, n, d ), S0 α = S0 α (n, d, n, d ) where the union is over all (n 1, n, d 1, d ) and (n 1, n, d 1, d ) suh that the above onditions apply, and also. n n 1 n > n n 1 n Remark 5.7. It an happen that S 0 or S 0 is empty. For instane there may be α α no possible hoies of (n 1, n, d 1, d ) and (n 1, n, d 1, d ) whih satisfy all the required onditions. In this ase, the impliation of the next lemma is that one or both of the flip loi S α is empty. Lemma 5.8. There are maps, say v : S0 N s, whih map triples to their α α equivalene lasses. The images ontain the flip loi S α. Proof. The existene of the maps is lear. The seond statement, about the images of the maps, follows by Proposition 5.4. Indeed, suppose that T represents a point in S α and that 0 T T T 0

24 is an extension of the type desribed in proposition 5.4, with T a triple of type (n, d ) and T a triple of type (n, d ). Then (n, d ) and (n, d ) satisfy onditions (5.5) and (5.6). Furthermore, sine µ α (T ) < µ α (T ), we must have. Thus T is n n 1 n < n n 1 n ontained in v ( S 0 ). A similar argument shows that S α α is ontained in v ( S 0 ). α 5.. Codimension estimates and omparison of moduli spaes. Consider a ritial value α (α m, α M ) for triples of type (n, d). Fix (n, d ) = (n 1, n, d 1, d ) and (n, d ) = (n 1, n, d 1, d ) as in Definition 5.6. For simpliity we shall denote the moduli spaes of α -semistable triples of type (n, d ), respetively (n, d ), by N α = N α (n, d ) and N α = N α (n, d ). Proposition 5.9. If α > g then S α (n, d, n, d ) is a loally trivial fibration over N N, with projetive fibers of dimension α α χ(n, d, n, d ) 1. In partiular, S α (n, d, n, d ) has dimension 1 χ(n, d, n, d ) χ(n, d, n, d ) χ(n, d, n, d ), where χ(n, d, n, d ) et. are as in setion 3. The same is true for S α (n, d, n, d ) when α = g. Proof. From the defining properties of S α (n, d, n, d ) there is map S α (n, d, n, d ) N N (5.7) α α whih sends an extension 0 T T T 0 to the pair ([T ], [T ]), where [T ] denotes the lass represented by T and similarly for [T ]. We first examine the fibers of this map. Notie that T and T satisfy the hypothesis of Proposition 3.6 and therefore of Corollary 3.7. Notie moreover that, sine µ α (T ) < µ α (T ), it is not possible to have T = T. Thus (f. Corollary 3.7 and Proposition 3.5()) we have dim P(Ext 1 (T, T )) = dim Ext 1 (T, T ) 1 = χ(t, T ) 1 = χ(n, d, n, d ) 1, (5.8) whih is independent of T and T. Note that if α = g, T and T satisfy the hypothesis of Proposition 3.5()) for α, but not for α. It remains to establish that the fibration (5.7) is loally trivial. If the oprimality onditions GCD(n 1, n, d 1 d ) = 1 = GCD(n 1, n, d 1 d ) hold then the moduli spaes N and N are fine moduli spaes (f. [6]). That is, there are universal α α objets, say U and U, defined over N X and N X. These an be viewed as α α oherent sheaves of algebras (f. []), or more preisely as examples of the Q-bundles onsidered in [16]. Pulling these bak to N N X we an onstrut Hom(U, U ) α α (where we have abused notation for the sake of larity). Taking the projetion from N N X onto N N, we an then onstrut the first diret image sheaf. By α α α α 3