BALANCED METRICS ON TWISTED HIGGS BUNDLES MARIOGARCIA-FERNANDEZANDJULIUSROSS “My name is Claire Bennet and that was attempt number...I guess I’ve lost count.” Claire Bennet, Heroes 4 Contents 1 0 1. Introduction 1 2 2. Balanced metrics 5 n 3. Geometric Invariant Theory 8 a 4. Stability and the Existence of a Balanced Metric 14 J 5. Asymptotics of the Balanced Condition 17 0 6. Limits of balanced metrics 24 3 7. Generalizations 26 ] Appendix A. Weakly Geometric Metrics 28 G References 30 D . h t a m 1. Introduction [ By a twisted Higgs bundle on a Ka¨hler manifold X we shall mean a pair (E,φ) consisting of a holomorphic vector bundle E and a holomorphic bundle morphism 2 v φ: M ⊗E →E 8 0 for some holomorphic vector bundle M. Such objects were first considered by 1 Hitchin [21] when X is a curve and M is the tangent bundle of X, and also by 7 Simpson [37] for higher dimensional base. . 1 Forachoiceofpositiverealconstantc,thereisaHitchin-Kobayashicorrespondence 0 4 [2, 8, 21, 36] for such pairs, generalizing the Donaldson–Uhlenbeck–Yau Theorem 1 [10, 39] for vector bundles. This result states that (E,φ) is polystable if and only : if E admits a hermitian metric h solving the Hitchin equation v Xi iΛFh+c[φ,φ∗]=λId, (1.1) r where F denotes the curvature of the Chern connection of the hermitian metric, a h [φ,φ∗] = φφ∗ −φ∗φ with φ∗ denoting the adjoint of φ taken fibrewise and λ is a topological constant. Theaforementionedcorrespondenceisapowerfultooltodecidewhetherthereexists a solution of (1.1), but it provides little informationas to the actualsolution h. In this paper we study a quantization of this problem that is expressed in terms of finite dimensional data and “balanced metrics” that give approximate solutions to the Hitchin equation. To discuss details, suppose that X is projective, so carries an ample line bundle L which admits a positive hermitian metric h whose curvature is a K¨ahler form ω, L and also fix a hermitian metric on M. The hypothesis we will make throughout this paper on the vector bundle M is that it is globally generated (we expect that 1 2 M.GARCIA-FERNANDEZANDJ.ROSS this hypothesis can be removed). Writing E(k) := E ⊗Lk, we fix a sequence of positive rationals δ =δ =O(kn−1) and let k dimH0(E(k)) χ=χ = , k rk Vol(X,L) E which is a topologicalconstantof order O(kn). We recallthat the sections of E(k) give a natural embedding ι: X →G:=G(H0(E(k));rk ) E into the Grassmannian of rk -dimensional quotients of H0(E(k)). To capture the E Higgs field φ consider the composition φ =φ : H0(M)⊗H0(E(k))→H0(M ⊗E(k))→φ H0(E(k)) ∗ ∗,k wherethe firstmap is the naturalmultiplication. Notice that H0(M)is hermitian, sinceitcarriestheL2-metricinducedbythehermitianmetriconM andthevolume form determined by ω. Thus given a metric on H0(E(k)) (by which we mean a metric induced from a hermitian inner product) there is an adjoint (φ )∗: H0(E(k))→H0(M)⊗H0(E(k)). ∗ From this we define an endomorphism of H0(E(k)) by δ[φ ,(φ )∗] P :=χ−1 Id+ ∗ ∗ , 1+|||φ∗|||2 ! where |||φ |||2 :=tr((φ )∗φ ) (see §2.2 for details). Observe that P depends on the ∗ ∗ ∗ choice of metric on H0(E(k)) since the adjoint (φ )∗ does. ∗ Definition 1.1. We say that a metric on H0(E(k)) is balanced if for some or- thonormal basis s=(s ) we have j ωn (s ,s ) =P , (1.2) l j ι∗hFS n! jl ZX where h denotes the Fubini-Study metric on G and P =(P ) in this basis. FS jl Definition 1.2. A hermitian metric h on E is said to be a balanced metric for (E,φ) at level k if it is the pullback of the induced Fubini-Study metric for some balanced metric on H0(E(k)), i.e. h=h−k⊗ι∗h L FS InthiscasewerefertothemetriconH0(E(k))asthecorrespondingbalancedmetric. One verifies easily that if (1.2) holds for some orthonormal basis then it holds for any orthonormal basis. In fact, the left hand side of (1.2) is simply the matrix of the L2-metric induced by ι∗h . Thus when φ = 0 this is precisely the standard FS definition of a balanced metric on E as considered by Wang [42, 43]. The two main results of this paper focus on different aspects of this definition. Firstwe will show that a balancedmetric admits an interpretationas the zero of a moment map. Thus the existence of such a metric should be thought of as a kind of stability condition, and we show this is the case: Theorem1.3. AssumethatM isglobally generated. AtwistedHiggsbundle(E,φ) is Gieseker-polystable if and only if for all k sufficiently large it carries a balanced metric at level k. SecondweinvestigatehowbalancedmetricsrelatetosolutionstotheHitchinequa- tion. This turns out to be a much more complicated and interesting problem than for the case φ=0 [45]. BALANCED METRICS ON TWISTED HIGGS BUNDLES 3 Theorem 1.4. Assume that M is globally generated. Suppose h is a sequence of k hermitian metrics on E which converges (in C∞ say) to h as k tends to infinity. Suppose furthermore that h is balanced at level k and that the sequence of corre- k sponding balanced metrics on H0(E(k)) is “weakly geometric”. Then h is, after a possible conformal change, a solution of Hitchin equations. By the weakly geometric hypothesis we mean that the operator norm of φ is uni- ∗ formly bounded over k, and its Frobenius norm is strictly O(kn). This assumption isquitenaturalforaslongasφ6=0itholds,forinstance,ifthissequenceofmetrics is “geometric” by which we mean it is the L2-metric induced by some hermitian metric on E. 1.1. Proofs and techniques: TherearethreemainpartsoftheproofofTheorem 1.3. In the first partwe identify a complex parameterspace for twisted Higgs bun- dles (§2) carryinga positive symplectic structure and a moment mapthat matches the balanced condition. In the second part we extend a classical result of Gieseker [17]to characterizestability oftwistedHiggsbundles in termsofGeometric Invari- antTheory(Theorems3.5and3.6). The proofis thencompletedby anadaptation of Phong-Sturm’s refinement [29] of Wang’s result in the case φ = 0 [43]. The positivityofthe symplecticstructureandthe linearization(usedintheGITresult) turn out to be the main obstacles to undertake our construction for general M. The proof of Theorem 1.4 starts with the observation that the balanced condition, which appears as a condition involving finite-dimensional matrix groups, interacts with the K¨ahler geometry of X via the identity (Ps′)(·,s′) =Id, (1.3) j j Hk j X wherethes′ formanorthonormalbasisfortheL2-metricinducedbyH =h ⊗hk. j k k L Using the weakly geometric hypothesis, we are able to prove in Theorem 5.3 an asymptotic expansion for the endomorphism P around χ−1Id, which relates the left hand side endomorphim to the Bergman function B = s′(·,s′) . k j j Hk j X Equation(1.3)combinedwiththeHormanderestimateimpliesthentheasymptotic condition (in L2-norm) B +ckn−1[φ,φ∗]=χId+O(kn−2). (1.4) k With this at hand, the key tool for the proof of Theorem 1.4 is the asymptotic expansion of the Bergman kernel [9, 14, 26, 38, 47, 48], which says that 1 S 1 k−nB =Id+ ΛF + ω Id +O (1.5) k k h 2 k2 (cid:18) (cid:19) (cid:18) (cid:19) where S is the scalar curvature of ω. For vector bundles without a Higgs field, ω Theorem1.4 followsalmostimmediately fromthis expansion(as observedby Don- aldson). With the introduction of the Higgs field the proof is much more involved, essentially for the following reason: given a holomorphic map φ: E → F between hermitian vector bundles, no information is lost when one considers instead the pushforward φ : H0(E(k)) → H0(F(k)) for k sufficiently large. However the ad- ∗ jointφ∗: F →E isnotholomorphic,andsoone cannotdothe samething (atleast not with the space of holomorphic sections). The natural object to consider in- steadis the adjointof(φ )∗: H0(F(k))→H0(E(k)) takenwith respectto induced ∗ L2-metrics, and we shall prove that that this adjoint captures all the information that we need. Thus we have a method for quantizing the adjoint of a holomorphic bundle morphism, which is a tool that we hope will be of use elsewhere. 4 M.GARCIA-FERNANDEZANDJ.ROSS ForvectorbundleswithouttheHiggsfieldφ,apertubationargumentofDonaldson [11] gives the converse to Theorem 1.4. We expect the same argument can be applied to non-zero φ and to show that a solution to the Hitchin equation gives a sequence of balanced metrics that is weakly geometric (and plan to take this up in a sequel). 1.2. Comparison with Other Work: Ourmotivationfor this study comes from work of Donagi–Wijnholt [13, §3.3] concerning balanced metrics for twisted Higgs bundles on surfaces with M = K−1, which in turn was motivated by physical X quantities whose calculation depended on detailed knowledge of the solutions of the Hitchin equations. In this case the equations go under the name of Vafa- Witten equations and are particularly interesting [20, 46], arising directly from the study of supersymmetric gauge theories in four dimensions [40]. In the work [13] the authors consider the equation B +ckn−1[φ,φ∗]=χId+O(kn−2). (1.6) k as the defining condition for the balanced metrics. This equation, however, was to be taken “pro forma” rather than as part of any general framework. We will see that our definition of balanced agrees (and refines) that of Donagi–Wijnholt, and thus puts this work into the theory of moment-maps. We stress that the work herecanonlybeappliedtothe Vafa–WittenequationsifK−1 isgloballygenerated X (which obviously holds on Calabi-Yau manifolds for instance) but expect that it is possible to relax this hypothesis. Another interesting arena for the application of our results is the theory of co-Higgs bundles [30, 31],in which M = TX∗, allows further interesting examples where the globally generated assumption is satisfied. ArelatednotionofbalancedmetricwasintroducedbyJ.Kellerin[18], forsuitable quiver sheavesarising from dimensionalreduction consideredin [3], but as pointed outin[2]thisdoesnotallowtwistingintheendomorphismandthusdoesnotapply to twisted Higgs bundles. We remark also that our definition differs from that of L. Wang for which the analogue of Theorem 1.3 was missing [41, Remark p.31]. We will also discuss in §7 further possible extensions, at which point the precise relationship between these different notions becomes clearer. By being finite dimensional approximations to solutions to the Hitchin equations (or to the Hermitian-Yang-Mills equation in the case φ = 0), balanced metrics are amenable to numerical techniques. We expect that a version of Donaldson’s approximation theorem [12] should hold in this setting. If this is the case then Donaldson’siterativetechniquescanreasonablybeappliedinthesettingoftwisted Higgs bundles (as proposed by Donagi-Wijnholt). In particular one should be able tousethis to approximatethe Weyl-Petersonmetrics onthe moduli spaceofHiggs bundlesandvorticesbyadaptingtheideasin[19],butnoneofthiswillbeconsidered further in this paper. 1.3. Organization: We start in §2 with a discussion of the parameter space for twisted Higgs bundles that we will use, and give the details of the definition of a balanced metric. We then show that the existence of a balanced metric has an interpretationasazeroofamomentmaponthisparameterspace. Wethendiscuss in §3 the stability of a twisted Higgs bundle and its connection with Geometric Invariant Theory. In §4 we give a direct proof of the necessity of stability for the existenceofabalancedmetric,whichis infactsimplerthanexisting proofsevenin the case of vector bundles, and then give the proof of the first Theorem. Finally, in §5 and §6 we take up the relationship between the balanced condition and the Hitchin equation. Acknowledgements: We wish to thank Bo Berndtsson, Julien Keller and Luis A´lvarez-Co´nsulandMartijnWijnholtforhelpfulcommentsanddiscussions. During BALANCED METRICS ON TWISTED HIGGS BUNDLES 5 this project JR has been supported by an EPSRC Career Acceleration Fellowship andMGFbytheE´colePolytechniqueF´ed´eraldeLausanne,theHausdorffResearch InstituteforMathematics(Bonn)andtheCentreforQuantumGeometryofModuli Spaces (Aarhus). 2. Balanced metrics 2.1. A Parameter Space for twisted Higgs bundles. Let X be a smooth projective manifold and L an ample line bundle on X. Suppose also that M is a fixed holomorphic vector bundle on X. The following objects were introduced in [5, 28]. Definition 2.1. A twisted Higgs bundle (E,φ) consists of a holomorphic vector bundle E and a holomorphic bundle morphism φ: M ⊗E →E. Twisted Higgs bundles also go under the name of Hitchin pairs. A morphism betweentwistedHiggsbundles(E ,φ )and(E ,φ )isabundlemorphismα: E → 1 1 2 2 1 E suchthatα◦φ =φ ◦(id ⊗α)(notethebundleM isthesameforbothpairs), 2 1 2 M and this defines what it means for two twisted Higgs bundles to be isomorphic. Theautomorphismgroupof(E,φ)willbedenotedAut(E,φ), and(E,φ)issaidbe simple if Aut(E,φ)=C. We let E(k)=E⊗Lk, and denote the Hilbert polynomial by c (L)n P (k)=χ(E(k))=rk kn 1 +O(kn−1) E E n! ZX where rk is the rank of E. E Definition 2.2. We saythat(E,φ) is Gieseker-(semi)stable if foranyproper sub- sheaf F ⊂E such that φ(M ⊗F)⊂F we have P (k) P (k) F E (≤) for all k ≫0. rk rk F E We say (E,φ) is Gieseker-polystable if E = E and φ = ⊕φ where (E ,φ ) is i i i i i Gieseker-stable and P /rk =P /rk for all i [33]. Ei Ei E E L HencethisistheusualdefinitionforGiesekerstabilityonlyrestrictingtosubsheaves invariant under φ. Similarly one can define Mumford-(semi)stability by replacing thepolynomialsP /rk withtheslopesdeg(E)/rk . Thentheusualimplications E E E [22, 1.2.13] between Mumford and Gieseker (semi)stability hold, and the Hitchin- Kobayashicorrespondencefor twisted Higgs bundles (see e.g. [2]) is to be taken in the sense of Mumford-polystability. There are a number of ways that one can parameterise decorated vector bundles [34]. Since we will assume throughout that M is globally generated, we can work with the following rather simple setup. Definition 2.3. Given a vector space U we let Z :=Z(U):=Hom(H0(M)⊗U,U) and Z :=P(Z⊕C). Definition 2.4. Let φ =φ be the linear map defined by ∗ ∗,k φ : H0(M)⊗H0(E(k))→H0(M ⊗E(k))→φ H0(E(k)) ∗ where the first map is the natural multiplication (in the following we will omit this multiplication map from the notation where it cannot cause confusion). Thus φ ∈Z(H0(E(k)) which we identify also with [φ ,1]∈Z. ∗ ∗ 6 M.GARCIA-FERNANDEZANDJ.ROSS Toputthisintothecontextwewishtouse,supposewehaveatwistedHiggsbundle (E,φ)andanisomorphismH0(E(k))≃CN givenbyabasissforH0(E(k)). Then underthisisomorphismφ∗ ∈Z :=Z(CNk)andthesectionsofE giveanembedding ι : X →G s where G denotes the Grassmannian of rkE dimensional quotients of CNk. Definition 2.5. Define the embedding f =f : X →Z×G by f(x)=(φ ,ι (x)). s ∗ s ThegroupGL actsontherighthandsideinanaturalway,reflectingthedifferent N choices of s, and one can easily check that pairs (φ,E) and (φ˜,E˜) are isomorphic if and only if the associated embeddings (for any choices of basis) lie in the same GL orbit. N 2.2. Balanced Metrics. Fixahermitianmetrich onM andpositivehermitian M metric h with curvature ω. These induce an L2-metric on the space H0(M) by L ωn ksk2 := |s|2 . L2 hM n! ZX Also fix the topological constant h0(E(k)) χ:=χ(k)= , (2.1) rk Vol(X) E with Vol(X):= 1 c (L)n, so by Riemann-Roch n! X 1 R χ=kn(1+O(1/k)). We also fix a δ =δ(k)>0 depending on a positive integer k (in the application we have in mind δ =ℓkn−1 for some chosen constant ℓ>0). Nowsuppose we choosea hermitianinner productonH0(E(k)). Thenthe domain and target of H0(M)⊗H0(E(k)) −−−φ∗−→ H0(E(k)). (2.2) are hermitian (induced by this chosen inner product and the fixed L2-metric on H0(M)). Define |||φ |||2 :=tr((φ )∗φ ), ∗ ∗ ∗ where (φ )∗ denotes the adjoint of φ . ∗ ∗ Definition 2.6. Set [φ ,(φ )∗]=φ (φ )∗−(φ )∗φ . (2.3) ∗ ∗ ∗ ∗ ∗ ∗ and define an endomorphism P of H0(E(k)) by δ[φ ,(φ )∗] P :=χ−1 Id+ ∗ ∗ . (2.4) 1+|||φ∗|||2 ! Remark 2.7. Hereandbelow weuse the followingabuseofnotation. By ametric on a vector space we shall always mean one that arises from a hermitian inner product. If U,V have given metrics and f: U ⊗V → U is a linear map we will denote the induced map U →U ⊗V∗ also by f. So the adjoint f∗ can be thought of either as a map U → U ⊗V or as a map U ⊗V∗ → U. Thus the commutator [f,f∗]=ff∗−f∗f is a well-defined map U →U. Definition2.8. WesaythatametriconH0(E(k))isbalanced ifforanorthonormal basis s the embedding ι and quantized Higgs field φ satisfy s ∗ ωn (sj,sl)ι∗shFS n! =Plj ∈iu(N), (2.5) ZX BALANCED METRICS ON TWISTED HIGGS BUNDLES 7 where h denotes the Fubini-Study metric on the universal quotient bundle over FS G and P are the components of P in this basis. A hermitian metric h on E is a lj balanced metric for (E,φ) at level k if h=h⊗(−k)⊗ι∗h L s FS whereh istheFubini-StudymetriccomingfromabalancedmetriconH0(E(k)). FS If such a metric h exists then we say (E,φ) is balanced at level k and refer to the balanced metric on H0(E(k)) as the corresponding balanced metric. 2.3. Balancedmetricsaszerosofamomentmap. Wenextinterpretbalanced metrics in terms of a moment map. Take U =CNk and Z as in Definition 2.3. Definition 2.9. We let S ⊂C∞(X,Z×G) denote the space of embeddings f : X →Z×G, for different choice ofbasis s. We s define a form on S by δ ωn Ω| (V ,V )= V y V y ω + ω ∧ , (2.6) f 1 2 2 1 G χVol(X) Z n! ZX (cid:18) (cid:18) (cid:19)(cid:19) where V ∈ T S ∼= H0(X,f∗T(Z ×G)) and ω and ω denote the Fubini-Study j f Z G metrics on Z and G. Lemma 2.10. The form Ω is closed, positive and U(N)-invariant. There exists a moment map for the U(N)-action on (S,Ω), given by i ωn iδ [φ ,(φ )∗] µ(f )=− (s ,s ) + ∗ ∗ ∈u(N), (2.7) s 2ZX j l fs∗hFS n! 2χ 1+|||φ∗|||2!lj where h denotes the Fubini-Study metric on G. FS Proof. The first part follows from the closedness, positivity and invariance of ω Z and ω (see [43, Remark 3.3] and cf. [15, Remark 2.3]). G Now let µ : G→u(N)∗ and µ : Z →u(N)∗ be the moment maps for the U(N)- G Z action on G and Z respectively. Then δ ωn µ(f )= f∗ µ + µ , s s G χVol(X) Z n! ZX (cid:18) (cid:19) is the map we require. Now, by definition of the action i hµ (A),ζi=− trA∗(AA∗)−1Aζ G 2 for every ζ ∈ u(N), where we think of a point in G as an rk ×N matrix A. We E observe that f∗(A∗(AA∗)−1A)ωn = (s ,s ) ωn, s j l fs∗hFS (cid:18)ZX (cid:19)lj ZX and that µ is constant on X, which proves the statement. (cid:3) Z Corollary 2.11. A twisted Higgs bundle (E,φ) is balanced at level k if and only if there exists a basis s of H0(E(k)) such that f is a solution of the moment map s equation iχ µ(f )=− Id. s 2 Proof. This is precisely the definition of the balanced condition. (cid:3) 8 M.GARCIA-FERNANDEZANDJ.ROSS 2.4. A further characterization of the balanced condition. In addition to the moment map interpretation of the balanced condition, we have the following characterizationin terms of metrics on E and H0(E(k)). Proposition 2.12. (E,φ) is balanced at level k if and only if there exists a pair (h,(·,·)) consisting of a hermitian metric h on E and hermitian inner product (·,·) on H0(E(k)) such that if P is the operator defined by (·,·) =(P·,·) with (·,·) Hk Hk denoting the L2-metric induced by to H =h⊗hk then k L Id= (Ps′)(·,s′) and, j j Hk j X (2.8) δ P =χ−1 Id+ [φ ,(φ )∗] . 1+|||φ∗|||2 ∗ ∗ ! Here (s′) is an orthonormal basis for (·,·) and the adjoint (φ )∗ and Frobenius j Hk ∗ norm |||φ |||2 are taken with respect to (·,·). ∗ Remark 2.13. Note that the first condition in (2.8) is independent of the choice of L2-orthonormalbasis s′. j Remark 2.14. When φ = 0 the two equations become P = χ−1Id and B := k s′(·,s′) = χId where B is the Bergman function of H . In this case the j j Hk k k existence of a balancedmetric is equivalentto one for whichthe Bergmanfunction iPs constant (for then one can take (·,·) to be the induced L2-metric). Proof. The proof is based on two facts. First, given a basis s = (s ,...,s ) of 1 N H0(E(k)), the pull-back of the Fubini-Study metric h on the universal quotient FS bundle over G(CN;r) is given by ι∗h =(B−1·,·) , for B = s (·,s ) s FS Hk l l Hk l X andanarbitrarychoiceofhermitianmetricH onE(k). Second,givenaninvertible k endomorphismP ofH0(E(k))thatishermitianwithrespecttothehermitianmetric induced by s, the basis s′ =P−1/2s = (P−1/2) s satisfies j j l lj l s (·,s ) = P s′P(·,s′) = (Ps′)(·,s′) (2.9) j j Hk jl j l Hk l l Hk j jl l X X X We proceed to the proof. For the ‘only if’ part, take H = ι∗h , with s the k s FS balancedbasisanddenoteby (·,·) the inducedL2-metriconH0(E(k)). Observe Hk that the balanced condition implies the relation (P·,·)=(·,·) , Hk with P as in (2.8) and hence s′ = P−1/2s is an orthonormal basis for (·,·) . j j Hk The result follows from(2.9) and the fact that H is pull-back of the Fubini-Study k metric, that gives Id= s (·,s ) . j j j Hk For the ‘if’ part, choose an orthonormal basis (s′) for (·,·) and consider s = P j Hk j P1/2s′, that provides an orthonormal basis for (·,·) = (P−1·,·) . We claim that j Hk (s )isabalancedbasis. Thisfollowsfrom(2.9)andthefirstequationin(2.8),that j give H =ι∗h . k s FS (cid:3) 3. Geometric Invariant Theory 3.1. Further Properties of twisted Higgs bundles. We collect some further propertiesofatwistedHiggsbundle(E,φ: M⊗E →E)againundertheassumption that M is globally generated. Abusing notation we shall let φ also denote the induced map M ⊗E(k)→E(k) obtained by tensoring with the identity. BALANCED METRICS ON TWISTED HIGGS BUNDLES 9 Lemma 3.1. If (E,φ) is Gieseker stable then it is simple. Proof. The proof is the same as the case for bundles [22, 1.2.7], since if α: E →E is a morphismof twistedHiggsbundles then φ(ker(α)⊗M)⊂ker(α) andsimilarly for im(α). (cid:3) Thenextlemmasaysthatφ completelycapturesthe morphismφ. Overanypoint ∗ x∈X we denote by e : H0(E(k))→E(k) 1,x e : H0(M)⊗H0(E(k))→M ⊗E(k+l). 2,x the evaluation maps, that are surjective for k sufficiently large. Lemma 3.2. The map φ 7→ φ is a bijection between bundle morphisms φ: M ⊗ ∗ E → E and linear maps α: H0(M)⊗H0(E(k)) → H0(E(k)) that for all x ∈ X satisfy α(ker(e ))⊂ker(e ). 2,x 1,x Proof. Asimple diagramchaseshowsthatif α=φ thenα satisfiesthis condition. ∗ In the other direction, suppose that α is a linear map that satisfies α(ker(e ,x))⊂ 2 ker(e ) for all x∈X. Then we can define φ˜: M ⊗E(k)→E(k) by saying if ζ ∈ 1,x M⊗E(k) pick an s∈H0(M)⊗H0(E(k)) with s(x)=ζ andset φ˜(x):=α(s)(x). x The assumed condition implies this is independent of choice of s, and so φ˜ gives a holomorphic bundle map that induces φ: M ⊗E →E obtained by tensoring with id . Clearly then α=φ and this gives the required bijection. (cid:3) L−k ∗ This correspondence respects subobjects, as made precise in the next lemma. Definition 3.3. (1) We say a subsheaf F ⊂E is invariant under φ if φ(M ⊗F)⊂F. (2) We say a subspace U ⊂H0(E(k)) is invariant under φ if 0 ∗ φ (H0(M)⊗U )⊂U . ∗ 0 0 Lemma 3.4. (1) If F ⊂E is invariant under φ then H0(F(k)) is invariant under φ . ∗ (2) Suppose U′ ⊂ H0(E(k)) is invariant under φ . Then the subsheaf F of E ∗ generated by U′⊗L−k is invariant under φ. (3) Let U be a subspace of U for j = 1,2 and let F be the subsheaf of E j j generated by U ⊗L−k. If φ (H0(M)⊗U )⊂U then φ(M⊗G )⊂φ(G ). j ∗ 1 2 1 2 Proof. The statement (1) is clear, for if s ∈ H0(F(k)) and s ∈ H0(M) then M φ (s⊗s )∈H0(F(k)) since F is invariant under φ. ∗ M The statement (2) follows from (3) letting U =U =U′. We first prove (3) in the 1 2 case that G and G are subbundles of E. Let ζ ∈ G (k) for some x ∈ X and 1 2 F 1 x ζ ∈ M . By definition there is a u ∈ U so that u(x) = ζ . Moreover as M is M x 1 F globally generated there is an s ∈H0(M) with s (x)=ζ . Now by hypothesis M M M φ (s ⊗u)∈U and so as U generates G (k) we have ∗ M 1 1 2 φ(ζ ⊗ζ )=φ (s ⊗u)(x)∈G (k) . M F ∗ M 2 x Thus φ(M ⊗G (k))⊂G (k) so φ(M ⊗G )⊂G as claimed. 1 2 1 2 The case that G and G are merely subsheaves follows from this. For both G 1 2 1 and G are necessarily torsion free, so there is a Zariski open set U over which G 2 1 and G are subbundles, and the above gives φ(M ⊗G ) ⊂ G | . But since U is 2 1 2 U dense this implies that in fact φ(M ⊗G )⊂G ) (as can be seen by looking at the 1 2 localization of the corresponding modules). (cid:3) 10 M.GARCIA-FERNANDEZANDJ.ROSS 3.2. An extension of a result of Gieseker. For any vector bundle E of rank r there is the associated multiplication map T : ΛrH0(E(k))→H0(det(E(k))). E WeletHom :=Hom(ΛrH0(E(k),H0(det(E(k)). AclassicalresultofGieseker[17] k states that for all k sufficiently large, E is Gieseker (semi)stable if and only if the point [T ]∈P(Hom )=:P E k is (semi)stable in the sense of Geometric Invariant Theory. More precisely, fix- ing a vector space U of dimension N := h0(E(k)) and choice of isomorphism k U ≃ H0(E(k)) the orbit of [T ] is independent of this choice, and E is Gieseker E (semi)stable if and only the points in the orbitare (semi)stable with respect to the linearised SL(U) action on OP(Homk)(1). Here we abuse notation somewhat since the space H0(det(E(k))) also depends on E, but this is easily circumvented by treating P(Hom ) as a suitable projective bundle over Pic(X) (or alternatively by k restricting attention to bundles E with a given determinant). The purpose of this section is to extend this result to twisted Higgs bundles. Such extensions are quite standard, and have been achieved in various contexts (e.g. [23, 28, 32]) with perhaps the most general being [33]. We include the details for completeness, and since they are somewhat simpler in the specific case we are considering. Fix a twisted Higgs bundle (E,φ: E⊗M →E) with M globally generatedand let Z = Hom(H0(M)⊗H0(E(k)),H0(E(k)) be the parameter space as considered in §2.1. We recall that φ also denotes the image of φ under the natural inclusion ∗ ∗ Z ⊂ Z = P(Z ⊕C). The group SL(U) acts on the product Z ×P, and admits a natural linearization to the line bundle L := O (ǫ)⊠OP(1) which is ample for Z ǫ>0. Theorem 3.5. There is an ǫ > 0 such that for all rational ǫ ≥ ǫ k−1 the fol- 0 0 lowing holds: if the twisted Higgs bundle (E,φ) is Gieseker (semi/poly)-stable then (φ ,T )∈Z×P is (semi/poly)-stable. ∗ E Theorem 3.6. Suppose a twisted Higgs bundle (E,φ) is not (semi/poly)-stable. Then for all k sufficiently large and all positive ǫ the point (φ ,T )∈Z×P is not ∗ E (semi/poly)-stable. Remark 3.7. We shall apply the previous theorems with δ ǫ= χ which satisfies the hypothesis as δ and χ are strictly of orders O(kn−1) and O(kn) respectively. The proofs are a standard application of the Hilbert-Mumford criterion, and we shall as far as possible follow the approach taken in [33] and [22]. A non-trivial oneparametersubgroupλofSL(U)determinesaweightdecompositionU =⊕ U n n where U is the eigenspace of weight n. Define U =⊕ U . Fixing an isomor- n ≤n i≤n i phism H0(E(k))=U, let ρ: U ⊗L−k →E be the natural evaluation map which is surjective for k ≫ 0. We let F be ≤n the saturation of ρ(U ⊗L−k) ⊂ E and set F := F /F . Then the one- ≤n n ≤n ≤n−1 parametersubgroupactsonF withweightn. We observealsothatthe saturation n assumption implies r :=rk ≥1. n Fn