Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. I: Generalities and the one-dimensional case
aa r X i v : . [ m a t h . DG ] A p r Approximate Hermitian-Yang-Mills structuresand semistability for Higgs bundles.I: Generalities and the one-dimensional case.
S. A. H. Cardona ∗ SISSA - Via Bonomea 265 - 34136, Trieste - ItalySeptember 25, 2018
Abstract
We review the notions of (weak) Hermitian-Yang-Mills structure andapproximate Hermitian-Yang-Mills structure for Higgs bundles. Then, weconstruct the Donaldson functional for Higgs bundles over compact K¨ahlermanifolds and we present some basic properties of it. In particular, weshow that its gradient flow can be written in terms of the mean curvatureof the Hitchin-Simpson connection. We also study some properties ofthe solutions of the evolution equation associated with that functional.Next, we study the problem of the existence of approximate Hermitian-Yang-Mills structures and its relation with the algebro-geometric notion ofsemistability and we show that for a compact Riemann surface, the notionof approximate Hermitian-Yang-Mills structure is in fact the differential-geometric counterpart of the notion of semistability. Finally, we reviewthe notion of admissible Hermitian structure on a torsion-free Higgs sheafand define the Donaldson functional for such an object.
In complex geometry, the Hitchin-Kobayashi correspondence asserts that thenotion of (Mumford-Takemoto) stability, originally introduced in algebraic ge-ometry, has a differential-geometric equivalent in terms of special metrics. In itsclassical version, this correspondence is established for holomorphic vector bun-dles over compact K¨ahler manifolds and says that such bundles are polystable ifand only if they admit an Hermitian-Einstein structure. This correspondenceis also true for Higgs bundles.The history of this correspondence starts in 1965, when Narasimhan andSeshadri [12] proved that a holomorphic bundle on a Riemann surface is stableif and only if it corresponds to a projective irreducible representation of thefundamental group of the surface. Then, in the 80’s Kobayashi [8] introduced ∗ Electronic address: [email protected] In the literature Hermitian-Einstein, Einstein-Hermite and Hermitian-Yang-Mills are allsynonymous. Sometimes, also the terminology Hermitian-Yang-Mills-Higgs is used [5].
Acknowledgements
First to all, the author would like to thank his thesis advisor, Prof. U. Bruzzo,for his constant support and encouragement and also for suggesting this prob-lem. I would like to thank also The International School for Advanced Studies(SISSA) for graduate fellowship support. Finally, I would like to thank O. Iena,G. Dossena, P. Giavedoni and Carlos Marin for many enlightening discussionsand for some helpful comments. The results presented in this article will bepart of my Ph.D. thesis in the Mathematical-Physics sector at SISSA.
We start with some basic definitions. Let X be an n -dimensional compactK¨ahler manifold with ω its K¨ahler form and let Ω X be the cotangent sheaf to X , i.e., it is the sheaf of holomorphic one-forms on X . A Higgs sheaf E over X is a coherent sheaf E over X , together with a morphism φ : E → E ⊗ Ω X of O X -modules, such that the morphism φ ∧ φ : E → E ⊗ Ω X vanishes. The mor-phism φ is called the Higgs field of E . A Higgs sheaf E is said to be torsion-free(resp. reflexive, locally free, normal, torsion) if the sheaf E is torsion-free (resp.reflexive, locally free, normal, torsion). A Higgs subsheaf F of E is a subsheaf F of E such that φ ( F ) ⊂ F ⊗ Ω X . A Higgs bundle E is just a Higgs sheaf inwhich the sheaf E is a locally free O X -module.Let E and E be two Higgs sheaves over a compact K¨ahler manifold X . Amorphism between E and E is a map f : E −→ E such that the diagram E φ / / f (cid:15) (cid:15) E ⊗ Ω Xf ⊗ Id (cid:15) (cid:15) E φ / / E ⊗ Ω X commutes. We will denote such a morphism by f : E −→ E . A sequence ofHiggs sheaves is a sequence of their corresponding coherent sheaves where eachmap is a morphism of Higgs sheaves. A short exact sequence of Higgs sheaves(also called an extension of Higgs sheaves or a Higgs extension [3], [6]) is definedin the obvious way.We define the degree deg E and rank rk E of a Higgs sheaf simply as the de-gree and rank of the sheaf E . If the rank is positive, we introduce the quotient3 ( E ) = deg E / rk E and call it the slope of the Higgs sheaf E . In a similar wayas in the ordinary case (see for instance [10], [11], [13], [19]) there is a notionof stability for Higgs sheaves, which depends on the K¨ahler form and makesreference only to Higgs subsheaves [2], [3], [4], [5], [6]. Namely, a Higgs sheaf E is said to be ω -stable (resp. ω -semistable) if it is torsion-free and for anyHiggs subsheaf F with 0 < rk F < rk E and torsion-free quotient, one has theinequality µ ( F ) < µ ( E ) (resp. ≤ ). We say that a Higgs sheaf is ω -polystable if itdecomposes into a direct sum of ω -stable Higgs sheaves all having the same slope.Let E = ( E, φ ) be a Higgs bundle of rank r over X and let ω be the K¨ahlerform of X . Using the Chern connection D ( E,h ) of E and the Higgs field φ onedefines the Hitchin-Simpson connection on E by: D ( E ,h ) = D ( E,h ) + φ + ¯ φ h , (1)where ¯ φ h is the adjoint of the Higgs field with respect to the Hermitian structure h , that is, it is defined by the formula h ( ¯ φ h s, s ′ ) = h ( s, φs ′ ) with s, s ′ sectionsof the Higgs bundle. The curvature of the Hitchin-Simpson connection is thengiven by R ( E ,h ) = D ( E ,h ) ◦ D ( E ,h ) and hence R ( E ,h ) = R ( E,h ) + D ′ ( E,h ) ( φ ) + D ′′ ( E,h ) ( ¯ φ h ) + [ φ, ¯ φ h ] . (2)We say that the pair ( E , h ) is Hermitian flat, if the curvature R ( E ,h ) vanishes.We denote by Herm( E ) the space of Hermitian forms in E and by Herm + ( E ) thespace of Hermitian structures (i.e., positive definite Hermitian forms) in E . Forany Hermitian structure h it is possible to identify Herm( E ) with the tangentspace of Herm + ( E ) at that h (see [10] for details). That isHerm( E ) = T h Herm + ( E ) . (3)If v denotes an element in Herm( E ), one defines the endomorphism h − v of E by the formula v ( s, s ′ ) = h ( s, h − vs ′ ) , (4)where s, s ′ are sections of E . We define a Riemannian structure in Herm + ( E )via this identification. Namely, for any v, v ′ in Herm( E ) we define( v, v ′ ) h = Z X tr( h − v · h − v ′ ) ω n /n ! . (5)The Higgs field φ can be considered as a section of End E ⊗ Ω X and hencewe have a natural dual morphism φ ∗ : E ∗ → E ∗ ⊗ Ω X . From this it follows that E ∗ = ( E ∗ , φ ∗ ) is a Higgs bundle. On the other hand, if Y is another compactK¨ahler manifold and f : Y → X is a holomorphic map, the pair defined by f ∗ E = ( f ∗ E, f ∗ φ ) is also a Higgs bundle. We have also some natural propertiesassociated to tensor products and direct sums. In particular we have Proposition 2.1
Let E and E two Higgs bundles with Higgs fields φ and φ respectively. Then (i) The pair E ⊗ E = ( E ⊗ E , φ ) is a Higgs bundle with φ = φ ⊗ I + I ⊗ φ . (ii) If pr i : E ⊕ E → E i with i = 1 , denote the natural projections, then E ⊕ E = ( E ⊕ E , φ ) is a Higgs bundle with φ = pr ∗ φ + pr ∗ φ .
4n a similar form as in the ordinary case [10], [13], [11], we have a notionof Hermitian-Einstein structure for Higgs bundles [2], [3]. Let us consider theusual star operator ∗ : A p,q → A n − q,n − p and the operator L : A p,q → A p +1 ,q +1 defined by Lϕ = ω ∧ ϕ , where ϕ is a form on X of type ( p, q ). Then we define,as usual, Λ = ∗ − ◦ L ◦ ∗ : A p,q → A p − ,q − . Consider now a metric h inHerm + ( E ) (i.e., h is an Hermitian structure on E ), associated with this metricwe have a Hitchin-Simpson curvature R ( E ,h ) . We can define the mean curvatureof the Hitchin-Simpson connection, just by contraction of this curvature withthe operator i Λ. In other words, K ( E ,h ) = i Λ R ( E ,h ) . (6)The mean curvature is an endomorphism in End( E ). We say that h is a weakHermitian-Yang-Mills structure with factor γ for E if K ( E ,h ) = γ · I (7)where γ is a real function defined on X and I is the identity endomorphismon E . From this definition it follows that if h is a weak Hermitian-Yang-Millsstructure with factor γ for E , then the dual metric h ∗ is a weak Hermitian-Yang-Mills structure for the dual bundle E ∗ and also, that any metric h on aHiggs line bundle is necessarily a weak Hermitian-Yang-Mills structure. As inthe ordinary case, also for Higgs bundles we have some simple properties relatedwith the notion of weak Hermitian-Yang-Mills structure, in particular, from theusual formulas for the curvature of tensor products and direct sums we have thefollowing Proposition 2.2 (i) If h and h are two weak Hermitian-Yang-Mills struc-tures with factors γ and γ for Higgs bundles E and E , respectively, then h ⊗ h is a weak Hermitian-Yang-Mills structure with factor γ + γ for thetensor product bundle E ⊗ E . (ii) The metric h ⊕ h is a weak Hermitian-Yang-Mills structure with factor γ for the Whitney sum E ⊕ E if and only if both metrics h and h are weakHermitian-Yang-Mills structures with the same factor γ for E and E , respec-tively. If we have a weak Hermitian-Yang-Mills structure in which the factor γ = c is constant, we say that h is an Hermitian-Yang-Mills structure with factor c for E . From Proposition 2.2 and this definition we get Corollary 2.3
Let h ∈ Herm + ( E ) be a (weak) Hermitian-Yang-Mills structurewith factor γ for the Higgs bundle E . Then (i) The induced Hermitian metric on the tensor product E ⊗ p ⊗ E ∗ ⊗ q is a (weak)Hermitian-Yang-Mills structure with factor ( p − q ) γ . (ii) The induced Hermitian metric on V p E is a (weak) Hermitian-Yang-Millsstructure with factor pγ for every p ≤ r = rk E . In general, if h is a weak Hermitian-Yang-Mills structure with factor γ , theslope of E can be written in terms of γ . To be precise, we obtain If we consider a local frame field { e i } ri =1 for E and a local coordinate system { z α } nα =1 of X , the components of the mean curvature are given by K ij = ω α ¯ β R ijα ¯ β . roposition 2.4 If h ∈ Herm + ( E ) is a weak Hermitian-Yang-Mills structurewith factor γ , then µ ( E ) = 12 nπ Z X γ ω n . (8) Proof:
Let R be the Hitchin-Simpson curvature of E , then in general we havethe identity in R ∧ ω n − = K ω n . (9)Now, by hypothesis h is a weak Hermitian-Yang-Mills structure with factor γ ,then taking the trace of (9) and integrating over X we obtain deg E = r nπ Z X γ ω n , (10)where r is the rank of E . Q.E.D.Consider now a real positive function a = a ( x ) on X , then h ′ = ah definesanother Hermitian metric on E . Since h ′ is a conformal change of h , we have inparticular ¯ φ h ′ = ¯ φ h . Then, from (9) we obtain K ′ ω n = in ( R ′ + [ φ, ¯ φ h ′ ]) ∧ ω n − = (cid:0) K ′ + i Λ[ φ, ¯ φ h ] (cid:1) ω n . (11)Now, defining (cid:3) = i Λ d ′′ d ′ (see [10] for details) we have K ′ = K + (cid:3) (log a )and hence using the identity (11) we get K ′ ω n = K ω n + (cid:3) (log a ) ω n . (12)From this we conclude the following Lemma 2.5
Let h be a weak Hermitian-Yang-Mills structure with factor γ for E and let a be a real positive definite function on X , then h ′ = ah is a weakHermitian-Yang-Mills structure with factor γ ′ = γ + (cid:3) (log a ) . Making use of Lemma 2.5 we can define a constant c which plays an impor-tant role in the definition of the Donaldson functional. Such a constant c is anaverage of the factor γ of a weak Hermitian-Yang-Mills structure. Namely Proposition 2.6 If h ∈ Herm + ( E ) is a weak Hermitian-Yang-Mills structurewith factor γ , then there exists a conformal change h ′ = ah such that h ′ is anHermitian-Yang-Mills structure with constant factor c , given by c Z X ω n = Z X γ ω n . (13) Such a conformal change is unique up to homothety.Proof:
Let c be as in (13), then Z X ( c − γ ) ω n = 0 . (14) We consider here the integral i π R X tr R ∧ ω n − . Notice that only the (1 ,
1) part of thecurvature R makes a real contribution in such an integral and since tr [ φ, ¯ φ ] is identically zero,that integral must be the degree of the holomorphic bundle E and hence it is equal to deg E .
6t is sufficient to prove that there is a function u satisfying the equation (cid:3) u = c − γ , (15)where, as we said before (cid:3) = i Λ d ′′ d ′ . Because if this holds, then by applyingLemma 2.5 with the function a = e u the result follows.Now, from Hodge theory we know that the equation (15) has a solution ifand only if the function c − γ is orthogonal to all (cid:3) -harmonic functions. Since X is compact, a function is (cid:3) -harmonic if and only if it is constant. But (14)says that c − γ is orthogonal to the constant functions and hence the equation(15) has a solution u . Finally, the uniqueness follows from the fact that (cid:3) -harmonic functions are constant. Q.E.D.Since every weak Hermitian-Yang-Mills structure can be transformed into anHermitian-Yang-Mills structure using a conformal change of the metric, with-out loss of generality we avoid using weak structures and work directly withHermitian-Yang-Mills structures. As we have seen in the preceding section, if we have an Hermitian-Yang-Millsstructure with factor c , this constant can be evaluated directly from (8) and wehave c = 2 π µ ( E )( n − X . (16)On the other hand, regardless if we have an Hermitian-Yang-Mills structure ornot on E , we can always define a constant c just by (16). Introduced in sucha way, c depends only on c ( E ) and the cohomology class of ω and not on themetric h . We define the length of the endomorphism K − cI by the formula |K − cI | = tr [( K − cI ) ◦ ( K − cI )] . (17)We say that a Higgs bundle E over a compact K¨ahler manifold X admits an approximate Hermitian-Yang-Mills structure if for any ǫ > h (which depends on ǫ ) such thatmax |K − cI | < ǫ . (18)From the above definition it follows that E ∗ admits an approximate Hermitian-Yang-Mills structure if E does. This notion satisfies some simple properties withrespect to tensor products and direct sums. Proposition 3.1 If E and E admit approximate Hermitian-Yang-Mills struc-tures, so does their tensor product E ⊗ E . Furthermore if µ ( E ) = µ ( E ) , sodoes their Whitney sum E ⊕ E .Proof: Assume that E and E admit approximate Hermitian-Yang-Mills struc-tures with factors c and c respectively and let ǫ >
0. Then, there exist h and h such thatmax X |K − c I | < ǫ √ r , max X |K − c I | < ǫ √ r , r , r and I , I are the ranks and the identity endomorphisms of E and E respectively. Now, let K be the Hitchin-Simpson mean curvature of E ⊗ E associated with the metric h = h ⊗ h . Then, by defining c = c + c and I = I ⊗ I it follows |K − cI | = |K ⊗ I + I ⊗ K − ( c + c ) I I ⊗ I |≤ | ( K − c I ) ⊗ I | + | I ⊗ ( K − c I ) |≤ √ r |K − c I | + √ r |K − c I | < ǫ and hence the tensor product E ⊗ E admits an approximate Hermitian-Yang-Mills structure.On the other hand, if µ ( E ) = µ ( E ), necesarily the constants c and c coincide. Then, taking this time c = c = c , I = I ⊕ I and K = K ⊕ K , wehave |K − cI | = |K ⊕ K − c I ⊕ I | = p tr ( K − c I ) + tr ( K − c I ) ≤ |K − c I | + |K − c I | . From this inequality it follows that E ⊕ E admits an approximate Hermitian-Yang-Mills structure. Q.E.D. Corollary 3.2 If E admits an approximate Hermitian-Yang-Mills structure, sodo the tensor product bundle E ⊗ p ⊗ E ∗⊗ q and the exterior product bundle V p E whenever p ≤ r . Finally, in a similar way as in the classical case, we have a version of theBogomolov-L¨ubke inequality also for Higgs bundles admiting an approximateHermitian-Yang-Mills structure (see [10], [21] for details). To be precise, weobtain
Theorem 3.3
Let E be a Higgs bundle over a compact K¨ahler manifold X andsuppose that E admits an approximate Hermitian-Yang-Mills structure, then Z X (cid:2) r c ( E ) − ( r − c ( E ) (cid:3) ∧ ω n − ≥ . (19) Proof:
Assume that E admits an approximate Hermitian-Yang-Mills structure.Let ǫ > h ǫ is a metric on E satisfying (18). Then, we have closed2 k -forms c k ( E , h ǫ ) representing the k -th Chern classes. From [10], Ch.IV, weobtain(2 r c ( E , h ǫ ) − ( r − c ( E , h ǫ ) ) ∧ ω n − ( n − (cid:2) r ( | R ǫ | − | K ǫ | ) + σ ǫ − | ρ ǫ | (cid:3) ω n n !where the quantities on the right-hand side are associated to the metric h ǫ andare given by | K ǫ | = tr K ǫ and σ ǫ = tr K ǫ and | R ǫ | = X i,j,α,β | ( R ǫ ) ijα ¯ β | , | ρ ǫ | = X i,α,β | ( R ǫ ) iiα ¯ β | . r | R ǫ | ≥ | ρ ǫ | , and hence integrating over X we obtain Z X (2 r c ( E , h ǫ ) − ( r − c ( E , h ǫ ) ) ∧ ω n − ( n − ≥ Z X (cid:2) σ ǫ − r | K ǫ | (cid:3) ω n n ! . (20)Since h ǫ is an approximate Hermitian-Yang-Mills structure, we have ǫ > |K ǫ − cI | = |K ǫ | − c σ ǫ + c r . (21)On the other hand, |K ǫ | = tr (cid:2) ( K ǫ + i Λ[ φ, ¯ φ ǫ ]) · ( K ǫ + i Λ[ φ, ¯ φ ǫ ]) (cid:3) = | K ǫ | + 2 i Λ tr (cid:2) K ǫ · [ φ, ¯ φ ǫ ] (cid:3) + ( i Λ) tr (cid:2) [ φ, ¯ φ ǫ ] (cid:3) = | K ǫ | + 2 i Λ tr (cid:2) K ǫ · [ φ, ¯ φ ǫ ] (cid:3) . Now, K ǫ = cI + ǫ A with A a self-adjoint endomorphism of E and hence we canestimate the term involving the trace in the last expression astr (cid:2) K ǫ · [ φ, ¯ φ ǫ ] (cid:3) = c tr [ φ, ¯ φ ǫ ] + ǫ tr (cid:2) A · [ φ, ¯ φ ǫ ] (cid:3) = ǫ η (22)where the (1 , η = tr (cid:2) A · [ φ, ¯ φ ǫ ] (cid:3) . Consequently |K ǫ | = | K ǫ | + 2 ǫ ( i Λ η ) . (23)Finally, from (21) and (23) it follows σ ǫ − r | K ǫ | > ( σ ǫ − cr ) + f ( ǫ )where f ( ǫ ) = rǫ (2 ( i Λ η ) − ǫ ). Then, by replacing this last expression in (20) weconclude Z X (2 r c ( E , h ǫ ) − ( r − c ( E , h ǫ ) ) ∧ ω n − ( n − > Z X f ( ǫ ) ω n n ! . (24)Now, the integral on the left-hand side is independent of the metric h ǫ . Onthe other hand, the above inequality holds for all ǫ > f ( ǫ ) → ǫ →
0. Therefore, one has the inequality (19) if E admits an approximateHermitian-Yang-Mills metric. Q.E.D.We want to construct a functional L on Herm + ( E ), that will be called Don-aldson’s functional and whose gradient is related with the mean curvature of theHitchin-Simpson connection. The construction of this functional is in certainway similar to the ordinary case. However, there are some differences, which inessence are due to the extra terms involving the Higgs field φ in the expressionfor the curvature (2). Given two Hermitian structures h, k in Herm + ( E ), we connect them by a curve h t , 0 ≤ t ≤
1, in Herm + ( E ) so that k = h and h = h . We set Q ( h, k ) = log(det( k − h )) , Q ( h, k ) = i Z tr( v t · R t ) dt , (25)9here v t = h − t ∂ t h t and R t denotes the curvature of the Hitchin-Simpson con-nection associated with h t . Notice that Q ( h, k ) does not involve the curve (infact, it is the same functional of the ordinary case). On the other hand, thedefinition of Q ( h, k ) uses explicitly the curve and differs from the ordinary casebecause of the extra terms in (2). We define the Donaldson functional by L ( h, k ) = Z X h Q ( h, k ) − cn Q ( h, k ) ω i ∧ ω n − / ( n − , (26)where c is the constant given by c = 2 π µ ( E )( n − X . (27)Notice that the components of (2 ,
0) and (0 ,
2) type of R t do not contributeto L ( h, k ). This means that, in practice, it is enough to consider in the defini-tion of Q ( h, k ) just components of (1 , .The following Lemma and the subsequent Proposition are straightforwardgeneralizations of a result of Kobayashi (see [10], Ch.VI, Lemma 3.6) to theHiggs case. Part of the proof is similar to the proof presented in [10], howeversome differences arise because of the term involving the commutator in theHitchin-Simpson curvature. Lemma 4.1
Let h t , a ≤ t ≤ b , be any differentiable curve in Herm + ( E ) and k any fixed Hermitian structure of E . Then, the (1,1)-component of i Z ba tr( v t · R t ) dt + Q ( h a , k ) − Q ( h b , k ) (28) is an element in d ′ A , + d ′′ A , .Proof: Following [10], we consider the domain ∆ in R defined by∆ = { ( t, s ) | a ≤ t ≤ b , ≤ s ≤ } (29)and let h : ∆ → Herm + ( E ) be a smooth mapping such that h ( t,
0) = k , h ( t,
1) = h t for a ≤ t ≤ b , let h ( a, s ) and h ( b, s ) line segments curves from k to h a and respectively from k to h b . We define the endomorphisms u = h − ∂ s h , v = h − ∂ t h and we put R = d ′′ ( h − d ′ h ) + [ φ, ¯ φ h ] (30)and Ψ = i tr[ h − ˜ dh R ] , where ˜ dh = ∂ s h ds + ∂ t h dt is considered as the exteriordifferential of h in the domain ∆. It is convenient to rewrite Ψ in the formΨ = i tr[( u ds + v dt ) R ] . (31) In other words, in computations involving integration over X , we can always replace thecurvature by R , t = R t + [ φ, ¯ φ h t ]. Notice that we have a simple expression for line segments curves from k to h t given by h ( t, s ) = sh t + (1 − s ) k . Z ∆ ˜ d Ψ = Z ∂ ∆ Ψ . (32)The right hand side of the above expression can be computed straightforwardfrom definition. In fact, after a short computation we obtain Z ∂ ∆ Ψ = i Z ba tr( v t · R , t ) dt + Q , ( h a , k ) − Q , ( h b , k ) . (33)Therefore, we need to show that the left hand side of (32) is an element in d ′ A , + d ′′ A , , and hence, it suffices to show that ˜ d Ψ ∈ d ′ A , + d ′′ A , .Now, from the definition of Ψ we have˜ d Ψ = i tr[ ˜ d ( u ds + v dt ) R − ( u ds + v dt ) ˜ d R ]= i tr[( ∂ s v − ∂ t u ) R − u ∂ t R + v ∂ s R ] ds ∧ dt . On the other hand, a simple computation shows that ∂ t u = − vu + h − ∂ t ∂ s h , ∂ s v = − uv + h − ∂ s ∂ t h , (34) ∂ t R = d ′′ D ′ v + [ φ, ∂ t ¯ φ h ] , ∂ s R = d ′′ D ′ u + [ φ, ∂ s ¯ φ h ] . (35)Replacing these expressions in the formula for ˜ d Ψ and writing R = R + [ φ, ¯ φ h ]we obtain˜ d Ψ = i tr [( vu − uv ) R − u d ′′ D ′ v + v d ′′ D ′ u ] ds ∧ dt + i tr (cid:2) v [ φ, ∂ s ¯ φ h ] − u [ φ, ∂ t ¯ φ h ] + ( vu − uv )[ φ, ¯ φ h ] (cid:3) ds ∧ dt . The first trace in the expression above does not depend on Higgs field φ (in fact,it is the same expression that is found in [10] for the ordinary case). The secondtrace is identically zero. In order to prove this, we need first to find explicitexpressions for ∂ t ¯ φ h and ∂ s ¯ φ h . Now (omitting the parameter t for simplicity)we know from [2] that¯ φ h s + δs = u − ¯ φ h s u = ¯ φ h s + u − [ ¯ φ h s , u ] (36)where u is a selfadjoint endomorphism such that h s + δs = h s u . Now h s + δs = h s + ∂ s h s · δs + O ( δs ) (37)and hence, at first order in δs , we obtain u = 1 + u · δs and consequently ∂ s ¯ φ h = [ ¯ φ h , u ]. In a similar way we obtain the formula ∂ t ¯ φ h = [ ¯ φ h , v ]. Therefore,using these relations, the Jacobi identity and the cyclic property of the trace,we see that the second trace is identically zero. On the other hand, the terminvolving the curvature R can be rewritten in terms of u, v and their covariantderivatives. So, finally we get˜ d Ψ = − i tr[ v D ′ d ′′ u + u d ′′ D ′ v ] ds ∧ dt . (38)As it is shown in [10], defining the (0,1)-form α = i tr[ v d ′′ u ] we obtain˜ d Ψ = − [ d ′ α + d ′′ ¯ α + i d ′′ d ′ tr( vu )] ds ∧ dt (39)11nd hence ˜ d Ψ is an element in d ′ A , + d ′′ A , . Q.E.D.As a consequence of Lemma 4.1 we have an important result for piecewisedifferentiable closed curves. Namely, we have Proposition 4.2
Let h t , α ≤ t ≤ β , be a piecewise differentiable closed curvein Herm + ( E ) . Then i Z βα tr (cid:16) v t · R , t (cid:17) dt = 0 mod d ′ A , + d ′′ A , . (40) Proof:
Let α = a < a · · · < a p = β be the values of t where h t is notdifferentiable. Now take a fixed point k in Herm + ( E ). Then, Lemma 4.1 appliesfor each triple k, h a j , h a j +1 with j = 0 , , ..., p − Corollary 4.3
The Donaldson functional L ( h, h ′ ) does not depend on the curvejoining h and h ′ .Proof: Clearly, from the definition of Q Q ( h, h ′ ) + Q ( h ′ , h ) = 0 . (41)If γ and γ are two differentiable curves from h to h ′ and we apply Proposition4.2 to γ − γ , we obtain Q , ( h, h ′ ) + Q , ( h ′ , h ) = 0 mod d ′ A , + d ′′ A , , (42)and the result follows by integrating over X . Q.E.D. Proposition 4.4
For any h in Herm + ( E ) and any constant a > , the Don-aldson functional satisfies L ( h, ah ) = 0 .Proof: Clearly Q ( h, ah ) = log det[( ah ) − h ] = − r log a . Now, let b = log a and consider the curve h t = e b (1 − t ) h from ah to h . For thiscurve v t = − bI and we have R , t = d ′′ ( h − t d ′ h t ) + [ φ, ¯ φ t ] = d ′′ ( h − d ′ h ) + [ φ, ¯ φ t ] , where ¯ φ t is an abbreviation for ¯ φ h t . Therefore, the (1,1)-component of Q ( h, ah )becomes Q , ( h, ah ) = i Z tr( v t · R , t ) dt = i Z tr (cid:2) − b ( R + [ φ, ¯ φ t ]) (cid:3) dt = − ib tr R and hence, from the above we obtain cn Z X Q ( h, ah ) ω ∧ ω n − / ( n − − crb vol X , Z X Q ( h, ah ) ∧ ω n − / ( n − − ib ( n − Z X tr R ∧ ω n − = − πb ( n − E and the result follows from the definition of the constant c . Q.E.D.12 emma 4.5 For any differentiable curve h t and any fixed point k in Herm + ( E ) we have ∂ t Q ( h t , k ) = tr( v t ) , (43) ∂ t Q , ( h t , k ) = i tr( v t · R , t ) mod d ′ A , + d ′′ A , . (44) Proof:
Since k does not depend on t , we get ∂ t Q ( h t , k ) = ∂ t log(det k − ) + ∂ t log(det h t ) = ∂ t log(det h t ) = tr( v t ) . Considering b in (28) as a variable, and differentiating that expression with re-spect to b , we obtain the formula. Q.E.D.By using the above Lemma, we have a formula for the derivative with respectto t of Donaldson’s functional ddt L ( h t , k ) = Z X h i tr( v t · R , t ) − cn tr( v t ) ω i ∧ ω n − ( n − Z X [tr( v t · K t ) − c tr( v t )] ω n n != Z X tr [( K t − cI ) v t ] ω n n ! . Since v t = h − t ∂ t h t and we can consider the endomorphism K t as an Hermitianform by defining K t ( s, s ′ ) = h t ( s, K t s ′ ), for any fixed Hermitian metric k andany differentiable curve h t in Herm + ( E ) we obtain ddt L ( h t , k ) = ( K t − c h t , ∂ t h t ) , (45)where K t is considered here as a form. For each t , we can consider ∂ t h t ∈ Herm( E ) as a tangent vector of Herm + ( E ) at h t . Therefore, the differential d L of the functional evaluated at ∂ t h t is given by d L ( ∂ t h t ) = ddt L ( h t , k ) , (46)and hence, the gradient of L (i.e., the vector field on Herm + ( E ) dual to the form d L with respect to the invariant Riemannian metric introduced before) is givenby ∇L = K − ch . From the above analysis we conclude the following Theorem 4.6
Let k be a fixed element in Herm + ( E ) . Then, h is a critical pointof L if and only if K − c h = 0 , i.e., if and only if h is an Hermitian-Yang-Millsstructure for E . In order to derive some properties of L it is convenient to divide the Hichin-Simpson connection (see [2], [3]) in the form D ′ h = D ′ h + ¯ φ h and D ′′ = D ′′ + φ .In fact, using the above decomposition it is not difficult to show that all criticalpoints of L correspond to an absolute minimum. Notice that from the definition (4), the endomorphism K t can be written formally as K t = h − t K t ( · , · ) where K t ( · , · ) denotes this time the mean curvature as a form. Therefore,we can express the derivative of the functional as an inner product of the forms K t − c h t and ∂ t h t as in (5). heorem 4.7 Let k be a fixed Hermitian structure of E and ˜ h a critical pointof L ( h, k ) , then the Donaldson functional attains an absolute minimum at ˜ h .Proof: Let h t , ≤ t ≤ , be a differentiable curve such that h = ˜ h , then wecan compute straightforward the second derivative of L d dt L ( h t , k ) = ddt Z X tr [( K t − cI ) v t ] ω n n != Z X tr [ ∂ t K t · v t + ( K t − cI ) ∂ t v t ] ω n n ! . Since h is a critical point of the functional, then K t − cI = 0 at t = 0 , andhence d dt L ( h t , k ) (cid:12)(cid:12) t =0 = Z X tr( ∂ t K t · v t ) ω n n ! (cid:12)(cid:12) t =0 . (47)On the other hand, ∂ t K t can be written in terms of the endomorphism v t in thefollowing way D ′′ D ′ v t = D ′′ ( D ′ v t + [ ¯ φ t , v t ])= D ′′ D ′ v t + [ φ, D ′ v t ] + D ′′ [ ¯ φ t , v t ] + [ φ, [ ¯ φ t , v t ]] , and since ∂ t φ t = [ ¯ φ t , v t ] we get ∂ t R , t = ∂ t R t + [ φ, ∂ t ¯ φ t ] = D ′′ D ′ v t + [ φ, [ ¯ φ t , v t ]] . (48)Therefore, taking the trace with respect to ω (i.e., applying the i Λ operator) weobtain i Λ D ′′ D ′ v t = i Λ ∂ t R , t = ∂ t K t . (49)Hence, replacing this in the expression for the second derivative of L we find d dt L ( h t , k ) (cid:12)(cid:12) t =0 = Z X tr( i Λ D ′′ D ′ v t · v t ) ω n n ! (cid:12)(cid:12) t =0 = kD ′ v t k t =0 , (50)(that is, h must be at least a local minimum of L ). Now suppose in additionthat h is an arbitrary element in Herm + ( E ) and joint them by a geodesic h t ,and hence ∂ t v t = 0 . Therefore, for a such a geodesic we have d dt L ( h t , k ) = Z X tr( ∂ t K t · v t ) ω n n ! . (51)Following the same procedure we have done before, but this time at t arbitrary,we get for 0 ≤ t ≤ d dt L ( h t , k ) = kD ′ v t k h t ≥ t on the right hand side via D ′ , wewrite a subscript h t in the norm) and it follows that L ( h , k ) ≤ L ( h , k ). Nowif we assume that h is also a critical point of L , we necessarily obtain theequality. Therefore, it follows that the minimum defined for any critical pointof L is an absolute minimum. Q.E.D.14et k be a fixed Hermitian structure, then any Hermitian metric h will beof the form ke v for some section v of End( E ) over X . We can join k to h bythe geodesic h t = ke tv where 0 ≤ t ≤ v t = h − t ∂ t h t = v isconstant, i.e., it does not depend on t ). Now, in the proof of Theorem 4.7 wegot an expression for the second derivative L ( h t , k ) for any curve h t . Namely d dt L ( h t , k ) = Z X tr [ ∂ t K t · v t + ( K t − cI ) ∂ t v t ] ω n n ! . (53)Notice that in our case, the chosen curve is such that h = k , since it is also ageodesic ∂ t v t = 0 we have d dt L ( h t , k ) = Z X tr( ∂ t K t · v ) ω n n ! = kD ′ v k h t . (54)Therefore, following [13], the idea is to find a simple expression for kD ′ v k h t orequivalently for kD ′′ v k h t and to integrate it twice with respect to t . We cando this using local coordinates, in fact, at any point in X we can choose a localframe field so that h = I and v = diag( λ , ..., λ r ) . In particular, using such alocal frame field we have h ijt = e − λ j t δ ij , and hence (after a short computation)we obtain kD ′′ v k h t = Z X r X i,j =1 e ( λ i − λ j ) t |D ′′ v ij | ω n − ( n − . (55)Now, at t = 0 the functional L ( h t , k ) vanishes and since k = h is not necessarilyan Hermitian-Yang-Mills structure, we have ddt L ( h t , k ) (cid:12)(cid:12) t =0 = Z X tr [( K − cI ) v ] ω n n ! . (56)Then, by integrating twice (54) we obtain L ( h t , k ) = t Z X tr [( K − cI ) v ] ω n n ! + Z X r X i,j =1 ψ t ( λ i , λ j ) |D ′′ v ij | ω n − ( n − ψ t is a function given by ψ t ( λ i , λ j ) = e ( λ i − λ j ) t − ( λ i − λ j ) t − λ i − λ j ) . (58)In particular, at t = 1 the expression (57) corresponds (up to a constant term)to the definition of Donaldson’s functional given by Simpson in [2]. Notice alsothat if the initial metric k = h is Hermitian-Yang-Mills, the first term of theright hand side of (57) vanishes and the functional coincides with the Donaldsonfunctional used by Siu in [13]. For the construction of Hermitian-Yang-Mills structures, the standard procedureis to start with a fixed Hermitian metric h and try to find from it an Hermitianmetric satisfying K = cI using a curve h t , 0 ≤ t < ∞ (in other words, we try15o find that metric by deforming h through 1-parameter family of Hermitianmetrics) and we expect that at t = ∞ , the metric will be Hermitian-Yang-Mills.At this point, it is convenient to introduce the operator ˜ (cid:3) h = i Λ D ′′ D ′ h ,which depends on the metric h . Using it, we can rewrite (49) as ∂ t K t = ˜ (cid:3) t v t , where the subscript t reminds the dependence of the operator on the metric h t .As we said before, to get an Hermitian-Yang-Mills metric we want to make K − cI vanish. Therefore, a natural choice is to go along the global gradientdirection of the functional given by the global L -norm of K t − cI . Therefore,taking the derivative of this functional we obtain ddt kK t − cI k = Z X ∂ t K t · ( K t − cI )) ω n n != 2 Z X tr (cid:0) ˜ (cid:3) t v t · ( K t − cI ) (cid:1) ω n n != 2 Z X tr (cid:0) v t · ˜ (cid:3) t K t (cid:1) ω n n ! , and the equation that naturally emerges (i.e., the associated steepest descentcurve) is v t = − ˜ (cid:3) t K t , or equivalently h − t ∂ t h t = − i Λ D ′′ D ′ h K t . (59)Since K t is of degree two, the right hand side of the above equation becomes aterm of degree four and hence we get at the end a nonlinear equation of degreefour. To do the analysis, it is easier to deal with an equation of lower degree.In fact, this is one of the reasons for introducing the Donaldson functional.Following the same argument we did before, but this time using the functional L ( h t , k ) with k fixed, in place of the functional kK t − cI k , we end up with anonlinear equation of degree two (the heat equation), to be more precise, weobtain directly from (45) the equation ∂ t h t = − ( K t − ch t ) , (60)where this time, K t represents the associated two form, and not an endomor-phism . Simpson has shown that also for the Higgs case, we have always solu-tions of the above non linear evolution equation. This was proved in [2] for thenon-compact case satisfying some additional conditions. That proof covers thecompact K¨ahler case without any change. Then, from [2] we have the following Theorem 5.1
Given an Hermitian structure h on E , the non-linear evolutionequation ∂ t h t = − ( K t − ch t ) (61) has a unique smooth solution defined for ≤ t < ∞ . In the rest of this section, we study some properties of the solutions of theevolution equation. In particular, we are interested in the study of the meancurvature when the paramater t goes to infinity. Notice that the equivalent equation involving endomorphisms will be v t = − ( K t − cI ) . roposition 5.2 Let h t , ≤ t < ∞ , be a 1-parameter family of Herm + ( E ) satisfying the evolution equation. Then (i) For any fixed Hermitian structure k of E , the functional L ( h t , k ) is a mono-tone decreasing function of t ; that is ddt L ( h t , k ) = −kK t − cI k ≤ (ii) max |K t − cI | is a monotone decreasing function of t ; (iii) If L ( h t , k ) is bounded below, i.e., L ( h t , k ) ≥ A > −∞ for some real constant A and ≤ t < ∞ , then max X |K t − cI | → as t → ∞ . (63) Proof:
From the proof of Lemma 4.5 we know that ddt L ( h t , k ) = ( K t − c h t , ∂ t h t ) . (64)Since h t is a solution of the evolution equation, we get ddt L ( h t , k ) = − ( K t − c h t , K t − c h t ) = −kK t − c h t k (65)and (i) follows from the definition of the Riemannian structure in Herm + ( E )(considered this time as a metric for endomorphisms). The proofs of (ii) and(iii) are similar to the proof in the classical case [10], but we need to work thistime with the operator ˜ (cid:3) h = i Λ D ′′ D ′ h instead of the operator (cid:3) h = i Λ D ′′ D ′ h .In fact, from this definition ˜ (cid:3) v t = ∂ t K t and since v t = − ( K t − cI ) , we get( ˜ (cid:3) + ∂ t ) K t = 0 . (66)On the other hand, D ′′ D ′ |K t − cI | = D ′′ D ′ tr( K t − cI ) = 2 tr(( K t − cI ) D ′′ D ′ K t ) + 2 tr( D ′′ K t · D ′ K t )and by taking the trace with respect to ω we get˜ (cid:3) |K t − cI | = 2 tr(( K t − cI ) ˜ (cid:3) K t ) + 2 i Λ tr( D ′′ K t · D ′ K t )= − K t − cI ) ∂ t K t ) − |D ′′ K t | = − ∂ t |K t − cI | − |D ′′ K t | . So, finally we obtain ( ∂ t + ˜ (cid:3) ) |K t − cI | = − |D ′′ K t | ≤ heorem 5.3 Let E be a Higgs bundle over a compact K¨ahler manifold X with K¨ahler form ω . Then we have the implications (i) → (ii) → (iii) for thefollowing statements: (i) for any fixed Hermitian structure k in E , there exists a constant B such that L ( h, k ) ≥ B for all Hermitian structures h in E ; (ii) E admits an approximate Hermitian-Yang-Mills structure, i.e., given ǫ > there exists an Hermitian structure h in E such that max X |K − ch | < ǫ ; (68) (iii) E is ω -semistable .Proof: Assume (i). Then the funcional is bounded below by a constant A . Inparticular L ( h t , k ) ≥ A for h t , 0 ≤ t < ∞ , a one-parameter family of Hermitianstructures satisfying the evolution equation. Therefore, from (63) it follows thatgiven ǫ > t such that max X |K t − cI | < ǫ for t > t . (69)This shows that (i) implies (ii). On the other hand, (ii) → (iii) has been provedby Bruzzo and Gra˜na-Otero in [4]. Q.E.D. We need some results which allow us to solve some problems about Higgs bun-dles by induction on the rank. This section is essentially a natural extension toHiggs bundles of the classical case, which is explained in detail in [10].Let 0 / / E ′ ι / / E p / / E ′′ / / h in E induces Hermitian structures h ′ and h ′′ in E ′ and E ′′ respectively. We have also a second fundamental form A h ∈ A , (Hom( E ′ , E ′′ )) and its adjoint B h ∈ A , (Hom( E ′′ , E ′ )) where, asusual, B ∗ h = − A h . In a similar way, some properties which hold in the ordinarycase, also hold in the Higgs case. Proposition 6.1
Given an exact sequence (70) and a pair of Hermitian struc-tures h, k in E . Then the function Q ( h, k ) and the form Q ( h, k ) satisfies thefollowing properties: (i) Q ( h, k ) = Q ( h ′ , k ′ ) + Q ( h ′′ , k ′′ ) , (ii) Q ( h, k ) = Q ( h ′ , k ′ ) + Q ( h ′′ , k ′′ ) − i tr[ B h ∧ B ∗ h − B k ∧ B ∗ k ]mod d ′ A , + d ′′ A , .Proof: (i) is straightforward from the definition of Q . On the other hand, (ii)follows from an analysis similar to the ordinary case. Notice that given ǫ >
0, any metric h = h t with t > t in principle satisfies (69). h we have a splitting of the exact sequence by C ∞ -homomorphisms µ h : E → E ′ and λ h : E ′′ → E . In particular, B h = µ h ◦ d ′′ ◦ λ h . (71)We consider now a curve of Hermitian structures h = h t , 0 ≤ t ≤ h = k and h = h . Corresponding to h t we have a family of homomorphisms µ t and λ t . We define the homomorphism S t : E ′′ → E ′ given by λ t − λ = ι ◦ S t . (72)A short computation (see [10], Ch.VI) shows that ∂ t B t = d ′′ ( ∂ t S t ) and choosingconvenient orthonormal local frames for E ′ and E ′′ , we know that the endomor-phism v t can be represented by the matrix v t = (cid:18) v ′ t − ∂ t S t − ( ∂ t S t ) ∗ v ′′ t (cid:19) where v ′ t , v ′′ t are the natural endomorphisms associated to h ′ t , h ′′ t respectively.Now, from the ordinary case we have R t = (cid:18) R ′ t − B t ∧ B ∗ t D ′ B t − D ′′ B ∗ t R ′′ t − B ∗ t ∧ B t (cid:19) , where R ′ t and R ′′ t are the Chern curvatures of E ′ and E ′′ associated to the metrics h ′ t and h ′′ t respectively. Now R , t = R t + [ φ, φ t ] and since E ′ and E ′′ are Higgssubbundles of E , we obtain a simple expression for the (1,1)-component of theHitchin-Simpson curvature R , t = (cid:18) R ′ , t − B t ∧ B ∗ t D ′ B t − D ′′ B ∗ t R ′′ , t − B ∗ t ∧ B t (cid:19) , where R ′ , t = R ′ t + [ φ, φ t ] E ′ and R ′′ , t = R ′′ t + [ φ, φ t ] E ′′ . Hence, at this pointwe can compute the tracetr( v t · R , t ) = tr( v ′ t · R ′ , t ) + tr( v ′′ t · R ′′ , t )+tr( ∂ t S t · D ′′ B ∗ t ) − tr(( ∂ t S t ) ∗ · D ′ B t )+tr( v ′ t · B t ∧ B ∗ t ) − tr( v ′′ t · B ∗ t ∧ B t ) . The last four terms are exactly the same as in the ordinary case. Finally we getthat, modulo an element in d ′ A , + d ′′ A , tr( v t · R , t ) = tr( v ′ t · R ′ , t ) + tr( v ′′ t · R ′′ , t ) − ∂ t tr( B t ∧ B ∗ t ) . (73)Then, multiplying the last expression by i and integrating from t = 0 to t = 1we obtain (ii). Q.E.D.From Proposition 6.1 we get an important result for the compact case when µ ( E ) = µ ( E ′ ) = µ . Indeed, in that case we have also µ ( E ′′ ) = µ . Then, byintegrating Q ( h, k ) and Q ( h, k ) over X , and since − i tr( B ∧ B ∗ ) ∧ ω n − = | B | ω n /n ! (74)we have the same constant c for all functionals L ( h, k ) , L ( h ′ , k ′ ) and L ( h ′′ , k ′′ )and we obtain the following 19 orollary 6.2 Given an exact sequence (70) over a compact K¨ahler manifold X with µ ( E ) = µ ( E ′ ) and a pair of Hermitian structures h and k in E , thefunctional L ( h, k ) satisfies the following relation L ( h, k ) = L ( h ′ , k ′ ) + L ( h ′′ , k ′′ ) + k B h k − k B k k . (75)In the one-dimensional case, when X is a compact Riemann surface, thenotion of stability (resp. semistability) does not depend on the K¨ahler form ω ,therefore we can establish our results without make reference to any ω . At thispoint we can establish a boundedness property for the Donaldson functional forsemistable Higgs bundles over Riemann surfaces. To be precise we have Theorem 6.3
Let E be a Higgs bundle over a compact Riemann surface X . Ifit is semistable, then for any fixed Hermitian structure k in Herm + ( E ) the set {L ( h, k ) , h ∈ Herm + ( E ) } is bounded below.Proof: Fix k and assume that E is semistable. The proof runs by induction onthe rank of E . If it is stable, then by [2] there exists an Hermitian-Yang-Millsstructure h on it, and we know that Donaldson’s functional must attain anabsolute minimum at h , i.e., for any other metric h L ( h, k ) ≥ L ( h , k ) (76)and hence the set is bounded below. Now, suppose E is not stable, then amongall proper non-trivial Higgs subsheaves with torsion-free quotient and the sameslope as E we choose one, say E ′ , with minimal rank. Since µ ( E ′ ) = µ ( E ) thissheaf is necessarily stable . Now let E ′′ = E / E ′ , then using Lemma 7.3 in [10]it follows that µ ( E ′′ ) = µ ( E ) and E ′′ is semistable and hence we have thefollowing exact sequence of sheaves0 / / E ′ / / E / / E ′′ / / E ′ and E ′′ are both torsion-free. Since dim X = 1 they are also locally freeand hence the sequence is in fact an exact sequence of Higgs bundles. Assumenow that h is an arbitrary metric on E , then by applying the preceding Corollaryto the metrics h and k we obtain L ( h, k ) = L ( h ′ , k ′ ) + L ( h ′′ , k ′′ ) + k B h k − k B k k , (78)where h ′ , k ′ and h ′′ , k ′′ are the Hermitian structures induced by h, k in E ′ and E ′′ respectively. If the rank of E is one, it is stable and hence L ( h, k ) is boundedbelow by a constant which depends on k . Now, by the inductive hypothesis, If E ′ is not stable, there exists a proper Higgs subsheaf F ′ of E ′ with µ ( F ′ ) ≥ µ ( E ′ ) andsince F ′ is clearly a subsheaf of E and this is semistable we necessarily obtain µ ( F ′ ) = µ ( E ),which is a contradiction, because E ′ was chosen with minimal rank. In fact, if E ′′ is not semistable, then there exists a proper Higgs subsheaf H of E ′′ with µ ( H ) > µ ( E ′′ ). Then, using Lemma 7.3 in [10] we have µ ( E ′′ ) > µ ( E ′′ / H ). Defining K as thekernel of the morphism E −→ E ′′ / H , we get the exact sequence0 / / K / / E / / E ′′ / H / / µ ( E ) = µ ( E ′′ ), using again the same Lemma in [10] we conclude that µ ( K ) > µ ( E ),which contradicts the semistability of E . ( h ′ , k ′ ) and L ( h ′′ , k ′′ ) are bounded below by constants depending only on k ′ and k ′′ respectively. Then L ( h, k ) is bounded below by a constant dependingon k . Q.E.D.As a consequence of the above result, we get that in the one-dimensionalcase all three conditions in the main theorem are equivalent. As a consequencewe obtain the following Corollary 6.4
Let E be a Higgs bundle over a compact Riemann surface X .Then E is semistable if and only if E admits an approximate Hermitian-Yang-Mills structure. This equivalence between the notions of approximate Hermitian-Yang-Mills struc-tures and semistability is one version of the so called
Hitchin-Kobayashi corre-spondence for Higgs bundles. As a consequence of the Corollary 6.4 we see thatin the one-dimensional case, all results about Higgs bundles written in termsof approximate Hermitian-Yang-Mills structures can be traslated in terms ofsemistability. In particular we have
Corollary 6.5 If E and E are semistable Higgs bundles over a compact Rie-mann surface X , then so is the tensor product E ⊗ E . Furthermore if µ ( E ) = µ ( E ) , so is the Whitney sum E ⊕ E . Corollary 6.6 If E is semistable Higgs bundle over a Riemann surface X , thenso is the tensor product bundle E ⊗ p ⊗ E ∗⊗ q and the exterior product bundle V p E whenever p ≤ r . The equivalence between the existence of approximate Hermitian-Yang-Millsstructures and semistability is also true in higher dimensions. However, sincetorsion-free sheaves over compact K¨ahler manifolds with dim X ≥ A natural notion of a metric on a torsion-free sheaf is that of admissible Her-mitian structure. This was first introduced by S. Bando and Y.-T. Siu in [19].In their article, they proved first the existence of admissible structures on anytorsion-free sheaf, and then obtained an equivalence between the stability of atorsion-free sheaf and the existence of an admissible Hermitian-Yang-Mills met-ric on it, thus extending the Hitchin-Kobayashi correspondence to torsion-freesheaves.Admissible structures were used again by I. Biswas and G. Schumacher [5]to prove an extended version of the correspondence of Bando and Siu to theHiggs case. In this last section, we briefly discuss some of these notions.Let E be a torsion-free Higgs sheaf over a compact K¨ahler manifold X . Thesingularity set of E is the subset S = S ( E ) ⊂ X where E is not locally free. Asis well known, S is a complex analytic subset with codim S ≥
2. Following [5],[19], an admissible structure h on E is an Hermitian metric on the bundle E | X \ S (i) The Chern curvature R of h is square-integrable, and (ii) The corresponding mean curvature K = i Λ R is L -bounded.Let consider now the natural embedding of E into its double dual E ∨∨ ;since S ( E ∨∨ ) ⊂ S ( E ), an admissible structure on E ∨∨ restrics to an admissiblestructure on E . An admissible structure h is called an admissible Hermitian-Yang-Mills structure if on X \ S the mean curvature of the Hitchin-Simpsonconnection is proportional to the identity. In other words if K = K + i Λ[ φ, ¯ φ h ] = c · I (79)is satisfied on X \ S for some constant c , where I is the identity endomorphismof E . It is important to note here that, in contrast to an admissible metric, thecondition defining an admissible Hermitian-Yang-Mills metric depends on theHiggs field.Let E be a torsion-free Higgs sheaf over a compact K¨ahler manifold X . Sinceits singularity set S is a complex analytic subset with codimension greater orequal than two, X \ S satisfies all assumptions Simpson [2] imposes on the basemanifold and hence we can see E | X \ S as a Higgs bundle over the non-compactK¨ahler manifold X \ S .Following Simpson [2], Proposition 3.3 (see also [5], Corollary 3.5) it fol-lows that a torsion-free Higgs sheaf over a compact K¨ahler manifold with anadmissible Hermitian-Yang-Mills metric must be at least semistable. However,as Biswas and Schumacher have shown in [5], this is just a part of a strongerresult. To be more precise, they proved the Hitchin-Kobayashi correspondencefor Higgs sheaves. This result can be written as Theorem 7.1
Let E be a torsion-free Higgs sheaf over a compact K¨ahler man-ifold X with K¨ahler form ω . Then, it is ω -polystable if and only if there existsan admissible Hermitian-Yang-Mills structure on it. Let h be an admissible metric on E , then K h is L -bounded. On the otherhand, from [5], Lemma 2.6, we know the Higgs field φ is also L -bounded on X \ S (in particular it is square integrable). From this we conclude that K h = K h + i Λ[ φ, ¯ φ h ] (80)is L -bounded and hence, for any admissible metric h on the torsion-free Higgssheaf E we must have Z X \ S |K h | ω n < ∞ . (81)Let Herm + ( E X \ S ) be the space of all smooth metrics on E X \ S satisfying thecondition (81) and suppose that h and k are two metrics in the same connected Notice that since X is compact, by [2], Proposition 2.1, it satisfies the assumptions on thebase manifold that Simpson introduced. Now, from this and [2], Proposition 2.2, it followsthat X \ S also satisfies the assumptions. + ( E X \ S ). Then h = ke v for some endomorphism v of E | X \ S and following Simpson [2], we can write the Donaldson functional as L ( ke v , k ) = Z X \ S tr [ v ( K k − cI )] ω n n ! + Z X \ S r X i,j =1 ψ ( λ i , λ j ) |D ′′ v ij | ω n − ( n − ψ is given by ψ ( λ i , λ j ) = e ( λ i − λ j ) − ( λ i − λ j ) − λ i − λ j ) . (83)We define the Donaldson functional on the Higgs sheaf E just as the corre-sponding functional (82) defined on the Higgs bundle E | X \ S . In [2], Simpsonestablished an inequality between the supremum of the norm of the endomor-phism v relating the metrics h and k and the Donaldson functional for Higgsbundles over (non necessarily) compact K¨ahler manifolds; this result can beimmediately adapted to Higgs sheaves as follows: Corollary 7.2
Let k be an admissible metric on a torsion-free Higgs sheaf E over a compact K¨ahler manifold X with K¨ahler form ω and suppose that sup X \ S |K k | ≤ B for certain fixed constant B . If E is ω -stable, then there existconstants C and C such that sup X \ S | v | ≤ C + C L ( ke v , k ) (84) for any selfadjoint endomorphism v with tr v = 0 and sup X \ S | v | < ∞ and suchthat sup X \ S |K ke v | ≤ B . In a future work, we will study more in detail admissible metrics and Don-aldson’s functional for torsion-free Higgs sheaves and the equivalence betweensemistability and the existence of approximate Hermitian-Yang-Mills metricsfor Higgs bundles in higher dimensions.
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