Nonabelian Hodge Theory

Carlos T.Simpson

Laboratoire de Topologie et Geometrie, U.F.R.M.I.G. Université Paul Sabatier, 118, route de Narbonne, F-31062 Toulouse-Cedex, France

Introduction

Classically, Hodge theory and related constructions provided extra structure to abelian topological invariants of the usual topological spaces associated to alge­braic varieties over (C. I would like to explain how, in analogy with these abelian constructions, certain nonabelian topological invariants of complex algebraic varieties have extra structures.

We will be concerned with two related invariants, the space R of framed 71-dimensional representations of %\(X,x), and M, its universal categorical quo­tient by the action of G£(n). These are related to the nonabelian cohomology H1(X,G^(n)), which is the set of isomorphism classes of representations, and which has a structure of non-Hausdorff space. The moduli space M is the set of Jordan equivalence classes of representations, the universal Hausdorff space to which the cohomology space maps. The representation space R is the coho­mology of X relative to a choice of base point x. We will interpret R and M as nonabelian analogues of the abelian cohomology H1(X,(C).

The Nonabelian Hodge Theorem

Suppose X is smooth and projective over C A harmonic bundle on X is a C00

vector bundle E with differential operators d and d, and algebraic operators 9 and 8 (operators from E to one-forms with coefficients in E), such that the following hold. There exists a metric K so that ô + ô is a unitary connection and 8 + 8 is self adjoint. And if we set D = d + d + 8 + 8 and D" = d + 8, then D2 = 0 and (D")2 = 0. With these conditions, (E,D) is a vector bundle with flat connection, and (E,d,9) is a Higgs bundle: a holomorphic vector bundle with holomorphic section 0 e H°(End(E) ® ß^) such that 8 A 9 = 0. Fix a class of Kahler metric for X. A Higgs bundle is stable (resp. semistable) if, for any coherent subsheaf F cz E preserved by 9 with 0 < rk(F) < rk(F), we have

deg(F)/rk(F) < (resp. <) deg(£)/rk(£).

Proceedings of the International Congress n f A A Q i U m a l i r i a n c Tfi/rdr» Tarati 1 QOfi

748 Carlos T. Simpson

Theorem 1. There is a natural equivalence between the categories of harmonic bun­dles on X and semisimple flat bundles (or representations of %\(X)). There is a natural equivalence between the categories of harmonic bundles and direct sums of stable Higgs bundles with vanishing Chern classes. The resulting correspondence be­tween representations and Higgs bundles can be extended to an equivalence between the categories of all representations of%\(X), and all semistable Higgs bundles with vanishing Chern classes.

The first part of the theorem, relating harmonic bundles and representations, is due to Codette [2] and Donaldson [6]. It is a generalization of the theorem of Eells and Sampson - a harmonic bundle is equivalent to a representation of the fundamental group in G£(n) together with an equivariant harmonic map from the universal cover of X to G£(n)/U(n). This equivalence uses the Bochner formula of Siu [18]. The part relating harmonic bundles and Higgs bundles is a generalization of the theorem of Narasimhan and Seshadri [11] (when 0 = 0 a Higgs bundle is just a vector bundle, and the corresponding representation is unitary). Part of this generalization involves the higher dimensional Hermitian-Yang-Mills theory for stable vector bundles developed by Donaldson [5] and Uhlenbeck, Yau [19]. Hitchin [9] gave the definition of stable Higgs bundles, and the proof of their correspondence with irreducible flat bundles, when X is a curve. These two directions are combined for the general statement in [14]. The last statement of the theorem comes from a formality result for the collection of complexes which control extensions [17].

A generalization to the case of noncompact curves is carried out in [16]. The noncompact higher dimensional case is still open (Codette has some recent work on nonabelian L2 cohomology in the higher dimensional case). For the remainder of the present discussion we will stick with the assumption that X is compact.

The set of flat bundles is analogous to the abelian de Rham cohomology, while the set of Higgs bundles (E, 8) is analogous to the abelian Dolbeault cohomology, Hx(®x) © H°(Qx). The first two parts of Theorem 1 may be interpreted as giving harmonic representatives for certain nonabelian de Rham and Dolbeault coho­mology classes. The fact that the notion of harmonic representative (harmonic bundle) is the same in both cases, is the analogue of the abelian Kahler identity Ad = 2A-Q.

Variations of Hodge Structure. There is a natural <C* action on the category of semistable Higgs bundles with vanishing Chern classes, defined by t : (E, 8) »-> (E,t8).

Lemma 2. The semisimple representations which are fixed by this action of (C* are exactly those which underly complex variations of Hodge structure.

Proof. If (E, 8) is fixed, then there is an action of C on E, corresponding to a decomposition E = 0 E? with 8 : Ep -» E?'1 ® Ql

x. The spaces Ep will be mutually orthogonal with respect to the harmonic metric [14]. A harmonic bundle with orthogonal decomposition which is preserved by d and shifted once by 8 is exactly a complex variation of Hodge structure. •

Nonabelian Hodge Theory 749

Rigid Representations. A representation Q of m (X) is rigid if any nearby represen­tation is conjugate to it. The correspondence given by Theorem 1 is continuous on the moduli space of semisimple representations (see below). Thus, if a semisimple representation is rigid, then it must be fixed by (ST, and Lemma 2 implies that it comes from a complex variation of Hodge structure. This can be strengthened:

Proposition3. Suppose Q is a rigid semisimple representation ofm(X). Then there is a (^-variation of Hodge structure FQ such that Q is a direct factor of the monodromy representation of VQ®Q (that monodromy is a sum of conjugates of Q). If Q is two dimensional, then the variation of Hodge structure FQ can be assumed to have Hodge types (1,0) and (0,1) only.

For an Aadic analogue, suppose X is defined over K cz (C. If a representation Q is rigid, and integral at a prime £, then the resulting representation into PG£(n,Qj>) descends to a representation of the algebraic fundamental group nfg(XK') for some finite extension K'/K. These results suggest the following conjectures (which are actually special cases of the conjectures in [15]).

Conjecture 4 (rigid => motivic). Suppose Q is a rigid semisimple representation of n\(X). Then Q is a direct factor in the monodromy representation of a motive (i.e. family of varieties) over X.

Conjecture 5 (rigid => integral). Suppose Q is a rigid semisimple representation of ny(X). Then Q is isomorphic to a representation with coefficients which are algebraic integers.

Note that the first would imply the second. By Proposition 3, a rigid integral two dimensional representation comes from a family of abelian varieties, so in this case the second statement would imply the first. In the case of two dimensional representations, one can show that a Zariski dense representation into S/(2) is either rigid, or else it is equal in PS7(2) to a representation pulled back (via an algebraic map) from a curve with orbifold structure. So Conjecture 5 would imply that any such two dimensional representation is pulled back, either from a curve with orbifold structure, or a subspace of the moduli space of abelian varieties.

Restrictions on Fundamental Groups. The nonabelian Hodge theorem, and re­lated considerations, have led to restrictions on which groups can be fundamental groups of smooth projective varieties (or really, of any compact Kahler mani­folds). This subject begins with Siu's rigidity theorem [18]. Siu's method has been used in [1], [2], and [13]. A typical example is:

Theorem 6 (Carlson, Toledo). Suppose F is a discrete cocompact subgroup of SO(n, 1) (n > 2). Then F is not the fundamental group of a compact Kahler mani­fold.

750 Carlos T. Simpson

The monodromy group of a variation of Hodge structure must be a Hodge group [17]. For connected simple groups, the Hodge groups ate all except the complex groups, and those isogenous to S/(n,IR) (n > 3), SU*(2n) (n > 3), SO(p,q) (p, q odd), Fö(6)5 or ^6(-26)- By work of many people, almost all lattices in real groups are known to be rigid, so some further nonexistence results can be derived from Proposition 3 (again, indirectly, from Siu's Bochner formula).

Theorem 7. Suppose G is a real group which is not a Hodge group. Suppose F c= G is a rigid lattice. Then F is not the fundamental group of a compact Kahler manifold. If F is a group of the above form, and ifFf is a semi-direct product of F with any other group (in which F is the quotient), then F' is not the fundamental group of a smooth projective variety.

Siu's technique has been used to obtain other topological information. In a related direction, Codette has used harmonic maps to prove some rigidity results about lattices in real groups. With a generalization of the Bochner technique applying to spaces with quaternionic structures, he proves that any cocompact lattice F in Sp(n, 1) (n > 2), or F^"20, is superrigid over archimedean fields [3].

Moduli Spaces

Fix a number n. The Betti versions of our topological invariants are easy to define. RB(X,X) is the space of representations of the fundamental group %\(X,x) on the fixed space C". It is an affine variety defined by the generators and relations of the fundamental group. The group G£(n, (C) acts. There is a universal categorical quotient MB(X), sometimes called the character variety. When X is a smooth projective variety, R and M have algebraic de Rham and Dolbeault realizations, as provided by the following theorem.

Theorem 8. Suppose X is a smooth projective variety over (C, with a point x e X. There exist algebraic varieties RDR PC x), RD0\(X,X), MDR(X) , and M^0\(X), quasiprojective over <C, which are moduli spaces as follows. RDR is a fine moduli space for the set of(E, V, ß) where (E, V) is a vector bundle with integrable algebraic connection and ß : Ex = <En is a frame. RDOI is a fine moduli space for the set of (E,8,ß) where (E,8) is a semistable Higgs bundle with vanishing Chern classes, and ß is a frame for Ex. The group G«f(n,(C) acts on RDR and RDOI* and MDR and Mpoi are the universal categorical quotients. Their points parametrize semisimple objects.

The construction of the moduli space of Higgs bundles was treated in Hitchin's paper [9], and later by Nitsure [12], in the case when X is a curve. The higher dimensional case is treated in a preprint by the author (containing some errors -particularly with regard to the next proposition - which have subsequently been corrected).

Proposition 9. The Riemann-Hilbert correspondence between bundles with integrable connection and representations provides isomorphisms of the associated complex an­alytic spaces, (RDR)an = (RB)an and (MDR)an = (MB)an.

Nonabelian Hodge Theory 751

The Riemann-Hilbert correspondence is not algebraic. It provides an example of two different algebraic varieties whose associated complex analytic spaces are isomorphic, generalizing the example of Serre.

Proposition 10. The correspondence of Theorem 1 provides a homeomorphism be­tween the usual topological spaces underlying MDOI and MB. However, the extended correspondence between the points ofR^0\ and R& provided by the last part of The­orem 1, is not continuous.

This proposition was treated by Hitchin in his case [9]. Hitchin also discussed several important properties of the moduli space MDOI, all of which carry over to the case of higher dimensional X:

Theorem 11. There is a natural algebraic action of(E* on the space MDOI, repre­senting the action defined previously. There is a proper map from MDOI to a vector space, compatible with an action of (C* on the vector space having only positive weights. Thus if p is any point in MD0I> then the limit \imt^o tp exists in MDOI-

A corollary of this theorem, the continuity of Proposition 10, and the charac­terization of Lemma 2, is that any representation of m (X) can be deformed to a complex variation of Hodge structure. This provides the mechanism for proving the restrictions stated in the second half of Theorem 7.

The Gauss-Manin Connection. Suppose X —> S is a smooth projective family over a base scheme S, with section of base points x : S —> X. Then the above constructions give families of moduli spaces RDR (X/S, x) and M D R C Z / S ) over S. There is an analogue of the Gauss-Manin connection.

Theorem 12. Suppose X/S is smooth and projective with base point x(S). Let (S x S)A denote the formal neighborhood of the diagonal in S xS, similarly in S xS xS. There is a canonical isomorphism

<P :P\(RDR(X/S))\(SXS)* -> P*2(RDR(X/S))\{SXS)*>

such that on (SxSxS)A the cocycle condition pì$((p)pì2(<p) = Ph(ç) is satisfied. The same holds for the universal categorical quotient M^R(X/S). These isomorphisms are compatible with the local analytic trivializations which arise from the Riemann-Hilbert correspondence.

Proof. The category of vector bundles with connection on X/S can be interpreted as a category of crystals on X/S. If So cz S is a closed subscheme defined by a nilpotent ideal, and Xo is the inverse image of So, then the category of crystals on X/S is equivalent to the category of crystals on Xo/S. Hence the scheme &DR(X/S) over S depends functorially only on the family Zo —• So and the inclusion So cz S, although the existence of R^^(X/S) seems to depend on the existence of a family X —> S. This gives a crystal of schemes over the stratifying site of S, which is equivalent to the structure described in the theorem. D ,

752 Carlos T. Simpson

At points where S and RDR (X/S) are smooth, the structure provided by this theorem may be described more simply as a splitting %*TS —> T(RDR(X/S)) of the quotient of tangent bundles, satisfying an integrability condition. If S is an artinian scheme, it gives a natural isomorphism R D R ( X / S ) = RDR(XO) X S.

The Hyperkähler Structure, The identification given by Theorem 1 is not complex analytic. Thus, the same space is given several complex structures.

Theorem 13 (Hitchin). Denote by Ms, the set of smooth points of the moduli space. The isomorphism Mpol = M | is smooth. Let I and J denote the complex structures on Ms obtained from MQO1 and M | respectively. Then K = IJ is a complex structure, and this triple gives a quaternionic structure in the tangent space at any point. The resulting quaternionic manifold Ms has a natural hyperkähler structure.

This was pointed out by Hitchin [9] in the case when X is a curve; the general case has been treated in [4] and recently, in detail, by Fujiki [7]. We refer to Hitchin's paper for the definition of hyperkähler manifold.

The Space of ^-Connections. Here is a definition due to Deligne - the contents of this subsection were all described or outlined by him in correspondence [4]. Suppose l e C A X-connection on a vector bundle E is an operator V : E -» E ® Qx, such that V(ae) = Xd(a)e + aV(e). If X ^ 0 then a A-connection V corresponds to a connection A_1V. On the other hand, if X = 0 then a bundle with integrable A-connection is just a Higgs bundle (E, 8), 8 = V. So this definition provides a deformation from the notion of connection to the notion of Higgs bundle. Its importance comes from the resulting moduli space.

Theorem 14. Suppose X is smooth and projective over (C, with base point x. There exist quasiprojective algebraic varieties W+ —> A1 and Z + —> A1 which are moduli spaces for X-connections as follows. W+ is a fine moduli space for (X,E,V,ß) where X G A1, (F,V) is a vector bundle with integrable X-connection (semistable with vanishing Chern classes in the case X = 0), and ß is a frame for Ex. The group G£(n, (C) acts on W+ with universal categorical quotient Z + .

The fibers Wf, W$, Z+, and ZQ are respectively the moduli spaces RDR, RDO15 MDR, and JÎfooi- There are natural actions of (C* on W+ and Z + , defined by t : (X,E,V,ß) •-> (tX,E,iS7,ß), covering the standard action on A1.

Deligne thought of the space of A-connections as a way of approaching the notion of quaternionic structure for the singular space M. There are complex analytic spaces W and Z over P 1 characterized by the following properties. There are actions of (C* covering the standard action on P1 , and antilinear involutions G compatible with the action of (C* (and the map to P1) relative the involution t\-* (t)"1. There are identifications

w\Al^w+ z\Al^z+

Nonabelian Hodge Theory 753

compatible with the action of C*. On the fibers W\ = RB and Zi ^ MB, the involution a takes a representation to the complex conjugate representation.

A harmonic bundle E yields a family of holomorphic bundles (E, d + X8) with A-connections V = Xd + 8, indexed by A e A1. The resulting section A1 —• W+ is holomorphic and extends to a section P 1 —• W. Refer to these sections (and their projections in Z) as preferred sections. The involution o maps a preferred section to the preferred section corresponding to the complex conjugate representation. We get a trivialization Z = M x F1 , where M denotes the moduli space of harmonic bundles.

Proposition 15 (Deligne). The trivialization Z = M x P 1 is a homeomorphism, and on the set of smooth points it identifies Z s with the twistor space for the quaternionic structure of Ms.

Proof. There is a P 1 of complex structures q in the quaternions. The twistor space is Ms x P 1 endowed with a complex structure which makes the horizontal projective lines holomorphic, and which endows each fiber Ms x {q} with the complex structure q deduced from the quaternionic structure of Ms. One has to check that the identification Zq = Ms takes the complex structure of Zq to the complex structure q of Ms. D

Deligne proposes a way of defining the notion of quaternionic structure for a singular space such as M by some axioms for a twistor space such as Z.

Compactifications. The space of A-connections suggested by Deligne turns out to be very useful for compactifying the moduli spaces.

Lemma 16. Suppose z is a point ofZ+. Then the limit limf_>o tz exists in Z + . Also, the set of fixed points o / C in Z + is compact.

Proposition 17. Suppose Z + is any algebraic variety with an algebraic action of<D*, such that the limits lim^o tz exist, and such that the set of fixed points is compact. Let S+ CZ Z + denote the subset of points s such that limt^œ ts exists in Z+. Then there is a universal geometric quotient P = (Z+ — S+)/(L*, and P is a proper scheme (not necessarily projective).

We can apply this construction to the moduli space of A-connections Z + , to obtain a compactification of M DR.

Corollary 18. Let Z + be the space of X-connections. Let PDR = (Z + — S+)/(C* be the proper scheme defined above. Then MDR <= PDR is an open subset, providing a compactification of the space of bundles with integrable connection.

Proof. Note that the subset S+ cz Z + is contained in the fiber over 0 e A1. Let D = (Z£ — S+)/(C* cz P D R. It is a closed subset, and the complement is equal to ( Z + - Z + ) / ( C * ^ M D R . •

There is a similar compactification MDOI <= FDOI where the complement is the same divisor D. In fact, these fit together into a relative compactification of Z+ over A1.

754 Carlos T. Simpson

The Hodge Filtration. Here is another use for the space of A-connections. It can be thought of as the nonabelian analogue of the Hodge filtration on the de Rham moduli space. Suppose F is a finite dimensional vector space with an exhaustive decreasing filtration Fp. Define a locally free sheaf Ç(V,F) over A1 as the submodule of F (g) <£[t, r1] generated by the t~pFp.

Lemma 19. The construction Ç provides an equivalence between the category of vector spaces with exhaustive decreasing filtrations, and the category of locally free sheaves on A1 with action o/C* covering the standard action on A1.

The vector space V is recovered as the fiber of Ç(V,F) at 1, whereas the fiber at 0 is canonically the associated graded Gr^(F). Given a real structure, one can glue together £(V,F) and £,(V,F) using the involution ty-> (ï)_1

5 to obtain a locally free sheaf t;(V,F,F) on P1 with action of (C* and antilinear involution.

In view of this lemma, we can define a nonabelian filtration to be a scheme over A1 with action of (C*. The spaces W+ and Z + over A1, together with their (C* actions, may then be interpreted as the nonabelian Hodge filtrations of the. fibers RDR = W? and MDR = Zx

+. The spaces o(W+) and G(Z+) (which are the restrictions of W and Z to the affine neighborhood of infinity) are the complex conjugates of the Hodge filtrations. The Dolbeault spaces RDOI and MDoi become the associated graded spaces of the nonabelian Hodge filtrations (just as for abelian cohomology). Lemma 16 is analogous to saying that the Hodge filtration is concentrated in positive degrees.

These definitions are compatible with the mixed Hodge structure defined by Morgan [10] and Hain [8] on the nilpotent completion of the fundamental group. Let A = (C7Ci)A be the completion of the group algebra at the augmentation ideal. They show that this has a Hodge filtration F giving a mixed Hodge structure (the real structure being the completion of the real group algebra, and the weight filtration given by powers of the augmentation ideal). By Lemma 19, this data is equivalent to the sheaf of algebras Ç(A,F,F) together with its <C* action and involution.

Given a complete augmented algebra such as A, one can define a formal scheme R(A), the space of representations infinitesimally close to the trivial one. For A = (CTCI)A as above, the space R(A) is naturally isomorphic to the completion of RB at the point corresponding to the trivial representation.

Theorem20. The family of algebras £,(A,F,F) over P1 gives rise to a family of formal schemes R(Ç(A,F,F)) over P1. This family of formal schemes (with C* action and involution) is naturally isomorphic to the formal neighborhood of the trivial preferred section in W. The isomorphism is compatible with the one described above in the fiber over 1.

Conversely, Tannaka duality allows the mixed Hodge structure Ç(A, F, F) to be recovered from the data of the formal neighborhoods of the spaces W, together with some information on how they fit together for different ranks.

Nonabelian Hodge Theory 755

Speculations

It is natural to wonder how the results outlined above might fit into a notion of nonabelian Hodge structure. The action of (C* on the space of representations gives, by Tannaka duality, an action of (C* on the proalgebraic completion w\ (X) (the inverse limit of the Zariski closures of the images, over all finite dimensional representations of ni(X)). The action of S1 cz C* on the proreductive quotient m[ed satisfies some axioms which might be taken as a preliminary definition of pure Hodge structure on a group [17], The map S1 x 7ti(X) —• mfed is continuous when the right side is given the inverse limit of analytic topologies. And the action of —1 6 S1 is a Cartan involution of the real form (tn[ed)]R (inverse limit of real Zariski closures). The restrictions of the first half of Theorem 7 may be derived from these axioms (one would like to interpret all those kinds of topological restrictions as following from the nonabelian Hodge structure).

This definition refers to the continuous structure of the space of represen­tations only in the crudest way. Preferable would be a notion of nonabelian Hodge structure for the space of representations. We have defined the nonabelian Hodge filtration. What about weight filtrations and polarizations? Probably the hyperkähler metric will play a role. The mixed Hodge structure on (<Dn\)A will provide the power series for the nonabelian Hodge structure at the trivial repre­sentation (Theorem 20).

With or without a precise definition, we are afforded the pleasant opportunity of revisiting all the old glories of Hodge theory, and trying to provide their nonabelian analogues. The subject started with Griffiths' transversality result. Preliminary heuristic calculations seem to show that an analogous transversality relates the Gauss-Manin connection defined in Theorem 12, to the Hodge filtra­tion interpreted as the space Z+ . The lifts of vector fields from the base S, to the restriction of Z+(X/S) over A1 — {0}, will have poles of order < 1 along ZQ (and hence also along the divisor at infinity in the compactificaton).

This raises the question of whether there is a good notion of variation of nonabelian Hodge structure over S. One could try to replicate the works of Schmid and others about degenerating variations. Is there an appropriate sense in which the Gauss-Manin connection has regular singularities? And is there some type of limiting nonabelian Hodge structure? This type of theory would be in the service of dévissage, trying to obtain information about the topology of a space X by fibering X -> Y and studying the family of fundamental groups of the fibers. For example, the moduli space M(X) (with all of its structures) becomes a constant subspace of the varying family M(Xy). The wide array of strutures on M(Xy) seems to make this an unlikely event.

Let me close with some wild speculation on how this picture might fit into an overall view of Hodge theory. We have treated two topological invariants, the space R of framed representations, and the quotient M. For which topological invariants will there be such a theory? Suppose F : Top —» Sch is a functor from the category of topological spaces to the category of schemes. Are there some simple conditions which would guarantee that, for a smooth projective variety X, F(X) has a "Hodge structure"? Perhaps (to say something entirely

756 Carlos T. Simpson

unsubstantiated) there might be an action of (C* on the set of such functors F, and the functors which are fixed are of Hodge type.

References

1. Carlson, J.A., Toledo, D. : Harmonic mappings of Kahler manifolds to locally sym­metric spaces. Pubi. Math. I.H.E.S. 69 (1989) 173-201

2. Codette, K: Flat G-bundles with canonical metrics. J. Diff. Geom. 28 (1988) 361-382 3. Codette, K. : Archimedean superrigidity and hyperbolic geometry. Preprint 4. Deligne, P: Letter, March 20 (1989) 5. Donaldson, S.K.: Anti self dual Yang-Mills connections over complex algebraic

surfaces and stable vector bundles. Proc. London Math. Soc. 50 (1985) 1-26 6. Donaldson, S.K.: Twisted harmonic maps and self-duality equations. Proc. London

Math. Soc. 55 (1987) 127-131 7. Fujiki, A. : Hyperkähler structure on the moduli space of flat bundles. Preprint 8. Hain, R. : The de Rham homotopy theory of complex algebraic varieties, I. K-Theory

1 (1987) 271-324 9. Hitchin, N.J. : The self-duality equations on a Riemann surface. Proc. London Math.

Soc. 55 (1987) 59-126 10. Morgan, J. : The algebraic topology of smooth algebraic varieties. Pubi. Math. I.H.E.S.

48 (1978) 137-204 11. Narasimhan, M.S., Seshadri, CS.: Stable and unitary bundles on a compact Riemann

surface. Ann. Math. 82 (1965) 540-564 12. Nitsure, N. : Moduli space of semistable pairs on a curve. Proc. London Math. Soc.

(3) 62 (1991) 275-300 13. Sampson, J.H.: Applications of harmonic maps to Kahler geometry. Contemp. Math.

49 (1986) 125-133 14. Simpson, C.T.: Constructing variations of Hodge structure using Yang-Mills theory

and applications to uniformization. J.A.M.S. 1 (1988) 867-918 15. Simpson, C.T.: Transcendental aspects of the Riemann-Hilbert correspondence. Illi­

nois J. Math. 34 (1990) 368-391 16. Simpson, C.T: Harmonic bundles on noncompact curves. J.A.M.S. 3 (1990) 713-770 17. Simpson, C.T: Higgs bundles and local systems. Pubi. Math. I.H.E.S. (to appear) 18. Siu, Y.T.: Complex analyticity of harmonic maps and strong rigidity of complex

Kahler manifolds. Ann. Math. 112 (1980) 73-110 19. Uhlenbeck, K.K., Yau, S.T. : On the existence of Hermitian-Yang-Mills connections

in stable vector bundles. Comm. Pure and Appi. Math. 39-S (1986) 257-293