NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD INDRANIL BISWAS Abstract. Our aim is to review some recent results on holomorphic principal bundles over a compact Kähler manifold. We describe a necessary and sufficient condition for a holomorphic principal G bundle over a compact Riemann surface to admit a holomorphic connection. Another result of the existence of connections is the following: A principal G bundle E G over a compact Kähler manifold admits a Hermitian Einstein connection if and only if E G is polystable. Let G be a connected reductive linear algebraic group over C. Fix a parabolic subgroup P G without any simple factor and also fix a character χ of P such that χ is trivial on the center Z(G) G, and the restriction of χ to the parabolic subgroup of each simple factor of G/Z(G) defined by P is nontrivial and antidominant. Let E G be a principal G bundle over a connected projective manifold M. Then the following four statements are equivalent: (1) The G bundle E G is semistable and the second Chern class c 2 (ad(e G )) H 4 (M, Q) vanishes. (2) The associated line bundle L χ := (E G C χ )/P over E G /P for the character χ is numerically effective. (3) For every pair of the form (Y, ψ), where Y is a compact connected Riemann surface and ψ : Y M a holomorphic map, and every holomorphic reduction E P ψ E G of structure group to P of the principal G bundle ψ E G over Y, the associated line bundle E P (χ) = (E P C χ )/P over Y is of nonnegative degree. (4) For any pair (Y, ψ) as in (3), the G bundle ψ E G over Y is semistable. A generalization to principal bundles of the Atiyah-Krull-Schmidt theorem for vector bundles is described. 1. Preliminaries Let M be a compact connected Kähler manifold equipped with a Kähler form ω. Let G be a linear algebraic group defined over the 1

2 I. BISWAS field of complex numbers. (E G, f, p), where A principal G bundle over M is a triple (1) E G is a complex manifold, (2) f : E G G E G is a holomorphic map defining a right action of G on E G, and (3) p : E G M is a holomorphic submersion such that the following two conditions hold: (1) p f = p p 1, where p 1 is the natural projection of E G G to E G, and (2) the map to the fiber product Id EG f : E G G E G M E G is an isomorphism (note that the condition p f = p p 1 implies that the image of Id EG f is contained in the submanifold E G M E G E G E G ). Let E G be a holomorphic principal G bundle over M. Given any homomorphism ρ : G H of complex groups, the extension of structure group of E G to H is the principal H bundle (E G H)/G over M. In the construction of this quotient, the action of any g G sends any point (z, h) E G H to (zg, ρ(g) 1 h). If H is a closed complex subgroup of G, then a holomorphic reduction of structure group of E G to H is a complex submanifold E H E G such that the action of H on E G makes E H a principal H bundle over M. A holomorphic reduction of structure group of E G to H is given by a holomorphic section of the fiber bundle E G /H over M. Given a holomorphic section σ : M E G /H, the inverse image q 1 (σ(m)) E G, where q : E G E G /H is the quotient map, is a holomorphic reduction of structure group of E G to H. If F is a coherent analytic sheaf defined on a dense open subset U M such that the complement M \ U is a complex analytic space

PRINCIPAL BUNDLES OVER KÄHLER MANIFOLD 3 of (complex) codimension at least two, then the degree of F is defined to be degree(f ) := c 1 (ι F )ω d 1, where d = dim C M, and ι : U M is the inclusion map. Note that the condition that the codimension of M \ U is at least two ensures that the direct image ι F is a coherent analytic sheaf on M. A parabolic subgroup of a reductive group G is a closed complex subgroup P such that G/P is compact. So, G itself is a parabolic subgroup of G. A Levi subgroup of P is a connected reductive subgroup of P that projects isomorphically to the quotient P/R u (P ), where R u (P ) denotes the unipotent radical of P. Any Levi subgroup of a parabolic group is a centralizer of some torus of G, and conversely, the centralizer of any torus of G is a Levi subgroup of some parabolic subgroup of G. A. Ramanathan introduced the notion of a (semi)stable principal bundle which generalizes Mumford s definition of a (semi)stable vector bundle; we will recall Ramanathan s definitions. A holomorphic principal G bundle E G over M is called semistable (respectively, stable) if for every triple (P, U, σ), where M (1) P G is a maximal proper parabolic subgroup, (2) U M is a dense open subset such that the complement M \U is a complex analytic space of (complex) codimension at least two, (3) σ : U E G /P is a holomorphic reduction of structure group, over U, of the principal G bundle E G to the subgroup P one has (1) degree(σ T rel ) 0 (respectively, degree(σ T rel ) 0), where T rel is the relative tangent bundle over E G /P for the natural projection E G /P M (see [Ra]). A principal G bundle E G over M is called polystable if either E G is stable or there is a parabolic subgroup P of G and a holomorphic reduction E L(P ) E G of structure group of E G over M to a Levi subgroup L(P ) P such that

4 I. BISWAS (1) the principal L(P ) bundle E L(P ) is stable; (2) the extension of structure group of E L(P ) to P, constructed using the inclusion of L(P ) in P, is an admissible reduction of structure group of E G to P. (A holomorphic reduction of structure group E P E G to P is called admissible if for every character χ of P trivial on the center of G, the associated line bundle E P (χ) over M is of degree zero.) Let H be a complex Lie group. Let π : E H M be a holomorphic principal H bundle over M. For any open subset U M, by Γ(T π 1 (U)) H we will denote the space of all H invariant holomorphic vector fields on π 1 (U). Associating Γ(T π 1 (U)) H to U we obtain a presheaf on M. The sheafification of it is a locally free coherent analytic sheaf on M (the coherence property of it follows from the fact that H acts transitively on the fibers of π). The holomorphic vector bundle defined by this sheaf is called the Atiyah bundle, and it is denoted by At(E H ). Note that At(E H ) has a subbundle defined by the sheaf of H invariant holomorphic vertical vector fields on E H (vertical for the projection π). This subbundle is canonically identified with the adjoint vector bundle ad(e H ) := E H H h, where h is the Lie algebra of H; the canonical identification is constructed using the fact that h is identified with the space of all left invariant holomorphic vector fields on H. Therefore, we have an exact sequence (2) 0 ad(e H ) At(E H ) T M 0 of holomorphic vector bundles over M; the vector bundle T M is the holomorphic tangent bundle of M (see [At2] for the details). A complex connection on E H is a smooth splitting of (2). A holomorphic connection on E H is a holomorphic splitting of (2). The Lie bracket operation of vector fields gives a Lie algebra structure on the fibers of all the three vector bundles in (2). The curvature of a complex connection on E H is the obstruction for the corresponding splitting of (2) to be a splitting of Lie algebra bundles. It is easy to see that the curvature of a complex connection on E H is a smooth section

PRINCIPAL BUNDLES OVER KÄHLER MANIFOLD 5 of (Ω 2,0 M Ω 1,1 M ) ad(e H ). The curvature of a holomorphic connection is a holomorphic section of Ω 2,0 M ad(eh ). Fix a maximal compact subgroup K(G) G, where G, as before, is a complex reductive group. If E K(G) E G is a smooth reduction of structure group to K(G) of a holomorphic principal G bundle E G over M, then there is a unique smooth connection on E K(G) that induces a complex connection on E G. Note that smooth reduction of structure group of E G to K(G) is given by a smooth section of the smooth fiber bundle E G /K(G) over M. This (unique) induced complex connection on E G given by E K(G) is known as the Chern connection corresponding to E K(G). Let denote the Chern connection on E G corresponding to the reduction of structure group E K(G) E G. Let K( ) denote the curvature on ; it is a smooth section of Ω 1,1 M ad(e G). So we have (3) Λ ω K( ) C (ad(e G )), where Λ ω is the adjoint of multiplication of forms by the Kähler ω. Let z(g) denote the center of the Lie algebra g of G. Note that z(g) C (ad(e G )). The Chern connection is called a Hermitian Einstein connection if the smooth section Λ ω K( ) in (3) is contained in the image of z(g). 2. Connections on principal bundles 2.1. Existence of connection on a Riemann surface. First we consider the case where M is a compact Riemann surface. Let G be a complex reductive affine algebraic group. Theorem 2.1 (AB, Theorem 4.1). A holomorphic G bundle E G over M admits a holomorphic connection if and only if for every triple of the form (H, E H, λ), where (1) H is a Levi subgroup of G, (2) E H E G is a holomorphic reduction of structure group to H, and (3) λ is a character of H, the associated line bundle E H (λ) = (E H C)/H over M is of degree zero.

6 I. BISWAS If we set G = GL(n, C), then this gives the well known result of Atiyah Weil that a holomorphic vector bundle E over M admits a holomorphic connection if and only if each direct summand of E is of degree zero (see [At2], [We]). 2.2. Polystable bundles over a Kähler manifold. A holomorphic vector bundle E over a compact Kähler manifold (M, ω) admits a Hermitian Einstein connection if and only if E is polystable [UY], [Do]. Theorem 2.2 (ABi, Theorem 0.1). A principal G bundle E G over a compact Kähler manifold M admits a Hermitian Einstein connection if and only if E G is polystable. Moreover, a polystable G bundle admits a unique Hermitian Einstein connection. When M is a projective manifold, Theorem 2.2 was proved in [RS]. In [BS], a similar result is proved for principal bundles over M with an arbitrary (not necessarily reductive) algebraic group as the structure group. Let E G be a polystable principal G bundle over M. Take a homomorphism of algebraic groups ρ : G H where H is also a connected reductive group. Let E H denote the principal H bundle obtained by extending the structure group of E G using ρ. If ρ takes the connected component, containing the identity element, of the center of G inside the connected component, containing the identity element, of the center of H, then E H is polystable [AB, page 224, Theorem 3.9]. When M is a projective variety, this was proved earlier in [RR] (see [RR, page 285, Theorem 3.18]). We give a sufficient condition which ensures that E H is actually stable. Using the holonomy of the (unique) Hermitian Einstein connection on E G, it is possible to construct a holomorphic reduction of structure group of E G to a certain reductive subgroup of G. The reductive subgroup of G in question, which depends on E G, is the Zariski closure of the holonomy group. This reduction of structure group behaves well with respect to the extension of structure group. The following theorem is proved in [Bi2]:

PRINCIPAL BUNDLES OVER KÄHLER MANIFOLD 7 Theorem 2.3. The principal H bundle E H is stable if and only if the image in H of the reductive subgroup of G corresponding to E G is not contained in any proper parabolic subgroup of H. The above theorem has the following corollary: Corollary 2.4. If E G does not admit any holomorphic reduction of structure group to any proper reductive subgroup of G, and furthermore, ρ(g) is not contained in any proper parabolic subgroup of H, then the principal H bundle E H is stable. 3. Criterion for semistability 3.1. Criterion for principal bundles on a Riemann surface. Let M be a compact connected Riemann surface. We recall that a holomorphic vector bundle E over M is called semistable if and only if for all holomorphic subbundles V E of positive rank the inequality degree(v )/rank(v ) degree(e)/rank(e) holds. One can construct examples to see that it is not enough to check this inequality for subbundles of a fixed rank to conclude semistability. The above inequality is needed to be checked for subbundles of all ranks. Similarly, to check that a holomorphic principal G bundle E G over M is semistable, it is necessary to verify the inequality (1) for all holomorphic reduction of structure group of E G to all maximal proper parabolic subgroups. It is not enough if (1) is verified for holomorphic reductions of E G to a fixed maximal proper parabolic subgroup of G. On the other hand, if f : Y M is a holomorphic map from a compact connected Riemann surface Y, and E G is a holomorphic principal G bundle over X, then the pullback f E G is a semistable G bundle provided E G is semistable. Indeed, this follows immediately from the fact that the Harder Narasimhan reduction of f E G descends to the Harder Narasimhan reduction of E G. (See [AAB], [BH1] for Harder Narasimhan reduction of a principal bundle.) The following theorem is proved in [BB]: Theorem 3.1. Fix a maximal proper parabolic subgroup P G.

8 I. BISWAS A holomorphic principal G bundle E G over M is semistable if and only if for every holomorphic map f : Y M from some compact connected Riemann surface Y and every holomorphic reduction of structure group σ : Y E G /P to the fixed parabolic subgroup P, the inequality degree(σ T rel ) 0 holds, where T rel is the relative tangent bundle over f E G /P for the natural projection f E G /P Y. A holomorphic principal G bundle E G over M is semistable if and only if the relative anti-canonical line bundle top T rel over E G /P is numerically effective. Actually, in [BB] the theorem is proved for a more general parabolic subgroup, not necessarily maximal. In the special case where G = GL(n, C) and P is such that G/P = CP n 1, this theorem was proved by Y. Miyaoka (see [Mi]). 3.2. Criterion for principal bundles on a Projective manifold. Now we will assume M to be a complex projective manifold. Furthermore, the singular cohomology class represented by the Kähler form ω on M will be assumed to be rational. Let V be a holomorphic vector bundle over M. It is known that V is semistable if and only if the restriction of V to the general smooth curve obtained by intersecting hyperplanes of sufficiently high degree is semistable [MR]. It is natural to ask if there is a characterization of all the holomorphic vector bundles over M whose restriction to every complete irreducible curve is semistable. By the above mentioned result, such a vector bundle must be semistable. In [BB] it is proved that the restriction of a holomorphic vector bundle V over M to every complete curve irreducible curve in M is semistable if and only if V is semistable with c 2 (End(V )) = 0. The following more general theorem is proved in [BB]. Theorem 3.2. Let G be a connected reductive linear algebraic group over C. Fix a parabolic subgroup P G without any simple factor, and also fix a character χ of P such that (i) χ is trivial on the center Z(G) G, and

PRINCIPAL BUNDLES OVER KÄHLER MANIFOLD 9 (ii) the restriction of χ to the parabolic subgroup of each simple factor of G/Z(G) defined by P is nontrivial and antidominant. Let E G be a principal G bundle over a connected projective manifold M. Then the following four statements are equivalent: (1) The G bundle E G is semistable and the second Chern class c 2 (ad(e G )) H 4 (M, Q) vanishes. (2) The associated line bundle L χ := (E G C χ )/P over E G /P for the character χ is numerically effective. (3) For every pair of the form (Y, ψ), where Y is a compact connected Riemann surface and ψ : Y M a holomorphic map, and every holomorphic reduction E P ψ E G of structure group to P of the principal G bundle ψ E G over Y, the associated line bundle E P (χ) = (E P C χ )/P over Y is of nonnegative degree. (4) For any pair (Y, ψ) as in (3), the G bundle ψ E G over Y is semistable. In [BG], the principal bundle analog of the Flenner s restriction theorem and the Grauert Mülich restriction theorem are proved. Let E G be a principal G bundle over M whose restriction to every curve is a trivial G-bundle, then E G itself is trivial [BH2]. Actually, in [BH2] this is proved for projective varieties defined over an arbitrary algebraically closed field. 4. Atiyah-Krull-Schmidt reduction Let M be a compact complex manifold. A holomorphic vector bundle over M is called indecomposable if it is not holomorphically isomorphic to a direct sum of two holomorphic vector bundles of positive rank. Any holomorphic vector bundle E over M decomposes as a direct sum of indecomposable vector bundles. This follows immediately by

10 I. BISWAS induction on rank. If E = k E i = i=1 l F j, j=1 where all E i and F j are indecomposable vector bundles over M, then m = n, and furthermore, each E i is isomorphic to some F j [At1]. In other words, two representations of any vector bundle as a direct sum of indecomposable vector bundles are isomorphic up to a permutation of the direct summands. Let E be a holomorphic vector bundle of rank n over M. Let E GL(n,C) be the principal GL(n, C) bundle over M defined by E. So the fiber of E GL(n,C) over any point x M is the space of all linear isomorphisms of C n with the fiber E x. Giving a filtration of subbundles of E is equivalent to giving a holomorphic reduction of structure group of E GL(n,C) to some parabolic subgroup of GL(n, C), and giving a holomorphic decomposition of E is equivalent to giving a holomorphic reduction of structure group of E GL(n,C) to some Levi subgroup. Let G be a reductive affine algebraic group defined over C. We note that if H G is a Levi subgroup, and H H is a Levi subgroup of the reductive group H, then H is also a Levi subgroup of G. Let E G be a holomorphic principal G bundle over M. In [BBN2] and [BP] the following is proved: There is a Levi subgroup H G and a holomorphic reduction of structure group E H E G over M such that E H does not admit any holomorphic reduction of structure group to any proper Levi subgroup of H, and furthermore, if E H1, here H 1 G is a Levi subgroup, is another such holomorphic reduction which does not admit any further reduction to any proper Levi subgroup of H 1, then there is a pair (g, τ), where (1) g G, and (2) τ : E G E G is a holomorphic automorphism of the principal G bundle E G coin- such that g 1 Hg = H 1 and the two submanifolds E H g and E H1 cide.

PRINCIPAL BUNDLES OVER KÄHLER MANIFOLD 11 If we set G = GL(n, C), then the above theorem says the following: If we take a holomorphic vector bundle E over M and two decompositions of E into direct sum of indecomposable vector bundles, then there is a holomorphic automorphism of E that takes one decomposition to the other. Therefore, setting G = GL(n, C) in the above result we recover the earlier mentioned theorem of [At1]. 5. Ramified bundles Parabolic vector bundles over a Riemann surface were introduced in [MS]. In [BBN1] the generalization of them in the set-up of principal bundles was obtained, which are called as the ramified bundles. We recall that a principal G bundle is a fiber bundle with the group G acting freely transitively on each fiber. For a ramified bundle, the action of G fails to be free over a finite set of points of the Riemann surface. Let G be a complex connected affine algebraic group. Let M be a compact connected Riemann surface and D M a finite subset of it. A ramified G bundle over M with parabolic structure over D is given by the following data: a connected complex manifold E G with a projection ψ : E G M and an action of G on the right of E G such that (1) M = E G /G; (2) the projection ψ and the action of G together make ψ 1 (M \D) a principal G bundle over the complement M \ D; (3) for any point z ψ 1 (D), the corresponding isotropy subgroup, for the action of G on E G, is a finite cyclic group. If we set G = GL(n, C), then a ramified G bundle over M gives a parabolic vector bundle of rank n over M, and conversely a parabolic vector bundle of rank n gives a ramified GL(n, C) bundle. Let ψ : E G M be a ramified GL(n, C) bundle over M. Let x M be a point such that the isotropy group of any point z ψ 1 (x) for the action of G on E G is a cyclic group of order m, where m 2. Then x is a parabolic point for the parabolic vector bundle associated to E G. Furthermore, the parabolic weights of at x of the parabolic vector bundle are multiples of 1/m. The details of the correspondence

12 I. BISWAS between parabolic vector bundles of rank n and ramified GL(n, C) bundles are given in [BBN1]. As mentioned in the acknowledgments of [BBN1], the definition of ramified G bundles was inspired by Nori s work [No1] and [No2]. See [Bi1] for the definition of a connection on ramified bundles and some results related to it. Acknowledgments. I had a wonderful time collaborating with H. Azad, B. Anchouche, V. Balaji, U. N. Bhosle, U. Bruzzo, T. L. Gómez, Y. I. Holla, D. S. Nagaraj, A. J. Parameswaran, G. Schumacher and S. Subramanian in investigations of principal bundles. I record my gratitude and regards to all of them. References [At1] M. F. Atiyah: On the Krull Schmidt theorem with application to sheaves. Bull. Soc. Math. Fr. 84 (1956), 307 317. [At2] M. F. Atiyah: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85 (1957), 181 207. [AAB] B. Anchouche, H. Azad and I. Biswas: Harder Narasimhan reduction for principal bundles over a compact Kähler manifold. Math. Ann. 323 (2002), 693 712. [AB] H. Azad and I. Biswas: On holomorphic principal bundles over a compact Riemann surface admitting a flat connection. Math. Ann. 322 (2002), 333 346. [ABi] B. Anchouche and I. Biswas: Einstein Hermitian connections on polystable principal bundles over a compact Kähler manifold. Amer. Jour. Math. 123 (2001), 207 228. [BB1] I. Biswas and U. N. Bhosle: Mukai Sakai bound for equivariant principal bundles. Arkiv Math. 43 (2005). [BB2] I. Biswas and U. N. Bhosle: On the exceptional principal bundles over a surface. Bull. London Math. Soc. 36 (2004), 193 197. [BBN1] V. Balaji, I. Biswas and D. S. Nagaraj: Ramified G-bundles as parabolic bundles. Jour. Ramanujan Math. Soc. 18 (2003), 123 138. [BBN2] V. Balaji, I. Biswas, and D. S. Nagaraj: Krull-Schmidt reduction for principal bundles. Jour. Reine Angew. Math. (to appear). [BB] I. Biswas and U. Bruzzo: On semistable principal bundles over a complex projective manifold. Preprint. [BG] I. Biswas and T. L. Gómez: Restriction theorems for principal bundles. Math. Ann. 327 (2003), 773 792. [BH1] I. Biswas and Y. I. Holla: Harder Narasimhan reduction of a principal bundle. Nagoya Math. Jour. 174 (2004), 201 223. [BH2] I. Biswas and Y. I. Holla: Principal bundles whose restriction to curves are trivial. Math. Zeit. (to appear).

PRINCIPAL BUNDLES OVER KÄHLER MANIFOLD 13 [Bi1] I. Biswas: Connections on a parabolic principal bundle over a curve. Canadian Jour. Math. (to appear). [Bi2] I. Biswas: Stable bundles and extension of structure group. Diff. Geom. Appl. (to appear). [BP] I. Biswas and A. J. Parameswaran: On the equivariant reduction of structure group of a principal bundle to a Levi subgroup. Preprint. [BS] I. Biswas and S. Subramanian: Flat holomorphic connections on principal bundles over a projective manifold. Trans. Amer. Math. Soc. 356 (2004), 3995 4018. [BSc] I. Biswas and G. Schumacher: Kähler structure on moduli spaces of principal bundles. Preprint. [Do] S. K. Donaldson: Infinite determinants, stable bundles and curvature. Duke Math. Jour. 54 (1987), 231 247. [Mi] Y. Miyaoka: The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry. Sendai, 1985, 449 476, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. [MR] V. B. Mehta and A. Ramanathan: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258 (1982), 213 224. [MS] V. B. Mehta and C. S. Seshadri: Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248 (1980), 205 239. [No1] M. V. Nori: On the representations of the fundamental group scheme. Compos. Math. 33 (1976) 29 41. [No2] M. V. Nori: The fundamental group scheme. Proc. Ind. Acad. Sci. (Math. Sci.) 91 (1982), 73 122. [Ra] A. Ramanathan: Stable principal bundles on a compact Riemann surface. Math. Ann. 213 (1975), 129 152. [RR] S. Ramanan and A. Ramanathan: Some remarks on the instability flag. Tôhoku Math. Jour. 36 (1984), 269 291. [RS] A. Ramanathan and S. Subramanian: Einstein Hermitian connections on principal bundles and stability. Jour. Reine Angew. Math. 390 (1988), 21 31. [UY] K. Uhlenbeck and S.-T. Yau: On the existence of Hermitian-Yang-Mills connections on stable vector bundles. Commun. Pure Appl. Math. 39 (1986), 257 293. [We] A. Weil: Généralisation des fonctions abéliennes. Jour. Math. Pures Appl. 17 (1938), 47 87. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India E-mail address: indranil@math.tifr.res.in