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Holomorphic GL(2)-geometry on compact complex manifolds

We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds. We show that a compact K\"ahler manifold of complex even dimension higher than two admitting a holomorphic GL(2)-geometry is covered by a compact complex torus. We classify compact K\"ahler-Einstein manifolds and Fano manifolds bearing holomorphic GL(2)-geometries. Among the compact K\"ahler-Einstein manifolds we prove that the only examples bearing holomorphic GL(2)-geometry are those covered by compact complex tori, the three dimensional quadric and those covered by the three dimensional Lie ball (the non compact dual of the quadric).


Introduction
A holomorphic GL 2 (C) geometric structure on a complex manifold X of complex dimension n is a holomorphic point-wise identification between the holomorphic tangent space T X and homogeneous polynomials in two variables of degree (n − 1). More precisely, a GL 2 (C) geometric structure on X is a pair (E, ϕ), where E is a rank two holomorphic vector bundle on X and ϕ is a holomorphic vector bundle isomorphism of T X with the (n − 1)-fold symmetric product S n−1 (E) (see Definition 2.1). If E has trivial determinant (i.e. the holomorphic line bundle 2 E is trivial), then (E, ϕ) is called a holomorphic SL 2 (C) geometric structure.
The above definitions are the holomorphic analogues of the concepts of GL 2 (R) and SL 2 (R) geometries in the real smooth category (for the study of those geometries in the real smooth category we refer to [DG, FK, Kr] and references therein).
This article deals with the classification of compact complex manifolds admitting holomorphic GL 2 (C) and SL 2 (C) geometries.
Holomorphic GL 2 (C) and SL 2 (C) geometries on X are examples of holomorphic Gstructures (see [Kob2] and Section 2). They correspond to the reduction of the structural group of the GL n (C)-frame bundle of X to GL 2 (C) and SL 2 (C) respectively.
When the dimension n of X is odd, then a GL 2 (C) geometry produces a holomorphic conformal structure on X (see, for example, [DG], Proposition 3.2 or Section 2 here). Recall that a holomorphic conformal structure is defined by a holomorphic line bundle L over X and a holomorphic section of S 2 (T * X) ⊗ L, which is a L-valued fiberwise nondegenerate holomorphic quadratic form on X. Moreover, if n = 3, then a GL 2 (C)-geometry on X is exactly a holomorphic conformal structure. The standard flat example is the smooth quadric Q 3 in CP 4 defined by the equation Z 2 0 + Z 2 1 + Z 2 2 + Z 2 3 + Z 2 4 = 0. A SL 2 (C)-geometry on X defines a holomorphic Riemannian metric (i.e. a holomorphic section of S 2 (T * X) which is point-wise non degenerate [Du]) when n is odd. When n = 3, a SL 2 (C)-geometry is the same data as a holomorphic Riemannian metric.
When the dimension n of X is even, a holomorphic SL 2 (C)-geometry on X produces a nondegenerate holomorphic two form ω on X. Moreover, if X is a compact Kähler manifold, then ω is automatically closed, and hence ω is a holomorphic symplectic form, and hence X is a hyper-Kähler manifold (see [Be]).
When n is even, a GL 2 (C)-geometry on X produces a twisted nondegenerate two form ω on X, which means that there is a holomorphic line bundle L over X such that ω is a fiberwise nondegenerate holomorphic global section of Ω 2 X ⊗ L. We use the terminology of [Is1,Is2] and call ω a twisted holomorphic symplectic form.
Let us describe the results of this article. In Section 3 we prove Theorem 3.1 asserting that a compact Kähler manifolds of even complex dimension n ≥ 4 admitting a holomorphic GL 2 (C)-geometry is covered by a compact torus. A key ingredient of the proof is a result of Istrati [Is1,Is2] proving that Kähler manifolds bearing a twisted holomorphic symplectic form have vanishing first Chern class. By Yau's proof of Calabi's conjecture [Ya] these manifolds admit a Ricci flat metric and the canonical line bundle K X is trivial (up to a finite cover) [Bo1,Be]. Hence the GL 2 (C)-geometry is induced by an underlying SL 2 (C)-geometry. The proof of Theorem 3.1 involves showing that any compact Kähler manifold of complex dimension n ≥ 3 bearing a SL 2 (C)-geometry is covered by a compact complex torus. Recall that compact Kähler manifolds of odd complex dimension bearing a SL 2 (C)-geometry also admit a holomorphic Riemannian metric and hence has the associated holomorphic (Levi-Civita) affine connections; such manifolds are known to have vanishing Chern classes, [At2], and hence they admit covering by some compact complex torus [IKO].
Theorem 3.4 deals with manifolds X in Fujiki class C (i.e. holomorphic images of compact Kähler manifolds [Fu]) bearing a GL 2 (C)-geometry. Under the technical assumption that there exists a cohomology class [α] ∈ H 1,1 (X, R) which is numerically effective (nef) and has positive self-intersection (meaning X α 2m > 0, where 2m = dim C X), we prove that there exists a non-empty Zariski open subset Ω ⊂ X admitting a flat Kähler metric. We recall that simply connected non-Kähler manifolds in Fujiki class C admitting a holomorphic symplectic form were constructed in [Bo2,Gu1,Gu2].
In Section 4 we obtain the classification of compact Kähler-Einstein manifolds (Theorem 4.1) and of Fano manifolds (Theorem 4.2) bearing a GL 2 (C)-geometry. Theorem 4.1 states that any compact Kähler-Einstein manifold, of complex dimension at least three, bearing a GL 2 (C)-geometry is • either covered by a torus, • or biholomorphic to the three dimensional quadric Q 3 , • or covered by the three dimensional Lie ball D 3 (the noncompact dual of Q 3 , as Hermitian symmetric space).
In all these three situations the GL 2 (C)-geometry is the (flat) standard one.
Theorem 4.2 asserts that a Fano manifold bearing a holomorphic GL 2 (C)-geometry is isomorphic to the quadric Q 3 endowed with its standard GL 2 (C)-structure.
Theorem 4.1 and Theorem 4.2 belong to the same circle of ideas as the following known results. Kobayashi and Ochiai proved in [KO3] that compact Kähler-Einstein manifolds bearing a holomorphic conformal structures are the standard ones: quotients of tori, the smooth n-dimensional quadric Q n and the quotients of the non compact dual D n of Q n . Moreover the same authors proved in [KO2] that holomorphic G-structures, modeled on an irreducible Hermitian symmetric space of rank ≥ 2 (in particular, a holomorphic conformal structure), on compact Kähler-Einstein manifolds are always flat. Let us also mention the main result in [HM] which says that holomorphic irreducible reductive G-structures on uniruled projective manifolds are always flat. Consequently, a uniruled projective manifold bearing a holomorphic conformal structure is biholomorphic to the quadric Q n (see also [Ye]). The following generalization was proved in [BM]: all holomorphic Cartan geometries (see [Sh]) on manifolds admitting a rational curve are flat.
Let us clarify that the classification obtained in Theorem 4.1 and Theorem 4.2 does not use results coming from [BM,HM,KO2,KO3,Ye]: the methods are specific to the case of GL 2 (C)-geometry and unify the twisted holomorphic symplectic case (even dimension) and the holomorphic conformal case (odd dimension).
The last section discuss some related open problems. Let us emphasis one of those questions dealing with SL 2 (C)-geometries on non-Kähler manifolds.
Recall that Ghys constructed in [Gh] exotic deformations of quotients of SL 2 (C) by normal lattices. Those Ghys manifolds are non-Kähler and non-parallelizable, but they admit a non-flat holomorphic SL 2 (C)-geometric structure (i.e. a holomorphic Riemannian metric). Nevertheless, all Ghys holomorphic Riemannian manifolds are locally homogeneous. Moreover, it was proved in [Du] that holomorphic Riemannian metrics on compact complex threefolds are always locally homogeneous.
For higher complex odd dimensions we conjecture that SL 2 (C)-geometries on compact complex manifolds are always locally homogeneous. Some evidence for it was recently provided by the main result in [BD] showing that compact complex simply connected manifolds do not admit any holomorphic Riemannian metric (in particular, in odd dimension, they do not admit a holomorphic SL 2 (C)-geometry).
2. Holomorphic GL 2 (C) and SL 2 (C) geometries In this section we introduce the framework of GL 2 (C) and SL 2 (C) geometries on complex manifolds and describe the geometry of the quadric (model of the flat holomorphic conformal geometry) and that of its noncompact dual. We focus on the complex dimension three, the only case admitting a GL 2 (C)-geometry.
The holomorphic tangent bundle of a complex manifold Z will be denoted by T Z. The frame bundle for T Z will be denoted by R(Z). We recall that R(Z) consists of all linear isomorphisms from C d to the fibers of T Z, where d = dim C Z. The R(Z) is a holomorphic principal GL(d, C)-bundle over Z.
Definition 2.1. A holomorphic GL 2 (C)-structure (or GL 2 (C)-geometry) on a complex manifold X of complex dimension n ≥ 2 is a holomorphic bundle isomorphism T X ≃ S n−1 (E), where S n−1 (E) is the (n − 1)-th symmetric power of a rank two holomorphic vector bundle E over X. If E has trivial determinant, meaning the line bundle 2 E is holomorphically trivial, it is called a holomorphic SL 2 (C)-structure (or SL 2 (C)-geometry) on X.
Holomorphic GL 2 (C)-structures and SL 2 (C)-structures are particular cases of holomorphic irreducible reductive G-structures [Kob2,HM]. They correspond to the holomorphic reduction of the structure group of the frame bundle R(X) of X from GL n (C) to GL 2 (C) and SL 2 (C) respectively. It should be clarified that for a GL 2 (C)-structure, the corresponding group homomorphism GL 2 (C) −→ GL n (C) is given by the (n − 1)-th symmetric product of the standard representation of GL 2 (C). This n-dimensional irreducible linear representation of GL 2 (C) is also given by the induced action on the homogeneous polynomials of degree (n − 1) in two variables. For an SL 2 (C)-geometry, the corresponding homomorphism SL 2 (C) −→ GL n (C) is the restriction of the above homomorphism to SL 2 (C) ⊂ GL 2 (C).
The standard symplectic form on C 2 produces a nondegenerate quadratic form (respectively, nondegenerate alternating form) on the symmetric product S 2i (C 2 ) (respectively, S 2i−1 (C 2 )) for all i ≥ 1. Consequently, the above linear representation SL 2 (C) −→ GL n (C) preserves a nondegenerate complex quadratic form on C n if n is odd and a nondegenerate two (alternating) form on C n if n is even (see, for example, Proposition 3.2 and Sections 2 and 3 in [DG] or [Kr]). The above linear representation GL 2 (C) −→ GL n (C) preserves the line in (C n ) * ⊗ (C n ) * spanned by the above tensor; the action of GL 2 (C) on this line is nontrivial. Consequently, the GL 2 (C)-geometry (respectively, SL 2 (C)-geometry) on X induces a holomorphic conformal structure (respectively, holomorphic Riemannian metric) if the complex dimension of X is odd, and it induces a twisted holomorphic symplectic structure (respectively, a holomorphic nondegenerate two form) when the complex dimension of X is even.
A holomorphic SL 2 (C)-structure on a complex surface X is a holomorphic trivialization of the canonical bundle K X = 2 (T X) * .
The simplest nontrivial examples of GL 2 (C) and SL 2 (C) structures are provided by the complex threefolds. In this case a GL 2 (C)-structure is a holomorphic conformal structure on X, and an SL 2 (C)-structure is a holomorphic Riemannian metric on X. Indeed, this is deduced from the fact that the SL 2 (C)-representation on the three-dimensional vector space of homogeneous quadratic polynomials in two variables Consequently, a holomorphic isomorphism between PSL 2 (C) and the complex orthogonal group SO(3, C) is obtained. In other words, the discriminant, being nondegenerate, induces a holomorphic Riemannian metric on the threefold X. Notice that a holomorphic Riemannian metric coincides with a reduction of the structural group of the frame bundle R(X) to the orthogonal group O(3, C). On a double unramified cover of a holomorphic Riemannian threefold there is a reduction of the structural group of the frame bundle to the connected component of the identity SO(3, C). Hence in complex dimension three a SL 2 (C)-geometry is the same data as a holomorphic Riemannian metric, while a reduction of the frame bundle to PSL 2 (C) is the same data as a holomorphic Riemannian metric with an orientation.
Moreover, the GL 2 (C)-representation on the vector space in (2.1) preserves the line generated by the discriminant ∆ = b 2 − 4ac. This gives an isomorphism between GL 2 (C)/(Z/2Z) and the conformal group CO(3, C) = (O(3, C) × C * )/(Z/2Z) = SO(3, C) × C * . Consequently, the GL 2 (C)-structure coincides with a holomorphic reduction of the structure group of the frame bundle R(X) to CO(3, C). This holomorphic reduction of the structure group defines a holomorphic conformal structure on X. Notice that CO(3, C) being connected, the two different orientations of a three dimensional holomorphic Riemannian manifold are conformally equivalent.
Recall that flat conformal structures in complex dimension n ≥ 3 are locally modeled on the quadric The holomorphic automorphism group of Q n is PSO(n + 2, C).
Let us mention that Q n is identified with the real Grassmannian of oriented 2-planes in R n+2 . Consequently, Q n = SO(n + 2, R)/(SO(2, R) × SO(n, R)) (see, for instance, Section 1 in [JR2]). The action of SO(n + 2, R) on Q n is via holomorphic automorphisms. To see this, first note that the real tangent space at a point of Q n = SO(n + 2, R)/(SO(2, R) × SO(n, R)) is identified with the corresponding quotient of the real Lie algebras so(n + 2, R)/(so(2, R) ⊕ so(n, R)), and the stabilizer SO(2, R) × SO(n, R) acts on this quotient vector space through the adjoint representation. The complex structure of the tangent space (induced by the complex structure of Q 3 ) is given by the operator J which satisfies the condition that {exp(tJ)} t∈R is the adjoint action of the factor SO(2, R). Since SO(2, R) lies in the center of the stabilizer, the almost complex structure J is preserved by the action of the stabilizer. Consequently, the complex structure is SO(n + 2, R)-invariant.
The quadric Q n is a Fano manifold, and it is an irreducible Hermitian symmetric space of type BD I [Bes,p. 312]. For more about its geometry and that of its noncompact dual D n (as a Hermitian symmetric space) the reader is referred to [JR2, Section 1].
Theorem 4.2 implies that among the quadrics Q n , n ≥ 3, only Q 3 admits a holomorphic GL 2 (C)-structure. Moreover, Theorem 4.1 states that the only compact non-flat Kähler-Einstein manifolds bearing a holomorphic GL 2 (C)-structure are Q 3 and those covered by its noncompact dual D 3 .
Recall that a general result of Borel on Hermitian irreducible symmetric spaces shows that the non compact dual is always realized as an open subset of its compact dual.
We will give below a geometric description of the noncompact dual D 3 of Q 3 as an open subset in Q 3 which seems to be less known.
Consider the complex quadric form q 3,2 := Z 2 0 + Z 2 1 + Z 2 2 − Z 2 3 − Z 2 4 of five variables, and let Q ⊂ CP 4 be the quadric Q defined by the equation q 3,2 = 0. Then Q is biholomorphic to Q 3 . Let O(3, 2) ⊂ GL(5, R) be the real orthogonal group of q 3,2 , and denote by SO 0 (3, 2) the connected component of O(3, 2) containing the identity element. The quadric Q admits a natural holomorphic action of the real Lie group SO 0 (3, 2), which is not transitive, in contrast to the action of SO(5, R) on Q 3 . The orbits of the SO 0 (3, 2)-action on Q 3 coincide with the connected components of the complement Q 3 \ S, where S is the real hypersurface of Q 3 defined by the equation Notice that the above real hypersurface S contains all real points of Q. In fact, it can be shown that S Q coincides with the set of point m ∈ Q such that the complex line (m, m) is isotropic (i.e. it also lies in Q). Indeed, since q 3,2 (m) = 0, then the line generated by (m, m) lies in Q if and only if m and m are perpendicular with respect to the bilinear symmetric form associated to q 3,2 , or equivalently m ∈ S.
For any point m ∈ Q \ S, the form q 3,2 is nondegenerate on the line (m, m). To see this first notice that the complex line generated by (m, m), being real, may be considered as a plane in the real projective space RP 4 . The restriction of the (real) quadratic form q 3,2 to this real plane (m, m) vanishes at the points m and m which are distinct (because all real points of Q lie in S). It follows that the quadratic form cannot have signature (0, 1) or (1, 0) when restricted to the real plane (m, m). Consequently, the signature of the restriction of q 3,2 to this plane is either (2, 0) or (1, 1) or (0, 2). Each of these three signature types corresponds to an SO 0 (3, 2) orbit in Q.
Take the point m 0 = [0 : 0 : 0 : It is an open subset of Q biholomorphic to a bounded domain in C 3 ; it is the three dimension Lie ball (the bounded domain IV 3 in Cartan's classification).
The signature of q 3,2 on the above line (m 0 , m 0 ) is (0, 2). The signature of q 3,2 on the orthogonal part of (m 0 , m 0 ), canonically isomorphic to T m 0 Q, is (3, 0). Then the SO 0 (3, 2)orbit of m 0 in Q inherits an SO 0 (3, 2)-invariant Riemannian metric. The stabilizer of m 0 is SO(2, R) × SO(3, R). Here SO(2, R) acts on C 3 through the one parameter group exp(tJ), with J being the complex structure, while the SO(3, R) action on C 3 is given by the complexification of the canonical action of SO(3, R) on R 3 . Consequently, the action of U(2) = SO(2, R) × SO(3, R) is the unique irreducible action on the symmetric product S 2 (C 2 ) = C 3 . This action coincides with the holonomy representation of this Hermitian symmetric space D 3 ; as mentioned before, D 3 is the noncompact dual of Q 3 . The holonomy representation for Q 3 is the same.
Recall that the automorphism group of the noncompact dual D 3 is PSO 0 (3, 2); it is the subgroup of the automorphism group of Q that preserves D 3 (which lies in Q as the SO 0 (3, 2)orbit of m 0 ∈ Q). Consequently, any quotient of D 3 by a lattice in PSO 0 (3, 2) admits a flat holomorphic conformal structure induced by that of the quadric Q.
Notice that compact projective threefolds admitting holomorphic conformal structures (GL 2 (C)-structures) were classified in [JR1]. There are in fact only the standard examples: finite quotients of three dimensional abelian varieties, the smooth quadric Q 3 and quotients of its non compact dual D 3 . In [JR2], the same authors classified also the higher dimensional compact projective manifolds admitting a flat holomorphic conformal structure; they showed that the only examples are the standard ones.

GL 2 (C)-structures on Kähler and Fujiki class C manifolds
Every compact complex surface of course admits a holomorphic GL 2 (C)-structure by taking E in Definition 2.1 to be the tangent bundle itself. The situation is much more stringent in higher dimensions. The following result shows that a compact Kähler manifold of even dimension n ≥ 4 bearing a holomorphic GL 2 (C)-structure has trivial holomorphic tangent bundle (up to finiteétale cover).
Theorem 3.1. Let X be a compact Kähler manifold of even complex dimension n ≥ 4 admitting a holomorphic GL 2 (C)-structure. Then X admits a finite unramified covering by a compact complex torus.
Proof. Let E be a holomorphic vector bundle on X and T X ≃ S n−1 (E) an isomorphism with the symmetric product, giving a GL 2 (C)-structure on X. Then where L = ( 2 E) ⊗(n−1) . The above isomorphism between T X and (T X) * ⊗ L produces, when n is even, a holomorphic section ω ∈ H 0 (X, Ω 2 X ⊗ L) which is a fiberwise nondegenerate 2-form with values in L. Writing n = 2m, the exterior product is a nowhere vanishing section, where K X = Ω n X is the canonical line bundle on X. Consequently, we have K X ≃ (L * ) m , in particular, c 1 (X) = mc 1 (L). Any Hermitian metric on T X induces an associated Hermitian metric on L m , and hence produces an Hermitian metric on L.
Using de Rham decomposition theorem and Berger's classification of the irreducible holonomy groups of nonsymmetric Riemannian manifolds (see [Jo], Section 3.2 and Theorem 3.4.1 in Section 3.4) we deduce that the universal cover ( X, g) of (X, g) is a Riemannian product ( X, g) = (C l , g 0 ) × (X 1 , g 1 ) × · · · × (X p , g p ) , (3.1) where (C l , g 0 ) is the standard flat complete Kähler manifold and (X i , g i ) is an irreducible Ricci flat Kähler manifold of complex dimension r i ≥ 2, for every 1 ≤ i ≤ p. The holonomy of each (X i , g i ) is either SU(r i ) or the symplectic group Sp( r i 2 ), where r i = dim C X i (in the second case r i is even). Notice, in particular, that symmetric irreducible Riemannian manifolds of (real) dimension at least two are never Ricci flat. For more details, the reader can refer to [Be,Theorem 1] and [Jo,p. 124,Proposition 6.2.3].
The holomorphic GL 2 (C)-structure on X produces a holomorphic SL 2 (C)-structure on a finite unramified cover of X in (3.2). More precisely, T X = S n−1 (ϕ * E) and since the canonical bundle K X is trivial, the holomorphic line bundle 2 E is torsion. This implies that on a finite unramified cover of X (still denoted by X for notations convenience) we can (and will) assume that 2 E is holomorphically trivial.
The above SL 2 (C)-structure on X gives a holomorphic reduction R ′ ( X) ⊂ R( X) of the structure group of the frame bundle R( X) of X from GL n (C) to SL 2 (C); recall that the homomorphism SL 2 (C) −→ GL n (C) is given by the (n − 1)-th symmetric product of the standard representation, and this homomorphism is injective because n is even. There is a holomorphic tensor θ on X satisfying the condition that the SL 2 (C)-subbundle R ′ ( X) ⊂ R( X) consists of those frames that pointwise preserve θ. Indeed, this is a consequence of Chevalley's theorem asserting that there exists a finite dimensional linear representation W of GL n (C), and an element θ 0 ∈ W , such that the stabilizer of θ 0 is the image of the above homomorphism SL 2 (C) −→ GL n (C) (see [Hu,p. 80,Theorem 11.2], [DMOS,p. 40, Proposition 3.1(b)]; since SL 2 (C) does not have a nontrivial character, the line in [Hu,p. 80,Theorem 11.2] (or [DMOS,p. 40, Proposition 3.1(b)]) is fixed pointwise in our case). Now, since any irreducible representation W of the reductive group GL n (C) is a factor of a representation (C n ) ⊗p ⊗ ((C n ) * ) ⊗q , for some integers p, q ≥ 0 [DMOS, p. 40, Proposition 3.1(a)], the above element θ 0 gives rise to a holomorphic tensor θ ∈ H 0 ( X, (T X) ⊗p ⊗ (T * X) ⊗q ) (3.3) with p, q ≥ 0. By construction, θ is stabilized exactly by the frames lying in R ′ ( X).
Consider the Levi-Civita connection on X associated to the Ricci-flat Kähler metric ϕ * g in (3.2). It is known that the parallel transport for it preserves any holomorphic tensor on X [LT, p. 50, Theorem 2.2.1]. In particular, θ in (3.3) is parallel with respect to this connection. Consequently, the subbundle R ′ ( X) ⊂ R( X) defining the SL 2 (C)-structure (considered as a G-structure) is invariant under the parallel transport by the Levi-Civita connection for ϕ * g. This implies that the holonomy group of ϕ * g lies in the maximal compact subgroup of SL 2 (C). Hence the holonomy group of ϕ * g lies in SU(2).
From (3.2) it follows that the holonomy of ϕ * g is where Hol(g i ) is the holonomy of g i . As noted earlier, • either Hol(g i ) = SU(r i ), with dim C X i = r i ≥ 2, or • Hol(g i ) = Sp( r i 2 ), where r i = dim C X i is even. Therefore, the above observation, that Hol(ϕ * g) is contained in SU(2), and (3.4) together imply that (1) either ( X, ϕ * g) = (T l , g 0 ), or (2) ( X, ϕ * g) = (T l , g 0 ) × (X 1 , g 1 ), where X 1 is a K3 surface equipped with a Ricci-flat Kähler metric g 1 .
If ( X, ϕ * g) = (T l , g 0 ), then then proof of the theorem evidently is complete.
Therefore, we assume that where X 1 is a K3 surface equipped with a Ricci-flat Kähler metric g 1 . Note that l ≥ 2 (because l + 2 = n ≥ 4) and l is even (because n is so).
Fix a point t ∈ T l . The holomorphic vector bundle on X 1 obtained by restricting ϕ * E to {t} × X 1 ⊂ T l × X 1 will be denoted by F . Restricting the isomorphism note that (T X)| {t}×X 1 = T X 1 ⊕ O ⊕l X 1 . The vector bundle T X 1 is polystable of degree zero (we may use the Kähler structure g 1 on X 1 to define the degree), because X 1 admits a Kähler-Einstein metric. Hence T X 1 ⊕ O ⊕(l+1) X 1 is polystable of degree zero. This implies that End(T X 1 ⊕ O ⊕(l+1) X 1 ) is polystable of degree zero, because for any Hermitian-Einstein structure on T X 1 ⊕ O ⊕(l+1) X 1 (which exists by [UY]), the Hermitian structure on End(T X 1 ⊕ O ⊕(l+1) X 1 ) induced by it is also Hermitian-Einstein. On the other hand, the holomorphic vector bundle End(F ) is a direct summand of End(S l+1 (F )). Since End(S l+1 (F )) = End(T X 1 ⊕ O ⊕(l+1) X 1 ) is polystable of degree zero, this implies that the holomorphic vector bundle End(F ) is also polystable of degree zero.
Let H be a Hermitian-Einstein metric on F . The holonomy of the Chern connection on F associated to H is contained in SU(2), because 2 F = O X 1 (since 2 E is trivial). Let Hol(H) denoted the holonomy of the Chern connection on F associated to H. We note that H is connected because X 1 is simply connected. Therefore, either Hol(H) = SU(2) or Hol(H) = U(1) (since T X 1 is not trivial, from (3.6) it follows that F is not trivial and hence the holonomy of H cannot be trivial).
First assume that Hol(H) = SU(2). Let H denote the Hermitian-Einstein metric on S n−1 (F ) induced by H. Since Hol(H) = SU(2), the holonomy Hol(H) of the Chern connection on S n−1 (F ) associated to H is the image of the homomorphism given by the (n − 1)-th symmetric power of the standard representation. The action of h 0 (SU(2)) on C n , obtained by restricting the standard action of SU(n), is irreducible. From this it follows that H 0 (X 1 , S n−1 (F )) = 0 because any holomorphic section of S n−1 (F ) is flat with respect to the Chern connection on S n−1 (F ) associated to H [LT,p. 50 In view of this contradiction we conclude that Hol(H) = SU(2).

So assume that Hol(H) = U(1). Then
where L is a holomorphic line bundle on X 1 . Note that degree(L) = 0, because F is polystable. From (3.9) it follows that Now using (3.8) we conclude that H 0 (X 1 , L k ) = 0 for some nonzero integer k. Since degree(L) = 0, this implies that the holomorphic line bundle L is trivial; note that the Picard group Pic(X 1 ) is torsionfree. Hence S n−1 (F ) is trivial. Since T X 1 is not trivial, this contradicts (3.6) (see [At1,p. 315,Theorem 2]). Therefore, (3.5) can't occur. This completes the proof of the theorem.
Recall that a compact Kähler manifold of odd complex dimension bearing a holomorphic SL 2 (C)-structure also admits a holomorphic Riemannian metric and inherits of the associated holomorphic (Levi-Civita) affine connection. Those manifolds are known to have vanishing Chern classes [At2] and, consequently, all of them are covered by compact complex tori [IKO]. Therefore Theorem 3.1 has the following corollary.
Corollary 3.2. Let X be a compact Kähler manifold of complex dimension n ≥ 3 bearing a holomorphic SL 2 (C)-structure. Then X admits a finite unramified covering by a compact complex torus.
We will prove an analog of Theorem 3.1 for Fujiki class C manifolds. Recall that a compact complex manifold Y is in the class C of Fujiki if Y is the image of a Kähler manifold through a holomorphic map. By a result of Varouchas in [Va] we know that Fujiki class C manifolds are precisely those that are bimeromorphic to Kähler manifolds.
In order to use a result in [BDG] (Theorem A) we will need to make a technical assumption. Let us recall some standard terminology: Definition 3.3. Let X be a compact complex manifold of dimension n, and let [α] ∈ H 1,1 (X, R) be a cohomology class represented by a smooth, closed (1, 1)-form α.
The reader is referred to Section 2 of [BDG] for a discussion on Definition 3.3.
Theorem 3.4. Let X be a compact complex manifold in Fujiki class C bearing a holomorphic GL 2 (C)-structure. Assume that there exists a cohomology class [α] ∈ H 1,1 (X, R) which is nef and has positive self-intersection. Then the following two statements hold.

(i) If the complex dimension of X is even and at least 4, then there exists a non-empty
Zariski open subset Ω ⊂ X admitting a flat Kähler metric. (ii) If the complex dimension of X is odd and the first Chern class of X vanishes, then X admits a finite unramified cover which is a torus.
Proof. (i) Since the complex dimension of X is even, X inherits a twisted holomorphic symplectic form ω ∈ H 0 (X, Ω 2 X ⊗ L), defined by a a holomorphic line bundle L and a non degenerate L-valued holomorphic two form ω.
The proof of Theorem 2.5 in [Is1] shows that L inherits a holomorphic connection (see Remark 2.7 in [Is1]) and hence its curvature, representing the first Chern class of L, is a holomorphic two form. In particular, c 1 (L) admits a representative which is a two form on X of type (2, 0). On the other hand, starting with a Hermitian metric on L, the classical computation of the first Chern class using the associated Chern connection gives a representative of c 1 (L) which is a two form on X of type (1, 1). Since on Fujiki class C manifolds (just as for Kähler manifolds) forms of different types are cohomologous only in the trivial class this implies that c 1 (L) = 0 and, consequently, as in the proof of Theorem 3.1, c 1 (X) = 0.
Then Theorem A of [BDG] constructs a closed, positive (1, 1)-current on X, lying in the class [α], which induces a genuine Ricci-flat Kähler metric g on a non-empty Zariski open subset Ω ⊂ X. Furthermore, given any global holomorphic tensor θ ∈ H 0 (X, (T X) ⊗p ⊗ T * X) ⊗q ), with p, q ≥ 0, the restriction of θ to Ω is parallel with respect to the Levi-Civita connection on Ω associated to g.
As in the proof of Theorem 3.1, this implies that the holonomy group of Levi-Civita connection for g lies inside SU(2).
Take any u ∈ Ω. Using de Rham's local splitting theorem, there exists a local decomposition of an open neighborhood U u ⊂ Ω of u such that (U u , g) is a Riemannian product (U u , g) = (U 0 , g 0 ) × · · · × (U p , g p ) , (3.10) where (U 0 , g 0 ) is a flat Kähler manifold and (U i , g i ) is an irreducible Kähler manifold of complex dimension r i ≥ 2 for every 1 ≤ i ≤ p (the reader is referred to [GGK,Proposition 2.9] for more details on this local Kähler decomposition). Since g is Ricci flat, each (U i , g i ) is also Ricci flat. For every 1 ≤ i ≤ p, the holonomy Hol(g i ) of the Levi-Civita connection for g i satisfies the following: • either Hol(g i ) = SU(r i ), or • Hol(g i ) = Sp( r i 2 ) (in this case r i is even); see [GGK,Proposition 5.3]. The holonomy Hol(g) of the Levi-Civita connection for g is Hol(g i ) . (3.11) As observed above, Hol(g) lies inside SU(2). First assume that Hol(g) SU(2) .
Therefore, dim Hol(g) < 3. Now from (3.11) it follows that Hol(g) = 1 and there is no factor (U i , g i ) with 1 ≤ i ≤ p in the decomposition (3.10) (because dim Hol(g i ) ≥ 3 for every 1 ≤ i ≤ p). It follows that (U u , g) = (U 0 , g 0 ) and hence Ω admits the flat Kähler metric g.
Consider the homomorphism in (3.7). Consider C n as a SU(2)-representation using h 0 and the standard representation of SU(n). This SU(2)-representation is irreducible, in particular, there are no nonzero SU(2)-invariant in C n . From this it can be deduced that dim U 0 = 0, where U 0 is the factor in (3.10). Indeed, • the direct summand of T U u given by the tangent bundle T U 0 is preserved by the Levi-Civita connection on T U n corresponding to g, and • this direct summand of T U u given by T U 0 is generated by flat sections of T U u .
Since T U u does not have any flat section, we conclude that dim C U 0 = 0. Therefore, (U u , g) = (U 1 , g 1 ), and dim C U n = dim C U 1 = 2. This contradicts the assumption that dim C X ≥ 4. Hence the proof of (i) is complete.
(ii) Fix a GL 2 (C)-structure on X. As the dimension of X is odd, the GL 2 (C)-structure on X produces a holomorphic conformal structure on X. Since the first Chern class of X vanishes, by Theorem 1.5 of [To], there exists a finite unramified covering of X with trivial canonical bundle. Replacing X by this finite unramified covering we shall assume that K X is trivial.
As in the proof of Theorem 3.1, this implies that, up to a finite unramified cover, the GL 2 (C)-structure of X is induced by a SL 2 (C)-structure on X. In particular, X admits a holomorphic Riemannian metric. Now Theorem C of [BDG] says that X admits a finite unramified cover which is a compact complex torus.
If X is Kähler, the open subset Ω ⊂ X in Theorem 3.4 (point (i)) is the entire manifold. The proof of point (i) in Theorem 3.4 encapsulates an alternative proof to Theorem 3.1.
We note that there are simply connected non-Kähler manifolds in Fujiki class C that admit a holomorphic symplectic form. Such examples were constructed in [Bo2,Gu1,Gu2].
A compact complex manifold is called Moishezon if it is bimeromorphic to a projective manifold [Mo]. It is known that Moishezon manifolds bearing a holomorphic Cartan geometry (in particular, a holomorphic conformal structure [Sh]) are projective (see Corollary 2 in [BM]). Therefore a Moishezon manifold admitting a SL 2 (C)-structure is covered by an abelian variety.
We conjecture that the statement of Theorem 3.4 holds for all Fujiki class C manifolds. In particular, we conjecture that a Fujiki class C manifold bearing a SL 2 (C)-structure is covered by a compact complex torus.

GL 2 (C)-structures on Kähler-Einstein and Fano manifolds
Theorem 4.1. Let X be a compact Kähler-Einstein manifold, of complex dimension at least three, bearing a holomorphic GL 2 (C)-structure. Then one of the following three holds: (1) X admits an unramified covering by a compact complex torus; (2) X is the three dimensional quadric Q 3 equipped with its standard GL 2 (C)-structure; (3) X is covered by the three-dimensional Lie ball D 3 (the noncompact dual of the Hermitian symmetric space Q 3 ) endowed with the standard GL 2 (C)-structure.
Proof. Let n be the complex dimension of X. Let g be a Kähler-Einstein metric on X with Einstein factor e g .
If e g = 0, then X is Calabi-Yau. Up to a finite unramified covering we can assume that the canonical bundle K X is trivial [Bo1,Be], and hence X admits a holomorphic SL 2 (C)structure. Now Corollary 3.2 implies that X is covered by a compact complex torus torus.
Assume now e g = 0.
For any integers m, l, the Hermitian structure H on (T X) ⊗m ⊗ ((T X) * ) ⊗l induced by g also satisfies the Hermitian-Einstein condition. The Einstein factor e H for the Hermitian-Einstein metric H is (m − l)e g . In particular, e H = 0 when m = l. Consequently, any holomorphic section ψ of (T X) ⊗m ⊗ ((T X) * ) ⊗m is flat with respect to the Chern connection on (T X) ⊗m ⊗ ((T X) * ) ⊗m corresponding to the Hermitian-Einstein structure H [LT,p. 50,Theorem 2.2.1].
Fix a GL 2 (C)-structure on X. It produces a holomorphic reduction of the structure group of the frame bundle R(X) defined by a GL 2 (C) principal subbundle denoted by R ′ (X); recall that the homomorphism GL 2 (C) −→ GL n (C) is given by the (n − 1)-th symmetric product of the standard representation of GL 2 (C). This holomorphic reduction R ′ (X) induces a holomorphic reduction, to GL 2 (C), of the structure group of (T X) * , and hence we get a holomorphic reduction, to GL 2 (C), of the structure group of T X ⊗ (T X) * . This reduction of structure group of T X ⊗ (T X) * is given by a holomorphic tensor ψ (as in the proof of Theorem 3.1 this is a consequence of Chevalley's theorem). As noted above, ψ is parallel with respect to the Chern connection on T X ⊗ (T X) * associated to the Hermitian-Einstein metric H.
Let ad(R(X)) and ad(R ′ (X)) be the adjoint vector bundles for the principal bundles R(X) and R ′ (X) respectively. Note that ad(R(X)) = End(T X) = T X ⊗ (T X) * , and the Lie algebra structure of the fibers of ad(R(X)) is the Lie algebra structure of the fibers of End(T X) given by the usual Lie bracket (A, B) −→ AB − BA. We have a holomorphic inclusion of Lie algebra bundles ad(R ′ (X)) ֒→ ad(R(X)) = T X ⊗ (T X) * induced by the above holomorphic reduction of structure group R ′ (X) ⊂ R(X). As noted before, the Kähler-Einstein structure g produces a Hermitian-Einstein structure H on T X ⊗ (T X) * . In particular, the vector bundle ad(R(X)) is polystable of degree zero.
Since ad(R ′ (X)) is a subbundle of degree zero of the polystable vector bundle T X ⊗(T X) * of degree zero, the Hermitian-Einstein connection on T X ⊗ (T X) * given by H preserves this subbundle ad(R ′ (X)). From this it can be deduced that the Levi-Civita connection on T X corresponding to the Kähler-Einstein metric g induces a connection on the principal GL 2 (C)bundle R ′ (X). To see this we note the following general fact. Let A ⊂ B ⊂ C be Lie groups such that A is normal in C. Let E B be a principal B-bundle and E C = E B (C) the principal C-bundle obtained by extending the structure group of E B . Since A is normal in C, the quotient E B /A (respectively, E C /A) is a principal B/A-bundle (respectively, C/A-bundle). Let ∇ be a connection on E C such that the connection on the principal C/A-bundle E C /A induced by ∇ preserves the subbundle E B /A. Then ∇ preserves the subbundle E B ⊂ E C . In our situation, C = GL(n, C) and A is its center, while B is the image of GL(2, C).
The above observation, that the Levi-Civita connection for g induces a connection the principal GL 2 (C)-bundle R ′ (X), implies that the holonomy group of g lies in the subgroup GL 2 (C) ∩ U(n) = U(2) ⊂ U(n) .
Using de Rham local Riemannian decomposition (see Proposition 2.9 in [GGK] for a proof adapted to the Kähler case), and Berger's list of holonomy (see Proposition 3.4.1 in [Jo,p. 55]) of irreducible holonomies, we conclude that this holonomy, namely a subgroup of U(2), appears only for locally symmetric Hermitian spaces. The classification of the holonomies of locally symmetric Hermitian spaces shows that X is • either biholomorphic to the Hermitian symmetric space SO(5, R)/(SO(2, R)×SO(3, R)) [Bes,p. 312] (this is the case BD I in the list) (recall that SO(5, R)/(SO(2, R) × SO(3, R)) is biholomorphic to the quadric Q 3 ); • or X is covered by the bounded domain which is the noncompact dual (recall that the noncompact dual of Q 3 is D 3 = SO 0 (3, 2)/(SO(2, R) × SO(3, R))).
The holonomy group of the Kähler-Einstein metric of the quadric Q 3 is SO(2, R) × SO(3, R)) [Bes,p. 312] (case BD I in the list). Here SO(2, R) acts on C 3 by the one parameter group exp(tJ) with J being the complex structure, while the action of SO(3, R) on C 3 is given by the complexification of the standard action of SO(3, R)) on R 3 . Consequently, the action of the covering U(2) of SO(2, R) × SO(3, R) on C 3 coincides with the action on the second symmetric power S 2 (C 2 ) of the standard representation.
The GL 2 (C)-structure on the quadric Q 3 must be flat [KO, Kl, HM, Ye]. Since the quadric is simply connected this flat GL 2 (C)-structure coincides with the standard one [Oc, HM, Ye].
Also, the only holomorphic conformal structure on any compact manifold covered by the noncompact dual D 3 of the quadric is the standard one [Kl, Ye].
The next result deals with Fano manifolds. Recall that a Fano manifold is a compact complex projective manifold such that the anticanonical line bundle K −1 X is ample. Fano manifolds are known to be rationally connected [Ca2], [KMM], and they are simply connected [Ca1].
A basic invariant of a Fano manifold is its index, which is, by definition, the maximal positive integer l such that the canonical line bundle K X is divisible by l in the Picard group of X.
Theorem 4.2. Let X be a Fano manifold, of complex dimension n ≥ 3, that admits a holomorphic GL 2 (C)-structures. Then n = 3, and X is biholomorphic to the quadric Q 3 (the GL 2 (C)-structure being the standard one).

Proof. Let
T X ∼ −→ S n−1 (E) be a holomorphic GL 2 (C)-structure on X, where E is a holomorphic vector bundle of rank two on X. A direct computation shows that Hence the index of X is at least n(n−1) 2 . It is a known fact that the index of a Fano manifold Y of complex dimension n is at most n + 1. Moreover, the index is maximal (= n + 1) if and only if Y is biholomorphic to the projective space CP n , and the index equals n if and only if Y is biholomorphic to the quadric [KO1]. These imply that n = 3 and X is biholomorphic to the quadric Q 3 . As in the proof of Theorem 4.1, the results in [HM, KO, Oc] imply that the only GL 2 (C)-structure on the quadric Q 3 is the standard one.

Related open questions
In this section we collect some open questions on compact complex manifolds bearing a holomorphic GL 2 (C)-structure or a holomorphic SL 2 (C)-structure.
We think that the statement of Theorem 3.4 holds without the technical assumption described in Definition 3.3 (the existence of a cohomology class [α] ∈ H 1,1 (X, R) which is nef and has positive self-intersection).
In particular, we conjecture that an odd dimensional compact complex manifold X in Fujiki class C bearing a holomorphic SL 2 (C)-structure is covered by a compact torus. Notice that in this case X admits the Levi-Civita holomorphic affine connection corresponding to the associated holomorphic Riemannian metric, which implies, as in the Kähler case, that all Chern classes of X vanish [At2].
Recall that Ghys manifolds, constructed in [Gh] by deformation of quotients of SL 2 (C) by normal lattices, admit non-flat locally homogeneous holomorphic Riemannian metrics (or equivalently, SL 2 (C)-structures). It was proved in [Du] that all holomorphic Riemannian metrics on compact complex threefolds are locally homogeneous.
We conjecture that SL 2 (C)-structures on compact complex manifolds of odd dimension are always locally homogeneous.
GL 2 (C)-structures on compact Kähler manifolds of odd dimension.
Recall that these manifolds admit a holomorphic conformal structure. Flat conformal structures on compact projective manifolds were classified in [JR2]: beside some projective surfaces, there are only the standard examples.
As for the conclusion in Theorem 4.1 and in Theorem 4.2, we think that Kähler manifolds of odd complex dimension ≥ 5 and bearing a holomorphic GL 2 (C)-structure are covered by compact tori.