Holomorphic Cartan geometries and rational curves

We prove that any compact K\"ahler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact K\"ahler manifold.

14. Tame rational curves and fiber bundles 33 15. Proof of the main theorem 34 16. Open problems 34 References 35

Introduction
All manifolds will be connected. Unless specified otherwise, all manifolds and maps henceforth are assumed to be complex analytic, and all Lie groups, algebras, etc. are complex. We will prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. First we will define the notion of Cartan geometry and the appropriate notion of inheritance.
1.1. Definition of Cartan geometries. Throughout we use the convention that structure groups of principal bundles act on the right. Definition 1. If E → M is a principal right G-bundle, denote the G-action as r g e = eg, where e ∈ E and g ∈ G.
Definition 2. Let H ⊂ G be a closed subgroup of a Lie group, with Lie algebras h ⊂ g. A G/H-geometry, or Cartan geometry modelled on G/H, on a manifold M is a choice of C ∞ principal H-bundle E → M , and smooth 1-form ω ∈ Ω 1 (E) ⊗ g called the Cartan connection, so that: (1) r * h ω = Ad −1 h ω for all h ∈ H. (2) ω e : T e E → g is a linear isomorphism at each point e ∈ E.
(3) For each A ∈ g, define a vector field A on E by the equation A ω = A.
The vector fields A for A ∈ h generate the H-action on E H . Example 1. The principal H-bundle G → G/H is a Cartan geometry (called the model Cartan geometry). The Cartan connection is ω = g −1 dg, the left invariant Maurer-Cartan 1-form on G.

Freedom of rational curves.
Definition 5. A rational curve in a complex manifold M is a nonconstant holomorphic map P 1 → M . A rational curve f : P 1 → M is free if f * T M is spanned by its global sections.
In Section 3 on page 13 we will prove: Theorem 1. In a complex manifold M with holomorphic Cartan geometry, every rational curve is free.
Corollary 1. If a complex manifold blows down, then it admits no holomorphic Cartan geometry.
Example 2. Every cubic surface is the projective plane blown up at 6 points. The exceptional divisors of those 6 points are not free. Therefore no cubic surface has any holomorphic Cartan geometry.
Definition 6. Suppose that M is a compact complex manifold and that f : C → M is a compact complex curve (i.e., a nonconstant holomorphic map from a compact connected Riemann surface). It is well known that the following are equivalent: (1) There is a Hermitian metric on M and a bound A > 0 so that all deformations of f have area at most A in that metric. (2) For any Hermitian metric on M there is a bound A > 0, so that all deformations of f have area at most A in that metric. (3) Let X be the moduli space of stable maps; see [3,55,59]. Let [f ] ∈ X be the point corresponding to f . Then the path component of [f ] in X is compact; see [34] for proof. (4) Let Z be the cycle space of M and [f ] ∈ Z the point corresponding to f .
Then the path component of [f ] in Z is compact; see [5,43] p. 810. A compact complex curve which satisfies any, hence all, of these is tame. A complex manifold is (rationally) tame if every (rational) curve in it is tame.
Remark 2. Suppose that M → M ′ is a holomorphic fiber bundle with complete rational homogeneous fibers. Suppose that M is a compact Kähler manifold. Then M ′ is a compact Kähler manifold by pushing down a suitable power of the Kähler form on M .
Suppose that M → M ′ is a holomorphic fiber bundle with complete rational homogeneous fibers. Suppose that M is a compact complex manifold bimeromorphic to a compact Kähler manifold. Then M ′ is a compact complex manifold bimeromorphic to a compact Kähler manifold; Campana [14,12]. 1.4. The main theorem.
Definition 8. From now on, all Cartan geometries in this paper will be assumed holomorphic, i.e., G is a complex Lie group, H ⊂ G is a closed complex subgroup, E → M is a holomorphic principal H-bundle on a complex manifold M , and the Cartan connection ω is a holomorphic 1-form. Isomorphisms will therefore be biholomorphic.

Theorem 2. Suppose that
(1) G/H is a complex homogeneous space,  (2) M is a connected compact complex manifold and (3) M contains a tame rational curve and (4) M bears a G/H-geometry. Then M = G/H and the geometry is the model geometry. Moreover H ⊂ G is a maximal parabolic subgroup and G/H is a complete rational homogeneous variety.
Proof. By Theorem 3 on the preceding page, the geometry must drop to a G/H ′geometry, for H ′ = G or for a candidate H ′ strictly containing H. Since there is no candidate, the geometry must drop to a G/G-geometry, meaning to a point. The drop must be isomorphic to G/G, and therefore its lift must also be isomorphic to the lift of G/G, i.e., to G/H. The bundle mapping G/H → G/G must be a bundle of rational homogeneous varieties, so H ⊂ G must be a parabolic subgroup. If H is not maximal, say H ⊂ H ′ ⊂ G, then H ′ is parabolic, so a candidate subgroup.

Applications to locally homogeneous geometric structures.
Definition 11. Suppose that G/H is a homogeneous space. Let X = G/H and x 0 = 1 · H ∈ X. Suppose that M is a manifold. Pick a point m 0 ∈ M . Denote the universal covering space by for every γ ∈ π 1 (M ). Two (G, X)-structures are equivalent if they differ by for some g ∈ H; see [32].
Definition 12. Suppose that X = G/H and X ′ = G/H ′ with H ⊂ H ′ . Suppose that H ′ /H is connected. Suppose that M ′ is a connected manifold with chosen point m ′ 0 ∈ M ′ and with a (G, X ′ )-structure (δ ′ , h ′ ). Let be the universal covering map. Define a manifoldM and a mapδ : The group π 1 (M ′ ) acts onM ′ by deck transformations, and on X by h ′ , and so acts onM . Let M = π 1 (M ′ ) \M . So H ′ /H → M → M ′ is a fiber bundle, giving a morphism f : π 1 (M ) → π 1 (M ′ ), and we let h = h ′ • f . SinceM → M is a covering map, there is a factorization of the universal covering map h) is a (G, X)-structure on M , which we call the lift of (δ ′ , h ′ ). A (G, X)-structure drops to a (G, X ′ )-structure if it is isomorphic to the lift of that (G, X ′ )-structure.
Example 3. Every (G, X)-structure is precisely a flat G/H-geometry with a chosen point (see the definition of curvature in Section 2 on page 12, and see [63] for proof). Lifting or dropping (G, X)-structures is precisely lifting or dropping flat Cartan geometries.
For convenience in applications, we state the obvious consequence of our theorem in the category of (G, X)-structures.
M bears a (G, X)-structure and (4) in the induced complex structure, M contains a tame rational curve C.
Then the (G, X)-structure drops to a (G, X ′ )-structure on a compact manifold M ′ , so that Theorem 5. Suppose that (1) G/H is a complex homogeneous space, (2) M is a compact connected manifold, (3) M bears a (G, X)-structure and (4) in the induced complex structure, M is rationally tame.
Then the (G, X)-structure drops to a unique (G, X ′ )-geometry on a compact manifold M ′ , so that Theorem 6. If M is complex manifold and bears a nondegenerate plane field, and M contains a rational curve then the plane field is locally isomorphic to Cartan's example on G 2 /P 2 . If M is compact and either (1) M has finite fundamental group or (2) the rational curve is tame, then M = G 2 /P 2 with its standard G 2 -invariant rank 2 subbundle V ⊂ T M .
Proof. Cartan [18] associates to every nondegenerate plane field V ⊂ T M on any a Cartan geometry modelled on G 2 /P 2 , where G 2 is the exceptional simple Lie group of dimension 14, and P 2 is a certain maximal parabolic subgroup. The homogeneous space G 2 /P 2 is rationally primitive since P 2 is a maximal subgroup. The rest is proven in section 4 on page 15.
1.7. Applications to surface geometry.
M has a holomorphic G/H-geometry and (4) M contains a rational curve. Then the geometry is flat and the model X is precisely one of (1) P 2 , P 1 × C × or (5) P 1 × E, E any elliptic curve. If X = P 2 then M = P 2 and the geometry on M is the model geometry. If M is a Hirzebruch surface, not equal to P 2 , then M = X = P 1 × P 1 and the geometry on M is the model geometry.
Otherwise X = P 1 × (G/H ′ ), for a complex codimension 1 closed subgroup H ′ ⊂ G with H ⊂ H ′ . (Warning: the group G doesn't have to act faithfully on G/H.) Moreover, M is a ruled surface, M → C, over some compact Riemann surface C. The G/H-geometry on M is a (G, X)-structure, lifted from a (G, X ′ )-structure on C.
Proof. Compact complex surfaces are tame, so the rational curve is tame. The G/H-geometry must drop, say to some G/H ′ -geometry on a compact complex curve, or to a point. If to a point, then G/H must be a complete rational homogeneous surface, so P 2 or P 1 × P 1 , and M = G/H with its standard model geometry.
Otherwise, the complex Riemann surface G/H ′ is complex homogeneous, so precisely one of P 1 , C, C × , E. The bundle G/H → G/H ′ has 1-dimensional rational homogeneous fibers, so is a homogeneous bundle of rational curves. By the classification of complex homogeneous surfaces (see Huckleberry and Livorni [40]), G/H → G/H ′ is a trivial bundle. Geometries on curves are flat, since curvature is a semibasic 2-form. In our setting, the geometry is holomorphic, so the curvature is holomorphic: a semibasic (2, 0)-form. Therefore the geometry is a (G, X ′ )-structure.
This theorem reduces the problem of surface geometries to the cases where the surface is minimal, and therefore given (more or less explicitly) in the Enriques-Kodaira classification up to biholomorphism. 1.8. Applications to conformal geometry. In dimensions at least 3, the model of conformal geometry is the smooth quadric hypersurface G/H = PO (n + 2, C) /P ⊂ P n+1 .
In the special case n = 2, G/H = P 1 × P 1 , and a holomorphic conformal geometry is never a holomorphic Cartan geometry.
Corollary 3. A compact complex surface containing a rational curve bears a holomorphic P 1 × P 1 -geometry if and only if the surface and conformal geometry are constructed from the process given in theorem 7 on the preceding page out of a holomorphic projective connection (i.e., P 1 -geometry) on a compact complex curve.
Note that this observation easily obtains part of the classification theorem of Kobayashi and Ochiai [52].
Theorem 8. Suppose that M is a complex manifold of dimension at least 3, with a holomorphic conformal structure, and M contains a rational curve. Then the holomorphic conformal structure is flat. If M is compact and either (1) has finite fundamental group or (2) the rational curve is tame, then M is a smooth quadric hypersurface with its standard conformal structure.
Proof. A holomorphic conformal geometry is equivalent to a holomorphic Cartan geometry; see [17]. The homogeneous model, namely the smooth quadric hypersurface, is rationally primitive, since it is a rational homogeneous variety PO (n + 2, C) /P where P ⊂ PO (n + 2, C) is a maximal parabolic subgroup. Therefore if M is compact and there is a tame rational curve, the result follows from the main theorem. The rest is proven in section 4 on page 15.
This strengthens the work of Belgun [9], who proved that complex null geodesics cannot be rational except on conformally flat manifolds.
1.9. Applications to classification of geometric structures on Fano manifolds.
Definition 13. A complex manifold is called rationally connected if any two points of it lie in a rational curve.  Proof. Rationally connected implies bimeromorphic to a compact Kähler manifold (see Campana [13,p. 547 Theorem 4.5]) and therefore tame. Apply Theorem 3 on page 4. Theorem 9. Rigidity: the only holomorphic Cartan geometries on a complete rational homogeneous variety G/P are the flat G ′ /P ′ -geometries given by global biholomorphisms G/P → G ′ /P ′ .
Proof. Complete rational homogeneous varieties are rationally connected.
The isomorphisms between complete rational homogeneous varieties G/P and G ′ /P ′ with different groups G and G ′ are classified; see [2, p. 214], [67]. All such isomorphisms are equivariant under transitive holomorphic group actions.
Corollary 5. Suppose that M is a compact rationally tame complex manifold bearing a holomorphic Cartan geometry, modelled on a homogeneous space G/H. By Theorem 3 on page 4, there is a unique drop M → M ′ with rational homogeneous fibers so that M ′ has no rational curves. This drop M → M ′ is the maximal rationally connected fibration (see [53, p. 222]); in particular, that fibration is a smooth fiber bundle of complete rational homogeneous fibers over a smooth base, and the base contains no rational curves.  [41, p. 6] and [62].
A well known conjecture says that for a projective variety M , if κ M = −1, then M is uniruled. If M admits a holomorphic Cartan geometry which doesn't drop, then using Theorem 3, this conjecture implies that κ M ≥ 0.
Theorem 10 is similar to results of Frances [26] and Bader, Frances and Melnick [4].
1.12. Applications to projective connections. The following theorem generalizes the results of Jahnke and Radloff [44]. They require that the projective connection be torsion-free and that the manifold be Kähler; see Gunning [36].
Theorem 11. The only connected compact complex manifold bearing a projective connection and containing a tame rational curve is P n , and the only projective connection it bears is the standard flat one.
1.13. Applications to the geometry of ordinary differential equations. As usual, we treat points of P 1 as lines through 0 in C 2 , and we write O (n) for the bundle over P 1 whose fiber over a point L ∈ P 1 is the tensor power (L * ) ⊗n if n > 0, L ⊗n if n < 0 and C if n = 0. We will denote the total space of the line bundle O (n) also as O (n). Thus the global sections of O (n) → P 1 are the homogeneous polynomials Sym n C 2 * of degree n. Write points of O (n) as pairs (L, q), where L ∈ P 1 and q ∈ (L * ) ⊗n . Let Every surface with O (n)-geometry inherits a foliation, corresponding to the fiber bundle map, and inherits a family of curves which locally are identified with the global sections.
It is well known (see Lagrange [56,57], Fels [24,25], Dunajski and Tod [23], Godliński and Nurowski [31], Doubrov [21]) that every holomorphic scalar ordinary differential equation of order n + 1 ≥ 3 is locally determined by, and locally determines, an O (n)-geometry invariant under point transformations. Each solution of the differential equation is mapped by the developing map to a global section of O (n). Not every O (n)-geometry arises -even locally -from a holomorphic scalar ordinary differential equation, but we can ignore this issue.
Theorem 12. Suppose that M is a compact complex surface. Suppose that M bear an O (n)-geometry. Then M contains no rational curves. In particular, the universal covering space of each local solution of the ordinary differential equation develops to a rational curve in the model, but not by an isomorphism.
Proof. Elementary algebra shows that the O (n) is rationally primitive. But M is compact, so it is not isomorphic to the model. Therefore M contains no rational curves.
1.14. Applications to cominiscule geometries. Theorem 3 on page 4 generalizes the Main Theorem of Hwang & Mok [42, p. 55]. Hwang and Mok used the language of G-structures. We will provide a dictionary to make the connection between their result and ours clearer.
Definition 15. A complex homogeneous space G/H is called an irreducible compact Hermitian symmetric space (ICHSS), also called a cominiscule variety, if it is compact, with G and H connected, G semisimple and g/h an irreducible H-module; see Landsberg [58].
If G/H is a cominscule variety, then H ⊂ G is a maximal subgroup. In particular, cominiscule varieties are rationally primitive.
Definition 16. A Cartan geometry modelled on a cominiscule variety is called a cominiscule geometry or almost Hermitian symmetric geometry [6,7,8] or ICHSS geometry. Note that the cominiscule variety is treated as a complex homogeneous space G/H, so that G is the biholomorphism group, not the isometry group, of the ICHSS. Definition 19. An S-structure (following the terminology of Hwang and Mok [42]) is a G-structure where G ⊂ GL (n, C) is a reductive linear algebraic complex Lie group with a faithful irreducible complex representation on C n .
As explained byČap [16] (and proven by Tanaka [68] andČap and Schichl [17]), Cartan's method of equivalence associates to any holomorphic S-structure a holomorphic cominiscule geometry. The construction is local, in terms of differential invariants. The S-structure can be recovered from the cominiscule geometry. Not every cominiscule geometry arises in this way; those which do are called normal and can be characterized by certain curvature equations (seeČap and Schichl [17] for complete details).
Theorem 13 (Hwang-Mok [42]). Suppose that M is a uniruled smooth projective variety bearing a holomorphic S-structure. Then M is a cominiscule variety, and the holomorphic S-structure is the standard flat one.
Clearly we can improve this result: Theorem 14. Let M be a compact complex manifold containing a tame rational curve, and equipped with a holomorphic cominiscule geometry (not necessarily normal), say modelled on some cominiscule variety G/H. Then M = G/H, and the cominiscule geometry on M is the flat G-invariant model geometry.
Corollary 6. Suppose that M is a compact complex manifold containing a tame rational curve and dim C M ≥ 2.
(1) If T M = U ⊗ V , where U and V are holomorphic vector bundles of ranks p and q respectively, and p, q ≥ 2, then M = Gr (p, p + q) is a complex Grassmannian. Proof. Each hypothesis is equivalent to one form or another of S-structure.
The analoguous result for uniruled projective manifolds is due to Hwang and Mok [42].
If G is a semisimple Lie group, then G has a unique maximal connected solvable subgroup up to conjugacy; these are called Borel subgroups. In this section we will fix a Borel subgroup of G and consider the parabolic subgroups of G that contain that same Borel subgroup. Parabolic geometries will be modelled on G/P where P must contain that fixed Borel subgroup.
Theorem 15. Two holomorphic parabolic geometries on compact complex manifolds, one of which is rationally tame, admit a twistor correspondence just when they have an isomorphic drop.
Proof. We can assume that F in Definition 20 is the identity map, so we face holomorphic parabolic geometries E → M 0 and E → M 1 , modelled on G/P 0 and G/P 1 , say, on the same total space and with the same Cartan connection. Let B ⊂ P 1 ∩ P 2 be the Borel subgroup of G. Then E/B → M 0 and E/B → M 1 are holomorphic fiber bundles with rationally connected fibers, and so the G/Bgeometry on E/B must drop to a geometry, say E/H ′ , so that the fibers of E/H ′ contain the fibers of both E/B → M 0 and of E → M 1 , i.e., H ′ contains P 0 and P 1 .

Curvature
Definition 21. Let ω be the Cartan connection of a Cartan geometry E → M modelled on a homogeneous space G/H. Consider the g-valued 2-form Since ω is an isomorphism of T E with the trivial vector bundle on E with fiber g, the 2-form in (1) produces a function on E with values in g ⊗ 2 g. This function descends to a function K on E with values in g ⊗ Λ 2 (g/h) * called the curvature of the Cartan connection. A Cartan geometry is called flat if K = 0.
The Lie algebra g is equipped with the adjoint action of H. Let E × H g be the vector bundle over M associated to the principal H-bundle E H for the H-module g. Note that the curvature K is a holomorphic section of E × H g ⊗Ω 2 (M ). Consider the principal G-bundle E H × H G over M obtained by extending the structure group of the principal H-bundle E H using the inclusion of H in G. The form ω defines a holomorphic connection on E H × H G [63]. The section K is the curvature of this connection.
This section K of E × H g ⊗ Ω 2 (M ) will also be denoted by ∇ω.
The model is flat. The Frobenius theorem tells us that a Cartan geometry is locally isomorphic to its model just when it is flat.
Since the curvature 2-form is a holomorphic 2-form (valued in a holomorphic vector bundle), and there are no nonzero holomorphic 2-forms on a curve, every Cartan geometry on a curve is flat.
Suppose that G/H is a homogeneous space and E → M is a Cartan geometry with Cartan connection ω. For each point e ∈ E, the set h e of vectors A ∈ g for which A(e) ∇ω = 0 is a Lie algebra, called the apparent structure algebra, and h ⊂ h e is a subalgebra. It follows then that if the Cartan geometry drops to a Cartan geometry modelled on, say, G/H ′ , then h ′ ⊂ h e . Conversely [15, p. 15], if h ′ ⊂ g is a Lie subalgebra and if h ′ ⊂ h e for every point e ∈ E, then every point m ∈ M lies in a neighborhood in which there is a local isomorphism to a G/H-geometry which drops to a G/H ′ -geometry. So the local dropping problem is easily analyzed in terms of the apparent structure algebra. These observations will play no role in this paper, but have played a significant role in the literature [15].

Development of curves
Lemma 1 (Sharpe [63], p. 188, Theorem 3.15). The Cartan connection of any Cartan geometry π : E → M determines isomorphisms for any points m ∈ M and e ∈ E m ; thus T M is identified with the vector bundle Cartan connections ω 0 and ω 1 respectively, and X is a manifold or a smooth variety.
Development is an equivalence relation. For example, a development of an open subset is precisely a local isomorphism.
The graph of F in Definition 22 is an integral manifold of the Pfaffian system ω 0 = ω 1 on f * 0 E 0 × E 1 , and so F is the solution of a system of (determined or overdetermined) differential equations, and conversely solutions to those equations produce developments.
Remark 3. By Lemma 1 on the previous page, the developing map f 1 has differential f ′ 1 of the same rank as f ′ 0 at each point of X, and moreover f * 0 T M 0 = f * 1 T M 1 as vector bundles on X. In particular, any development of an immersion is an immersion.
Proposition 1 ( [60,63]). Suppose that E → M is a holomorphic G/H-geometry, and f 0 : C → M is a holomorphic map of a simply connected Riemann surface. Suppose that e 0 ∈ E lies above a point of the image of f 0 . Then there is a unique development Definition 23. We will say that e 0 is the frame of the development f 1 .
Remark 4. Every holomorphic vector bundle on P 1 is a direct sum of holomorphic line bundles, each of which has the form O (d), for some degree d [35]. These degrees are uniquely determined up to permutation. A vector bundle over P 1 is spanned by its global sections just when none of the degrees are negative. Recall T P 1 = O (2).
We now prove Theorem 1 on page 3.
Proof. The development of any rational curve to the model is free because the tangent bundle of G/H is a quotient of the trivial vector bundle over G/P with fiber g; the quotient map is given by the left translation action of G on G/H. By Remark 3, the original rational curve in M has the same ambient tangent bundle, and so is also free.

Lemma 2.
Suppose that E → M is a G/H-geometry, and f : P 1 → M is a rational curve. Then the holomorphic vector bundle f * E × H g → P 1 is trivial.
Proof. Clearly it suffices to prove the result for any development f 1 : P 1 → G/H. The vector bundle G× H g over G/H associated to the principal H-bundle G → G/H for the H-module g is identified with the trivial vector bundle over G/H with fiber g; this isomorphism is obtained from the isomorphism Proof. Since the rational curves are free, we can apply the Horikawa deformation theorem [38,39].
The following lemma is essentially theorem 2 of [42, p. 55]. Proof. By Theorem 1 on page 3, f is free. By Remark 4 on the preceding page, f * T M is a sum of nonnegative degree line bundles and T + is a vector subbundle. Since f is not constant, f ′ : is a direct sum of line bundles of negative degree, so does not admit any nonzero section. In particular, the projection is the zero homomorphism.

The local consequences of rational curves
Theorem 16. Suppose that E → M is a G/H-geometry, and M contains a rational curve. Then there is a Lie algebra h ′ containing h, dim h ′ > dim h, and a covering of M by open sets U α so that the restriction of the G/H-geometry to U α is locally isomorphic to the lift of a holomorphic Cartan geometry modelled on (G, h ′ ). Take a complex Lie subgroup H ′ ⊂ G containing H with Lie algebra h ′ . The manifold M is foliated by complex submanifolds of positive dimension on which the G/H-geometry restricts to a flat Cartan geometry modelled on (H ′ , h).
Remark 5. For the definition of a a holomorphic Cartan geometry modelled on a pair (G, h ′ ), see Sharpe [63].
Proof. Inductively define covariant derivatives Write the map E → M as π : E → M . Pick a point m ∈ M and a point e ∈ E so that π(e) = m. If v 1 ∈ T m M is a vector on which v 1ω = 0, we can take any vectors v 1 , w ∈ T m M . Take any vectors Therefore we can let be the associated section of Inductively, if ∇ p ω has null vector v 1 , we can define Take v 1 tangent to a rational curve in M ; for all vectors v 2 , w 1 , . . . , w p , for all p, precisely as in lemma 4 on page 14, since the ambient tangent bundle of any rational curve is spanned by its global sections, and the tangent bundle of the projective line is spanned by global sections, all of which vanish somewhere. Consequently, if A ∈ g projects to some nonzero vector v ∈ T m M tangent to a rational curve, then A K vanishes with its derivatives of all orders at every point of the fiber E m . Therefore A K vanishes everywhere. The result follows fromČap [16], p. 15, unnumbered theorem.
Corollary 8. Suppose that G/H is a complex-homogeneous space, and that h ⊂ g has maximal dimension among complex Lie subalgebras. Suppose that E → M is a G/H-geometry, and M contains a rational curve. Then the G/H-geometry is flat and locally isomorphic to the model. Theorem 17. Suppose that M is complex manifold with a holomorphic parabolic geometry modelled on a complete rational homogeneous variety G/P where P ⊂ G is a maximal parabolic subgroup. Suppose that M contains a rational curve. Then the parabolic geometry is locally isomorphic to the model. If M is compact and either (1) has finite fundamental group or (2) has a tame rational curve, then M is isomorphic to the model.
Proof. If a finite fundamental group, then a finite unramified covering space develops to the model G/P . But G/P is simply connected.
Our results on conformal geometry are now clear, modulo the proof of the main theorem.

Rational trees and families of rational trees
Definition 24. A tree of projective lines is a connected and simply connected reduced complex projective curve with only nodal singularities whose irreducible components are projective lines.
A rational tree in a complex space M is a nonconstant morphism f : T → M from a tree of projective lines.
Definition 25 (Bien, Borel, Kollár [10]). A family of rational trees through a point m 0 ∈ M in a complex manifold M is a proper flat morphism X → Y of complex spaces such that every fiber of X → Y is a tree of projective lines, a regular section s : Y → X and a morphism f : X → M , so that f (s(y)) = m 0 for all y ∈ Y . A family of rational trees is tame if all of its rational trees are tame. If Y is connected, then the family is tame if and only if any one of the trees in the family is tame. To be more explicit, we need to explain how to graft two families. Suppose that are two families of curves rooted at the same point m 0 ∈ M , say with sections s 1 : Y 1 → X 1 and s 2 : Let X ⊂ X 1 × X 2 be the set of points (x 1 , x 2 ) ∈ X 1 × X 2 such that either x 1 = s 1 (y 1 ) or x 2 = s 2 (y 2 ), where y 1 = p 1 (x 1 ) and y 2 = p 2 (x 2 ). Clearly X → Y is a proper family.
We can see that X is a union of two components, X 1 and X 2 (perhaps themselves not irreducible). Moreover we can identify X 1 ∩ X 2 = Y . Away from the "grafting points" x = (x 1 , x 2 ) = (s 1 (y) , s 2 (y)) for any y ∈ Y , and X is local isomorphic to X 1 or to X 2 . At the grafting points, we have an exact sequence with the middle and final entries being flat over O y,Y . Therefore the first entry is flat over O y,Y ; see [37, p. 254 Clearly F is a morphism of complex spaces.
Deformations of a rational tree consist in deformations of the root component which pass through the root m 0 , and then any deformation of the adjacent components which pass through that first component, etc. We can describe the tangent space explicitly then as consisting in a section of the normal bundle along each component, and a choice of vector in the ambient tangent space at each marked or nodal point, so that the vectors at nodes agree (modulo the tangent directions) with the sections of the normal bundle of the components through that node. In particular, the deformations of any given free rational tree, with fixed topology, form a smooth manifold near that rational tree. Proof. Every infinitesimal deformation of a rational tree consists precisely of a choice of section s of the normal bundle of each component P 1 , and choice of tangent vector v at each marked point p where components are connected, so that s(p) = v mod T p P 1 . So at smooth points of the space Y , we can use this description of the normal sheaf of a tree.
Each rational tree in our family has a "root component" mapped through m 0 . On the root component of a rational tree through m 0 , f * ∇ω = 0, by Corollary 9 on the preceding page.
Pick out just one component of a rational tree. Then f * ∇ω = 0 on the tangent vectors of that component because it is a rational curve. Therefore we need only evaluate f * ∇ω = 0 on the normal vectors of each component.
Pick an element v of the normal bundle of a component of a rational tree. On any component, if v is not null for f * ∇ω, then v must belong to a nonpositive degree line bundle in the normal bundle. The normal bundle, being a quotient of the ambient tangent bundle, is a sum of nonnegative degree line bundles. Therefore v must belong to a zero degree line bundle inside the normal bundle. Moreover, the curvature must not vanish at any point of that section, since the curvature splits into nonpositive degree summands, and the nonzero contributions must come from zero degree summands.
Suppose that we have already established that f * ∇ω = 0 on all sections of the normal bundle of one component C 0 of our tree. Consider a component C 1 which is adjacent to C 0 . Take a normal vector v 1 to C 1 . Take a section s 1 of the normal sheaf along C 1 extending the normal vector v 1 . At the nodal point p where C 0 meets C 1 , there is some tangent vector w representing an infinitesimal motion of the tree given by the section s 1 . By freeness of rational curves, there must be a section s 0 of the normal bundle of C 0 which gives a corresponding infinitesimal motion of C 0 so that the tree can infinitesimally move by these sections and remain connected, i.e., a section with velocity w at the node. By induction s 0 f * ∇ω = 0. Therefore w f * ∇ω = 0. But s 1 (p) = w, since w represents the same tangent vector on both components, so s 1 (p) f * ∇ω = 0, The curvature vanishes at some point of s 1 , and therefore at all points: v 1 f * ∇ω = 0. Apply induction to components.
Definition 29. Following [10] we denote the H-coset of the identity in G/H (the "origin" of G/H) as o H . Then the development of this family of rational trees to a family of rational trees in G/H is a development as a map, i.e., the map f 1 : X → G/H is a development of the map f 0 : X → M . Remark 6. The idea of the proof is simple; we will try to state it very roughly first. As we move a rational tree around in a family, the total space of the family feels no curvature, by our earlier results. So we can pretend there is no curvature, and by the Frobenius theorem only worry about whether we can get from place to place on submanifolds along which ω = g −1 dg. We have such submanifolds: develop the various curves through m 0 in our family. We will now make these ideas precise.
Proof. We only need to check that the equation g −1 dg = ω is satisfied on the graph of the isomorphism F associated to the development. It suffices to check this over an open subset of Y . We can assume that Y consists of a single irreducible component. We can then, if needed, expand Y to consist in a single irreducible component of the space of deformations of a rational tree given by a generic fiber of X → Y , since any family will map to the deformation space, and the development will extend. But we can again replace Y by any open set in Y . Therefore we can assume that Y is smooth and that X → Y is a holomorphic submersion away from the nodal points of each tree. We can assume that the section s : Y → X of our family maps every point y ∈ Y to a smooth point of the rational tree X y .
Consider on f * 0 E × G the Pfaffian system ω = g −1 dg; see [11] for the theory of Pfaffian systems. The orbits of the diagonal right H-action are Cauchy characteristics [11], so the Pfaffian system is pulled back from a unique Pfaffian system on Z = f * 0 E × H G [11, p. 9]. Let π : f * 0 E × G → Z be the quotient by H-action. The torsion of the Pfaffian system is precisely f * 0 ∇ω, which vanishes by Lemma 7.
By the Frobenius theorem, the smooth part of each component of Z is foliated by maximal connected integral manifolds. The maximal connected integral manifolds are permuted by left G-action, because the Pfaffian system is left invariant. Clearly X → Y is a union of irreducible components, say X i → Y , so that each X i → Y is a family of deformations of a single component of a rational tree. Each X i has fibers consisting of rational curves X i y . So X i → Y is a family of rational curves, and we can assume that X i → Y is a holomorphic fibration. Moreover, if two components, say X 0 → Y and X 1 → Y , represent components of a tree meeting at a node, then there is a point where they meet, say p 0 : Y → X 0 and p 1 : Y → X 1 so that p 0 (y) = p 1 (y) for every y ∈ Y . Replacing Y by an open subset of Y , we can assume that p 0 : Y → X 0 and p 1 : Y → X 1 are smooth morphisms.
We want to work on Foliate Z i by maximal connected integral manifolds of the Pfaffian system (by the Frobenius theorem). We can write each point z ∈ Z i as z = (y, x, e, g) H where y ∈ Y and x ∈ X i y , e ∈ E f0(x) and g ∈ G and f 1 (x) = go H . We have an obvious map Φ : X → Z given Φ(x) = (y, x, e, F (e)) H. if x ∈ X y , i.e., essentially Φ is just F . By definition of a development, over each X y this map Φ y : X y → Z is an integral curve of the Pfaffian system on Z. In particular, the image of each X i y under Φ lies in a unique maximal connected integral manifold of the Pfaffian system on Z i . We will prove that Φ is itself an integral variety of the Pfaffian system on Z.
There is a distinguished "root" component of each rational tree in our family: the component of X y containing the point s(y). Let's call this component X 0 y . In particular, if we replace Y by a smaller open subset of Y , then this specifies a component X 0 of X and a component Z 0 of Z.
All of the points x = s(y) get mapped to f (x) = m 0 . They are the "roots" of the various trees. Consider the map σ : Y → Z given by σ(y) = (y, s(y), e 0 , 1) H for every point y ∈ Y . Clearly σ is an integral manifold of the Pfaffian system on Z 0 . Since Y is connected, the image of σ must lie in a maximal connected integral manifold Λ 0 ⊂ Z 0 .
Each integral curve Φ| X 0 y : X 0 y → Z 0 strikes the graph of σ at the unique point Φ(s(y)) = σ(y), and so intersects Λ 0 . Being an integral curve of the Pfaffian system, this integral curve must lie inside Λ 0 . So Φ restricted to each curve X 0 y has graph inside Λ 0 . But then the graph of Φ restricted to the component X 0 lies inside Λ 0 , and so Φ| X 0 : X 0 → Z is an integral manifold of the Pfaffian system.
Consider two components, say X 0 and X 1 , meeting along a node, say at p 0 (y) ∈ X 0 and p 1 (y) ∈ X 1 , so p 0 (y) = p 1 (y) ∈ X. Consider the morphism ψ : Y → Z 0 ∩Z 1 given by y → Φ (p 0 (y)) for an arbitrary point e ∈ E p0(y) . By definition, the image of ψ lies in the image of Φ| X 0 and so lies in Λ 0 . Therefore ψ is an integral manifold of the Pfaffian system on Z 0 and so is an integral manifold of the (same) Pfaffian system on Z 1 . Therefore the image of ψ lies inside a maximal connected integral manifold, say Λ 1 ⊂ Z 1 , of the Pfaffian system on Z 1 . For each y ∈ Y , the curve Φ| X 1 y : X 1 y → Z 1 is an integral curve of the Pfaffian system. But this curve intersects the image of ψ at the point ψ(y) = Φ p 1 (y) . Therefore the image of Φ| X 1 y : X 1 y → Z 1 lies inside Λ 1 . Note that this holds true for every point y ∈ Y . By induction, using the same argument on the next component X i of X, there is some integral manifold Λ i ⊂ Z i containing the image of Φ| X i : The graph of F : f * 0 E → f * 1 G consists precisely in the points (y, x, e, g) so that x ∈ X y and e ∈ E f0(x) and g ∈ G f1(x) . In other words, the graph of F is the preimage in f * 0 E × f * 1 G of the image of Φ in Z. Therefore since Φ an integral manifold of the Pfaffian system on Z, the graph of F is an integral manifold of the Pfaffian system upstairs. So the graph of F on E| X 0 is an integral manifold of ω = g −1 dg meaning, F * g −1 dg = ω, i.e., F is a development of maps.

Deformation of stable rational trees and compactness
Definition 30. A rational tree with marked points is stable if it is nonconstant on every edge of valence at most two with no marked or nodal points, and on every leaf with at most one marked or nodal point [3,55,59].
Definition 31. If f : T → M is a rational tree in a complex manifold M , with some number of marked points, the stabilization of T is the tree obtained by identifying to a point any component of T on which f is constant and which has too few marked or nodal points to be stable.
Remark 7. Suppose that M is a compact complex manifold M all of whose rational curves are free. All strata of the compactified moduli space of stable rational trees passing through a point, with any number of marked points, have the "expected" dimensions. The space of stable tame rational trees is an analytic orbispace, as proven by Siebert [66] 1747. (The proof asks for M to be Kähler, but only uses tameness.) The connected components are compact; see [65, Theorem 0.1]. Each stratum of each connected component is a deformation space of stable tame rational trees with fixed topology, so a smooth manifold.
An analytic orbispace is locally the quotient of a complex space by a finite group. The quotient of a complex space by a finite group is a complex space [46, p. 198]. Therefore there is a quotient complex space of any analytic orbivariety, through which the regular maps to complex spaces factor. For the space of stable tame rational trees, this quotient space is a coarse moduli space for stable tame rational trees [66, p. 1754]. In particular, the evaluation maps at marked points (evaluating each stable map at each marked point) factor through to maps to the coarse moduli space. By the Remmert proper mapping theorem [20, p. 150], the image of any evaluation map has Zariski closed image in M . (2) The restriction of this orbifamily to U is a smooth maximal dimensional family (not just an orbifamily). Proof. By Lemma 9, the tangent space to the blade is precisely the set of vectors arising from infinitesimal deformations of the generic stable tame rational curve through m 0 .

Development and compactness
If M is a compact complex manifold bearing a holomorphic Cartan geometry, then we can develop the entire "orbifamily" of all stable tame rational trees (which could have infinitely many components), and the evaluation map of the development on each connected component has Zariski closed image in G/H. Proposition 3. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact complex manifold with a holomorphic G/H-geometry. If a stable tame rational tree through a point o H ∈ G/H is the development of a stable tame rational tree through m 0 , then every deformation of that stable tame rational tree through o H is the development of a stable tame rational tree through m 0 . Indeed the analytic orbivariety of stable tame rational trees through m 0 is biholomorphically mapped to a union of irreducible components of the analytic orbivariety of stable tame rational trees through o H via development through any given frame e 0 .
Proof. By the Horikawa deformation theorem [38,39] and the freedom of all rational curves in M , the deformations of any rational tree with fixed root are unobstructed and form a smooth moduli space near that tree. As explained in Lemma 6 on page 18, all rational trees in M through m 0 develop to rational trees in G/H. Stable trees develop to stable trees. Tame trees develop to tame trees. Distinct trees develop to distinct trees. As we vary trees holomorphically, their developments vary holomorphically. We therefore have an injective morphism between the space of deformations of any rational tree and the space of deformations of its development.
By Theorem 1 on page 3, stable rational trees are unobstructed, and the deformations through a point of a stable rational tree form a complex space of dimension given by the dimension of sections of the normal sheaf vanishing at m 0 . (There may still be some "bubbling" among these deformations, but only in lower dimension.) As we pointed out in Remark 3 on page 14, the normal sheaf of a stable rational curve is identified with the normal sheaf of any of its developments. Therefore the deformation space of a stable tree and of its development are complex spaces of equal dimension, with tangent spaces identified by development. The developing map, being injective and having injective differential, gives a local biholomorphism between these spaces. The image of this morphism is therefore open. So we have mapped the analytic orbivarieties of stable rational trees by an injective local biholomorphism.
During the process of convergence in Gromov-Hausdorff norm, i.e., in Siebert's analytic orbivariety, marked points on tame trees stay away from the nodes. Each stable tame rational tree in our family has a marked point mapping to m 0 . So the development of all of our marked stable tame trees remains defined and varies continuously in Gromov-Hausdorff norm, and smoothly away from the "bubbling", by smooth dependence of solutions of ordinary differential equations. We can develop all stable tame rational trees, mapping every point of the analytic orbivariety of m 0 to that of o H injectively and holomorphically. By compactness of connected components, the image is closed, so the map is onto each component that its image touches.  Proof. The curvature vanishes on the blade R (m 0 ). The development along any path is thus independent of homotopy modulo endpoints. Therefore there is a development of the universal covering space R of the blade. Trees are simply connected, so the lifting is clear. Existence and uniqueness for ordinary differential equations ensures f 1 = dev • f 0 .

Developing the blade
Lemma 10. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact complex manifold with a holomorphic G/H-geometry. Fix a point m 0 ∈ M and a frame e 0 ∈ E m0 . Let R be the universal covering space of the blade R (m 0 ). Then R is compact. Every tame rational tree through m 0 is the image of a unique tame rational tree through m 0 ∈ R. Every point of R lies on a tame rational tree through m 0 .
Proof. Since tame rational trees are simply connected, we can lift them from R = R (m 0 ) to R. By Lemma 9 on page 23, there is a compact orbifamily of stable tame rational curves whose image in M is precisely R, so that a Zariski open subset of that orbifamily is mapped smoothly submersively to a Zariski open subset of R. We can lift each element of this orbifamily to a stable tame rational tree in R, and by existence, uniqueness and smoothness of solutions of ordinary differential equations, these developments form an orbifamily in R. By compactness of the orbifamily, the image of the orbifamily in R is compact. By the Remmert proper mapping theorem, the image is a compact subvariety. The deformation theory of any stable tame rational curve is locally identical in R and in R, so the orbifamily has image containing a Zariski open subset of R. Because R is the universal covering of an irreducible variety R, R is also irreducible. Therefore the orbifamily has image R. So R is compact, and every point of R lies on a stable tame rational tree from this orbifamily.
Corollary 13. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact complex manifold with a holomorphic G/H-geometry. Then every blade in M has finite fundamental group. By Proposition 3 on page 23, every element of this family of stable tame rational trees is the development of a stable tame rational tree through m 0 .
Definition 32. Suppose that G/H is a complex homogeneous space. Suppose that M is a complex manifold with a holomorphic G/H-geometry. Fix a point m 0 ∈ M and a frame e 0 ∈ E m0 . For each rational tree f 0 : T → M through m 0 , its development f 1 : T → G/H. The points of f * 1 G can be written as pairs (t, g) where t ∈ T and g ∈ G and f 1 (t) = go H .
Let H ′ = H ′ (e 0 ) be the set of elements g ∈ G for which there is some rational tree f 0 through m 0 and some development f 1 and there is some Remark 9. We are eventually going to prove that the drop group H ′ is precisely the group H ′ which appears in our main theorem. Unfortunately there are some minor complications in case the group H is not connected, which will require us to consider a close relative to H ′ , which we introduce now. Let H ′ 0 = H ′ 0 (e 0 ) be the set of elements g ∈ G for which there is some rational tree f 0 through m 0 (whose development we will call f 1 ), and there is some point Proof. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact complex manifold with a holomorphic G/H-geometry. Fix a point m 0 ∈ M and a frame e 0 ∈ E m0 . Let H ′ be the drop group of e 0 . Take two rational trees f 1 : T 1 → G/H and f 2 : T 2 → G/H, both rooted at the point o H ∈ G/H. So each has a marked point, say p 1 ∈ T 1 and p 2 ∈ T 2 so that f 1 (p 1 ) = f 2 (p 2 ) = o H . We want simply to imagine sliding the tree T 1 along the tree T 2 , so that the root of T 1 traces out the points of T 2 . If we can do this, then for any point in H ′ coming from T 1 , say g 1 , and any point in H ′ coming from T 2 , say g 2 , we will slide over to find that g 1 g 2 ∈ H ′ .
If a point lies on the development of a rational tree, then it lies on the development of a stable rational tree, since we can stabilize by cancelling out constant maps on components, which has no effect on development. So we can assume that our rational trees are stable.
Map F 2 : X 2 → G/H by F 2 (q 2 , g 2 , r 1 ) = f 2 (r 1 ) . Say that (r 1 , q 2 , g 2 ) ∼ (r 2 , q 2 , g 2 ) if r 1 = p 1 and r 2 = q 2 , i.e., graft the root of tree T 1 to the point q 2 . Let X = (X 1 ⊔ X 2 ) / ∼. We can identify X with the variety X ⊂ T 1 × T 2 × Y consisting of points (r 1 , r 2 , q 2 , g 2 ) so that r 1 = p 1 or r 2 = q 2 . Therefore X → Y is clearly a proper family. The algebraic variety X splits into two irreducible components, say X = X 1 ∪X 2 , given by X 1 = (r 2 = q 2 ) and X 2 = (r 1 = p 1 ). Note that X 1 ∩X 2 ∼ = Y . Away from the "grafting point" (p 1 , q 2 , q 2 , g 2 ) clearly the map X → Y is locally identified with either X 1 → Y or X 2 → Y , so is flat. Near the "grafting point" we have an exact sequence with the middle and final entries being flat over O y,Y . Therefore the first entry is flat over O y,Y ; see [37, p. 254, Proposition 9.1A]. So X → Y is flat.
The maps F 1 and F 2 agree along the identified points so descend to a morphism F : X → G/H. We take as section s : Y → X the mapping s (q 2 , g 2 ) = (p 1 , p 2 , q 2 , g 2 ) .
Clearly we have constructed a family of stable rational trees through o H . If the original trees were tame, then so is the family.
Lemma 13. The drop group is closed under taking inverses, as is the restricted drop group.
Proof. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact complex manifold with a holomorphic G/H-geometry. Fix a point m 0 ∈ M and a frame e 0 ∈ E m0 . Let H ′ be the drop group of e 0 . Again we can assume that all rational trees are stable. Take a stable tame rational tree f : T → G/H, say with root at p 0 ∈ T . Let Y = f * G, and let X = T × f * G, with the obvious map X → Y . Map h : X → G/H by h (p, q, g) = g −1 f (p). Then h (p, p 0 , 1) = f (p), so h deforms f . Also if some point g 1 o H lies in the image of f , say g 1 o H = f (p 1 ), then p → h (p, p 1 , g 1 ) is a tree rooted at p 1 with h (p 1 , p 1 , g 1 ) = o H and h (p 0 , p 1 , g 1 ) = g We need a slight variation on the Borel-Remmert theorem.
Proposition 4. If X is a rationally connected compact homogeneous complex manifold, then X is a complete rational homogeneous variety X = G/P .
Suppose that X is a complex manifold with a holomorphic unramified covering π : G/P → X by a rational homogeneous variety G/P . Then π is a biholomorphism.
Proof. We can assume that G is the biholomorphism group of G/P without loss of generality. Let Γ = π 1 (X); the group Γ acts as holomorphic deck transformations, so Γ ⊂ G. But every element of G must act on G/P with a fixed point [69]. Therefore Γ = {1}.
Suppose that E → M is a holomorphic Cartan geometry on a compact complex manifold. Then the drop group of a frame is constant as we vary the frame through any connected component of E, as is the restricted drop group.
Proof. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact complex manifold with a holomorphic G/H-geometry E → M . For each point m 0 ∈ M and frame e 0 ∈ E m0 , let H ′ 0 (e 0 ) be the restricted drop group of e 0 . By freedom of rational trees, as we vary e 0 = e 0 (t), and correspondingly vary m 0 = m 0 (t), we can deform any tame rational tree so that a marked point on it passes through m 0 (t), and develop f t via the frame e 0 (t) to a family of tame rational trees through o H . These trees through o H form a family in o H . One member of that family is a development via e 0 (0), and so by Proposition 3 on page 23 the entire family is a development of a family g t : T t → M via e 0 (0). Therefore the entire family lies inside the homogeneous space H ′ 0 (e 0 (0)) o H . Therefore H ′ 0 (e 0 (t)) ⊂ H ′ 0 (e 0 (0)) for any t in our family. Since the choice of t and 0 are arbitrary, H ′ 0 (e 0 (t)) is constant. Suppose we pick two points m 0 and m 1 in the same blade, and two frames e 0 ∈ E m0 and e 1 ∈ E m1 . Since m 0 and m 1 lie in the same blade, under development via e 0 , e 0 is taken to 1 ∈ H ′ , while e 1 is taken to some point of H ′ , say g. By left invariance, development by e 1 is just development by e 0 followed by left translation by g −1 . Since right action on H ′ commutes with left action, the right action of H ′ on E computed via either development is the same.
It is not immediately clear whether or not the action of H ′ turns E → E/H ′ into a principal H ′ -bundle; nonetheless we can see how the Cartan connection behaves.

Lemma 22.
Under the hypotheses of the previous proposition, conditions (1), (2) and (3)  Proof. Condition (2): the Cartan connection is a linear isomorphism on each tangent space. This is clear because the Cartan connection is unchanged from the original Cartan geometry E → M . Condition (3): the H ′ -action is generated by the vector fields A for A ∈ h ′ . This is true in the model, i.e., on G, and therefore on E because the H ′ -action is identified with the H ′ -action on H ′ ⊂ G. We only have to check condition (1) By the Cartan family equation and using the equation we see that it suffices to check whether i.e., whether A ∇ω = 0, which we know already from corollary 11 on page 24. Proof. Let π : E → M be the bundle map. At each point e ∈ E, the Cartan connection ω gives a real linear isomorphism ω e : T e E → g. This isomorphism makes E into an almost complex manifold, using the complex linear structure on g. This almost complex structure is integrable just exactly when dω is a multiple of ω, equivalenty, no ω ∧ ω terms in dω, equivalenty, when the curvature has no g⊗ C Λ 0,2 (g/h) terms. Should this occur, the 1-form ω is then holomorphic just when dω has no ω ∧ ω terms, that is the curvature is complex linear, in g ⊗ C Λ 2,0 (g/h). Clearly the 1-form ω + h given by ω e + h : T e E → g/h is semibasic for π.
Let Ω e be the linear map Ω e : Clearly Ω e is a linear isomorphism for each e ∈ E.
Use Ω e to define a complex structure on T m M . We know that r * g ω = Ad(g) −1 ω for all g ∈ H. Therefore Ω e transforms by a complex linear isomorphism if we move e while fixing the underlying point m = p(e). Therefore M has a unique almost complex structure for which π : E → M is holomorphic. But π is a submersion, so the almost complex structure is a complex structure. The morphism E → M is therefore a holomorphic principal bundle.
Proposition 6. Suppose that G/H is a complex homogeneous space. Suppose that M is a connected compact complex manifold with a holomorphic G/H-geometry E → M . Suppose that M contains a tame rational curve. Pick a point m 0 ∈ M and frame e 0 ∈ E m0 . Let H ′ be the drop group of the Cartan geometry. The G/H-geometry on M drops to a G/H ′ -geometry on a Kähler manifold M ′ so that M → M ′ is a holomorphic fiber bundle, with fibers biholomorphic to H ′ /H.
Proof. By development, we identify each H ′ -orbit with H ′ , and therefore H ′ acts freely on E.
Consider an H ′ -orbit H ′ e 0 ⊂ E. By development of a blade R (m 0 ) ⊂ M , we see that the blade is identified with H ′ o H ⊂ G/H and the H ′ -orbit H ′ e 0 identified with H ′ . Since H ′ contains H 0 , H ′ is a union of path components in the preimage in G of H ′ o H ⊂ G/H. Therefore the orbit H ′ e 0 is a union of path components of the preimage in E of R (m 0 ) ⊂ M . Therefore H ′ e 0 is closed in E. Therefore each H ′ -orbit in E is a closed set and a submanifold biholomorphic to H ′ . At this step of our proof, we need a small lemma: Then R is a closed subset in E × E.
Proof. Take a convergent sequence of points e n → e ∈ E and a convergent sequence of points e n g n → e ′ ∈ E with g n a sequence in H ′ . We want to prove that e ′ = eg for some g ∈ H ′ . Write the map E → M as π : E → M . Let m n = π (e n ) , m = π (e) , m ′ = π (e ′ ).
By definition of H ′ , we can find a sequence of stable tame rational trees f n : T n → M , with f n passing through m n , so that there is a path in f * n E from e n to e n g n . By Proposition 2 on page 23, there is a compact connected component of the orbifamily of stable rational trees through o H ∈ G/H, so that this family maps surjectively to the blade H ′ o H ⊂ G/H. By development, the blades of M are all identified with H ′ o H , so we can assume that all of the maps f n come from developments of this family via the various frames e n . By compactness of the orbivariety, a subsequence of these f n converges in Gromov-Hausdorff sense to a limiting stable rational tree f : T → M with a path from e to e ′ . The development of f via e must lie in H ′ o H , so we see that e ′ = eg for some g ∈ H ′ .
Returning to the proof of the theorem, clearly R is an immersed submanifold of E × E, since the map (e, g) → (e, eg) is a local biholomorphism to R. Moreover, this map is 1-1, since H ′ acts freely. Therefore R ⊂ E × E is a closed set and a submanifold. Therefore E/H ′ admits a unique smooth structure as a smooth manifold for which E → E/H ′ is a smooth submersion (see Abraham  This map φ is a local diffeomorphism, because φ is H ′ -equivariant and ρ•φ(u, g) = u is a submersion. Moreover, φ is 1-1 because H ′ acts freely. Every element of ρ −1 U lies in the image of our section s modulo H ′ -action, so clearly φ is onto. Therefore E → M ′ is locally trivial, so a principal H ′ -bundle. The Cartan connection of E → M clearly is a Cartan connection for E → M ′ . By Lemma 23 on page 31, this Cartan geometry is holomorphic.
14. Tame rational curves and fiber bundles Proposition 7. Suppose that M and M ′ are two compact complex manifolds. Suppose that M → M ′ is a holomorphic fiber bundle, with complete rational homogeneous fibers. Suppose that M is rationally tame. Then M ′ is rationally tame.
Proof. Pick a rational curve f : P 1 → M ′ . Suppose that the fibers of M → M ′ are isomorphic to G/P , for G a connected complex semisimple Lie group and P ⊂ G a parabolic subgroup. Consider the pullback bundle E G/P = f * M → P 1 , a holomorphic fiber bundle of rational homogeneous varieties. Let E G → P 1 be the associated principal bundle. The group G is the connected component of Bihol(G/P ); see Knapp [48]. Because P 1 is simply connected, this bundle is a principal G-bundle, not merely a Bihol(G/P )-bundle. By Grothendieck's classification of holomorphic principal bundles on P 1 with reductive structure group [35], E G can be constructed as follows: let H ⊂ G be a Cartan subgroup, and α : C × → H a complex Lie group homomorphism. Then the quotient of C 2 − 0 × G by the action λ(z, g) = λz, α(λ) −1 g .
Therefore E G/P has as global section: the quotient of the pairs (z, P ), since H ⊂ P . So there is a rational curve in M mapping to the rational curve in M ′ . Compactness of the cycle space of M therefore implies that of M ′ .

Proof of the main theorem
We now prove Theorem 3 on page 4.
Proof. By Proposition 6 on page 32, if there is a rational curve in M , then we can drop, say to some geometry E → M ′ . Since every rational curve lies in a blade in M , all rational curves in M lie in the fibers of M → M ′ . By proposition 7 on the preceding page, M ′ is rationally tame. If M ′ contains a rational curve, we can repeat this process, dropping to some geometry E → M ′′ , etc. At each step, we reduce dimension by at least one, i.e., dim M ′ ≤ dim M − 1, since the connected components of the fibers of M → M ′ are rational homogeneous varieties containing rational curves. Therefore after finitely many steps, we must arrive at a drop which contains no rational curves, say M → M , say a drop to a G/H-geometry.
At each stage in the process, the connected components of the fibers of M → M ′ , and of M ′ → M ′′ , etc. are rational homogeneous varieties, and in particular are rationally connected. So the connected components of the fibers of M → M are rationally connected and compact. These fibers are homogeneous spaces, being copies of H/H. They are rationally tame, being complex submanifolds of M . Therefore the fibers are rational homogeneous varieties by proposition 4 on page 28. But then the fibers of M → M must lie inside the blades of M , and therefore lie on rational curves in M , so M = M ′ , i.e., we need precisely one step in this process to acheive the result that M ′ contains no rational curves.
Suppose that M → M ′′ is some other drop, and that M ′′ contains no rational curves. The fibers of M → M ′ are rational homogeneous varieties, so rationally connected [53].

Open problems
Dumitrescu classified the holomorphic Cartan geometries admitted by smooth compact complex curves; see Dumitrescu [22].
Problem 16.1. The classification of compact complex surfaces admitting holomorphic Cartan geometries is open. One would also like to classify the holomorphic Cartan geometries they admit. See Kobayashi and Ochiai [49,50,51,52] and Klingler [47] for the classification of "normal" holomorphic G/H-geometries on compact complex surfaces when G/H is a rational homogeneous variety and G is a semisimple complex Lie group.
Problem 16.2. The classification of compact complex 3-folds admitting cominiscule geometries is still open, even among the smooth projective 3-folds. Conjecture 2. Suppose that G/H is a complex homogeneous space. Suppose that M is a compact Kähler manifold admitting a holomorphic G/H-geometry. Then there is a flat holomorphic G/H-geometry on M , i.e., a holomorphic (G, X)-structure, where X = G/H. Conjecture 3. Suppose that G/H is a complex homogeneous space which is not biholomorphic to a product P 1 × (G 0 /H 0 ). Suppose that M is a compact Kähler manifold admitting a holomorphic G/H-geometry. Then there is a closed real Lie subgroup G ′ ⊂ G, with an open orbit U ′ ⊂ G/H, and there is a cocompact lattice Γ ′ ⊂ G ′ , so that M = Γ\U ′ .
In particular, the image of the developing map on the universal covering space of M should be a biholomorphism to some homogeneous open set U ′ ⊂ G/H. On the other hand if G/H = P 1 [30] then it is known that the developing map can be more complicated.
Remark 10. Andrei Mustaţǎ pointed out that our Theorem 3 on page 4 is reminiscent of a speculation of Pandharipande [61, p. 1]. Pandharipande's speculation is that rationally connected varieties all of whose rational curves are free might be rational homogeneous varieties.