Lemma 14. Let p: P→Z be an algebraic principal bundle for a connected semisimple complex algebraic group H , where Z is smooth and irreducible. Suppose P carries an algebraic flat principal connection whose monodromy is Zariski dense in H …

Put h= H , and let ξ∈H^0( P,Ω^1_ P ⊗ h) be the absolute connection form. It has kernel the horizontal distribution, is the identity on fundamental vertical vectors, and satisfies dξ+12[ξ,ξ]=0, R_g^*ξ=Ad(g^ -1 )ξ (g∈H). Restrict to a nonempty smooth open subset R^⊂R on which ξ|_ TR has constant rank r . The germ U meets this subset, because it is Zariski dense in R . We may replace U by a small connected smooth germ there without changing its Zariski closure. Set K=(ξ|_ TR^ ) , and consider the algebraic Grassmann map β:R^(r, h), xξ(T_xR). For a local holomorphic vector field V in K , Cartan's formula and give L_V(ξ|_ R^ ) =ι_Vdξ+d(ι_Vξ)=0. The local flow of V consequently preserves both the restricted connection form and its image subspace; hence dβ(V)=0 . Since TU⊂ K|_U , the map β is constant on U . Its algebraicity and the Zariski density of U make it constant on R^ . Denote its value by b⊆ h . This fixed subspace is a Lie subalgebra. In fact, for b_1,b_2∈ b , locally choose tangent fields V_1,V_2 on R^ with ξ(V_i)=b_i . Such lifts exist because ξ|_ TR^ is a surjection onto the constant bundle b . Evaluating yields ξ([V_1,V_2])=[b_1,b_2], so [b_1,b_2]∈ b . Suppose b≠ h . Let A be the connected analytic subgroup of H with Lie algebra b , and let B be its Zariski closure. Then B is a connected algebraic subgroup and is proper in H . To prove the latter point, if B=H , the fact that A preserves b under its adjoint action would imply that H preserves b : the stabilizer of this subspace is algebraic. Thus b would be an ideal of the semisimple Lie algebra h . Every such ideal is a sum of simple factors and integrates to a closed connected normal algebraic subgroup of H . That subgroup is A , by uniqueness of a connected analytic subgroup with a given Lie algebra. A proper b therefore cannot have Zariski-dense A , a contradiction. Consider the irreducible algebraic subset R_B=R·B⊆ P. It dominates Z , since p(U) is dense, and it is invariant under the right action of B . The equivariance in shows that, at a general smooth point, ξ(T R_B)⊆ B. Indeed, on the image of the multiplication map R^×B→ P , tangent vectors coming from R^ have connection values in Ad(B) b⊆ B , and tangent vectors coming from B have values in B . Generic smoothness of this dominant map onto R_B proves . The B -orbits in a fiber have dimension B . Domination of Z therefore gives R_B≥ Z+ B . Conversely, and ξ= Z give the reverse inequality. Equality holds, and at general smooth points TR_B=ξ^ -1 ( B). In particular the whole horizontal distribution is tangent to R_B . This is an algebraic tangency statement on all of R_B : if a local algebraic horizontal vector field differentiates an equation of R_B , the result vanishes on its dense smooth locus and hence on R_B . Thus the ideal sheaf of R_B is preserved by horizontal vector fields. Their local holomorphic flows preserve R_B in both directions. Choose a nonempty smooth open Z_0⊆Z over which the general fiber of R_B→Z is nonempty of dimension B . Horizontal transport along a path in Z_0 is defined throughout the path: locally it is the fundamental solution of a holomorphic linear system after a faithful representation of H . The preceding ideal invariance shows that it preserves R_B . In particular the monodromy of Z_0 preserves the nonempty closed subset (R_B)_z of the H -torsor P_z . Deleting a proper algebraic subset of a smooth complex variety does not decrease the original monodromy image: the map π_1(Z_0,z)→π_1(Z,z) is surjective, since loops can be perturbed off a subset of real codimension at least two. This monodromy is therefore Zariski dense in H . A closed subset of an H -torsor invariant under a Zariski-dense translation subgroup is invariant under H , and is either empty or the whole torsor. But (R_B)_z is nonempty and has dimension B< H . This contradiction proves b= h . Finally the generic kernel of ξ|_ TR contains TU and hence has dimension at least U . Its generic image has dimension H , so rank--nullity gives.