# On Uniformization of Complex Manifolds: The Role of Connections. (MN-22)

ROBERT C. GUNNING
Pages: 150
https://www.jstor.org/stable/j.ctt130hk2c

1. Front Matter
(pp. None)
2. PREFACE
(pp. i-i)
R. C. Gunning
(pp. ii-ii)
4. §1. Introduction
(pp. 1-5)

The general uniformization theorem for Riemann surfaces is one of the most remarkable results in complex analysis, and is at the center of a circle of problems which are still very actively being investigated. An interest in extending this theorem to complex manifolds of higher dimensions has long been manifest, and indeed there have been several extensions of one or another aspect of the general uniformization theorem. As has been observed in other cases, some theorems in classical complex analysis appear as the accidental concurrence in the one-dimensional special case of rather separate phenomena in the general case; so a...

5. Part I: Description of the pseudogroups
• §2. The group of k-jets and its Lie algebra.
(pp. 6-15)

Consider the set of all germs of complex analytic mappings from the origin to the origin in the space ℂnof n complex variables. The k-jet of such a germ f, denoted by jkf, is defined to consist of the terms of order ≤ k in the Taylor expansion of the germ f ; but since all these germs are assumed to take the origin to the origin the conventional usage will be slightly modified in that the constant terms in the Taylor expansion, the terms of order = 0, will not be considered as part of the k-jet. Upon...

• §3. The pseudogroups defined by partial differential equations.
(pp. 16-20)

The definition and classification of the pseudogroups defined by families of partial differential equations are rather straightforward matters once the preceding general machinery has been developed. An analytic family of partial differential equations of order k in the analytic local homeomorphisms from ℂnto ℂncan be thought of merely as being an analytic subvariety A ⊆ Gk(n,ℂ). Of course this is a somewhat restrictive definition, since such families of partial differential equations do not involve the actual values of the mappings but only the derivatives of orders 1 through k of the component functions of the mappings, and the...

• §4. The classification of tangentially transitive pseudogroups: algebraic aspects
(pp. 21-38)

The detailed classification of pseudogroups will only be attempted here for the special case of the tangentially transitive Lie pseudogroups, those for which all the defining groups A ⊆ Gk(n,ℂ) have the property that j1A = G1(n,ℂ); these are the pseudogroups for which there are no restrictions imposed on the values of the Jacobian matrices of the mappings. The classification apparently involves determining all the integrable subgroups A ⊆ Gk(n,ℂ) with j1A = G1(n,ℂ), for all k ≥ 1, and then determining which of these subgroups describe the same pseudogroups; but it is actually a considerably simpler matter than might...

• §5. The classification of tangentially transitive pseudogroups: analytic aspects
(pp. 39-52)

There are the four general classes of possible defining groups for tangentially transitive Lie pseudogroups, described by the four classes of equations (51), (52), (53), and (54) respectively; and within each class the defining groups are parametrized by a linear space of tensors A. It is useful to introduce a notion of equivalence among the defining groups in each class separately; but the definition and elementary properties of this relation are formally almost the same in the different cases, so for convenience will only be discussed in detail for the class given by equation (51). In that case the defining...

6. Part II: Description of the connections
• §6. Pseudogroup structures and their associated connections
(pp. 53-67)

As is of course well known, an m-dimensional topological manifold M is a Hausdorff space each point of which has an open neighborhood homeomorphic to an open subset of ℝm. A coordinate covering [symbol] = {U2} of such a manifold is a covering of M by open subsets Uα⊆ M, for each of which there is a homeomorphism zα: Uα→ Vαbetween Uαand an open subset Vα⊆ ℝm; the sets Uαare called coordinate neighborhoods, and the homeomorphisms zαare called local coordinates. The compositions${{\text{f}}_{\alpha\beta}}={{\text{z}}_{\alpha}}\circ \text{z}_{\beta}^{-1}:{{\text{z}}_{\beta}}({{\text{U}}_{\alpha}}\cap {{\text{U}}_{\beta}})\to {{\text{z}}_{\alpha}}({{\text{U}}_{\alpha}}\cap {{\text{U}}_{\beta}})$are then homeomorphisms between subsets of Vβand...

• §7. Complex analytic affine connections
(pp. 68-78)

To begin the more detailed discussion of some properties and applications of the complex analytic connections associated with the various pseudogroup structures, consider the complex analytic affine connections. As might be expected from the terminology, these are essentially just the complex analytic analogues of the classical affine connections in differential geometry; but there is one point of difference which must be kept in mind. If$\{{{\text{s}}_{\alpha}}\}=\{\text{s}_{\alpha{{\text{j}}_{1}}{{\text{j}}_{2}}}^{\text{i}}\}$is a complex analytic affine connection then recalling (24) and (44) the defining equation (77) can be written out explicitly in the form$\caption {(89)} {{\Sigma }_{\text{k}}}\frac{\partial {{\text{z}}_{\beta\text{i}}}}{\partial {{\text{z}}_{\alpha\text{k}}}}\frac{{{\partial }^{2}}{{\text{z}}_{\alpha\text{k}}}}{\partial {{\text{z}}_{\beta{{\text{j}}_{\text{1}}}}}\partial {{\text{z}}_{\beta{{\text{j}}_{\text{2}}}}}}=\text{s}_{\beta{{\text{j}}_{\text{1}}}{{\text{j}}_{2}}}^{\text{i}}-{{\Sigma }_{\text{k}}}\frac{\partial {{\text{z}}_{\beta\text{i}}}}{\partial {{\text{z}}_{\alpha\text{k}}}}\text{s}_{\alpha{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}}^{\text{k}}\frac{\partial {{\text{z}}_{\alpha{{\text{k}}_{1}}}}}{\partial {{\text{z}}_{\beta{{\text{j}}_{1}}}}}\frac{\partial {{\text{z}}_{\alpha{{\text{k}}_{2}}}}}{\partial {{\text{z}}_{\beta{{\text{j}}_{2}}}}};$and that is the complex analytic analogue of the familiar...

• §8. Complex analytic projective connections.
(pp. 79-94)

Turning next to the complex analytic projective connections, there is a well developed but perhaps not so well known classical theory of projective connections [11]; a particularly readable recent survey of that theory can be found in [28]. Here too the complex analytic projective connections are just the complex analytic analogues of the classical projective connections, indeed more so than in the case of the complex analytic affine connections since for the projective connections symmetry is normally presumed. For the case n = 1 complex analytic projective connections were discussed in [20], and it was shown there that all such...

• §9. Complex analytic canonical connections
(pp. 95-100)

The complex analytic canonical connections are in many ways the simplest of the three classes of connections considered here. As was already essentially noted, a complex analytic canonical connection$\{\text{s}_{\alpha}^{'}\}=\{\text{s}_{\alpha{{\text{j}}_{1}}{{\text{j}}_{2}}}^{{{'}^{\text{i}}}}\}$can be written in the form (116) in each coordinate neighborhood Uαwhere the functions${{\text{x}}_{\alpha\text{j}}}={{\Sigma }_{\text{k}}}\text{s}_{\alpha\text{kj}}^{{{'}^{\text{k}}}}$are complex analytic in Uα; and the condition that$\text{s}_{\alpha}^{'}$is a canonical connection is equivalent to (117). Introducing the complex analytic 1-form φαin the coordinate neighborhood Uαdefined by$\caption {(132)} {{\varphi}_{\alpha}}({{\text{z}}_{\alpha}})={{\Sigma }_{\text{k}}}{{\text{ x}}_{\alpha\text{k}}}({{\text{z}}_{\alpha}})\text{d}{{\text{z}}_{\alpha}},$the condition (117) can be rewritten$\caption {(138)} \text{d}\ \log \,{{\vartriangle }_{\alpha\beta}}={{\varphi}_{\beta}}-{{\varphi}_{\alpha}}\quad \text{in}\ {{\text{U}}_{\alpha}}\cap {{\text{U}}_{\beta}}$where Δαβ= det{∂zαi/∂zβj} ; thus a complex analytic canonical connection can be described equivalently...

7. Part III: Complex analytic surfaces
• §10. Complex flat canonical structures on surfaces
(pp. 101-108)

A discussion of complex analytic pseudogroup structures on one-dimensional compact complex manifolds can be found in [20] ; although there remain several open problems in that case, they mostly have to do with more detailed properties of complex projective structures, and will not be considered here. Turning next to two-dimensional compact complex manifolds, it is convenient to change from the order in which the various pseudogroups have been discussed and to begin with the simplest case, that of the complex flat canonical structures.

If the two-dimensional compact complex manifold M admits a complex flat canonical structure then as already noted...

• §11. Complex affine structures on surfaces.
(pp. 109-122)

Although complex affine structures are more complicated than flat canonical structures, in part because of the nonlinearity of the defining partial differential equations, there is nonetheless a great deal known about such structures. Complex affine structures are of course subordinate to complex flat canonical structures, so that to determine which complex manifolds admit complex flat canonical structures it is only necessary to run through the list of complex manifolds with complex flat canonical structures and see which admit this finer structure. Thus the only compact complex surfaces that can possibly admit complex affine structures are those with Chern classes c1...

• §12. Complex projective structures on surfaces
(pp. 123-136)

Complex projective structures, the general structures in the case of one-dimensional manifolds, have been the least investigated of the structures considered here in the two-dimensional case. Turning now to the discussion of these structures, recall from Theorem 5 that on a two-dimensional manifold which admits a complex analytic projective connection the Chern classes must satisfy$\caption {(152)} {{\text{c}}_{2}}=\frac{1}{3}{{\text{c}}_{1}}^{2}.$

The first stage of the discussion is to determine in general terms which two-dimensional compact complex manifolds M satisfy this topological restriction (152). Unlike the conditions considered previously this topological restriction does not preclude the possibility that the surface M contains exceptional curves of...

8. Bibliography
(pp. 137-142)
9. Back Matter
(pp. 143-143)