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I've recently been learning some complex geometry, mostly for my own edification. In the course of my studies I came across the following statement:

If $X$ is a Riemann surface then a choice of complex structure is equivalent to choice of conformal structure.

That is, complex structures and conformal structures on a Riemann surface are in bijection.

I haven't seen an analog of this statement for higher dimensional manifolds. Given that the intuition that one develops in 1-complex variable falls apart alarmingly fast in several complex variables, I suspect that complex and conformal structures are not equivalent in higher dimensions.

My question is as follows. Are complex structures and conformal structures inequivalent in dimension >1? If so how can I understand this (why aren't they)? Furthermore, does the correspondence fall apart in any controlled way. For instance does conformal structure imply a complex structure but that the converse no longer holds? Or are conformal and complex structures simply incomparable in higher dimensions?

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First, note that "conformal structure" makes sense for any number of real dimensions, while "complex structure" only makes sense for even-dimensional manifolds.

Conformal and complex structures are very different in (real, even) dimensions greater than two. Just think about the structures that you get on the tangent spaces:

  • The tangent space to a $2n$-dimensional manifold with a complex structure is an $n$-dimensional complex vector space.

  • The tangent space to a $2n$-dimensional manifold with a conformal structure is a $2n$-dimensional real vector space equipped with a notion of angle (i.e. an inner product up to scaling).

These are very different structures. The $n$-dimensional complex vector space has a special collection of planes, namely the $1$-dimensional complex subspaces. Each nonzero vector is contained in a unique such plane, and you can only measure the angle between two vectors that lie in the same plane. The $2n$-dimensional conformal vector space has no special "planes", and you can measure the angle between any two nonzero vectors.

Thus a conformal structure does not give you a complex structure, and a complex structure does not give you a conformal structure.

At the level of groups, the structure group of the complex manifold is $\mathrm{GL}_n(\mathbb{C})$, while the structure group of the conformal manifold is $\mathrm{O}(2n)\times\mathbb{R}_+$. Neither of these groups is contained in the other.

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