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Michael
Michael
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Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A_k^n$, and that variety inherits the structure of $k$-manifold.

However, the above is not necessarily true when $k=\Bbb H$ is the skew-field of quaternions: the variety doesn't have to have $dim_{\Bbb R}$ divisible by $dim_{\Bbb R}\Bbb H$, let alone a quaternionic structure.

Therefore the question: what properties of (ring, skew-field, ...) $k$ cause a $\Bbb A_k$-variety have a natural manifold structure associate with $k$, whatever it is?

EDIT: based on the comment below I'd like to emphasize that I'm interested in the properties of $k$ that make the phrase "$k$-variety is a $k$-manifold" almost work, modulo singularities and such, or work for "nice enough", i.e. regular varieties.

Here's an example of a hunch: it's possible that the breakdown of "$k$-variety is a $k$-manifold" in the case of quaternions is due to the fact that $\Bbb H$ has "too many" automorphisms. Something along these lines: $k$ needs to have this and that property for its structure to be inherited from $\Bbb A_k$ to $T_p(V)$ for most points $p$ on a generic variety $V$.

Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A_k^n$, and that variety inherits the structure of $k$-manifold.

However, the above is not necessarily true when $k=\Bbb H$ is the skew-field of quaternions: the variety doesn't have to have $dim_{\Bbb R}$ divisible by $dim_{\Bbb R}\Bbb H$, let alone a quaternionic structure.

Therefore the question: what properties of (ring, skew-field, ...) $k$ cause a $\Bbb A_k$-variety have a natural manifold structure associate with $k$, whatever it is?

Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A_k^n$, and that variety inherits the structure of $k$-manifold.

However, the above is not necessarily true when $k=\Bbb H$ is the skew-field of quaternions: the variety doesn't have to have $dim_{\Bbb R}$ divisible by $dim_{\Bbb R}\Bbb H$, let alone a quaternionic structure.

Therefore the question: what properties of (ring, skew-field, ...) $k$ cause a $\Bbb A_k$-variety have a natural manifold structure associate with $k$, whatever it is?

EDIT: based on the comment below I'd like to emphasize that I'm interested in the properties of $k$ that make the phrase "$k$-variety is a $k$-manifold" almost work, modulo singularities and such, or work for "nice enough", i.e. regular varieties.

Here's an example of a hunch: it's possible that the breakdown of "$k$-variety is a $k$-manifold" in the case of quaternions is due to the fact that $\Bbb H$ has "too many" automorphisms. Something along these lines: $k$ needs to have this and that property for its structure to be inherited from $\Bbb A_k$ to $T_p(V)$ for most points $p$ on a generic variety $V$.

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Michael
Michael
  • 2.2k
  • 34
  • 42

What turns $k$-variety into $k$-manifold?

Here's what I have in mind: for $k=\Bbb C$ algebraic equations with $k$-coefficients lead to a $k$-variety in $\Bbb A_k^n$, and that variety inherits the structure of $k$-manifold.

However, the above is not necessarily true when $k=\Bbb H$ is the skew-field of quaternions: the variety doesn't have to have $dim_{\Bbb R}$ divisible by $dim_{\Bbb R}\Bbb H$, let alone a quaternionic structure.

Therefore the question: what properties of (ring, skew-field, ...) $k$ cause a $\Bbb A_k$-variety have a natural manifold structure associate with $k$, whatever it is?