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stankewicz
stankewicz
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There are a number of things floating around here.

First among them is the first excellent point that Marino made that the finite generation of group of rational points of an abelian variety over a field K is only true for global fields. So let's say we're working over $\mathbf{C}$, where any positive dimensional variety has uncountably many points.

Second is the other excellent point of Marino that $\mathbf{C}^\times$ is not compact, so it can't fit with the definition of an abelian variety as a complete, connected group variety.

Third, it's much stronger to say that an abelian variety over the complex numbers is $\mathbf{C}^n/\Lambda$ topologically than group-theoretically. But in fact much more is true. Analytically, an abelian variety is isomorphic to $\mathbf{C}^n/\Lambda$. This comes from showing that the exponential map from the tangent space at the identity is in fact surjective, followed by figuring out the kernel. Details on this can be found in Milne's notes on abelian varieties or the first chapter of Mumford's book. In fact, even if we relax down to $C^\infty$ isomorphisms (let alone homeomorphisms) we could say that an abelian variety is isomorphic to $\mathbf{C}^n/\mathbf{Z}^{2n}$.

There are a number of things floating around here.

First among them is the first excellent point that Marino made that the group of rational points of an abelian variety over a field K is only true for global fields. So let's say we're working over $\mathbf{C}$, where any positive dimensional variety has uncountably many points.

Second is the other excellent point of Marino that $\mathbf{C}^\times$ is not compact, so it can't fit with the definition of an abelian variety as a complete, connected group variety.

Third, it's much stronger to say that an abelian variety over the complex numbers is $\mathbf{C}^n/\Lambda$ topologically than group-theoretically. But in fact much more is true. Analytically, an abelian variety is isomorphic to $\mathbf{C}^n/\Lambda$. This comes from showing that the exponential map from the tangent space at the identity is in fact surjective, followed by figuring out the kernel. Details on this can be found in Milne's notes on abelian varieties or the first chapter of Mumford's book. In fact, even if we relax down to $C^\infty$ isomorphisms (let alone homeomorphisms) we could say that an abelian variety is isomorphic to $\mathbf{C}^n/\mathbf{Z}^{2n}$.

There are a number of things floating around here.

First among them is the first excellent point that Marino made that the finite generation of group of rational points of an abelian variety over a field K is only true for global fields. So let's say we're working over $\mathbf{C}$, where any positive dimensional variety has uncountably many points.

Second is the other excellent point of Marino that $\mathbf{C}^\times$ is not compact, so it can't fit with the definition of an abelian variety as a complete, connected group variety.

Third, it's much stronger to say that an abelian variety over the complex numbers is $\mathbf{C}^n/\Lambda$ topologically than group-theoretically. But in fact much more is true. Analytically, an abelian variety is isomorphic to $\mathbf{C}^n/\Lambda$. This comes from showing that the exponential map from the tangent space at the identity is in fact surjective, followed by figuring out the kernel. Details on this can be found in Milne's notes on abelian varieties or the first chapter of Mumford's book. In fact, even if we relax down to $C^\infty$ isomorphisms (let alone homeomorphisms) we could say that an abelian variety is isomorphic to $\mathbf{C}^n/\mathbf{Z}^{2n}$.

Source Link
stankewicz
stankewicz
  • 3.6k
  • 26
  • 39

There are a number of things floating around here.

First among them is the first excellent point that Marino made that the group of rational points of an abelian variety over a field K is only true for global fields. So let's say we're working over $\mathbf{C}$, where any positive dimensional variety has uncountably many points.

Second is the other excellent point of Marino that $\mathbf{C}^\times$ is not compact, so it can't fit with the definition of an abelian variety as a complete, connected group variety.

Third, it's much stronger to say that an abelian variety over the complex numbers is $\mathbf{C}^n/\Lambda$ topologically than group-theoretically. But in fact much more is true. Analytically, an abelian variety is isomorphic to $\mathbf{C}^n/\Lambda$. This comes from showing that the exponential map from the tangent space at the identity is in fact surjective, followed by figuring out the kernel. Details on this can be found in Milne's notes on abelian varieties or the first chapter of Mumford's book. In fact, even if we relax down to $C^\infty$ isomorphisms (let alone homeomorphisms) we could say that an abelian variety is isomorphic to $\mathbf{C}^n/\mathbf{Z}^{2n}$.