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Let $X$ be a complex toric variety and let $X_\mathbb R$ be its real part, that is the $X_\mathbb R$ consists of all the real valued points of $X$.

I would like to learn a little more about the real part and would appreciate some references.

Specifically I was looking at the following theorem (Theorem 4.1.3 of this book)-

Theorem : We have the exact sequence $$M\to \bigoplus_{\rho\in\Delta(1)}\mathbb Z D_\rho\to\text{ Cl }(X)\to 0$$ where the first map is $m\to\text{ div }(\chi^m)$ and the second sends a $T_N$ - invariant divisor to its divisor class in $\text{ Cl }(X)$. Furthermore, we have a short exact sequence $$0\to M\to\bigoplus_{\rho\in\Delta(1)}\mathbb Z D_\rho\to\text{ Cl }(X)\to 0$$ iff $\{ u_\rho\ |\ \rho\in\Delta(1)\}$ spans $N\otimes_\mathbb Z \mathbb R$.

Where $N$ is a lattice, $M$ is its dual, $T_N=\text{ Hom}_\mathbb Z(M,\mathbb C^*)$ is the torus, $\Delta$ is a fan in $N$, $\Delta(1)$ is the set of one dimensional cones (edges) in $\Delta$, $u_\rho$ is the primitive vector along the edge $\rho$, $D_\rho$ is the $T_N$ - invariant prime divisor given by the closure of the orbit corresponding to $\rho$, and $X$ is the toric variety given by the fan $\Delta$.

Now my question is whether I can prove this theorem for $X_\mathbb R$? For this I need a proper understanding of what prime divisors and the divisor class etc are. For example are prime divisors in $X_\mathbb R$ the real points of prime divisors of $X$ or are there other prime divisors of $X_\mathbb R$? I haven't been very successful in finding reading material online for this (real part of a toric variety) and would appreciate some suggestions. Fulton's book has very little.

Thank you.

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  • $\begingroup$ A complex variety doesn't have a notion of real points. You need a real variety for that. Divisors on it are Galois-invariant divisors on the complexification. $\endgroup$
    – Qiaochu Yuan
    Apr 28, 2016 at 17:04
  • $\begingroup$ There is a notion of the real part of a toric variety. So this notion can't be defined for a variety over $\mathbb C$? $\endgroup$
    – R_D
    Apr 28, 2016 at 17:14
  • $\begingroup$ Toric varieties have a lot more structure than just the structure of a complex variety. $\endgroup$
    – Qiaochu Yuan
    Apr 28, 2016 at 18:34
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    $\begingroup$ (contd.) In other words, you may want to ask yourself whether you are interested in such real toric varieties as the circle $\{X^2+Y^2=Z^2\}\subseteq\mathbb{P}^2$ or the pointless conic $\{X^2+Y^2+Z^2=0\}\subseteq\mathbb{P}^2$, which are defined by lattices and fans with a nontrivial Galois action. If not, the theory of "split" real toric varieties should essentially mimic the complex case. But if yes, you need to take into account Galois descent/cohomology. Duncan's Twisted Forms of Toric Varieties seems like a nice read. $\endgroup$
    – Gro-Tsen
    May 1, 2016 at 7:05
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    $\begingroup$ Yes, I think in the split case the theorem you state will remain true over any field (perhaps "perfect field"? but certainly over the reals). And a cursory reading of the proof in Cox Little & Schenck suggests that even the proof carries over (I don't see where they might have used algebraic closedness, but maybe I missed something). $\endgroup$
    – Gro-Tsen
    May 1, 2016 at 7:36

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