example of toric varieties with nontrivial first cohomology group

Question

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even dimensional cohomology group can be found in terms of h-polynomial. What about the case when the polytope is not simplicial rational? Are there any examples where the $H^1(X,\mathbb{Q})$ is nontrivial?

Answer 1

• In general, the first cohomology group over $\mathbb{Q}$ is given by$$H^1(X, \mathbb{Q}) = \text{Hom}(\pi_1(X), \mathbb{Q}),$$where $\pi_1(X)$ is the fundamental group. The toric variety $X$ of a complete fan has trivial fundamental group, so$$\pi_1(X) = H^1(X, \mathbb{Q}) = 0.$$
• As I follow-up, I think that the higher odd cohomology can be nonzero when the toric variety of a polytope is not simplicial. It is possible that this reference here might have something useful.
  • A. Jordan. Homology and Cohomology of Toric Varieties, Konstanzer Schriften in Mathematik und Informatik Nr. 57, Februar 1998.
• We can have odd homology when the polytope is not smooth, the first example of this was found by one of Fulton's students (McConnell, I think) and is mentioned in Cox/Little/Schenck's Toric Varieties, probably in Chapter 10.

• You are correct that the odd-dimensional cohomology groups are zero for a smooth or a simplicial toric variety. In other cases, some odd-dimensional cohomology can be nonzero, but the specific group you asked about, namely $H^1(X, \mathbb{Q})$, is actually never anything but $0$ for compact toric varieties (it can be nonzero for noncompact toric varieties corresponding to noncomplete fans). The best way I know to prove that is to "dig into" the properties of the method of computing the cohomology via spectral sequences. This was developed by Fischli and Jordan, and the basics are presented in Chapter 12 of Cox/Little/Schenck's Toric Varieties. But Jordan's thesis presents more details and it is actually pretty readable. Reading it might be a good way to learn about spectral sequences and using the spectral sequence from the filtration by torus orbits if you have time and are interested. The thesis was never published, but it is online here. The specific result that shows $H^1(X, \mathbb{Q})$ is always zero is the statement in part (c) of Corollary 2.4.9 about the $E_2^{1, 0}$ term in the spectral sequence.

  • A. Jordan. Homology and Cohomology of Toric Varieties, Konstanzer Schriften in Mathematik und Informatik Nr. 57, Februar 1998.