We introduce an additive basis of the integral cohomology ring of the Peterson variety which reflects the geometry of certain subvarieties of the Peterson variety. We explain the positivity of the structure constants from a geometric viewpoint, and provide a manifestly positive combinatorial formula for them. We also prove that our basis coincides with the additive basis introduced by Harada-Tymoczko.
A complete nonsingular toric variety (called a toric manifold) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\operatorname{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda’s three-fold, the simplest non-projective toric manifold, is over $\operatorname{vc}(I^3)$. In this paper, we classify toric manifolds over $\operatorname{vc}(I^n)$$(n\ge 3)$ as varieties and as smooth manifolds. It consequently turns out that there are many non-projective toric manifolds over $\operatorname{vc}(I^n)$ but they are all diffeomorphic, and toric manifolds over $\operatorname{vc}(I^n)$ in some class are determined by their cohomology rings as varieties.
We study torsion in the integral cohomology of a certain family of 2n-dimensional orbifolds X with actions of the n-dimensional compact torus. Compact simplicial toric varieties are in our family. For a prime number p, we find a necessary condition for the integral cohomology of X to have no p-torsion. Then we prove that the necessary condition is sufficient in some cases. We also give an example of X which shows that the necessary condition is not sufficient in general.
We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over P if its quotient by the compact torus is homeomorphic to P as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds over an n-cube I n and blowing them up at a fixed point produces toric manifolds over vc(I n ) an n-cube with one vertex cut. They are all projective. On the other hand, Oda's 3-fold, the simplest non-projective toric manifold, is over vc(I 3 ). In this paper, we classify toric manifolds over vc(I n ) (n ≥ 3) as varieties and also as smooth manifolds. As a consequence, it turns out that (1) there are many non-projective toric manifolds over vc(I n ) but they are all diffeomorphic, and (2) toric manifolds over vc(I n ) in some class are determined by their cohomology rings as varieties among toric manifolds.
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