A fundamental result in the theory of toric varieties says that the categories of toric varieties (over the complex numbers C ) and fans are equivalent (see Oda [20]). This reduces the classification of toric varieties to that of fans. Among toric varieties, compact smooth toric varieties which we call toric manifolds are well studied and the classification as varieties is completed for some classes of toric manifolds (see Kleinschmidt [11], Oda [20], Sato [22] for example).

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Cohomological rigidity of real Bott manifolds

A real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. 
A real Bott manifold is a real toric manifold and admits a flat riemannian metric invariant under the natural action of an elementary abelian 2-group. We also prove that the converse is true, namely a real toric manifold which admits a flat riemannian metric invariant under the action of an elementary abelian 2-group is a real Bott manifold.

A1-homotopy groups, excision, and solvable quotients

We study some properties of A^1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A^1-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory. 
As a case study in the geometry of solvable group quotients, we investigate A^1-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the "next" non-vanishing A^1-homotopy group (beyond \pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost "as tractable" (in low degrees) as ordinary homotopy for large classes of interesting varieties.

A^1-homotopy groups, excision, and solvable quotients

On the A^1-fundamental groups of smooth toric varieties

LetX (Δ) be the real toric variety associated to a smooth fan Δ. The main purpose of this article is: (i) to determine the fundamental group and the universal cover ofX (Δ), (ii) to give necessary and sufficient conditions on Δ under which π1(X(Δ)) is abelian, (iii) to give necessary and sufficient conditions on Δ under whichX(Δ) is aspherical, and when Δ is complete, (iv) to give necessary and sufficient conditions forCΔ to be aK (π, 1) space whereCΔ is the complement of a real subspace arrangement associated to Δ.

On the fundamental group of real toric varieties

Let $X(\Delta)$ be the real toric variety associated to a smooth fan $\Delta$. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of $X(\Delta)$, (ii) to give necessary and sufficient conditions on $\Delta$ under which $\pi_1(X(\Delta))$ is abelian, (iii) to give necessary and sufficient conditions on $\Delta$ under which $X(\Delta)$ is aspherical, and when $\Delta$ is complete, (iv) to give necessary and sufficient conditions for $\cc_{\Delta}$ to be a $K(\pi,1)$ space where $\cc_{\Delta}$ is the complement of a real subspace arrangement associated to $\Delta$.

V. Uma

Papers

On the fundamental group of real toric varieties

On the fundamental group of real toric varieties