On a smooth compactification of PSL(n,C)/T

Abstract

Let $T$ be a maximal torus of $PSL(n,\mathbb{C})$. For $n\ge 4$, we construct a smooth compactification of $PSL(n,\mathbb{C})/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{PSL(n,\mathbb{C})}$ for a suitable choice of $T$-linearized ample line bundle on $\overline{PSL(n,\mathbb{C})}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of $PSL(n,\mathbb{C})/T$ is $PSL(n,\mathbb{C})$ itself.