Minimal rational curves on wonderful group compactifications

Brion, Michel; Fu, Baohua

Journal de l’École polytechnique - Mathématiques, Tome 2 (2015), pp. 153-170.

Soient $G$ un groupe algébrique simple et $X$ sa compactification magnifique. Nous montrons que $X$ possède une unique famille de courbes rationnelles minimales, et nous décrivons explicitement la sous-famille formée des courbes passant par un point général. Nous en déduisons une propriété de rigidité de $X$, lorsque $G$ n’est pas de type $A_1$ ou $C$.

Consider a simple algebraic group $G$ of adjoint type, and its wonderful compactification $X$. We show that $X$ admits a unique family of minimal rational curves, and we explicitly describe the subfamily consisting of curves through a general point. As an application, we show that $X$ has the target rigidity property when $G$ is not of type $A_1$ or $C$.

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