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Minimal rational curves on wonderful group compactifications. (Courbes rationnelles minimales sur les compactifications magnifiques des groupes.) (English. French summary) Zbl 1348.14124
Let \(G\) be a semisimple complex algebraic group of adjoint type, of rank \(\ell\), and consider it as a \(G\times G\)-variety, where \(G\times G\) acts on \(G\) by left and right multiplication. The wonderful compactification \(X\) of \(G\) is a smooth projective \(G\times G\)-variety containing \(G\) as open \(G\times G\)-orbit. The complement \(X\setminus G\) is union of \(\ell\) irreducible divisors \(D_1,\dots,D_\ell\) with normal crossings. In total, \(X\) contains \(2^\ell\) nonempty \(G\times G\)-orbits, all of the form \(\bigcap_{i\in I}D_i\setminus\bigcup_{i\not\in I}D_i\) for some \(I\subseteq\{1,\dots,\ell\}\). The wonderful compactification \(X\) of \(G\) is rational, Fano, with Picard number equal to \(\ell\).
The paper under review concerns the study of the space of rational curves on \(X\).
An irreducible component \(\mathcal K\) of the normalization of the space of rational curves of a uniruled projective complex manifold \(X\) is called a family of minimal rational curves on \(X\) if \(\mathcal K_x\), the subfamily through a point \(x\in X\), is nonempty and projective for \(x\) general.
The authors prove that the wonderful compactification \(X\) of \(G\) admits a unique family of minimal rational curves.
They consider a point \(x\) in the open \(G\times G\)-orbit of \(X\), prove that \(\mathcal K_x\) is smooth, isomorphic to the variety of minimal rational tangents at \(x\), and describe it explicitly.
As an application they show that \(X\), if \(G\) is not of type \(A_1\) or \(C\), has the target rigidity property: for any projective variety \(Y\), every deformation of a surjective morphism \(f: X\to Y\) comes from an automorphism of \(X\).
Reviewer: Paolo Bravi (Roma)

MSC:
14M27 Compactifications; symmetric and spherical varieties
14E08 Rationality questions in algebraic geometry
20G15 Linear algebraic groups over arbitrary fields
14L30 Group actions on varieties or schemes (quotients)
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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