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Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity. (English) Zbl 1260.14050
Given a complex vector space \(V\) and a Lie subalgebra \(\mathfrak g \subset \mathfrak{gl}(V)\) one can study the Lie algebra \(\mathfrak g\) via its \(k\)-th prolongations \(\mathfrak g^{(k)}\). By the classical work of E. Cartan, S. Kobayashi and T. Nagano we know that if \(\mathfrak g^{(2)} \neq 0\), then \(\mathfrak g = \mathfrak{gl}(V), \mathfrak{sl}(V), \mathfrak{sp}(V)\) or \(\mathfrak{csp}(V)\). Moreover if \(\mathfrak g^{(2)} = 0\), but \(\mathfrak g^{(1)} \neq 0\), then \(\mathfrak g \subset \mathfrak{gl}(V)\) is isomorphic to the isotropy representation on the tangent space at a base point of an irreducible Hermitian symmetric space of compact type which is not the projective space.
In their work on deformation rigidity of rational homogeneous spaces [Invent. Math. 160, No. 3, 591–645 (2005; Zbl 1071.32022)], J.-M. Hwang and N. Mok studied the prolongations of \(\mathfrak g \subset \mathfrak{gl}(V)\) associated to a projective variety \(S \subset \mathbb P(V)\). More precisely, given a projective variety \(S \subset \mathbb P(V)\) they consider the Lie algebra \(\mathfrak{aut}(\hat S) \subset \mathfrak{gl}(V)\) of infinitesimal linear automorphisms of the affine cone \(\hat S\) and studied the relation between the prolongations \(\mathfrak{aut}(\hat S)^{(k)}\) and the geometry of \(S\) and its rational curves. In particular they showed that if \(S \subset \mathbb P(V)\) is an irreducible nonsingular non-degenerate projective variety such that \(\mathfrak{aut}(\hat S)^{(2)} \neq 0\), then \(S\) is the projective space \(\mathbb P(V)\) itself.
In the paper under review, the authors complete the program initiated by Hwang and Mok by giving a classification of irreducible nonsingular non-degenerate projective varieties \(S \subset \mathbb P(V)\) such that \(\mathfrak{aut}(\hat S)^{(1)} \neq 0\). As in the earlier work on this topic the variety of minimal tangents VMRT associated to a covering family of minimal rational curves plays a crucial role. In fact the authors prove that if we have \(S \subset \mathbb P(V)\) as above, the VMRT \(S' \subset \mathbb P(V')\) is irreducible, nonsingular, non-degenerate and \(\mathfrak{aut}(\hat S')^{(1)} \neq 0\). The proof is then a (quite non-trivial) induction on the dimension, we refer to the paper for a detailed description of the numerous difficulties arising in the induction process.
The authors give an application of their main theorem to the problem of target rigidity introduced in the work J.-M. Hwang, S. Kebekus and T. Peternell [J. Algebr. Geom. 15, No. 3, 551–561 (2006; Zbl 1112.14014)] and studied by the second-named author in [Ann. Sci. Éc. Norm. Supér. (4) 40, No. 1, 179–189 (2007; Zbl 1124.32008); J. Reine Angew. Math. 637, 193–205 (2009; Zbl 1187.14050)]: Let \(S \subset \mathbb P(V)\) be an irreducible nonsingular non-degenerate linearly normal variety such that the secant variety \(Sec (S)\) is not the whole projective space \(\mathbb P(V)\), then any deformation \(f_t: Y \rightarrow Bl_S \mathbb P(V)\) of a surjective morphism \(f_0: Y \rightarrow Bl_S \mathbb P(V)\) comes from an automorphism of the blowup \(Bl_S \mathbb P(V)\).

14J40 \(n\)-folds (\(n>4\))
14M17 Homogeneous spaces and generalizations
14N05 Projective techniques in algebraic geometry
Full Text: DOI arXiv
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