×

zbMATH — the first resource for mathematics

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 \(Bl_S \mathbb P(V)\).

MSC:
14J40 \(n\)-folds (\(n>4\))
14M17 Homogeneous spaces and generalizations
14N05 Projective techniques in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Araujo, C.: Rational curves of minimal degree and characterization of projective spaces. Math. Ann. 335, 937–951 (2006) · Zbl 1109.14032 · doi:10.1007/s00208-006-0775-2
[2] Arnold, V.I., Ilyashenko, Yu.S.: Ordinary differential equations. In: Encyclopaedia Math. Sci. Dynamical Systems, I, vol. 1, pp. 1–148. Springer, Berlin (1988)
[3] Fu, B.: Inductive characterizations of hyperquadrics. Math. Ann. 340, 185–194 (2008) · Zbl 1131.14056 · doi:10.1007/s00208-007-0143-x
[4] Fulton, W., Harris, J.: Representation Theory. A First Course. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991) · Zbl 0744.22001
[5] Guillemin, V.: The integrability problem for G-structures. Trans. Am. Math. Soc. 116, 544–560 (1965) · Zbl 0178.55702
[6] Hwang, J.-M.: Geometry of minimal rational curves on Fano manifolds. ICTP Lect. Notes 6, 335–393 (2001) · Zbl 1086.14506
[7] Hwang, J.-M.: Deformation of holomorphic maps onto the blow-up of the projective plane. Ann. Sci. Éc. Norm. Super. (4) 40, 179–189 (2007) · Zbl 1124.32008
[8] Hwang, J.-M.: Unobstructedness of deformations of holomorphic maps onto Fano manifolds of Picard number 1. J. Reine Angew. Math. 637, 193–205 (2009) · Zbl 1187.14050
[9] Hwang, J.-M.: Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents. Ann. Sci. Éc. Norm. Super. (4) 43, 607–620 (2010) · Zbl 1210.14044
[10] Hwang, J.-M., Kebekus, S.: Geometry of chains of minimal rational curves. J. Reine Angew. Math. 584, 173–194 (2005) · Zbl 1084.14040 · doi:10.1515/crll.2005.2005.584.173
[11] Hwang, J.-M., Mok, N.: Uniruled projective manifolds with irreducible reductive G-structures. J. Reine Angew. Math. 490, 55–64 (1997) · Zbl 0882.22007
[12] Hwang, J.-M., Mok, N.: Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds. Invent. Math. 136, 209–231 (1999) · Zbl 0963.32007 · doi:10.1007/s002220050308
[13] Hwang, J.-M., Mok, N.: Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1. J. Math. Pures Appl. 80, 563–575 (2001) · Zbl 1033.32013
[14] Hwang, J.-M., Mok, N.: Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kähler deformation. Invent. Math. 160, 591–645 (2005) · Zbl 1071.32022 · doi:10.1007/s00222-004-0417-9
[15] Hwang, J.-M., Kebekus, S., Peternell, T.: Holomorphic maps onto varieties of non-negative Kodaira dimension. J. Algebr. Geom. 15, 551–561 (2006) · Zbl 1112.14014 · doi:10.1090/S1056-3911-05-00411-X
[16] Ionescu, P., Russo, F.: Conic-connected manifolds. J. Reine Angew. Math. 644, 145–157 (2010) · Zbl 1200.14078
[17] Kobayashi, S., Nagano, T.: On filtered Lie algebras and geometric structures I. J. Math. Mech. 13, 875–907 (1964) · Zbl 0142.19504
[18] Ottaviani, G.: Spinor bundles on quadrics. Trans. Am. Math. Soc. 307, 301–316 (1988) · Zbl 0657.14006 · doi:10.1090/S0002-9947-1988-0936818-5
[19] Pasquier, B.: On some smooth projective two-orbit varieties with Picard number 1. Math. Ann. 344, 963–987 (2009) · Zbl 1173.14028 · doi:10.1007/s00208-009-0341-9
[20] Peternell, T., Schneider, M.: Compactifications of \(\mathbb{C}\) n : a survey. In: Several Complex Variables and Complex Geometry, Part 2, Santa Cruz, CA, 1989. Proc. Sympos. Pure Math., vol. 52, pp. 455–466. Am. Math. Soc, Providence (1991) · Zbl 0745.32012
[21] Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Adv. Study Pure Math. 22, 413–494 (1993) · Zbl 0812.17018
[22] Zak, F.L.: Tangents and Secants of Algebraic Varieties. Translations of Mathematical Monographs, vol. 127. Am. Math. Soc., Providence (1993) · Zbl 0795.14018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.