Abstract
1-Flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhöfer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.
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References
Akhiezer, D.: Lie Group Actions in Complex Analysis. Vieweg Verlag, Braunschweig/Wiesbaden (1995)
Brion, M., Fu, B.: Minimal rational curves on wonderful group compactifications. J. Ec. Polytech. Math. 2, 153–170 (2015)
Bryant, R.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)
Buczynski, J.: Legendrian subvarieties of projective space. Geom. Dedic. 118, 87–103 (2006)
Fu, B., Hwang, J.-M.: Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity. Invent. Math. 189, 457–513 (2012)
Fu, B., Hwang, J.-M.: Isotrivial VMRT-structures of complete intersection type. Asian J. Math. 22, 333–354 (2018)
Grauert, H., Peternell, T., Remmert, R.: Several Complex Variables VII (Encyclopedia of Mathematics Science), vol. 74. Springer, Berlin (1994)
Humphreys, J.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. Springer, New York (1972)
Hwang, J.-M.: Geometry of minimal rational curves on Fano manifolds. School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 335–393, ICTP Lecture Notes, 6, Abdus Salam Int. Cent. Theoret. Phys. Trieste (2001)
Hwang, J.-M.: Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents. Ann. Scient. Ec. Norm. Sup. 43, 607–620 (2010)
Hwang, J.-M.: Geometry of varieties of minimal rational tangents. In: Current Developments in Algebraic Geometry, Mathematical Sciences Research Institute Publications, vol. 59, pp. 197–226. Cambridge University Press (2012)
Hwang, J.-M.: Varieties of minimal rational tangents of codimension 1. Ann. Scient. Ec. Norm. Sup. 46, 629–649 (2013)
Hwang, J.-M.: Legendrian cone structures and contact prolongations. In: Geometry, Lie Theory and Applications - The Abel symposium 2019, vol. 16, pp. 131–145 (2022)
Hwang, J.-M., Li, Q.: Characterizing symplectic Grassmannians by varieties of minimal rational tangents. J. Differ. Geom. 119, 309–381 (2021)
Hwang, J.-M., Li, Q.: Unbendable rational curves of Goursat type and Cartan type. J. Math. Pures Appl. 155, 1–31 (2021)
Hwang, J.-M., Mok, N.: Varieties of minimal rational tangents on uniruled projective manifolds. In: Several complex variables (Berkeley, CA, 1995–1996), pp. 351–389, Math. Sci. Res. Inst. Publ., 37, Cambridge University Press, Cambridge (1999)
Hwang, J.-M., Mok, N.: Automorphism groups of spaces of minimal rational curves on Fano manifolds of Picard number 1. J. Algebraic Geom. 13, 663–673 (2004)
Hwang, J.-M., Mok, N.: Birationality of the tangent map for minimal rational curves. Asian J. Math. 8, 51–63 (2004)
Hwang, J.-M., Neusser, K.: Cone structures and parabolic geometries. to appear in Math. Annalen. arXiv:2010.14958
Kebekus, S.: Families of singular rational curves. J. Alg. Geom. 11, 245–256 (2002)
Landsberg, J.M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78, 65–100 (2003)
Landsberg, J.M., Manivel, L.: Legendrian varieties. Asian J. Math. 11, 341–360 (2007)
Merkulov, S., Schwachhöfer, L.: Classification of irreducible holonomies of torsion-free affine connections. Annals Math. 150, 77–149 (1999)
Mok, N.: Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. In: Third International Congress of Chinese Mathematicians. Part 1, 2, 41-61, AMS/IP Stud. Adv. Math., 42, pt. 1, 2, American Mathematical Society, Providence (2008)
Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. Adv. Study Pure Math. 22, 413–494 (1993)
Zak, F.L.: Tangents and secants of algebraic varieties. Transl. Math. Monographs 127, American Mathematical Society, Providence (1993)
Acknowledgements
We would like to thank Lorenz Schwachhöfer for helpful discussion on Theorem 1.2 and the referee for helpful suggestions to improve the presentation of the paper.
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Dedicated to the memory of Nessim Sibony.
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This work was supported by the Institute for Basic Science (IBS-R032-D1).
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Hwang, JM., Li, Q. Minimal Rational Curves and 1-Flat Irreducible G-Structures. J Geom Anal 32, 179 (2022). https://doi.org/10.1007/s12220-022-00901-7
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DOI: https://doi.org/10.1007/s12220-022-00901-7