# ON DIFFERENTIAL INVARIANTS OF HYPERPLANE SYSTEMS ON NONDEGENERATE EQUIVARIANT EMBEDDINGS OF HOMOGENEOUS SPACES

• HONG, JAEHYUN (Department of Mathematical Sciences Seoul National University)
• Received : 2015.03.23
• Published : 2015.06.30
• 72 16

#### Abstract

Given a complex submanifoldM of the projective space $\mathbb{P}$(T), the hyperplane system R on M characterizes the projective embedding of M into $\mathbb{P}$(T) in the following sense: for any two nondegenerate complex submanifolds $M{\subset}\mathbb{P}$(T) and $M^{\prime}{\subset}\mathbb{P}$(T'), there is a projective linear transformation that sends an open subset of M onto an open subset of M' if and only if (M,R) is locally equivalent to (M', R'). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of linear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane systems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.

#### Keywords

homogeneous spaces;fundamental forms

#### References

1. D. N. Akhiezer, Equivariant completions of homogeneous algebraic varieties by homogeneous divisors, Ann. Global Anal. Geom. 1 (1983), no. 1, 49-78. https://doi.org/10.1007/BF02329739
2. D. N. Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, E27. Friedr. Vieweg and Sohn, Braunschweig, 1995.
3. J.-M. Hwang and K. Yamaguchi, Characterization of Hermitian symmetric spaces by fundamental forms, Duke Math. J. 120 (2003), no. 3, 621-634. https://doi.org/10.1215/S0012-7094-03-12035-9
4. J. M. Landsberg, On the infinitesimal rigidity of homogeneous varieties, Compositio Math. 118 (1999), no. 2, 189-201. https://doi.org/10.1023/A:1017161326705
5. J. M. Landsberg, Griffiths-Harris rigidity of compact Hermitian symmetric spaces, J. Differential Geom. 74 (2006), no. 3, 395-405. https://doi.org/10.4310/jdg/1175266232
6. J. M. Landsberg and L. Manivel, Construction and classification of complex simple Lie algebras via projective geometry, Selecta Math. 8 (2002), no. 1, 137-159. https://doi.org/10.1007/s00029-002-8103-5
7. J. M. Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. 78 (2003), no. 1, 65-100. https://doi.org/10.1007/s000140300003
8. J. M. Landsberg and C. Robles, Fubini-Griffiths-Harris rigidity and Lie algebra cohomology, Asian J. Math. 16 (2012), no. 4, 561-586. https://doi.org/10.4310/AJM.2012.v16.n4.a1
9. J. M. Landsberg and C. Robles, Fubini-Griffiths-Harris rigidity of homogeneous varieties, Int. Math. Res. Not. 2013 (2013), no. 7, 1643-1664. https://doi.org/10.1093/imrn/rns016
10. B. Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344 (2009), no. 4, 963-987. https://doi.org/10.1007/s00208-009-0341-9
11. T. Sasaki, K. Yamaguchi, and M. Yoshida, On the rigidity of differential systems modelled on Hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces, Adv. Stud. Pure Math. 25 CR-geometry and overdetermined systems (1997), 318-354.
12. Y. Se-Ashi, On differential invariants of integrable finite type linear differential equations, Hokkaido Math. J. 17 (1988), no. 2, 151-195. https://doi.org/10.14492/hokmj/1381517803