[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.j140263

DEFORMATION RIGIDITY OF ODD LAGRANGIAN GRASSMANNIANS

Park, Kyeong-Dong (School of Mathematics Korea Institute for Advanced Study)

Publication Information

Journal of the Korean Mathematical Society / v.53, no.3, 2016 , pp. 489-501 More about this Journal

Abstract

In this paper, we study the rigidity under $K{\ddot{a}}hler$ deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case $Gr_{\omega}$ (n, 2n+1) of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.

Keywords

odd Lagrangian Grassmannian; deformation rigidity; variety of minimal rational tangents; prolongation of a linear Lie algebra;

Citations & Related Records

Reference

1	D. N. Akhiezer, Lie Group Actions in Complex Analysis, Vieweg+Teubner Verlag, Wiesbaden, 1995.
2	B. Fu and J.-M. Hwang, Classification of non-degenerate projective varieties with nonzero prolongation and application to target rigidity, Invent. Math. 189 (2012), no. 2, 457-513. DOI
3	J.-M. Hwang, Geometry of minimal rational curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), 335-393, ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001.
4	J.-M. Hwang, Deformation of the space of lines on the 5-dimensional hyperquadric, preprint, 2010.
5	J.-M. Hwang and N. Mok, Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kahler deformation, Invent. Math. 131 (1998), no. 2, 393-418. DOI
6	J.-M. Hwang and N. Mok, Varieties of minimal rational tangents on uniruled projective manifolds, Several complex variables (Berkeley, CA, 1995-1996), 351-389, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999.
7	J.-M. Hwang and N. Mok, Deformation rigidity of the rational homogeneous space associated to a long simple root, Ann. Sci. Ecole Norm. Sup. (4) 35 (2002), no. 2, 173-184. DOI
8	J.-M. Hwang and N. Mok, Deformation rigidity of the 20-dimensional $F_4$ -homogeneous space associated to a short root, Algebraic transformation groups and algebraic varieties, 37-58, Encyclopaedia Math. Sci., 132, Springer, Berlin, 2004.
9	J.-M. Hwang and N. Mok, Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under Kahler deformation, Invent. Math. 160 (2005), no. 3, 591-645. DOI
10	S. Kebekus, Families of singular rational curves, J. Algebraic Geom. 11 (2002), no. 2, 245-256. DOI
11	K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Grundlehren der mathematischen Wissenschaften 283, Springer, Berlin-Heidelberg-New York, 1986.
12	K. Kodaira and J. Morrow, Complex Manifolds, Holt, Rinehart and Winston, Inc., New York, 1971.
13	J. M. Landsberg and L. Manivel, On the projective geometry of rational homogeneous varieties, Comment. Math. Helv. 78 (2003), no. 1, 65-100. DOI
14	I. A. Mihai, Odd symplectic flag manifolds, Transform. Groups 12 (2007), no. 3, 573-599. DOI
15	Y. Miyaoka and S. Mori, A numerical criterion for uniruledness, Ann. of Math. 124 (1986), no. 1, 65-69. DOI
16	S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 110 (1979), no. 3, 593-606. DOI
17	S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. USA 86 (1989), no. 9, 3000-3002. DOI
18	B. Pasquier and N. Perrin, Local rigidity of quasi-regular varieties, Math. Z. 265 (2010), no. 3, 589-600. DOI
19	K.-D. Park, Varieties of minimal rational tangents on Fano manifolds, PhD thesis, Seoul National University, 2014.
20	B. Pasquier, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann. 344 (2009), no. 4, 963-987. DOI
21	F. L. Zak, Tangents and secants of algebraic varieties, Translations of Mathematical Monographs, Vol. 127, Amer. Math. Soc., Providence, Rhode Island, 1993.