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Characterization of Hermitian symmetric spaces by fundamental forms
 621–634. LIE ALGEBRA COHOMOLOGY, AND RIGIDITY 27
"... We show that an equivariantly embedded Hermitian symmetric space in a projective space which contains neither a projective space nor a hyperquadric as a component is characterized by its fundamental forms as a local submanifold of the projective space. Using some invarianttheoretic properties of th ..."
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We show that an equivariantly embedded Hermitian symmetric space in a projective space which contains neither a projective space nor a hyperquadric as a component is characterized by its fundamental forms as a local submanifold of the projective space. Using some invarianttheoretic properties of the fundamental forms and Seashi’s work on linear differential equations of finite type, we reduce the proof to the vanishing of certain Spencer cohomology groups. The vanishing is checked by Kostant’s harmonic theory for Lie algebra cohomology.
Disproof of Modularity of Moduli Space of CY 3folds of Double covers of P3 ramified along eight planes
 in General Positions, math.arXiv,0709.1051
"... Abstract. We prove that the moduli space of CalabiYau 3folds coming from eight planes of P 3 in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole Sp(20, R). We construct an interesting submoduli, which we ..."
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Abstract. We prove that the moduli space of CalabiYau 3folds coming from eight planes of P 3 in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole Sp(20, R). We construct an interesting submoduli, which we call hyperelliptic locus, over which the weight 3 QHodge structure is the third wedge product of the weight 1 QHodge structure on the corresponding hyperelliptic curve. The nonextendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved. 1.
Differential geometry of submanifolds of projective space
 2006 IMA WORKSHOP “SYMMETRIES AND OVERDETERMINED SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS"
, 2006
"... These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The ..."
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Cited by 5 (3 self)
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These are lecture notes on the rigidity of submanifolds of projective space “resembling” compact Hermitian symmetric spaces in their homogeneous embeddings. The
T.Yatsui, Parabolic Geometries associated with Differential Equations of Finite Type, Progress in Mathematics 252 (From Geometry to Quantum Mechanics
 Department of Mathematics,, Faculty of Science,, Hokkaido University,, Sapporo
"... Abstract. We present here classes of parabolic geometries arising naturally from Seashi’s principle to form good classes of linear differential equations of finite type, which generalize the cases of second and third order ODE for scalar function. We will explicitly describe the symbols of these d ..."
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Cited by 4 (2 self)
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Abstract. We present here classes of parabolic geometries arising naturally from Seashi’s principle to form good classes of linear differential equations of finite type, which generalize the cases of second and third order ODE for scalar function. We will explicitly describe the symbols of these differential equations. The model equations of these classes admit nonlinear contact transformations and their symmetry algebras become finite dimensional and simple. 1.
0 ON THE MONODROMY OF THE MODULI SPACE OF CALABIYAU THREEFOLDS COMING FROM EIGHT PLANES IN P3
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FubiniGriffithsHarris rigidity and Lie algebra cohomology preprint arXiv:0707.3410
"... Abstract. We prove a general rigidity theorem for represented semisimple Lie groups. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in Pg) is extrinsically rigid to third order (with the exception of g = a1). In contrast, we show t ..."
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Abstract. We prove a general rigidity theorem for represented semisimple Lie groups. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in Pg) is extrinsically rigid to third order (with the exception of g = a1). In contrast, we show that the adjoint variety of SL3C and the Segre product Seg(P 1 ×P n) are flexible at order two. In the SL3C example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry. We extend machinery developed by Hwang and Yamaguchi, Seashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques. 1.
EXTERIOR DIFFERENTIAL SYSTEMS, LIE ALGEBRA COHOMOLOGY, AND THE RIGIDITY OF HOMOGENOUS VARIETIES
"... Abstract. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an ..."
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Abstract. These are expository notes from the 2008 Srni Winter School. They have two purposes: (1) to give a quick introduction to exterior differential systems (EDS), which is a collection of techniques for determining local existence to systems of partial differential equations, and (2) to give an exposition of recent work (joint with C. Robles) on the study of the FubiniGriffithsHarris rigidity of rational homogeneous varieties, which also involves an advance in the EDS technology.
CONTACT GEOMETRY OF SECOND ORDER I KEIZO YAMAGUCHI
"... In [C1] and [C2], E.Cartan studied involutive systems of second order partial differential equations for a scalar function with 2 or 3 independent variables, following the tradition of the geometric theory of partial differential equations developed by Monge, Jacobi, Lie, Darboux, Goursat and others ..."
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In [C1] and [C2], E.Cartan studied involutive systems of second order partial differential equations for a scalar function with 2 or 3 independent variables, following the tradition of the geometric theory of partial differential equations developed by Monge, Jacobi, Lie, Darboux, Goursat and others. In fact he investigated the contact equivalence and the
Contents
, 2005
"... The aim of this article is to present and reformulate systematically what is known about surfaces in the projective 3space, in view of transformations of surfaces, and to complement with some new results. Special emphasis will be laid on line congruences and Laplace transformations. A line congruen ..."
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The aim of this article is to present and reformulate systematically what is known about surfaces in the projective 3space, in view of transformations of surfaces, and to complement with some new results. Special emphasis will be laid on line congruences and Laplace transformations. A line congruence can be regarded as a transformation connecting one focal surface with the other focal surface. A Laplace transformation is regarded as a method of constructing a new surface from a given surface by relying on the asymptotic system the surface is endowed with. The most interesting object in this article is a class of projectively minimal surfaces. We clarify the procedure of getting new projectievely minimal surfaces from a given one, which was proved by F. Marcus, as well as the procedure of