Differential Geometry of Submanifolds of Projective Space

Abstract

Abstract. These are lecture notes on the rigidity of submanifolds of projective space "resembling" compact Hermitian symmetric spaces in their homogeneous embeddings. The results of [16,20,29,18,19,10,31] are surveyed, along with their classical predecessors. The notes include an introduction to moving frames in projective geometry, an exposition of the Hwang-Yamaguchi ridgidity theorem and a new variant of the Hwang-Yamaguchi theorem. Overview• Introduction to the local differential geometry of submanifolds of projective space.• Introduction to moving frames for projective geometry.• How much must a submanifold X ⊂ P N resemble a given submanifold Z ⊂ P M infinitesimally before we can conclude X ≃ Z? • To what order must a line field on a submanifold X ⊂ P N have contact with X before we can conclude the lines are contained in X? • Applications to algebraic geometry.• A new variant of the Hwang-Yamaguchi rigidity theorem.• An exposition of the Hwang-Yamaguchi rigidity theorem in the language of moving frames. Representation theory and algebraic geometry are natural tools for studying submanifolds of projective space. Recently there has also been progress the other way, using projective differential geometry to prove results in algebraic geometry and representation theory. These talks will focus on the basics of submanifolds of projective space, and give a few applications to algebraic geometry. For further applications to algebraic geometry the reader is invited to consult chapter 3 of [11] and the references therein.Due to constraints of time and space, applications to representation theory will not be given here, but the interested reader can consult [23] for an overview. Entertaining applications include new proofs of the classification of compact Hermitian symmetric spaces, and of complex simple Lie algebras, based on the geometry of rational homogeneous varieties (instead of root systems), see [22]. The applications are not limited to classical representation theory. There are applications to Deligne's conjectured categorical generalization of the exceptional series [25], to Vogel's proposed Universal Lie algebra [27], and to the study of the intermediate Lie algebra Notations, conventions. I mostly work over the complex numbers in the complex analytic category, although most of the results are valid in the C ∞ category and over other fields, even characteristic p, as long as the usual precautions are taken. When working over R, some results become more complicated as there are more possible normal forms. I use notations and the ordering of roots as in [2] and label maximal parabolic subgroups accordingly, e.g., P k refers to the maximal parabolic obtained by omitting the spaces corresponding to the simple root α k . v 1 , ..., v k denotes the linear span of the vectors v 1 , ..., v k . If X ⊂ PV is a subset,X ⊂ V denotes the corresponding cone in V \0, the inverse image of X under the projection V \0 → PV . and X ⊂ PV denotes the Zariski closure of X, the zero set of all the homogeneous polynomials vanis...

Fubini-Griffiths-Harris rigidity and Lie algebra cohomology

“…In [10], Hwang and Yamaguchi used methods of Tanaka and Se-ashi [25] as developed in [24] to establish an extrinsic rigidity result for CHSS subject to partial vanishing of the first Lie algebra cohomology groups. The methods of [10] were translated to the language of EDS in [18]. In brief, for CHSS, if one quotients the kernel of the Spencer differential by admissible normalizations in the fiber one arrives naturally at the Lie algebra cohomology group H 1 (g − , g ⊥ ).…”

Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties

“…Corollary 3.4. [28] Let X ⊂ PV be a cominuscule variety, other than a quadric hypersurface. Let Y ⊂ PW be an unknown variety such that dim Y = dim V , and such that for…”

“…The proof of this result uses two facts: that the higher fundamental forms of cominuscule varieties are the (full) prolongations of the second, and that any variety with such fundamental forms must be the homogeneous model (which follows from Theorem 3.3). See [28] for details.…”