On rigidity of locally symmetric spaces

On the deformation rigidity of smooth projective symmetric varieties with Picard number one

Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kähler deformations of smooth projective symmetric varieties with Picard number one.

Griffiths-Harris rigidity of compact Hermitian symmetric spaces

I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.

Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces

I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler mani…

覆盖构造的局部刚性与曲线模空间的Weil–Petersson子簇

Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves

本文证明了曲线模空间$\u039c_{g,n}$的Weil–Petersson全测地子簇及覆盖构造是局部刚性的，该结论由一类轨形映射的一般刚性结果推导得到。

Epsilon local rigidity and numerical algebraic geometry

A well-known combinatorial algorithm can decide generic rigidity in the plane by determining if the graph is of Pollaczek-Geiringer-Laman type. Methods from matroid theory have been used to prove other interesting results, again under the assumption of generic configurations. However, configurations arising in applications may not be generic. We present Theorem 5 and its corresponding Algorithm 1 which decide if a configuration is epsilon-locally rigid, a notion we define. A configuration which is epsilon-locally rigid may be locally rigid or flexible, but any continuous deformations remain wi…

On the Validity or Failure of Gap Rigidity for Certain pairs of Bounded Symmetric Domains

In our previous work "Characterization of certain homorphic geodesic cycles on Hermitian locally symmetric manifolds of the noncompact type" in "Modern methods in Complex Analysis" Annals of Math. Studies 138 (1995) 85-118, we formulated a conjecture: the so called "gap phenomenon". The purpose of the article is two-fold. We give a counterexample to the gap phenomenon in the most general situation. We give new examples of situations in which the gap phenomenon holds and a unified conceptual, hopefully definitive, presentation of these situations. In the last section of the article, we survey s…