[Paper Review] Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces

Robert L. Bryant|ArXiv.org|Jun 24, 2000

Algebraic Geometry and Number Theory11 references19 citations

TL;DR

This paper investigates the rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces, particularly Grassmannians, by constructing non-involutive holomorphic exterior differential systems whose integral varieties correspond to subvarieties with specific homology classes. The key contribution is a characterization of when such cycles cannot be smoothed, leading to classification results for holomorphic bundles on Kähler manifolds with vanishing Chern classes.

ABSTRACT

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 manifolds that are generated by their global sections but that have certain polynomials in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0, etc.).

Motivation & Objective

• To understand why certain singular subvarieties in Grassmannians, such as Schubert cycles, cannot be homologous to smooth subvarieties.
• To develop a holomorphic exterior differential system (EDS) that characterizes compact complex subvarieties of given codimension with trivial intersection pairing against a fixed subvariety.
• To analyze the structure of these EDS ideals in Hermitian symmetric spaces, especially Grassmannians and exceptional spaces like E VII.
• To apply the results to classify holomorphic vector bundles on compact Kähler manifolds that are generated by global sections and have vanishing second or third Chern classes.
• To determine when such bundles must be pullbacks from curves or decompose into line bundles and trivial bundles.

Proposed method

• Construct a holomorphic exterior differential system $\mathcal{I}$ on a compact Hermitian symmetric space $M$ such that any $p$-dimensional subvariety $V \subset M$ with $[V] \cap [W] = 0$ is an integral variety of $\mathcal{I}$.
• Analyze the non-involutivity of $\mathcal{I}$ and use this to describe its $p$-dimensional integral varieties explicitly via representation-theoretic and combinatorial methods.
• Use the poset of ideals in the exterior algebra of the tangent space to classify integral varieties, particularly Schubert cycles, associated with specific nodes in the Hasse diagram.
• Apply the theory to Grassmannians $\mathrm{Gr}(m,n)$, focusing on Schubert cycles of codimension $k=2,3$ and their homological non-smoothability.
• Use results from Thom and Hartshorne–Rees–Thomas to show that certain homology classes in $\mathrm{Gr}(3,6)$ are not representable by smooth submanifolds.
• Study exceptional Hermitian symmetric spaces like $\mathsf{E\,VII}$ using computational tools such as simpLie to analyze the ideal posets and their integral varieties.

Experimental results

Research questions

• RQ1 Under what conditions is a Schubert cycle in a Grassmannian not homologous to a smooth subvariety?
• RQ2 Can the holomorphic exterior differential system $\mathcal{I}$ associated with a fixed subvariety $W$ be used to characterize all subvarieties $V$ with $[V] \cap [W] = 0$?
• RQ3 What is the structure of the integral varieties of non-involutive holomorphic EDS on Hermitian symmetric spaces, particularly in $\mathrm{Gr}(3,6)$ and $\mathsf{E\,VII}$?
• RQ4 How do vanishing Chern classes of globally generated holomorphic bundles on compact Kähler manifolds constrain their geometric structure?
• RQ5 Are all irreducible integrals of a given ideal $\mathcal{I}_{(p,q)}$ in $\mathsf{E\,VII}$ contained within specific Schubert cycles of higher dimension?

Key findings

• In $\mathrm{Gr}(3,6)$, the Schubert cycle $\sigma(W)$ of codimension 2 is not homologous to any smooth subvariety, as shown by Hartshorne, Rees, and Thomas using Thom's results on non-representability of homology classes.
• For $\mathrm{Gr}(2,5)$, the codimension 2 cycle $\sigma(W)$ is homologous to a difference of two smooth subvarieties but not to any single smooth subvariety.
• The holomorphic EDS $\mathcal{I}$ associated with a fixed subvariety $W$ is almost never involutive, yet its $p$-dimensional integral varieties can still be described explicitly.
• When $c_2(F) = 0$ for a holomorphic bundle $F$ on a compact Kähler manifold generated by global sections, $F$ is either a pullback from a curve or decomposes as $L \oplus T$ with $L$ a line bundle and $T$ trivial.
• In $\mathsf{E\,VII}$, the ideal $\mathcal{I}_{(6,6)}$ has irreducible integrals of dimension 10, corresponding to Schubert cycles at node $(10,0)$, and the dimension of the representation is 43,758, significantly smaller than its neighbor.
• The ideal $\mathcal{I}_{(10,0)}$ in $\mathsf{E\,VII}$ has irreducible integrals of dimension 17, corresponding to the node $(17,8)$, with the representation dimension being 100,386, less than one-eightieth of $\Lambda^{10}(\mathfrak{m})$.