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The Fubini--Study metric on an `odd' Grassmannian is rigid

Stuart James Hall

Abstract

Following the ideas of Gasqui and Goldschmidt, we give an explicit description of the infinitesimal Einstein deformations admitted by the Fubini--Study metric on complex Grassmannians $G_{m}(\mathbb{C}^{n+m})$ with $m,n\geq 2$. The deformations were first shown to exist by Koiso in the 1980s but it has remained an open question as to whether they can be integrated to give genuine deformations of the Fubini--Study metric. We show that when $n+m$ is odd, the answer is no.

The Fubini--Study metric on an `odd' Grassmannian is rigid

Abstract

Following the ideas of Gasqui and Goldschmidt, we give an explicit description of the infinitesimal Einstein deformations admitted by the Fubini--Study metric on complex Grassmannians with . The deformations were first shown to exist by Koiso in the 1980s but it has remained an open question as to whether they can be integrated to give genuine deformations of the Fubini--Study metric. We show that when is odd, the answer is no.
Paper Structure (25 sections, 25 theorems, 140 equations)

This paper contains 25 sections, 25 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. The main theorem and method of proof
  3. The recent work of Nagy, Semmelmann, and Schwahn
  4. Revisiting Koiso's original example of $\mathbb{CP}^{2n}\times\mathbb{CP}^{1}$
  5. Background on Hermitian variations of Kähler--Einstein metrics
  6. First order variations
  7. Second order variations
  8. Koiso's obstruction in complex coordinates
  9. Strategy for demonstrating the rigidity of Grassmannians
  10. An explicit description of the variations of the Grassmannian
  11. Tautological bundles for the Grassmannian
  12. Trivialising the tautological vector bundles
  13. Induced metrics on the canonical bundles and the Fubini--Study metric
  14. Seqsquilinear forms and the action of $SU_{n+m}$
  15. Eigenfunctions of the Laplacian
  16. ...and 10 more sections

Key Result

Theorem A

Let $m,n\in \mathbb {N}$ with $m,n\geq 2$ and $n+m$ odd. The Fubini--Study metric on $G_{m}(\mathbb{C}^{n+m})$ is rigid.

Theorems & Definitions (33)

  • Theorem A
  • Definition 2.1: Infinitesimal Einstein Deformation
  • Theorem 2.2: Koiso, Theorem 1.1 in KoiOsaka1 - see also GGbook Proposition 2.40
  • Lemma 2.3: Koiso, Lemma 4.7 in KoiOsaka2
  • Lemma 2.4: Koiso, Lemma 4.3 in KoiOsaka2
  • Lemma 2.5
  • proof
  • Lemma 3.1: Zeroth-order identities
  • Lemma 3.2: Derivatives of Hermitian metrics
  • Lemma 3.3: Generalised Euler Sequence
  • ...and 23 more