[Paper Review] The double Cayley Grassmannian

Laurent Manivel|arXiv (Cornell University)|Apr 1, 2020

Advanced Algebra and Geometry17 references4 citations

TL;DR

This paper introduces the double Cayley Grassmannian (DG), a smooth projective symmetric variety of Picard number one that compactifies the exceptional group $G_2$ under the action of $G_2 \times G_2$. It is constructed as the zero locus of a general section of a rank-7 homogeneous vector bundle on the spinor variety $\mathrm{Spin}_{14}/P_7$, and exhibits properties analogous to the Cayley Grassmannian but in a doubled form. The key contribution is proving that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid.

ABSTRACT

We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of Spin(14). We call it the double Cayley Grassmannian because quite remarkably, it exhibits very similar properties to those of the Cayley Grassmannian (the other symmetric variety of type G2), but doubled in the certain sense. We deduce among other things that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid.

Motivation & Objective

• To study the geometry of the double Cayley Grassmannian (DG), a symmetric variety compactifying $G_2$ under $G_2 \times G_2$ action.
• To establish that DG shares structural and cohomological properties with the Cayley Grassmannian, but in a doubled form.
• To prove that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid, extending a rigidity result to the full class.

Proposed method

• Construct DG as the zero locus of a general section of a rank-7 homogeneous vector bundle on the spinor variety $\mathrm{Spin}_{14}/P_7$, using the action of $\mathrm{Spin}_{14}$ on half-spin representations.
• Use the multiplicative double-point property of the spinor variety to analyze the geometry and cohomology of DG.
• Apply the theory of wonderful compactifications and blowups along closed orbits to relate DG to the wonderful compactification of $G_2$.
• Analyze the variety of minimal rational tangents (VMRT) at general points of DG to identify it as the adjoint variety $X_{\mathrm{ad}}(G_2)$ inside $\mathbb{P}\mathfrak{g}_2$.
• Use incidence correspondences and fiber computations to deduce that the Picard number of the variety of lines in DG is one.
• Leverage known results on cohomology of blowups and vector bundles to infer rigidity properties despite the inaccessibility of full cohomology of DG.

Experimental results

Research questions

• RQ1 How does the double Cayley Grassmannian $DG$ relate to the Cayley Grassmannian $CG$ in terms of geometric and representation-theoretic structure?
• RQ2 What is the precise construction of $DG$ as a zero locus in a homogeneous space, and how does it reflect the doubled symmetry of $G_2 \times G_2$?
• RQ3 Does the double Cayley Grassmannian admit a natural compactification as a wonderful variety, and how does its blowup relate to the full automorphism group?
• RQ4 What is the topological Euler characteristic of $DG$, and how does it compare to that of $CG$?
• RQ5 Are all smooth projective symmetric varieties of Picard number one infinitesimally rigid, and can this be proven via geometric and representation-theoretic tools?

Key findings

• The double Cayley Grassmannian $DG$ is constructed as the zero locus of a general section of a rank-7 homogeneous vector bundle on the spinor variety $\mathrm{Spin}_{14}/P_7$, and is Fano of index 4.
• The topological Euler characteristic of $DG$ is $\chi_{\text{top}}(DG) = 6^2 = 36$, in contrast to $\chi_{\text{top}}(CG) = \binom{6}{2} = 15$, reflecting the 'doubled' nature of $DG$.
• The variety $DG$ admits three $G_2 \times G_2$-orbits, with the closed orbit isomorphic to $\mathbb{Q}_5 \times \mathbb{Q}_5$, and the complement of the open orbit being a hyperplane section.
• The variety of minimal rational tangents (VMRT) at a general point of $DG$ is isomorphic to the adjoint variety $X_{\mathrm{ad}}(G_2)$, embedded in $\mathbb{P}\mathfrak{g}_2$, confirming its homogeneous structure.
• Blowing up the closed orbit of $DG$ yields the wonderful compactification of $G_2$, with two exceptional divisors isomorphic to $\mathbb{P}(C \boxtimes C') \to \mathbb{Q}_5 \times \mathbb{Q}_5$ and $\mathbb{P}(N \boxtimes N') \to X_{\mathrm{ad}}(G_2) \times X_{\mathrm{ad}}(G_2)$.
• All smooth projective symmetric varieties of Picard number one are infinitesimally rigid, as proven via the geometry of $DG$ and its blowup, extending a rigidity property to the entire class.