$D$-affinity of Quadrics Revisited

TL;DR

We address the problem of D-affinity for smooth quadrics in positive characteristic and prove that in characteristic $p\ge3$ the even-dimensional quadric $Q_{2m}$ with $m\ge2$ is not $D$-affine. The strategy reduces $H^{i}(X,\mathscr{D}_{X})=0$ for $i>0$ to a Frobenius-stability criterion on extensions, and the authors exhibit a non-splitting extension involving spinor bundles that remains non-split after Frobenius pushforward, yielding $H^{1}(Q_{2m},\mathscr{D}_{Q_{2m}})\neq0$. As a corollary, the Grassmannian $Gr(2,4)$ is not $D$-affine in $p\ge3$, providing a minimal-dimension non-$D$-affine flag variety in positive characteristic and complementing Langer’s result for odd-dimensional quadrics. The paper further connects to a broader program on $D$-affine flag varieties via fibrations, and develops the necessary machinery around spinor bundles, ACM bundles, and Frobenius decompositions to analyze $D$-affinity in this setting.

Abstract

Let $K$ be aa algebraically closed field of characteristic $p\geq3$ and let $Q_{n}\subset\mathbb{P}^{n+1}_{K}$ be a smooth quadric hypersurface. We show that if $n=2m\geq4$ then $Q_{n}$ is not $D$-affine. In particular, we show the grassmannian ${Gr}(2,4)$ is not $D$-affine, which gives an example of a non $D$-affine flag variety of minimal possible dimension in characteristic $p\geq3$. Our result complements previous work of A. Langer, who showed that if $p\geq n=2m+1$ then $Q_{n}$ is $D$-affine.

Figures (1)

• Figure 1: Dynkin diagram of type $D_{m+1}$

Theorems & Definitions (40)

• Theorem 1.1
• Remark 1.2
• Corollary 1.4
• Corollary 1.5
• Remark 1.6
• Corollary 1.7
• Remark 1.8
• Proposition 1.9
• Lemma 2.1
• proof