\begin{equation} \DeclareMathOperator\Gr{Gr} \DeclareMathOperator\LGr{LGr} \DeclareMathOperator\OGr{OGr} \DeclareMathOperator\SGr{SGr} \DeclareMathOperator\Kzero{K_0} \DeclareMathOperator\index{i} \DeclareMathOperator\rk{rk} \end{equation}

Grassmannian.info

A periodic table of (generalised) Grassmannians.

Orthogonal Grassmannian $\OGr(4,9)$

Y Z Y Z Y Z Y Z
There exist other realisations of this Grassmannian:
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 1 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 2 \\ \mathrm{b}_{ 10 } &= 2 \\ \mathrm{b}_{ 12 } &= 2 \\ \mathrm{b}_{ 14 } &= 2 \\ \mathrm{b}_{ 16 } &= 1 \\ \mathrm{b}_{ 18 } &= 1 \\ \mathrm{b}_{ 20 } &= 1 \end{align*}
Basic information
dimension
10
index
8
Euler characteristic
16
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 1$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 2$, $\mathrm{b}_{ 10 } = 2$, $\mathrm{b}_{ 12 } = 2$, $\mathrm{b}_{ 14 } = 2$, $\mathrm{b}_{ 16 } = 1$, $\mathrm{b}_{ 18 } = 1$, $\mathrm{b}_{ 20 } = 1$
$\mathrm{Aut}^0(\OGr(4,9))$
$\mathrm{PSO}_{ 10 }$
$\pi_0\mathrm{Aut}(\OGr(4,9))$
$1$
$\dim\mathrm{Aut}^0(\OGr(4,9))$
45
Projective geometry
minimal embedding

$\OGr(4,9)\hookrightarrow\mathbb{P}^{ 15 }$

degree
12
Hilbert series
1, 16, 126, 672, 2772, 9504, 28314, 75504, 184041, 416416, 884884, 1782144, 3426384, 6325632, 11267532, 19442016, 32605881, 53300016, 85131970, 133138720, ...
Projective duality

The other spinor tenfold in the dual half-spin projective space.

codimension
5
degree
12
self-dual
yes

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\OGr(4,9))$:

Quantum cohomology
small
generically semisimple [Chaput–Manivel–Perrin]

Minuscule/cominuscule case, including quadric realisations.

big
generically semisimple [Li–Liu–Yu–Zhang, Theorems 1.2 and 1.4]

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(4,9))$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Kuznetsov, Section 6.2]

Lefschetz block sizes: 2, 2, 2, 2, 2, 2, 2, 2

Block sizes $(2, 2, 2, 2, 2, 2, 2, 2)$; 16 objects.

The structure sheaf and the dual tautological bundle, with eight twists.

HP dual: the other spinor tenfold in the dual half-spin representation, for the rectangular decomposition. [Kuznetsov, Section 6.2]

$k$-Fanoness

2-Fano: $\mathrm{ch}_i(\mathrm{T}_{ \OGr(4,9) })\cdot Z>0$ for every nonzero effective $i$-dimensional cycle $Z$ and $1\leq i\leq 2$. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Kuznetsov, A. G. Hyperplane sections and derived categories. Izv. Math. 70 (2006), 447–547. MR2238172.
  2. Pierre-Emmanuel Chaput; Laurent Manivel; Nicolas Perrin. Quantum cohomology of minuscule homogeneous spaces III : semi-simplicity and consequences. Canad. J. Math. 62 (2010), 1246–1263. arXiv:0710.1224.
  3. Carolina Araujo; Roya Beheshti; Ana-Maria Castravet; Kelly Jabbusch; Svetlana Makarova; Enrica Mazzon; Libby Taylor; Nivedita Viswanathan. Higher Fano Manifolds. Rev. Un. Mat. Argentina 64 (2022), 103–125. arXiv:2110.02339.
  4. Alexander Samokhin; Wilberd van der Kallen. Highest weight category structures on $\operatorname{rep}(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$. Preprint (2024). arXiv:2407.13653.
  5. Changzheng Li; Zhaoyang Liu; Tony Yue Yu; Chi Zhang. Convergence and semisimplicity of big quantum cohomology of flag varieties. Preprint (2026). arXiv:2610.28418.

All references.