\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 $\OGr(2,9)$

Y Z Y Z Y Z Y Z
Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 2 \\ \mathrm{b}_{ 8 } &= 3 \\ \mathrm{b}_{ 10 } &= 3 \\ \mathrm{b}_{ 12 } &= 3 \\ \mathrm{b}_{ 14 } &= 3 \\ \mathrm{b}_{ 16 } &= 2 \\ \mathrm{b}_{ 18 } &= 2 \\ \mathrm{b}_{ 20 } &= 1 \\ \mathrm{b}_{ 22 } &= 1 \end{align*}
Basic information
dimension
11
index
6
Euler characteristic
24
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 3$, $\mathrm{b}_{ 14 } = 3$, $\mathrm{b}_{ 16 } = 2$, $\mathrm{b}_{ 18 } = 2$, $\mathrm{b}_{ 20 } = 1$, $\mathrm{b}_{ 22 } = 1$
$\mathrm{Aut}^0(\OGr(2,9))$
$\mathrm{SO}_{ 9 }$
$\pi_0\mathrm{Aut}(\OGr(2,9))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,9))$
36
Projective geometry
minimal embedding

$\OGr(2,9)\hookrightarrow\mathbb{P}^{ 35 }$

degree
528
Hilbert series
1, 36, 495, 4004, 22932, 102816, 383724, 1241460, 3581721, 9406540, 22837815, 51872184, 111260240, 227056896, 443584944, 833797368, 1514323161, 2666844180, 4567885399, 7629616764, ...
Exceptional collections
  • Kuznetsov constructed a full exceptional sequence in 2008, see MR2434094.
Quantum cohomology

The small quantum cohomology is generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(2,9))$ are given by:

Homological projective duality