\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 $\OGr(2,10)$

Betti numbers
\begin{align*} \mathrm{b}_{ 0 } &= 1 \\ \mathrm{b}_{ 2 } &= 1 \\ \mathrm{b}_{ 4 } &= 2 \\ \mathrm{b}_{ 6 } &= 3 \\ \mathrm{b}_{ 8 } &= 4 \\ \mathrm{b}_{ 10 } &= 4 \\ \mathrm{b}_{ 12 } &= 5 \\ \mathrm{b}_{ 14 } &= 5 \\ \mathrm{b}_{ 16 } &= 4 \\ \mathrm{b}_{ 18 } &= 4 \\ \mathrm{b}_{ 20 } &= 3 \\ \mathrm{b}_{ 22 } &= 2 \\ \mathrm{b}_{ 24 } &= 1 \\ \mathrm{b}_{ 26 } &= 1 \end{align*}
Basic information
dimension
13
index
7
Euler characteristic
40
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 3$, $\mathrm{b}_{ 8 } = 4$, $\mathrm{b}_{ 10 } = 4$, $\mathrm{b}_{ 12 } = 5$, $\mathrm{b}_{ 14 } = 5$, $\mathrm{b}_{ 16 } = 4$, $\mathrm{b}_{ 18 } = 4$, $\mathrm{b}_{ 20 } = 3$, $\mathrm{b}_{ 22 } = 2$, $\mathrm{b}_{ 24 } = 1$, $\mathrm{b}_{ 26 } = 1$
$\mathrm{Aut}^0(\OGr(2,10))$
$\mathrm{PSO}_{ 10 }$
$\pi_0\mathrm{Aut}(\OGr(2,10))$
$\mathbb{Z}/2\mathbb{Z}$
$\dim\mathrm{Aut}^0(\OGr(2,10))$
45
Projective geometry
minimal embedding

$\OGr(2,10)\hookrightarrow\mathbb{P}^{ 44 }$

degree
1716
Hilbert series
1, 45, 770, 7644, 52920, 282744, 1241460, 4671810, 15520791, 46521475, 127891764, 326602640, 782658240, 1774339776, 3830960880, 7921074996, 15757146405, 30275519505, 56374390534, 102023945100, ...
Exceptional collections
  • Kuznetsov–Smirnov constructed a full exceptional sequence in 2020, see arXiv:2001.04148.
Quantum cohomology

The small quantum cohomology is not generically semisimple.

The big quantum cohomology is generically semisimple.

The eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(2,10))$ are given by:

Homological projective duality