\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 $\OGr(2,15)$

Y Z Y Z Y Z 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 } &= 4 \\ \mathrm{b}_{ 14 } &= 4 \\ \mathrm{b}_{ 16 } &= 5 \\ \mathrm{b}_{ 18 } &= 5 \\ \mathrm{b}_{ 20 } &= 6 \\ \mathrm{b}_{ 22 } &= 6 \\ \mathrm{b}_{ 24 } &= 6 \\ \mathrm{b}_{ 26 } &= 6 \\ \mathrm{b}_{ 28 } &= 5 \\ \mathrm{b}_{ 30 } &= 5 \\ \mathrm{b}_{ 32 } &= 4 \\ \mathrm{b}_{ 34 } &= 4 \\ \mathrm{b}_{ 36 } &= 3 \\ \mathrm{b}_{ 38 } &= 3 \\ \mathrm{b}_{ 40 } &= 2 \\ \mathrm{b}_{ 42 } &= 2 \\ \mathrm{b}_{ 44 } &= 1 \\ \mathrm{b}_{ 46 } &= 1 \end{align*}
Basic information
dimension
23
index
12
Euler characteristic
84
Betti numbers
$\mathrm{b}_{ 0 } = 1$, $\mathrm{b}_{ 2 } = 1$, $\mathrm{b}_{ 4 } = 2$, $\mathrm{b}_{ 6 } = 2$, $\mathrm{b}_{ 8 } = 3$, $\mathrm{b}_{ 10 } = 3$, $\mathrm{b}_{ 12 } = 4$, $\mathrm{b}_{ 14 } = 4$, $\mathrm{b}_{ 16 } = 5$, $\mathrm{b}_{ 18 } = 5$, $\mathrm{b}_{ 20 } = 6$, $\mathrm{b}_{ 22 } = 6$, $\mathrm{b}_{ 24 } = 6$, $\mathrm{b}_{ 26 } = 6$, $\mathrm{b}_{ 28 } = 5$, $\mathrm{b}_{ 30 } = 5$, $\mathrm{b}_{ 32 } = 4$, $\mathrm{b}_{ 34 } = 4$, $\mathrm{b}_{ 36 } = 3$, $\mathrm{b}_{ 38 } = 3$, $\mathrm{b}_{ 40 } = 2$, $\mathrm{b}_{ 42 } = 2$, $\mathrm{b}_{ 44 } = 1$, $\mathrm{b}_{ 46 } = 1$
$\mathrm{Aut}^0(\OGr(2,15))$
$\mathrm{SO}_{ 15 }$
$\pi_0\mathrm{Aut}(\OGr(2,15))$
$1$
$\dim\mathrm{Aut}^0(\OGr(2,15))$
105
Projective geometry
minimal embedding

$\OGr(2,15)\hookrightarrow\mathbb{P}^{ 104 }$

degree
832048
Hilbert series
1, 105, 4080, 88179, 1270815, 13537524, 113859200, 791224200, 4695002676, 24385860460, 113015849856, 474507071220, 1827174287820, 6518164602264, 21722800137600, 68110029598100, 202120945079625, 570619024282125, 1539370437368400, 3983638152414375, ...
Projective duality

The projective dual is a hypersurface.

codimension
1
degree
84
self-dual
no

[Tevelev, Theorems 9.19 and 10.22 (arXiv version)] [Tevelev, Theorems 7.1–7.2 (arXiv version)] All Grassmannians.

Exceptional collections

Full exceptional collections in $\mathbf D^{\mathrm b}(\OGr(2,15))$:

Quantum cohomology
small
generically semisimple [Chaput–Perrin]

Adjoint, non-coadjoint case.

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

Convergent near the large-radius limit.

Eigenvalues of quantum multiplication by $\mathrm{c}_1(\OGr(2,15))$:

Homological projective duality

Full minimal rectangular Lefschetz collection. [Kuznetsov, Theorem 7.1]

Lefschetz block sizes: 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Block sizes $(7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7)$; 84 objects.

Symmetric powers of the dual tautological bundle and a spinor bundle.

No HP-dual model recorded.

$k$-Fanoness

Fano but not 2-Fano. [Araujo et al., Theorems 1.3 and 1.4]

References
  1. Kuznetsov, Alexander. Exceptional collections for Grassmannians of isotropic lines. Proc. Lond. Math. Soc. (3) 97 (2008), 155–182. MR2434094.
  2. Chaput, P. E.; Perrin, N. On the quantum cohomology of adjoint varieties. Proc. Lond. Math. Soc. (3) 103 (2011), 294–330. arXiv:0904.4824. MR2821244.
  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.